SMART TO THE CORE
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TEACHER’S EDITION
Educational Bootcamp
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GRADE 5 EDUCATIONAL BOOTCAMP
®
grade 5
Teacher’s Edition
TEACHER’S EDITION BOOKLET GRADE 5 Building Depth of Knowledge (DOK)
THIS BOOKLET INCLUDES: Core Skills Activities - the activities within each mission progresses from DOK 1 to DOK 3 to master each standard. Each Core Skills Activity is a set of standards-based basic skills practice questions. Practice Drill Questions - practice problems requiring the application of skills and real-world problem-solving. Each set of mission questions progresses from DOK 1 to DOK 3.
DIGITAL COMPONENT: Five-Star Challenge - standards-based assessments that measure students’ depth of knowledge, including their ability to reason abstractly, create models, write arguments and critique strategies.
MATH BOOTCAMP® - SMART TO THE CORE TEACHER’S EDITION - GRADE 5 (FLORIDA) Copyright© 2023 by Educational Bootcamp. First Edition. All rights reserved. Publisher: J&J Educational Boot Camp, Inc. Content Development: Educational Bootcamp J&J Educational Boot Camp, Inc. www.educationalbootcamp.com jandj@educationalbootcamp.com No part of this publication may be reproduced, transmitted, or stored in a retrieval system, in whole or in part, in any form or by any means, electronic or mechanical, including photocopying, recording, or otherwise, without written permission of Educational Bootcamp. Educational Bootcamp and Math Bootcamp are registered trademarks of J&J Educational Boot Camp, Inc. Printed in the United States of America For information regarding the CPSIA on printed material call: 203-595-3636 and provide reference # LANC 807255 MATH BOOTCAMP SMART TO THE CORE | EDUCATIONAL BOOTCAMP®, Inc.
MISSION
FL CODE
BENCHMARKS FOR EXCELLENT STUDENT THINKING
PAGES
NUMBER SENSE AND OPERATIONS MISSION 1 Understanding Place Value MISSION 2 Reading and Writing Multi-Digit Numbers with Decimals MISSION 3 Composing and Decomposing MultiDigit Numbers
MA.5.NSO.1.1
Express how the value of a digit in a multi-digit number with decimals to the thousandths changes if the digit moves one or more places to the left or right.
1-8
MA.5.NSO.1.2
Read and write multi-digit numbers with decimals to the thousandths using standard form, word form and expanded form.
9 - 16
MA.5.NSO.1.3
Compose and decompose multi-digit numbers with decimals to the thousandths in multiple ways using the values of the digits in each place. Demonstrate the compositions or decompositions using objects, drawings and expressions or equations.
MISSION 4 Plotting and Ordering Multi-Digit Numbers
MA.5.NSO.1.4
Plot, order and compare multi-digit numbers with decimals up to the thousandths.
25 - 32
MISSION 5 Rounding Multi-Digit Numbers
MA.5.NSO.1.5
Round multi-digit numbers with decimals to the thousandths to the nearest hundredth, tenth or whole number.
33 - 40
MISSION 6 Multiplying Multi-Digit Whole Numbers
MA.5.NSO.2.1
Multiply multi-digit whole numbers including using a standard algorithm with procedural fluency.
41 - 48
MISSION 7 Dividing Multi-Digit Whole Numbers
MA.5.NSO.2.2
Divide multi-digit whole numbers, up to five digits by two digits, including using a standard algorithm with procedural fluency. Represent remainders as fractions.
49 - 56
57 - 64
MISSION 8 Adding and Subtracting Multi-Digit Numbers
MA.5.NSO.2.3
Add and subtract multi-digit numbers with decimals to the thousandths, including using a standard algorithm with procedural fluency.
MISSION 9 Multiplying and Dividing Multi-Digit Numbers with Decimals
MA.5.NSO.2.5
Multiply and divide a multi-digit number with decimals to the tenths by one-tenth and one-hundredth with procedural reliability.
MA.5.NSO.2.4
MISSION 10 Dividing Two Whole Numbers as a Fraction MISSION 11 Adding and Subtracting Fractions with Unlike Denominators ii
17 - 24
Explore the multiplication and division of multi-digit numbers with decimals to the hundredths using estimation, rounding and place value. FRACTIONS
65 - 72
MA.5.FR.1.1
Given a mathematical or real-world problem, represent the division of two whole numbers as a fraction.
73 - 80
MA.5.FR.2.1
Add and subtract fractions with unlike denominators, including mixed numbers and fractions greater than 1, with procedural reliability.
81 - 88
MATH BOOTCAMP SMART TO THE CORE | EDUCATIONAL BOOTCAMP®, Inc.
MISSION
FL CODE
BENCHMARKS FOR EXCELLENT STUDENT THINKING
PAGES
MISSION 12 Multiplying Fractions by Fractions
MA.5.FR.2.2
Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.
89 - 96
MISSION 13 Predicting the Relative Size of the Product
MA.5.FR.2.3
When multiplying a given number by a fraction less than 1 or a fraction greater than 1, predict and explain the relative size of the product to the given number without calculating.
97 - 104
MISSION 14 Dividing Unit Fractions by Whole Numbers
MA.5.FR.2.4
Extend previous understanding of division to explore the division of a unit fraction by a whole number and a whole number by a unit fraction.
105- 112
ALGEBRAIC REASONING MISSION 15 Solve Multi-Step Real-World Problems
Solve multi-step real-world problems involving any combination of the four operations with whole numbers, including problems in which remainders must be interpreted within the context.
113 - 120
MISSION 16 Solve Real-World Problems Involving Fractions
MA.5.AR.1.2
Solve real-world problems involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1.
121 – 128
MISSION 17 Solve Real-World Problems Involving Division
MA.5.AR.1.3
Solve real-world problems involving division of a unit fraction by a whole number and a whole number by a unit fraction.
129 - 136
MA.5.AR.2.1
Translate written real-world and mathematical descriptions into numerical expressions and numerical expressions into written mathematical descriptions.
137 - 144
MA.5.AR.2.2
Evaluate multi-step numerical expressions using order of operations.
145 - 152
MA.5.AR.2.3
Determine and explain whether an equation involving any of the four operations is true or false.
MISSION 18 Translate Real-World Problems Into Numerical Expressions MISSION 19 Using the Order of Operations MISSION 20 True or False Equations and Determining the Unknown MISSION 21 Writing Equations to Determine the Unknown MISSION 22 Rules for Describing Patterns as an Expression MISSION 23 Mean, Median, Mode, and Range iii
MA.5.AR.1.1
MA.5.AR.2.4
MA.5.AR.3.1
MA.5.AR.3.2
153 - 160
Given a mathematical or real-world context, write an equation involving any of the four operations to determine the unknown whole number with the unknown in any position.
161 - 168
Given a numerical pattern, identify and write a rule that can describe the pattern as an expression.
169 - 176
Given a rule for a numerical pattern, use a two-column table to record the inputs and outputs.
177 - 184
MATH BOOTCAMP SMART TO THE CORE | EDUCATIONAL BOOTCAMP®, Inc.
MISSION
FL CODE
FLORIDA BENCHMARK
PAGES
MEASUREMENET MISSION 24 Converting Measurement Units MISSION 25 Solving Real-World Problems Involving Money
MA.5.M.1.1
MA.5.M.2.1
Solve multi-step real-world problems that involve converting measurement units to equivalent measurements within a single system of measurement.
185 - 192
Solve multi-step real-world problems involving money using decimal notation.
193 - 200
GEOMETRIC MISSION 26 Classifying Triangles and Quadrilaterals
MA.5.GR.1.1
MISSION 27 Classifying ThreeDimensional Figures
MA.5.GR.1.2
MISSION 28 Finding the Perimeter and Area of a Rectangle
MA.5.GR.2.1
MISSION 29 Using Volume as an Attribute of ThreeDimensional Figures MISSION 30 Finding the Volume of a Right Rectangular Prism MISSION 31 Solving Real-World Problems Involving Volume MISSION 32 Plotting and Labeling Ordered Pairs MISSION 33 Plotting Points Representative of Real-World Problems
MA.5.GR.3.1
MA.5.GR.3.2
MA.5.GR.3.3
MA.5.GR.4.1
MA.5.GR.4.2
REASONING
Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.
201 - 208
Identify and classify three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.
209 - 216
Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas. Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes. Find the volume of a right rectangular prism with wholenumber side lengths using a visual model and a formula. Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem. Identify the origin and axes in the coordinate system. Plot and label ordered pairs in the first quadrant of the coordinate plane. Represent mathematical and real-world problems by plotting points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation.
217 - 224
225 - 232
233 - 240
241 - 248
249 - 256
257 - 264
DATA ANALYSIS & PROBABILITY MISSION 34 Creating Tables, Graphs, and Line Plots
MA.5.DP.1.1
Collect and represent numerical data, including fractional and decimal values, using tables, line graphs or line plots.
MISSION 35 Classifying ThreeDimensional Figures
MA.5.DP.1.2
Interpret numerical data, with whole-number values, represented with tables or line plots by determining the mean, mode, median or range.
iv
MATH BOOTCAMP SMART TO THE CORE | EDUCATIONAL BOOTCAMP®, Inc.
265 - 272
273 - 280
UNDERSTANDING PLACE VALUE MA.5.NSO.1.1 Express how the value of a digit in a multi-digit number with decimals to the thousandths changes if the digit moves one or more places to the left or right. Compare values to show that a digit in a place represents 10 times what it would represent in the place to its right.
HUNDRED THOUSANDS
TEN THOUSANDS
THOUSANDS
HUNDREDS
TENS
ONES
700,000
70,000
7,000
700
70
7
10 times
10 times
10 times
10 times
10 times
Compare 7,000 and 700: The value of the 7 in 7,000 is 10 times the value of the 7 in 700. Compare 700 and 70. The value of the 7 in 700 is 10 times the value of the 7 in 70.
Compare values to show that a digit in a place represents 1/10 what it represents in the place to its LEFT.
ONES
TENTHS
HUNDREDTHS
7
7
7
1
/10
1
/10
THOUSANDTHS
7 1
/10
Compare 0.07 and 0.007: The value of the 7 in 0.07 is 1/10 times the value of the 7 in 0.007. Compare 0.7 and 0.07. The value of the 7 in 0.7 is 1/10 times the value of the 7 in 0.07. 1
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Write the value of each underlined digit.
6,093
14,537
53,501.1
6,000
4,000
500
5.089
3.950
0.534
0.009
0.9
0.03
DIRECTIONS: Use place value patterns to move digits to the left. #
Multi-Digit Number
Moving the Digit 1 Place to the Left
400
4,000 70,000 300,000 80,000,000
7,000 30,000 8,000,000
#
Multi-Digit Number
Moving the Digit 2 Places to the Left
400
4 70 300 80,000
7,000 30,000 8,000,000
DIRECTIONS: Use place value patterns to move digits to the right. #
Number
Moving the Digit 1 Place to the Right
6,000
#
Number
Moving the Digit 3 Places to the Right
60,000
6,000
6,000,000
9,000,000
90,000,000
9,000,000
9,000,000,000
300,000
3,000,000
300,000
300,000,000
COPYING IS STRICTLY PROHIBITED BY LAW
2
DIRECTIONS: Write the value of each underlined digit.
63,764,096
102,348,7549
19,235,089
60,000,000
2,000,000
30,000
25.087
4.563
8.674
0.007
0.5
0.07
DIRECTIONS: Use place value patterns to move digits to the left. #
Multi-Digit Number
Moving the Digit 1 Place to the Left
40,000
4,000 80,000 1,000 900,000
800,000 10,000 9,000,000
#
Multi-Digit Number
Moving the Digit 2 Places to the Left
40,000
400 8,000 100 90,000
800,000 10,000 9,000,000
DIRECTIONS: Use place value patterns to move digits to the right. #
Number
Moving the Digit 1 Place to the Right
700,000 5,000,000 20,000
3
#
Number
Moving the Digit 3 Places to the Right
7,000,000
70,000
70,000,000
50,000,000 200,000
5,000
5,000,000
20,000
20,000,000
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 There were 2,000 entrepreneurs who attended a conference in May. In June, the attendance for the same conference was 10 times more than it was in May. What was the total number of people who attended the conference in June? 200
2,000
20,000
2,000,000
Muhammad ordered a box of 1,500 business cards. He left locations. How many cards did Joe leave at each store? 15
150
of the cards at each of his store
1,500
15,000
Which statement correctly compares 0.78 and 78?
0.78 is
the value of 78.
0.78 is
78 is 1,000 times the value of 0.78.
the value of 78.
78 is 10 times the value of 0.78.
Esmeralda is selling lemonade for $1.45 per cup. If she sells 10 cups, how much money will she make? $0.145
$145.00
$14.50
$14.05
Maxwell and Zane are comparing the weights of their pets. Maxwell has a lizard that weighs 1.76 pounds, and Zane has a dog that weighs 17.6 pounds. Which of the following correctly compares the weights of the pets? Zane’s dog weighs 100 times more than Maxwell’s lizard. Zane’s dog weighs 10 times more than Maxwell’s lizard. Zane’s dog weighs
of the lizard’s weight.
Zane’s dog weighs
of the lizard’s weight. COPYING IS STRICTLY PROHIBITED BY LAW
4
DIRECTIONS: Write the value of each underlined digit.
63,764,096
102,348,7549
19,235,089
60,000,000
2,000,000
30,000
5.089
3.950
0.534
0.009
0.9
0.03
DIRECTIONS: Use place value patterns to move digits to the left. #
Multi-Digit Number
Moving the Digit 1 Place to the Left
5,000
500 8,000 300 700,000
80,000 3,000 7,000,000
#
Multi-Digit Number
Moving the Digit 2 Places to the Left
500
5 80 300 7,000
8,000 30,000 700,000
DIRECTIONS: Use place value patterns to move digits to the right. #
5
Number
Moving the Digit 1 Place to the Right
5,000
#
Number
Moving the Digit 3 Places to the Right
50,000
5,000
5,000,000
8,000
80,000
8,000
8,000,000
20,000
200,000
200
200,000
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What number is 100 times the value of 3.792? 37.92
What is
0.3792
379.2
3792.0
1.24
124
1,240
of 12.4? 0.124
Monty is doing a science experiment about friction. He rolls a toy car on different surfaces to see how far it travels and records his observations below. Which of the following is true about the distances the car traveled on the tile and the grass?
SURFACE
INCHES TRAVELED
The car traveled 100 times farther on the tile than it did on the grass.
Tile
14.52
The car traveled 10 times farther on the tile than it did on the grass.
Carpet
3.14
Grass
1.452
Concrete
7.03
The car traveled 100 times farther on the grass than it did on the tile. The car traveled 10 times farther on the grass than it did on the tile.
What makes the number sentence below true? x _____ = 61.07 6,107
6.107
610.7
0.6107
Niko raised $250 selling chocolate bars for his travel baseball team. If he sold 100 chocolate bars, what was the price of each bar? $0.25
$1.50
$2.50
COPYING IS STRICTLY PROHIBITED BY LAW
$25 6
1
What is
of 12.4? 0.124
1.24
124
1,240
What number makes the number sentence below true? 9.923 × _____ = 992.3 1,000
100
10
Zac and Emily made the same painting on different canvas lengths. Which of the following statements is true about the length of the painting? Zac’s painting is 100 times smaller than Emily’s painting.
NAMES
LENGTH (Inches)
Zac
3.75
Emily
37.5
Zac’s painting is 10 times larger than Emily’s painting. Zac’s painting is
times Emily’s painting.
Zac’s painting is
times Emily’s painting.
Leila bought 10 cartons of eggs at $1.85 each. How much does she have to pay? $0.18
$18.50
$18.55
$185.00
Which statement correctly compares 640 and 6.4? 640 is 10 times the value of 6.4.
6.4 is
times the value of 640.
6.4 is 10 times the value of 640. 640 is 7
times the value of 6.4. MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
Isabella saved $3,000 to buy a new camera. She went to the market and spent How much did she spend?
$300
Place a check mark (✓) next to all of the equations that are not correct.
$30,000
$3,000
$300,000
✓ 0.54 × 10 = 5.4 ✓
2.5 × 10 = 0.25
10 × 1.37 = 13.7 ✓
✓
of her savings.
10 × 0.09 = 0.9
10 × 0.603 = 6.03
When Mina started her company, she sold 56.5 cases of paper per week. After many years in the business, she now sells 10 times that amount. How many cases of paper are currently sold per week? 5.65 cases
565 cases
5,650 cases
56,500 cases
The world’s smallest fish is about 7.9 millimeters long. In the fish tank, it looks 10 times as long. How long does the fish appear in the tank? 0.079 millimeters
0.79 millimeters
790 millimeters
79 millimeters
Darice is conducting a science experiment on light. She shines a flashlight through different materials to see if the light passes through or is blocked, recording her observations below. Which of the following is true about the speed of light passing through the precious stone and the air? The light traveled 10 times faster through the air than it did through the precious stone.
MATERIAL
SPEED OF LIGHT
Vacuum
299,000,000
Air
299,000,000
Water
225,000,000
Precious Stone
29,900,000
The light traveled 100 times faster through the air than it did through the precious stone. The light traveled 100 times faster through the water than it did through the air. The light traveled 10 times faster through the water than it did through the vacuum.
COPYING IS STRICTLY PROHIBITED BY LAW
8
READING AND WRITING MULTI-DIGIT NUMBERS WITH DECIMALS MA.5.NSO.1.2 Read and write multi-digit numbers with decimals to the thousandths using standard form, word form and expanded form. To write decimals through the thousandths place value, use a place value chart. Example: 347.392
PLACE VALUE CHART HUNDREDS
TENS
ONES
TENTHS
HUNDREDTHS THOUSANDTHS
3
4
7
3
9
2
3 × 100
4 × 10
7×1
3× 1 10
9× 1 100
1 2× 1,000
Written Form: Three hundred forty-seven and three hundred ninety-two thousandths To read decimals up to the thousandth place, it is important to know the value of each digit in the decimal. Start with the digit in the tenths place, then move to the hundredths place, and finally, to the thousandths place. The digits to the right of the decimal point represent fractions of a whole number, and the further right the digit is, the smaller the fraction will be. One strategy to help you understand is to read the number in parts before and after commas and/or decimal points.
PLACE VALUE CHART
9
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Write the following numbers in word form. #
NUMBER
WORD FORM
421.2
Four hundred twenty-one and two tenths
7.6
Seven and six tenths
49.235
Forty-nine and two hundred thirty-five thousandths
2.005
Two and five thousandths
0.15
Fifteen hundredths
DIRECTIONS: Write the expanded notation for the numbers below. #
NUMBER
EXPANDED NOTATION
500.03
(5 x 100) + (0 x 10) + (0 x 1) + (0 x 1/10) + (3 x 1/100)
56.065
(5 x 10) + (6 x 1) + (0 x 1/10) + (6 x 1/100) + (5 x 1/1,000)
1.236
(1 x 1) + (2 x 1/10) + (3 x 1/100) + (6 x 1/1,000)
123.45
(1 x 100) + (2 x 10) + (3 x 1) + (4 x 1/10) + (5 x 1/100)
3.333
(3 x 1) + (3 x 1/10) + (3 x 1/100) + (3 x 1/1,000)
DIRECTIONS: Write the following numbers in standard form. #
NUMBER
STANDARD FORM
One hundred thirty-eight and two hundredths
138.02
Two thousand three hundred forty-one and one tenths
2,341.1
Four hundred fifty two and forty-two hundredths
452.42
Five thousand two hundred sixty-three and four tenths
5,263.4
Fourteen and six tenths
14.6
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10
DIRECTIONS: Write the following numbers in word form. #
NUMBER
WORD FORM
0.123
One hundred twenty-three thousandths
3.333
Three and three hundred thirty-three thousandths
0.456
Four hundred fifty-six thousandths
23.456
Twenty-three and four hundred fifty-six thousandths
1,000.1
One thousand and one tenth
DIRECTIONS: Write the expanded notation for the numbers below. #
NUMBER
EXPANDED NOTATION
12.345
(1 x 10) + (2 x 1) + (3 x 1/10) + (4 x 1/100) + (5 x 1/1,000)
999.99
(9 x 100) + (9 x 10) + (9 x 1) + (9 x 1/10) + (9 x 1/100)
1,000.1
(1 x 1,000) + (0 x 100) + (0 x 10) + (0 x 1) + (1 x 1/10)
0.123
(0 x 1) + (1 x 1/10) + (2 x 1/100) + (3 x 1/1,000)
458.356
(4 x 100) + (5 x 10) + (8 x 1) + (3 x 1/10) + (5 x 1/100) + (6 x 1/1,000)
DIRECTIONS: Write the following numbers in standard form. #
11
WORD FORM
STANDARD FORM
Fifteen and six hundred seventy-eight thousandths
15.678
One hundred three and five hundred sixty-seven thousandths
103.567
Seventy and fifteen thousandths
70.015
Eleven and one hundred eleven thousandths
11.111
One hundred twenty-two and thirty-four hundredths
122.34
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 The mass of a feather is 0.138 gram. What digit is in the hundredths place of 0.138? 0
1
3
8
A sticker has a diameter of 0.725 millimeter. What is the value of the digit 5 in the number 0.725? five ones
five tenths
five hundredths
five thousandths
Select all of the ways to write the decimal 0.257. ✓ (2 ×
) + (5 ×
) + (7 ×
(2 ×
) + (5 ×
) + (7 ×
(2 × 0) + (5 ×
) + (7 ×
(2 ×
) + (5 ×
) + (7 ×
(2 ×
) + (5 ×
) + (7 ×
) ) ) ) )
✓ (2 × 0.1) + (5 × 0.01) + (7 × 0.001) What is the standard form of the following number? (8 × 100) + (7 × 10) + (1 × 1) + 9 × ( 870.19
871.9
) 80,071.9
800.719
Which of the following represents the written form of the number 125.37?
One hundred twenty-five and thirty-seven tenths One hundred twenty-five and thirty-seven hundredths One hundred twenty-five and thirty-seven thousandths One hundred twenty-five and thirty-seven hundreds COPYING IS STRICTLY PROHIBITED BY LAW
12
DIRECTIONS: Write the following numbers in word form. #
NUMBER
WORD FORM
348.208
Three hundred forty-eight and two hundred eight thousandths
789.012
Seven hundred eighty-nine and twelve thousandths
456.789
Four hundred fifty-six and seven hundred eighty-nine thousandths
500.03
Five hundred and three hundredths
999.999
Nine hundred ninety-nine and nine hundred ninety-nine thousandths
DIRECTIONS: Write the expanded notation for the numbers below. #
NUMBER
EXPANDED NOTATION
345.678
(3 x 100) + (4 x 10) + (5 x 1) + (6 x 1/10) + (7 x 1/100) + (8 x 1/1,000)
7.777
(7 x 1) + (7 x 1/10) + (7 x 1/100) + (7 x 1/1,000)
987.654
(9 x 100) + (8 x 10) + (7 x 1) + (6 x 1/10) + (5 x 1/100) + (4 x 1/1,000)
1,234.56
(1 x 1,000) + (2 x 100) + (3 x 10) + (4 x 1) + (5 x 1/10) + (6 x 1/100)
32.005
(3 x 10) + (2 x 1) + (0 x 1/10) + (0 x 1/100) + (5 x 1/1,000)
DIRECTIONS: Write the following numbers in standard form. #
13
WORD FORM
STANDARD FORM
Six and nine hundred ninety-nine thousandths
6.999
Thirteen and five hundred seventy-nine thousandths
13.579
Fifteen thousand, four hundred eighty-six and thirty-two hundredths
15,486.32
Three hundred thirty-three thousand, seven hundred seventy-seven and seven tenths
333,777.7
Four hundred seventeen and two hundred three thousandths
417.203
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Daichi measured the reaction time of a participant in his experiment as 0.834 of a second. What digit is in the tenths place of 0.834? 0
8
3
4
The total mass of a paper clip is 1.047 gram. What is the value of the digit 4 in the number 1.047? four ones
four tenths
four hundredths
four thousandths
What is the standard form of eight hundred thirty-nine thousandths? 800.039
839.0
0.83
0.839
A fifth-grade teacher asked the students in her classroom to write the numbers forty-five and two hundred thirty-six thousandths in expanded form. Which of the following is the correct response? (4 × 10) + (5 × 1) + (2 × 1 ) + (36 ×
)
(4 × 10) + (5 × 1) + (2 ×
) + (3 ×
) + (6 ×
(4 × 10) + (5 × 1) + (2 ×
) + (0 ×
) + (36 ×
(4 × 10) + (5 × 1) + (2 ×
) + (3 ×
) + (6 ×
) ) )
Select all of the ways to write the number 0.257. ✓ (0 × 1) + (2 ×
) + (5 ×
) + (7 ×
)
✓ Two hundred fifty-seven thousandths ✓ 0.2 + 0.05 + 0.007 Two hundred fifty-seven and one thousandths 257 ✓ 1,000 COPYING IS STRICTLY PROHIBITED BY LAW
14
The length of a carpenter ant is six-tenths of a centimeter. What is six-tenths written in standard form? 0.6
6.01
6.10
610
The average mass of a herd of Asian elephants is about thirty-three thousand, fifty-eight thousandths of a pound. Which of the following represents thirty-three thousand, fifty-eight thousandths? (33 × 1,000) + (3 × 100) + (0 × 10) + (0 × 1) + (58 ×
)
(33 × 1,000) + (3 × 100) + (3 × 10) + (0 × 1) + (5 ×
) + (8 ×
)
(33 × 1,000) + (3 × 100) + (0 × 10) + (0 × 1) + (0 ×
) + (5 ×
) + (8 ×
)
(33 × 1,000) + (0 × 100) + (0 × 10) + (0 × 1) + (0 ×
) + (5 ×
) + (8 ×
)
What is the word form of 4.719? Forty-seven hundred nineteen thousandths Four and seven hundred nineteen thousandths Four and seven hundred nine thousandths Four and seven thousand nineteen
What is the standard form of (1 × 10) + (2 × 1) + (3 × 123.45
1234.5
12.345
) + (4 ×
) + (5 ×
)?
112.345
Amir wanted to find the thickness of a dime in inches. He recorded his measurement of its thickness as (5 × ) + (3 × ). What is the thickness of the dime in written form? 0.53 15
5.3
53.0
5.53
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Which of the following is the standard form of the number three and four hundred seventyone thousandths?
3.471
3.174
3,471
347.1
Which of the following is the expanded form of the number eighty and ninety-six hundredths? (8 × 10) + (9 ×
) + (6 ×
(8 × 10) + (9 ×
) + (6 ×
) )
(8 × 10) + (9 ×
) + (6 ×
(8 × 10) + (9 ×
) + (6 ×
) )
Which of the following is the word form of the number 205.17? Two hundred five and seventeen tenths Two hundred five and seventeenths Two hundred five and seventeen hundredths Two hundred five and seventeen thousandths
Which of the following is the standard form of the number five thousand ninety and four tenths? 5,090.040
5,090.004
5,090.04
5,090.4
Which of the following is the expanded form of the number 100.23? (1 × 100) + (2 ×
) + (3 ×
)
(1 × 100) + (2 ×
) + (3 ×
)
(1 × 100) + (2 ×
) + (3 ×
)
(1 × 100) + (2 ×
) + (3 ×
)
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COMPOSING AND DECOMPOSING MULTI-DIGIT NUMBERS MA.5.NSO.1.3 Compose and decompose multi-digit numbers with decimals to the thousandths in multiple ways using the values of the digits in each place. Demonstrate the compositions or decompositions using objects, drawings and expressions or equations. To compose a number, we combine the values of its individual digits in their respective places. Each digit in a multi-digit number represents a specific place value such as ones, tens, hundreds, thousandths, and so on. By adding these values together, we can create a complete number. Example: Compose the number 0.735. The digit 7 is in the tenths place, representing 7 tenths (0.7). The digit 3 is in the hundredths place, representing 3 hundredths (0.03). Finally, the digit 5 is in the thousandths place, representing 5 thousandths (0.005). HUNDREDS
Zero
TENS
Zero
ONES
TENTHS
HUNDREDTHS THOUSANDTHS
0
7
3
5
Zero Ones
Seven Tenths (0.7)
Three Hundredths (0.03)
Five Thousandths (0.005)
By adding these values together, we compose the number 0.735.
Each digit represents a specific place value, such as ones, tens, hundreds, thousandths, and so on. Understanding the value of each digit in its respective place is essential to effectively decompose numbers. Example: When decomposing the number 3.725, we can identify the value of each digit in terms of its place value. The digit 3 represents three whole units (3), the digit 7 represents seven tenths (0.7), the digit 2 represents two hundredths (0.02), and the digit 5 represents five thousandths (0.005). HUNDREDS
Zero
TENS
Zero
ONES
TENTHS
HUNDREDTHS THOUSANDTHS
3
7
2
5
Three Ones
Seven Tenths (0.7)
Two Hundredths (0.02)
Five Thousandths (0.005)
Breaking down the number into its individual place values, we decompose it into:
3 + 0.7 + 0.02 + 0.005 17
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DIRECTIONS: Decompose 8.397 in four different ways. #
VALUE IN WORD FORM 8 ones + 3 tenths + 9 hundredths + 7 thousandths 83 tenths + 97 thousandths 8 ones + 397 thousandths 8,397 thousandths
NOTE: There are additional ways to decompose 8.397.
DIRECTIONS: Decompose 47.25 in four different ways. #
VALUE IN WORD FORM 4 tens + 7 ones + 2 tenths + 5 hundredths 47 ones + 2 tenths + 5 hundredths 472 tenths + 5 hundredths 4,725 hundredths
NOTE: There are additional ways to decompose 47.25.
DIRECTIONS: Decompose 36.714 in four different ways. #
VALUE IN WORD FORM 3 tens + 6 ones + 7 tenths + 1 hundredths + 4 thousandths 36 ones + 71 hundredths + 4 thousandths 367 tenths + 14 thousandths 36,714 thousandths
NOTE: There are additional ways to decompose 36.714.
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DIRECTIONS: Decompose 32.435 in four different ways. #
VALUE IN WORD FORM 3 tens + 2 ones + 4 tenths + 3 hundredths + 5 thousandths 32 ones + 4 hundredths + 35 thousandths 324 tenths + 35 thousandths 32,435 thousandths
NOTE: There are additional ways to decompose 32.435.
DIRECTIONS: Decompose 523.76 in four different ways. #
VALUE IN WORD FORM 5 hundreds + 2 tens + 3 ones + 7 tenths + 6 hundredths
52 tens + 3 ones + 7 tenths + 6 hundredths 523 ones + 76 hundredths 52,376 hundredths NOTE: There are additional ways to decompose 523.76.
DIRECTIONS: Decompose 745.23 in four different ways. #
VALUE IN WORD FORM 7 hundreds + 4 tens + 5 ones + 2 tenths + 3 hundredths 7 hundreds + 45 ones + 23 hundredths 745 ones + 2 tenths + 3 hundredths
745,230 thousandths NOTE: There are additional ways to decompose 745.23.
19
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1 Which of the following is the standard form of 71 hundreds + 40 ones + 19 thousandths? 714.019
7,140.109
7,740.019
7,140.19
2 Which of the following is one way to express 1,408.026? 14 hundreds + 8 ones + 26 thousandths 1 thousands + 4 hundreds + 8 ones + 2 tenths + 6 thousandths 1 thousands + 4 hundreds + 8 ones + 2 tenths + 6 hundredths 1 thousands + 408 ones + 2 tenths + 6 thousandths
3 Which of the following is another way to express 5 thousand + 7 hundreds + 3 ones + 9 tenths + 2 hundredths?
57 hundreds + 3 ones + 92 thousandths 57 hundreds + 3 ones + 92 hundredths 57 hundreds + 3 ones + 902 hundredths 57 hundreds + 3 ones + 92 tenths
4 Which of the following is not the same as the rest? 25 hundreds + 40 ones + 16 hundredths 2 thousands + 5 hundreds + 4 tens + 1 hundredths + 6 thousandths 2 thousands + 5 hundreds + 4 tens + 16 hundredths 2 thousands + 5 hundreds + 4 tens + 1 tenths + 6 hundredths
5 Which of the following is not one way to express 840.104? 8 hundreds + 4 tens + 1 tenths + 4 thousandths 8 hundreds + 40 ones + 1 tenths + 4 thousandths 8 hundreds + 4 tens + 10 hundredths + 4 thousandths 8 hundreds + 40 ones + 104 hundredths COPYING IS STRICTLY PROHIBITED BY LAW
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DIRECTIONS: Compose the number 5.326. Compose Using an Expression
Compose Using a Place Value Chart
5.326 = 5 + 0.3 + 0.02 + 0.006 OR 5
5.326 = 5 + 0.326
3
2
DIRECTIONS: Decompose the number 45.926. #
VALUE IN WORD FORM 4 tens + 5 ones + 9 tenths + 2 hundredths + 6 thousandths 45 ones + 92 hundredths + 6 thousandths 459 tenths + 26 thousandths
45,926 thousandths DIRECTIONS: Decompose the number 582.784. #
VALUE IN WORD FORM 5 hundreds + 8 tens + 2 ones + 7 tenths + 8 hundredths + 4 thousandths 5 hundreds + 8 tens + 2 ones + 78 hundredths + 4 thousandths 582 ones + 784 thousandths 582,784 thousandths
DIRECTIONS: Decompose the number 837.245. #
VALUE IN WORD FORM
8 hundreds + 3 tens + 7 ones + 2 tenths + 4 hundredths + 5 thousandths 8 hundreds + 3 tens + 7 ones + 24 hundredths + 5 thousandths 837 ones + 245 thousandths 837,245 thousandths 21
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6
1 Which of the following is not the same as the rest? 17 tens + 2 tenths + 5 hundredths
17 tens + 25 tenths
1 tens + 7 ones + 2 tenths + 5 hundredths
1 tens + 7 ones + 25 hundredths
2 Which of the following is not the same as the rest? 8 thousand + 204 ones + 13 thousandths 8 thousand + 2 hundreds + 4 ones + 1 hundredths + 3 thousandths 8 thousand + 24 ones + 1 hundredths + 3 thousandths 82 thousand + 4 ones + 1 hundredths + 3 thousandths
3 Which of the following is not one way to express 5,680.37? 5 thousands + 6 hundreds + 80 ones + 3 hundredths + 7 thousandths 5 thousands + 6 hundreds + 8 tens + 3 tenths + 7 hundredths 5 thousands + 60 tens + 80 ones + 3 tenths + 7 hundredths 5 thousands + 6 hundreds + 8 tens + 37 hundredths
4 Which of the following is not the same as the rest? 4 thousands + 3 hundreds + 2 ones + 58 thousandths 4 thousands + 3 hundreds + 2 ones + 5 tenths + 8 thousandths 4 thousands + 30 tens + 2 ones + 5 tenths + 8 thousandths 4 thousands + 3 hundreds + 2 ones + 508 thousandths
5 Which of the following is not the same as the rest? 2 hundreds + 5 tens + 4 ones + 706 thousandths
2 hundreds + 5 tens + 4 ones + 7 tenths + 6 thousandths 2 hundreds + 5 tens + 4 ones + 70 hundredths + 6 thousandths 2 hundreds + 5 tens + 4 ones + 7 hundredths + 6 thousandths COPYING IS STRICTLY PROHIBITED BY LAW
22
What is the value of the number 17.25? 17 tens + 2 tenths + 5 hundredths
17 tens + 25 tenths
1 tens + 7 ones + 2 tenths + 5 hundredths
1 tens + 7 ones + 25 tenths
Which of the following is the standard form of 23 hundreds + 40 ones + 15 thousandths? 234.015
2,340.105
2,340.015
2,340.15
Which of the following is not the same as the rest? 4 thousands + 3 hundreds + 2 ones + 58 thousandths 4 thousands + 3 hundreds + 2 ones + 5 tenths + 8 thousandths 4 thousands + 30 tens + 2 ones + 5 tenths + 8 thousandths
4 thousands + 3 hundreds + 2 ones + 508 thousandths
Which of the following is another way to express 5 thousand + 7 hundreds + 3 ones + 9 tenths + 2 hundredths? 57 hundreds + 3 ones + 92 thousandths 57 hundreds + 3 ones + 92 hundredths 57 hundreds + 3 ones + 902 hundredths 57 hundreds + 3 ones + 92 tenths
Which of the following is the standard form of 86 hundreds + 40 ones + 19 thousandths? 864.019
8,640.109
8,640.019
8,640.19
Which of the following is not the same as the rest?
23
17 tens + 2 tenths + 5 hundredths
17 tens + 25 tenths
1 tens + 7 ones + 2 tenths + 5 hundredths
1 tens + 7 ones + 25 hundredths
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Which of the following is represented by the model below?
300 ones + 47 hundredths
30 ones + 47 hundredths
3 ones + 47 hundredths
3 ones + 47 tenths
What decimal is represented by the model below?
0.38
3.008
3.08
3.8
Which of the following is represented by the model below?
2.38
2.308
2.08
2.333
Which of the following is not the same as the rest? 34 tens + 8 tenths
30 ones + 4 tenths
3 tens + 48 tenths
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PLOTTING AND ORDERING MULTI-DIGIT NUMBERS
MA.5.NSO.1.4 Plot, order and compare multi-digit numbers with decimals up to the thousandths.
Use plotting techniques on a number line to order and compare multi-digit numbers. Start by selecting an appropriate scale for the number line or coordinate plane, placing the whole number part of the decimal in its corresponding position. Then, determine the location of the decimal part by dividing the remaining space between whole numbers into equal intervals based on the place value of the decimal. Example: Plot the number 625.75 on a number line. To plot the number 625.75 on a number line, determine the whole numbers that come before and after the number. Place those two numbers on opposite sides of the number line. First 625
626
Next, find the midpoint between the numbers. Plot the number in the middle of the number line. Second
625.00
625.50
626.00
Then, find the midpoints between these numbers. Plot the numbers on the number line. In this instance, the number 625.75 can be found between 625.5 and 626. Third 625.00
625.25
625.50
625.75
626.00
In addition to using place value to compare the values of digits in each place, you can also use the number line. The number line is a visual representation of numbers that can help you compare their values. To use the number line to compare two numbers, plot them on the line, then look at their relative positions. The number to the right is always greater than the number to the left. Example: To compare 625.75 to 625.67, first plot both points on a line using units that are close to the numbers being compared. Next, locate each point on the line and determine which is farther to the right. First 625.75
625.65
625.85
Second
625.65
625.75
625.70
625.80
625.85
625.8
625.85
Third 625.65
625.66
625.67
625.68
625.69
625.70
625.71
625.72
625.73
625.74
625.75
In this case, 625.75 is farther to the right than 625.67, so it is greater. 25
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
each pair of numbers below using comparison DIRECTIONS: Compare symbols (>, <, or =).
47.59 288.56
44.59
5.78
5.80
70.4
53.3
53.23
147.4
280.56
70.6 147.40
DIRECTIONS: Select all the numbers as indicated below.
Numbers greater than 0.545
Numbers greater than 0.72
Numbers less than 0.667
Numbers less than 0.08
0.554
0.702
0.667
0.081
0.455
0.720
0.665
0.080
0.556
0.07
0.668
0.078
0.565
0.723
0.686
0.079
0.655
0.732
0.866
0.073
0.555
0.722
0.777
0.087
0.444
0.719
0.607
0.077
DIRECTIONS: Use the number line below to compare (>, <, or =).
POINT A
POINT C
O.76
0.760
POINT F
POINT B
0.12
0.21
POINT C
POINT F
0.25
0.28
POINT D
POINT E
0.58
0.45
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26
each pair of numbers below using comparison DIRECTIONS: Compare symbols (>, <, or =).
148.213
148.213
78.56
288.755
288.75
258.562
78.65 258.563
189.63
189.603
303.147
300.147
DIRECTIONS: Select all the numbers as indicated below.
Numbers greater than 84.956
Numbers greater than 1,826.83
Numbers less than 6.667
Numbers less than 777.897
84.856
1,826.8
6.688
777.797
84.957
1,826.9
6.067
777.896
84.95
1,826.888
6.7
777.899
84.956
1,826.73
6.657
777.8
84.96
1,826.823
6.707
777.9
84.999
1,826.833
6.667
777.887
84.965
1,826.84
6.66
777.898
DIRECTIONS: Use the number line below to compare (>, <, or =).
27
POINT A
POINT B
O.76
POINT E
POINT D
POINT B
0.12
POINT B
POINT C
POINT D
0.25
POINT B
POINT B
POINT F
0.46
POINT D
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1 Three decimals are plotted on the number line. They are represented by the letters A, B, and C. Which of the following decimals could not be located between A and B on the number line? A 24
25
26
B
C
27
28
29
25.53
26.8
25.95
26.943
2 Which of these numbers is the least? 44.302
44.35
44.384
44.3
3 Which of these numbers is the greatest? 58.469
58.694
58.496
58.649
4 Which of these statements is true? 11.285 < 11.258
11.258 = 11.285
11.285 > 11.258
11.258 > 11.285
5 Augustus measured the daily temperature last week and recorded his data on the table. Which day had the highest recorded temperature?
Monday
DAY
TEMPERATURE
Monday
16.87° F
Tuesday
16.78° F
Wednesday
16.86° F
Thursday
16.68° F
Tuesday
Wednesday
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Thursday 28
each pair of numbers below using comparison DIRECTIONS: Compare symbols (>, <, or =).
56.900
56.9
278.56
365.77
365.707
7,125.808
278.5 7,125.9
789.553
789.535
4,300.749
4,301.749
DIRECTIONS: Select all the numbers as indicated below.
Numbers greater than 49.53
Numbers greater than 7,852.364
Numbers less than 8.95
49.521
7,852.374
8.895
Numbers less than 125.603 125.063
49.531
7,852.354
8.912
125.61
49.60
7,852.4
8.956
49.50
7,852.32
8.99
125.60 125.56
49.053
7,852.53
8.9
125.78
49.547
7,852.38
8.983
125.654
49.65
7,852.367
8.82
125.06
DIRECTIONS: Use the number line below to compare (>, <, or =).
29
POINT A
POINT B
O.76
POINT E
POINT D
POINT B
0.12
POINT B
POINT C
POINT D
0.25
POINT B
POINT B
POINT F
0.46
POINT D
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Select the symbol that makes the following statement true. 488.645 ? 488.546
>
=
<
None
2 Which of the following sets of decimals is listed in descending order? 95.75, 95.7, 95.175
95.175, 95.75, 95.7
95.7, 95.175, 95.75
95.175, 95.75, 95.7
3 Which of the following sets of decimals is listed in ascending order? 168.2, 168.025, 168.25
168.2, 168.25, 168.025
168.025, 168.2, 168.25
168.25, 168.025, 168.2
4 The table below shows the thickness of Susan’s books. Which one of Susan’s books is the thinnest? BOOK
THICKNESS
Blue
4.58 inches
Green
4.058 inches
Red
4.08 inches
Yellow
4.508 inches
Blue book
Green book
Red book
Yellow book
5 During a science experiment, Zayn measured the weight of four rocks. The table on the right shows the weight of the four rocks. Which rock was the heaviest?
Basalt
ROCK
WEIGHT
Basalt
300.45 ounces
Granite
300.405 ounces
Marble
300.54 ounces
Slate
300.504 ounces
Granite
Marble
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Slate 30
1 ITEM
PRICE
Eraser
Eraser
$0.32
Pen
$0.43
Pen
Pencil
$0.34
Pencil
Sharpener
$0.23
Sharpener
RIBBON
LENGTH
Blue
Blue
3.25 feet
Orange
3.205 feet
Pink
3.052 feet
Pink
Violet
3.025 feet
Violet
The table on the right shows the prices of school supplies. Which item has the highest price?
2 The table on the right shows the length of ribbons Carmi found in her drawer. What is the color of the shortest ribbon that Carmi found?
Orange
3 The table on the right shows the hair length of four students. Who has the longest hair?
STUDENT
LENGTH
Andrea
14.603 inches
Bernice
14.36 inches
Faith
14.63 inches
Faith
Victoria
14.063 inches
Victoria
Andrea Bernice
4 Which of these numbers will round to 962 when rounded to the nearest whole number? 961.89
960.999
961.032
962.589
5 Select the symbol that makes the following statement true.
45.78
> 31
=
?
45.780
<
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None
6 Select the symbol that makes the following statement true. 5,834.345
>
5,844.345
?
=
<
None
7 Which of the following sets of decimals is listed in descending order? 14.08, 14.18, 14.8
456.2, 456.12, 456.101
78.93, 78.9, 79.9
59.175, 59.75, 59.7
8 Which of the following sets of decimals is listed in ascending order? 809.32, 890.23, 889.1
781.2, 781.12, 781
365.2, 369.2, 356.2
489.12, 489.2, 489.222
9 The table on the right shows the mass of different fruits. Which fruit has the greatest mass?
FRUIT
MASS
Apple
152.568 grams
Banana
118 .9 grams
Orange
136.3 grams
Pineapple
Pineapple
250.2 grams
Apple
Banana Orange
10 The table on the right shows the heights of different buildings. Which building is the second tallest?
BUILDING
HEIGHT
Empire State
443.98 meters
Burj Khalifa
828.32 meters
Burj Khalifa
CN Tower
553.76 meters
CN Tower
Shanghai
632.002 meters
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Empire State
Shanghai
32
ROUNDING MULTI-DIGIT NUMBERS MA.5.NSO.1.5 Round multi-digit numbers with decimals to the thousandths to the nearest hundredth, tenth or whole number. Use place values to round decimals to any place. Example: Round 49.253 to the nearest tenths: 4 9 . 2 5 3 STEP 1: Underline the targeted digit to be rounded. STEP 2: Look at the number to the right of the underlined digit. If that number is 0 - 4, then the underlined digit stays the same. If the number is 5 - 9, then the underlined digit must be increased by 1. STEP 3: Change all the digits to the right of the targeted digit to zero. The rounded number becomes: 4 9 . 3 0 0 or 4 9 . 3
Use a number line to round decimals to any place. Example: Round 49.253 to the nearest tenths: 4 9 . 2 5 3 STEP 1: Underline the targeted digit to be rounded. STEP 2: Draw a number line with intervals at the far right and left sides of the line.
STEP 3: Keep the targeted digit the same. Change all of the digits to the right to zeros. The new number will be (49.200). Place this number on the far left side of the number line. 49.200
STEP 4: Increase the targeted digit by 1 number and change all the digits to the right of the targeted digit to zeros. 49.200
49.300
STEP 5: Add intervals to allow numbers in increasing order to be placed from 49.200 to 49.300 on the number line. Place a dot on the number line that represents the number to be rounded. Determine whether that number is closer to 49.200 or 49.300. 49.200
49.250
49.300
The number is closest to 4 9 . 3 0 0. Therefore, the number rounded to the nearest tenth is 49.3. 33
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DIRECTIONS: Round to the nearest tenth. 5,455.346 =
5,455.3
634.428 =
634.4
109.896 =
109.9
1,697.644 =
1,697.6
4,728.245 =
4,728.2
782.377 =
782.4
DIRECTIONS: Round to the nearest hundredth. 5,455.346 =
5,455.35
634.428 =
634.43
109.896 =
109.90
1,697.644 =
1,697.64
4,728.245 =
4,728.25
782.377 =
782.38
DIRECTIONS: Round to the nearest whole number. 5,455.346 =
5,455
634.428 =
634
109.896 =
110
1,697.644 =
1,698
4,728.245 =
4,728
782.377 =
782
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34
DIRECTIONS: Round to the nearest tenth. 6,227.528 =
6,227.5
389.497 =
389.5
426.623 =
426.6
7,362.523 =
7,362.5
2,148.157 =
2,148.2
885.331 =
885.3
DIRECTIONS: Round to the nearest hundredth. 6,227.528 =
6,227.53
389.497 =
389.50
426.623 =
426.62
7,362.523 =
7,362.52
2,148.157 =
2,148.16
885.336 =
885.34
DIRECTIONS: Round to the nearest whole number.
35
6,227.528 =
6,228
389.497 =
389
426.623 =
427
7,362.523 =
7,363
2,148.157 =
2,148
885.336 =
885
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of these numbers will round to 845 when rounded to the nearest whole number? 845.75
845.505
844.588
844.325
2 Which of these numbers will round to 128.4 when rounded to the nearest tenth? 128.329
128.345
128.482
128.435
3 Which of these numbers will round to 920.85 when rounded to the nearest hundredth? 920.855
920.858
920.845
920.843
4 Which of these numbers will not round to 620 when rounded to the nearest whole number? 620.655
619.855
620.355
619.635
5 Which of these numbers will not round to 58.2 when rounded to the nearest tenth? 58.195
58.165
58.235
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36
DIRECTIONS: Round to the nearest tenth. 7,503.802 =
7,503.8
500.637 =
500.6
920.467 =
920.5
3,537.621 =
3,537.6
5,721.482 =
5,721.5
574.292 =
574.3
DIRECTIONS: Round to the nearest hundredth. 7,503.802 =
7,503.80
500.637 =
500.64
920.467 =
920.47
3,537.621 =
3,537.62
5,721.482 =
5,721.48
574.292 =
574.29
DIRECTIONS: Round to the nearest whole number.
37
7,503.802 =
7,504
500.637 =
501
920.467 =
920
3,537.621 =
3,538
5,721.482 =
5,721
574.292 =
574
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of these numbers will not round to 420.35 when rounded to the nearest hundredth? 420.346
420.355
420.348
420.353
2 Joan bought a pumpkin for Halloween that weighs 12.45 pounds. What was the weight of the pumpkin when rounded to the nearest whole number? 13 pounds
10 pounds
12 pounds
15 pounds
3 There was a heavy snowstorm last night. It snowed 8.358 inches. Rounded to the nearest tenth, how many inches did it snow last night? 8.36 inches
8.35 inches
8.3 inches
8.4 inches
4 Enzo needs to download a file with a size of 845.228 megabytes. To the nearest hundredth, what is the size of the file that Enzo needs to download? 845.22 megabytes
845.23 megabytes
845.2 megabytes
845.3 megabytes
5 Which of the following rounding statements is true? 890.468 rounded to the nearest hundredth is 890.46. 647.159 rounded to the nearest hundredth is 647.200. 435.258 rounded to the nearest hundredth is 435.26. 288.639 rounded to the nearest hundredth is 288.600. COPYING IS STRICTLY PROHIBITED BY LAW
38
1 Which of the following rounding statements is true? 100.485 rounded to the nearest whole number is 100. 200.684 rounded to the nearest whole number is 200. 300.59 rounded to the nearest whole number is 300. 400.675 rounded to the nearest whole number is 400.
2 Which of the following rounding statements is true? 13.458 rounded to the nearest tenth is 13.4. 35.263 rounded to the nearest tenth is 35.3. 57.734 rounded to the nearest tenth is 57.8. 72.486 rounded to the nearest tenth is 72.4.
3 Peter bought four boards for his project. The table below shows the lengths of the four boards. Which of these statements is true? BOARD
LENGTH
P
3.455 feet
Q
3.625 feet
R
3.155 feet
S
3.425 feet
To the nearest tenth, board P is about 3.4 feet. To the nearest tenth, board Q is about 3.7 feet. To the nearest hundredth, board R is about 3.16 feet. To the nearest hundredth, board S is about 3.42 feet.
4 A marathon race is 42.195 kilometers long. To the nearest hundredth, what is the distance of the marathon?
39
42.195 kilometers
42.20 kilometers
42.19 kilometers
42.2196 kilometers
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
5 Which of these numbers will round to 698.23 when rounded to the nearest hundredth? 698.24
698.236
698.231
698.238
6 Nala went on a trip and bought a souvenir that weighed 5.987 pounds. What was the weight of the souvenir when rounded to the nearest hundredth of a pound? 5.99 pounds
5.98 pounds
5.9 pounds
5.999 pounds
7 A recipe requires 2.75 cups of flour. Meredith uses a little too much flour. Which of the following is most likely the amount Meredith used? 2.758 cups
2.57 cups
2.65 cups
2.75 cups
8 James wants to upload a video to a social media platform. The video file size is 1.237 gigabytes. To the nearest tenth, what is the size of the file? 1.32 gigabytes
1.2 gigabytes
1.237 gigabytes
1.23 gigabytes
9 Which of the following rounding statements is true? 156.893 rounded to the nearest hundredth is 156.894. 777.777 rounded to the nearest hundredth is 77.7. 549.351 rounded to the nearest hundredth is 549.36. 782.233 rounded to the nearest hundredth is 782.23.
10 Which of these numbers will not round to 705.26 when rounded to the nearest hundredth? 705.255
705.261
705.26 3
705.267 COPYING IS STRICTLY PROHIBITED BY LAW
40
MULTIPLYING MULTI-DIGIT WHOLE NUMBERS
MA.5.NSO.2.1 Multiply multi-digit whole numbers including using a standard algorithm with procedural fluency.
Use the standard algorithm to multiply multi-digit whole numbers.
STEP 1: Write the two numbers vertically, lining up the digits based on their place values (ones, tens, hundreds, etc.) STEP 2: Start with the rightmost digit of the second number (the one in the ones place) and multiply it by each digit in the first number, one by one. STEP 3: Write the products below each digit in the first number, aligning them based on their place values. STEP 4: Once you have multiplied the rightmost digit of the second number by all the digits in the first number, move to the next digit of the second number (the one in the tens place) and repeat the multiplication process with the digits in the first number. STEP 5: Continue this process for each digit in the second number, moving from right to left, until you have multiplied each digit by all the digits in the first number. STEP 6: After multiplying all the digits, add up the partial products, starting from the rightmost column, to obtain the final product. STEP 7: Carry over any values that exceed 9 to the next column when adding up the partial products. STEP 8: Write the final product below the line and ensure that the digits are aligned based on their place values.
NOTE: Always double-check your work and perform any necessary regrouping or adjustments to ensure accuracy.
41
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Multiply the multi-digit numbers below.
452 ×x 26 = 11,752
545 × 18 = 9,810
253 × 34 = 8,602
151 × 15 = 2,265
329 × 33 = 10,857
327 × 24 = 7,848
1,231 × 12 = 14,772
2,068 × 14 = 28,952
2,158 × 13 = 28,054
1,005 × 16 = 16,080
1,468 × 15 = 22,020
3,042 × 12 = 36,504
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42
DIRECTIONS: Multiply the multi-digit numbers below.
487 × 29 = 14,123
747 × 21 = 15,687
248 × 35 = 8,680
252 × 16 = 4,032
429 × 32 = 13,728
527 × 26 = 13,702
2,158 × 21 = 45,318
2,355 × 29 = 68,295
3,432 × 32 = 109,824
4,894 × 41 = 200,654
43
3,874 × 35 = 135,590
3,547 × 43 = 152,521
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Kara works at a candy factory. To prepare for the holiday season, the factory produces 5,829 pieces of candy per day. If this rate of production lasts for 14 days, how many pieces of candy will the factory have prepared for holiday shoppers? 29,145 pieces
81,600 pieces
5,843 pieces
81,606 pieces
2 What is 37 x 98? 3,526
3,626
135
629
3 Find the product of 501 and 476. 238,476
714
977
24,276
4 A ticket to the local theme park costs $115. Tahlia and six friends visited the park 6 times together in the past year. How much did they spend in total on tickets in the past year? $4,830
$3,450
$690
$1,035
5 The PTO ordered cupcakes for the year-end celebration. They want to make sure they ordered enough for each of the 821 students to have one. Each container contains 12 cupcakes, and they ordered 68 containers. Did they order enough cupcakes? Yes, they will have 5 extra cupcakes.
Yes, they will have exactly enough cupcakes. No, they will be short 5 cupcakes. No, they will be short 12 cupcakes.
COPYING IS STRICTLY PROHIBITED BY LAW
44
DIRECTIONS: Multiply the multi-digit numbers below.
45
874 × 36 = 31,464
587 × 28 = 16,436
476 × 37 = 17,612
749 × 53 = 39,697
642 × 57 = 36,594
547 × 63 = 34,461
5,245 × 45 = 236,025
4,627 × 48 = 222,096
3,877 × 39 = 151,203
7,682 × 58 = 445,556
6,475 × 59 = 382,025
8,762 × 69 = 604,578
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What number makes the equation below true?
1,689 x 15 = _____ 10,134
25,235
1,704
25,335
2 113,656 is the product of which expression below? 153 x 122
11 x 1,786
14,207 x 8
12,432 x 8
3 The City Center is preparing for a large outdoor concert downtown. The concert is sold out, with 6,000 total tickets purchased. The setup team has plans for 56 rows of 112 seats to be set up. How many seats will they have ready for the attendees? Will they have enough? They will have 7,320 seats which will be enough. They will have 1,232 seats, which will not be enough. They will have 128 seats, which will not be enough. They will have 6,272 seats, which will be enough.
4 What is 19,004 x 345? 6,556,780
6,556,380
1,425,300
1,415,300
5 Solomon is arranging student photos in the school yearbook. He decides they fit best on the pages when he puts them in rows of 7. If he places 9 rows on each page and uses this pattern on 23 of the pages, how many student photos will he have arranged?
1,449 student photos 1,315 student photos 1,494 student photos 1,288 student photos COPYING IS STRICTLY PROHIBITED BY LAW
46
1 Fredrick is stacking soup tins in his store. He stacks 17 tins in a row and makes 24 rows. How many tins are in his stack? 345
648
247
408
2 What is the product of 207 and 745? 154,215
164,315
257,215
264, 368
3 What is 45,578 × 124?
5,615,672
5,651,672
6,541,827
6,782, 007
4 What 2 numbers have a product of 256,443? 490 × 784
324 × 203
409 × 627
648 × 332
5 Patty and Carmela made 148 batches of 38 cookies each for an online order of 5,630 cookies. Did they bake enough cookies to complete the order? Yes, they will have 6 cookies left.
Yes, they will have exactly enough cookies. No, they will be short 6 cookies. No, they will be short 10 cookies.
47
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 A toy store received a shipment of 5,760 action figures. Each shelf can hold 48 action figures, and there are 12 shelves in each section. How many sections can be fully arranged? How many action figures will be left? 40 sections, 240 action figures left 10 sections, 0 action figures left 120 sections, 0 action figures left 160 sections, 720 action figures left
7 What number makes the equation below true?
1,785 × 12 = _____ 17,800
21,420
17,850
21,520
8 268,110 is the product of which expression below? 542 × 501
47,008 × 5
13 × 20,922
17,874 × 15
9 What is 19,004 x 345? 6,556,780
6,556,380
6,425,300
6,415,300
10 Jenny organizes a bookshelf with books of different sizes. She arranges them in stacks of 5 books each. If she has 12 stacks on each shelf and fills 9 shelves following this pattern, how many books has Sarah placed on the bookshelf?
540 books 548 books 40 books 60 books COPYING IS STRICTLY PROHIBITED BY LAW
48
DIVIDING MULTI-DIGIT WHOLE NUMBERS MA.5.NSO.2.2 Divide multi-digit whole numbers, up to five digits by two digits, including using a standard algorithm with procedural fluency. Represent remainders as fractions. Use the standard algorithm to divide multi-digit whole numbers.
STEP 1: Write the dividend (the number being divided) inside the division symbol and the divisor (the number doing the dividing) outside the division symbol. Make sure the digits are lined up based on their place values. STEP 2: Begin dividing by the leftmost digit of the divisor. If the divisor does not evenly divide into the first digit of the dividend, carry over the remaining value to the next digit. STEP 3: Divide the carried-over value along with the next digit of the dividend by the divisor. Continue this process until you have divided all the digits of the dividend. STEP 4: Write the quotient (the result of each division step) above the division symbol, aligning the digits based on their place values. STEP 5: Multiply the quotient by the divisor and subtract the result from the corresponding digits of the dividend. Write the difference below the line. STEP 6: Bring down the next digit from the dividend, and continue the process of dividing and subtracting until you have gone through all the digits of the dividend. STEP 7: If there are digits remaining in the dividend after the division process, add a decimal point and zeros as necessary to the quotient and continue dividing.
STEP 8: If there is a remainder after all the divisions, represent it as a fraction. Write the remainder as the numerator and the divisor as the denominator. Simplify the fraction if possible.
49
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Use a standard algorithm to solve the problems.
456 ÷ 23 = 199 19
789 ÷ 45 = 17 24
632 ÷ 56 = 11 16
357 ÷ 34 = 10 17
432 ÷ 23 = 18 18
534 ÷ 47 = 11 17
4,567 ÷ 36 = 126 31
8,765 ÷ 23 = 381
23
56
45
47
23
34
36
2 23
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3,456 ÷ 17 = 203 5
17
50
DIRECTIONS: Use a standard algorithm to solve the problems. 543 ÷ 89 = 6
874 ÷ 67 = 13 3
9 89
78
789 ÷ 67 = 11 52
9,876 ÷ 54 = 182 48
7,687 ÷ 27 = 284 19
25,432 ÷ 56 = 454 8
34,567 ÷ 28 = 1,234 15
45,678÷ 39 = 1,171
54
67
56
51
921 ÷ 78 = 11 63
67
27
28
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
9 39
1 What is the remainder when 28,496 is divided by 54? 19
21
38
42
2 What is the quotient?
14,540 ÷ 18 7 18 7 807 18 806
14 18 14 807 18 806
3 Uncle Bernard harvested a total of 15,460 apples. He plans to pack the apples in boxes of 75. How many full boxes can Uncle Bernard make? 206 full boxes
208 full boxes
209 full boxes
210 full boxes
4 The school administration has $45,000 to spend on varsity uniforms. Each set of uniforms costs $78. What is the greatest number of varsity uniform sets the school administration can buy? 575 sets
576 sets
577 sets
578 sets
5 Sara needs to distribute a total of 25,620 candies into several bags. The instruction she received is that each bag must be filled with 80 candies. How many more candies does Sara need so that there will be no leftover candies?
20 candies
25 candies
40 candies
60 candies COPYING IS STRICTLY PROHIBITED BY LAW
52
DIRECTIONS: Use a standard algorithm to solve the problems.
831 ÷ 18 = 46 3
769 ÷ 42 = 18 13
6,997 ÷ 26 = 269 3
87,654 ÷ 43 = 2,038
18
42
26
78,901 ÷ 46 = 1,715
53
698 ÷ 53 = 13 9
11 46
53
20 43
89,012 ÷ 42 = 2,119 14
42
65,432 ÷ 32 = 2,044
24 32
53,783 ÷ 23 = 2,338
9 23
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What is the remainder when 35,792 is divided by 84? 2
4
8
16
2 What is the quotient?
71,485 ÷ 65 50 65 1,099 30 65
40 65 1,099 20 65
1,099
1,099
3 A flower supplier needs to ship 12,860 roses to a neighboring town. If the flower supplier can pack 75 roses in each box, how many roses will be in the final partially-filled box?
65 roses
45 roses
35 roses
25 roses
4 What is the remainder when 25,462 is divided by 94? 2
16
41
82
5 What is the quotient?
36,731 ÷ 95 31 95 61 385 95 31 386 95 386 61 95 385
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54
1 What is the remainder when 88,104 is divided by 62? 1
2
4
8
2 What is the quotient?
75,385 ÷ 27 1 27 5 2,792 27 2,792
3 27 7 2,792 27 2,792
3 Hank needs to deliver 10,400 barrels of cooking oil to the market. If he can carry 45 barrels of cooking oil each trip, how many trips does Hank need to make to deliver all the barrels of cooking oil? 230 trips
232 trips
234 trips
235 trips
4 A toy store has $12,000 to spend on a shipment of teddy bears. Each teddy bear costs $25. What is the maximum number of teddy bears the toy store can purchase? 400 teddy bears
480 teddy bears
500 teddy bears
520 teddy bears
5 Jake needs to transfer a total of 18,345 gallons of oil into several containers. The instruction that Jake received is that each container must be filled with 35 gallons of oil. How many more gallons of oil does Jake need so that there will be no leftover oil?
55
5 gallons
15 gallons
25 gallons
30 gallons
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 What is the remainder when 28,927 is divided by 89? 2
4
8
16
7 What is the quotient?
59,420 ÷ 72 20 72 825 25 72
22 72 40 825 72
825
825
8 A toy manufacturer needs to pack 9,450 toy cars into boxes. If each box can hold 125 toy cars, how many toy cars will be in the final partially-filled box?
20 toy cars
75 toy cars
95 toy cars
90 toy cars
9 What is the remainder when 22,788 is divided by 89? 1
2
3
4
10 What is the quotient?
74,771 ÷ 95 787 6 95 787 9 95 787 17 95 787 23 95 COPYING IS STRICTLY PROHIBITED BY LAW
56
ADDING AND SUBTRACTING MULTI-DIGIT NUMBERS MA.5.NSO.2.3 Add and subtract multi-digit numbers with decimals to the thousandths, including using a standard algorithm with procedural fluency. Use a place value chart to add and subtract decimals. Example: 5.73 + 1.2 = ? STEP 1: Align the numbers vertically, making sure the decimal points are directly above each other. Use zeros to make sure all the numbers to be added have the same number of digits. STEP 2: Starting from the rightmost column (the hundredths place), perform the appropriate operation (addition or subtraction) for the digits in that column. If adding, add the digits together, considering any carried-over digits from the previous column. If subtracting, subtract the digits, considering any borrowed digits from the previous column. STEP 3: Move to the next column (tenths place) and perform the appropriate operation for the digits in that column, including any carried-over or borrowed digits from the previous step. If adding, add the digits together, considering any carried-over digits. If subtracting, subtract the digits, considering any borrowed digits. STEP 4: Repeat this process for each subsequent column, moving from right to left (ones place, etc.), performing the appropriate operation (addition or subtraction) for the digits in each column and carrying over or borrowing as necessary. STEP 5: Once you have completed the operations for all columns, write the final result below the line. STEP 6: If there are any carried-over or borrowed digits at the leftmost column, write them as well. STEP 7: Place the decimal point in the result directly below the decimal points in the numbers being added or subtracted.
NOTE: Check your work by performing the operations column-wise from right to left. The result should match the original calculation.
57
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Add the decimals using a standard algorithm. 3.25 + 2.56 = 5.81
7.24 + 0.69 = 7.93
1.12 + 0.434 = 1.554
6.42 + 0.89 = 7.31
2.978 + 3.06 = 6.038
1.2 + 2.36 = 3.56
DIRECTIONS: Subtract the decimals using a standard algorithm. 3.25 - 2.56 = 0.69
7.24 - 0.69 = 6.55
6.42 - 0.89 = 5.53
2.978 - 3.06 = 0.082
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1.12 - 0.434 = 0.686
1.2 - 2.36 = 1.16
58
DIRECTIONS: Add the decimals using a standard algorithm. 6.38 + 2.72 = 9.1
5.63 + 1.44 = 7.07
6.2 + 0.524 = 6.724
7.029 + 6.98 = 14.009
5.62 + 7.99 = 13.61
12.2 + 1.652 = 13.852
DIRECTIONS: Subtract the decimals using a standard algorithm.
59
6.38 - 2.72 = 3.66
5.63 - 1.44 = 4.19
6.2 - 0.524 = 5.676
7.029 - 6.98 = 0.049
4.802 - 2.002 = 2.8
10.3 - 4.57 = 5.73
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What is the difference of 27.168 and 9.829? 17.331
17.339
17.349
18.339
2 Find the sum of 8.904 and 18.944. 26.848
17.848
27.848
16.848
3 Daichi saved $50 from his birthday to buy a new helmet for his bike. The helmet that he chose cost $36.97. How much change will Daichi receive when he buys his helmet? $13.03
$14.13
$13.13
$14.03
4 Find the sum of seven hundred forty-two and one hundredth and eighty-one and three hundred sixty-four thousandths. 823.374
823.365
823.37
723.374
5 What is 7.641 - 3.905? 4.344
4.736
3.736
3.536
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60
DIRECTIONS: Add the decimals using a standard algorithm.
8.291 + 4.65 = 12.941
7.7 + 0.639 = 8.339
9.4 + 0.959 = 10.359
15.68 + 3.62 = 19.3
12.3 + 4.249 = 16.549
14.4 + 3.77 = 18.17
DIRECTIONS: Subtract the decimals using a standard algorithm.
61
8.291 - 4.65 = 3.641
7.7 - 0.639 = 7.061
9.4 - 0.959 = 8.441
8.6 - 3.496 = 5.104
12.3 - 4.249 = 8.051
14.4 - 3.77 = 10.63
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of the following is the sum of 422.621 and 94.05? 416.671
432.026
4,320.26
516.671
2 Which of the following makes the equation below true?
820.05 - 49.11 = ________ 881.94
770.94
869.16
131.115
3 What is the sum of 13.274, 64.099, and 321.74?
399.013
109.447
109.547
399.113
4 Dustin is practicing pitching for his big baseball game this weekend. His dad records the speed of Pitch Speed in Miles per Hour (mph) his pitches. The speeds of his warm-up pitches are Pitch 1 60.654 shown in the table. How much faster is Dustin’s fourth pitch than his first pitch?
Pitch 2 Pitch 3 Pitch 4
11.091 mph
12.091 mph
12.111 mph
12.191 mph
63.896 68.213 72.745
5 99.73 is the sum of which two numbers below? 54.718 and 43.212
63.15 and 36.78
55.926 and 43.804
71.93 and 18.60 COPYING IS STRICTLY PROHIBITED BY LAW
62
1 Selah bought a bracelet for $22.15 and a dress for $250.77. How much did she pay for both? $170.84
$200.04
$230.42
$272.92
2 Which of the following makes the equation below true? 450.78 - 17.09 = _________ 423.81
433.69
544.06
533.11
3 What is the sum of 56.007, 21.12, and 79.45?
156.577
145.677
174.258
187.477
4 Which two numbers below have a sum of 31.945?
20.78 and 10.78
4.487 and 27.458
5.708 and 25.088
15.001 and 15.045
5 The distance covered by the postal worker each day is given below. How many more miles did the postal worker cover on the 4th day than the 3rd day? Day 1 2 3 4 63
Distance (miles) 3.56 4.12 4.36 5.75
0.25
0.56
1.39
1.66
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 Find the sum of six hundred nine and twenty-two hundredths and one hundred forty and six tenths. 548.96
749.82
789.46
889.22
7 What is the difference of 59.419 and 21.711? 34.172
37.780
37.708
37.807
8 Find the sum of 42.062 and 9.599. 51.661
61.719
60.423
50.920
9 Which of the following is the sum of 756.983 and 89.60? 467.358
746.583
846.583
684.465
10 What is 19.324 - 11.729? 7.765
7.595
8.595
8.754
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64
MULTIPLYING & DIVIDING MULTI-DIGIT NUMBERS WITH DECIMALS MA.5.NSO.2.5 Multiply and divide a multi-digit number with decimals to the tenths by one-tenth and one-hundredth with procedural reliability. MA.5.NSO.2.4 Explore the multiplication and division of multi-digit numbers with decimals to the hundredths using estimation, rounding and place value. Estimate the product by rounding. Example: 3.25 × 1.75 = ? STEP 1: Estimate the product by rounding each decimal number to a whole number. This simplifies the calculation. (e.g., 3.25 ≈ 3 and 1.75 ≈ 2) STEP 2: Multiply the rounded whole numbers together using the standard multiplication algorithm (e.g., 3 × 2 = 6) STEP 3: Count the total number of decimal places in the original numbers being multiplied.
STEP 4: Place the decimal point in the product so the number of decimal places matches the count from step 3. Start from the rightmost digit (e.g., 3.25 × 1.75 ≈ 6). STEP 5: Adjust the decimal point position based on the level of estimation used in step 1. This step is optional but can improve the accuracy of the result.
Estimate the quotient by rounding. Example: 5.68 ÷ 2.34 = ? STEP 1: Estimate the quotient by rounding the dividend and divisor to the nearest whole numbers. This simplifies the calculation (e.g., 5.68 ≈ 6 and 2.34 ≈ 2). STEP 2: Divide the rounded dividend by the rounded divisor using the standard division algorithm (e.g., 6 ÷ 2 = 3). STEP 3: Count the total number of decimal places in the original dividend and divisor. STEP 4: Place the decimal point in the quotient so the number of decimal places matches the count from step 3. Start from the rightmost digit (e.g., 5.68 ÷ 2.34 ≈ 3). STEP 5: Adjust the decimal point position based on the level of estimation used in step 1. This step is optional but can improve the accuracy of the result.
65
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Multiply the decimals below. 28.2 × 12 = 338.4
0.01 × 25.68 = 0.2568
0.1 × 3.67 = 0.367
69.82 × 22 = 1,536.04
3.79 × 14 = 53.06
0.01 × 465 = 4.65
DIRECTIONS: Divide the decimals below.
28.2 ÷ 12 = 2.35
25.68 ÷ 0.01 = 2,568
3.67 ÷ 0.1 = 36.7
0.45 ÷ 0.15 = 3
4.2 ÷ 12 = 0.35
10.36 ÷ 3.7 = 2.8
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66
DIRECTIONS: Multiply the decimals below. 84.65 × 9 = 761.85
0.01 × 5.9 = 0.059
0.1 × 43.56 = 4.356
73.57 × 3.2 = 235.424
9.7 × 5.8 = 56.26
0.01 × 6.1 = 0.061
DIRECTIONS: Divide the decimals below.
67
623.2 ÷ 2 = 311.6
9.335 ÷ 0.01 = 933.5
7.9 ÷ 0.1 = 79
26.08 ÷ 8.15 = 3.2
5.65 ÷ 1.13 = 5
162.15 ÷ 35.25 = 4.6
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Look at the multiplication problem. 118 × 0.74 Which of the following is the most reasonable estimate for the product? 120
100
90
80
2 Look at the multiplication problem. 249 × 0.47 Which of the following expressions can be best used to estimate the product? 200 × 0.4
200 × 0.5
250 × 0.6
250 × 0.5
3 A multiplication problem is given below. 356 × 0.22
Which of these statements is true? The product is a little smaller than 89 because 0.22 is less than 1 , and 1 of 356 is 4 4 about 89. The product is a little bigger than 89 because 0.22 is less than 1 , and 1 of 356 is 89. 4 4 1 The product is a little smaller than 1,424 because 0.22 is less than , and 356 times 4 4 is 1,424. 1 The product is a little bigger than 1,424 because 0.22 is less than , and 356 times 4 is 4 1,424.
4 Look at the division problem. ______ ÷ 0.1 = 284 Select the number below that correctly completes the equation. 2.84
28.4
284
2,840
5 What is the quotient of 345.8 ÷ 0.1?
3.458
34.58
345.8
3,458 COPYING IS STRICTLY PROHIBITED BY LAW
68
DIRECTIONS: Multiply the decimals below. 76.35 × 15 = 1,145.25
0.01 × 648.23 = 6.4823
0.1 × 4.326 = 0.4326
283.2 × 41.3 = 11,696.16
69.23 × 2.4 = 166.152
0.01 × 2,157 = 21.57
DIRECTIONS: Divide the decimals below.
69
44.94 ÷ 7 = 6.42
86.93 ÷ 0.01 = 8,693
57.68 ÷ 0.1 = 576.8
15.75 ÷ 3.5 = 4.5
44.8 ÷ 5.6 = 8
66.25 ÷ 2.5 = 26.5
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Look at the division problem. 46.2 ÷ ____ = 4,620 Select the number that correctly completes the equation. 1 1 100 10 10 100
2 Look at the following multiplication problem. Select the number that correctly completes the equation. ____ × 1 = 7.325 100 7.325
73.25
732.5
7,325
3 Look at the division problem: 453 ÷ 0.89 Which of the following expressions can be best used to estimate the quotient? 400 ÷ 0.9
500 ÷ 0.9
450 ÷ 0.8
450 ÷ 0.9
4 Which of these statements is true? The product of 266 × 0.18 is about 40.
The product of 266 × 0.18 is about 50.
The product of 266 × 0.18 is about 60.
The product of 266 × 0.18 is about 70.
5 Look at the division problem. 1 ____ ÷ 10 = 326 Select the number that correctly completes the equation. 32,600
3,260
326
32.6 COPYING IS STRICTLY PROHIBITED BY LAW
70
1 What is the quotient of 782.5 ÷ 1 ? 100 7.825
78.25
7,825
78,250
2 Look at the division problem. 638.4 ÷ ____ = 63,840 Select the number that correctly completes the equation. 1 1 10 100 10
100
3 Look at the multiplication problem.
? × 1 = 1.96 10 Select the number that correctly completes the equation.
19.6
196
1,960
19,600
4 Look at the division problem. 368 ÷ 0.62 Which of the following expressions can be best used to estimate the quotient? 370 ÷ 0.6
360 ÷ 0.7
370 ÷ 0.7
360 ÷ 0.6
5 Look at the following multiplication problem. Which of the following expressions can be best used to estimate the product? 977 × 0.92
977 × 0.9 71
970 × 0.9
980 × 0.9
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
900 × 0.9
6 A division problem is given below. 228 ÷ 0.29 Which of these statements is true?
7
The quotient is a little smaller than 76 because 0.29 is less than 1 , and 1 of 228 is 76. 3 3 1 1 The quotient is a little bigger than 76 because 0.29 is less than , and of 228 is 76. 3 3 1 The quotient is a little smaller than 684 because 0.29 is less than 3 , and 228 times 3 is 684. The quotient is a little bigger than 684 because 0.29 is less than 1 , and 228 times 3 is 684. 3
Which of the following is true about the multiplication problem shown below? 184 × 0.77 The product is a little smaller than 138 because 0.77 is greater than 3 , and 3 of 184 is 138. 4 4 3 3 The product is a little bigger than 138 because 0.77 is greater than , and of 184 is 138. 4 4 3 The product is a little smaller than 245 because 0.77 is less than , and 184 times 3 is 4 4 about 245. The product is a little bigger than 245 because 0.77 is less than 3 , and 184 times 3 is 4 4 about 245.
8 Which of these statements is true? The product of 932 × 0.32 is about 220.
The product of 932 × 0.32 is about 310.
The product of 932 × 0.32 is about 300.
The product of 932 × 0.32 is about 350.
9 Look at the multiplication problem: 124 × 0.52 Which of the following is the most reasonable estimate for the product? 60
80
90
40
10 What is the product of 57.2 × 0.1? 0.572
5.72
572
5,720 COPYING IS STRICTLY PROHIBITED BY LAW
72
DIVIDING TWO WHOLE NUMBERS AS A FRACTION
MA.5.FR.1.1 Given a mathematical or real-world problem, represent the division of two whole numbers as a fraction.
When representing a real-world-problem as division, place the dividend over the divisor. STEP 1: Identify the two whole numbers involved in the problem. The dividend is the number being divided, and the divisor is the number dividing the dividend. STEP 2: Write the dividend as the numerator of the fraction. STEP 3: Write the divisor as the denominator of the fraction. STEP 4: Simplify the fraction, if possible, by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. This step is optional but can be done to express the fraction in its simplest form.
STEP 5: If the resulting fraction is an improper fraction (the numerator is greater than or equal to the denominator), convert it to a mixed number (a whole number with a fraction part).
Determining the dividend and the divisor in a real-world problem depends on the context and the specific details provided.
STEP 1: Look for words like "divide," "split," or "share." These words mean the problem is about dividing a number into parts. STEP 2: Check the sentence structure and order of the numbers. The dividend is the number being divided, while the divisor tells how many parts the dividend is split into. STEP 3: Think about what is happening in the problem and decide which number is the total and which is the size of each part. STEP 4: Identify what you know and do not know. If the problem gives you information on one quantity and asks for the other, the known quantity is usually the dividend, and the unknown quantity is the divisor.
73
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
key words that indicate the dividend, and DIRECTIONS: Underline circle the key words that are clues for the divisor. DIVIDEND
DIVISOR
Sarah baked 24 cookies and wants to divide them equally into 6 containers. How many cookies will be in each container?
24
6
A farmer has 45 watermelons and wants to distribute them evenly among 9 crates. How many watermelons will be in each crate?
45
9
There are 4 students in a class, and the teacher wants to distribute 12 books equally among them. How many books will each student receive?
12
4
A farmer wants to divide 20 pounds of apples equally among 5 baskets. How many pounds of apples will be in each basket?
20
5
A baker has to make 3 batches of cookies. He has only 7 cups of sugar to use for all of the batches. How many cups of sugar will be used in each batch?
7
3
50
32
A box containing 15 toy cars will be shared equally among 3 children. How many toy cars will each child receive?
15
3
A construction company has a budget of $3,600 for purchasing 12 sets of tools. What is the cost of each tool set?
3,600
12
An ice cream shop sold 80 ice cream cones in a day. If the cones were shared equally among 10 customers, how many cones did each customer receive?
80
10
A group of 6 friends wants to split $65 equally. How much money will each person get?
65
6
REAL-WORD PROBLEM
A classroom with 32 students has received a donation of 50 packs of pencils. If the packs are shared equally among the students, each student will receive how many pencils?
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74
key words that indicate the dividend, and DIRECTIONS: Underline circle the key words that are clues for the divisor. DIVIDEND
DIVISOR
A school has 120 pens that need to be divided equally among 12 students. How many pens will each student receive?
120
12
A bakery needs to make 10 cakes. The 10 cakes require 32 cups of flour. How many cups of flour will be used to make one cake?
32
10
150
10
80
4
A construction company wants to allocate 240 bricks equally among 8 workers. How many bricks will each worker receive?
240
8
A farmer wants to distribute 60 eggs equally into 12 cartons. How many eggs will be in each carton?
60
12
A company produced 245 bottles of juice and wants to distribute them equally among 8 stores. How many bottles of juice will each store receive?
240
8
A bakery has 12 boxes to fill with 120 cupcakes. How many cupcakes will be in each box if they are equally distributed?
120
12
A store received 400 T-shirts and wants to display them equally on 20 racks. How many T-shirts will be on each rack?
400
20
A charity organization collected 240 blankets and wants to distribute them equally among 12 shelters. How many blankets will each shelter receive?
240
12
REAL-WORD PROBLEM
A charity organization wants to support 10 homeless individuals by providing meals. In total, they have 150 meals to provide. How many meals will each person receive? A store owner has 4 jars he wants to fill with candy for display. If he has 80 candy pieces to distribute equally, how many pieces of candy will be in each jar?
75
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Yesterday, Noel downloaded 3 files that had a total size of 7 gigabytes. If each of the files that he downloaded was the same size, what was the size of one file he downloaded yesterday? 1 of a gigabyte 7 1 of a gigabyte 3
3 of a gigabyte 7 7 gigabytes 3
2 Hannah used 5 sticks of butter to make 2 batches of frosting. She used the same amount of butter in each batch. How much butter did Hannah use to make one batch of frosting? 1 of a stick 2 5 sticks 2
1 of a stick 5 2 of a stick 5
3 Marco used 15 yards of wallpaper to cover the 4 walls of his bedroom. He used the same amount of wallpaper on each of the walls. How much wallpaper did Marco use on each of the walls? 1 of a yard 15 1 of a yard 4
4 of a yard 15 15 yards 4
4 Six containers contained a total of 10 gallons of gas. If each container had the same amount of gas, how much gas was in one of the containers? 1 4 gallons 6 1 6 gallons 10
1 1 gallons 6 1 1 gallons 10
5 A group of 5 friends ordered 7 large pizzas. If they ate all of the pizzas, and each of them ate the same amount of pizza, how much pizza did each of them eat? 1 2 pizzas 5 1 1 pizzas 5
1 5 pizzas 7 1 1 pizzas 7 COPYING IS STRICTLY PROHIBITED BY LAW
76
DIRECTIONS: Underline key words that indicate the dividend, and circle the key words that are clues for the divisor.
77
REAL-WORD PROBLEM
DIVIDEND
DIVISOR
A toy store has 600 small toys and wants to divide them equally into 20 gift bags. How many small toys will be in each gift bag?
600
20
If a grocery store has 40 shelves and wants to distribute 480 cans of soda evenly, how many cans will be on each shelf?
480
40
A teacher has 540 stickers and wants to distribute them equally among 18 students. How many stickers will each student receive?
540
18
A bakery made 960 cookies and wants to package them equally into 40 boxes. How many cookies will be in each box?
960
40
A gardener has enough land to plant 18 rows of flowers. If he has 360 flowers, how many flowers will be in each row?
360
18
A company received 750 job applications and wants to divide them equally among 30 recruiters. How many job applications will each recruiter review?
750
30
A supermarket received 1,200 cans of soup and wants to display them equally on 60 shelves. How many cans of soup will be on each shelf?
1,200
60
A classroom has 450 math worksheets, and the teacher wants to distribute them equally among 15 students. How many math worksheets will each student receive?
450
15
A bakery made 600 loaves of bread and wants to package them equally into 50 bags. How many loaves of bread will be in each bag?
600
50
A pet store has 58 bags of dog food and wants to distribute them equally among 23 customers. How many bags of dog food will each customer receive?
58
23
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Amy used 2 feet of ribbon to make 12 bows. She used the same amount of ribbons for each bow. How much ribbon did Amy use to make one bow? 1 1 1 2 feet 3 12 feet 2 2 2 12 feet 12 of a foot
2 Dorothy used six cans of canned tomatoes to make five servings of a dish. She made sure that she used the same amount of canned tomatoes in each serving. How much canned tomatoes did she use in each serving? 5 of a can 1 5 cans 6 6 1 1 1 cans 1 cans 5 6
3 Sheila has 5 cups of blue frosting to use on 20 cupcakes. She wants to use the same amount of frosting on each of the cupcakes. How much frosting should she use on each cupcake? 2 5 3 cups 3 cups 6 20 5 1 of a cup 3 cups 20 5
4 Mr. Romero has 8 acres of land that he wants to divide equally among his 3 sons. How many acres of land should he give to each of his sons? 1 1 acres 3 3 of a acre 8
3 3 acres 6 2 2 acres 3
5 Victoria bought a 10-pound turkey that she will serve for Thanksgiving. If she is expecting 24 guests at Thanksgiving and wants each of her guests to have the same amount of turkey, how much turkey should she serve each guest?
10 pounds 24 10 of a pound 24
1
4 pounds 10 4 of a pound 10
2
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78
1 Daniel used a total of 2 rolls of stamps to send 25 letters. He used the same amount of stamps on each letter. How many stamps did he use on each letter? 2 of a roll 25 1 of a roll 2
1 rolls 2 1 10 rolls 25 12
2 Ananya used a total of 16 packs of beads to make 3 bracelets. She used the same amount of beads to make each bracelet. How many beads did she use to make 1 bracelet? 1 packs 16 1 of a pack 16
5
5
1 packs 3
1 of a pack 3
3 Joel used a total of 9 liters of gas to make 5 trips. If he used the same amount of gas on each trip, how many liters of gas did Joel use on each of the trips? 1 4 liters 5 5 of a liter 9
1 5 liters 9 4 of a liter 9
4 Five cylinders contained a total of 12 gallons of gas. If each cylinder contained the same amount of gas, how much gas was in 1 of the cylinders? 1 3 2 5 gallons 1 5 gallons 2 2 5 gallons
2 1 3 gallons
5 A group of seven friends ordered eleven cakes. If they ate all of the cakes, and each of them ate the same amount of cake, how much cake did each of them eat? 1 4 cakes 5
1 7 cakes 4
3 cakes 7
1 4 cakes 7
1 79
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 Ben used a total of 5 feet of ribbon for 9 gift packs. He used the same amount of ribbon for each gift pack. How much ribbon did Ben use for one gift pack? 5 of a foot 1 3 feet 9 9 9 3 feet 1 feet 5 5
7 Remy used 7 packets of dark chocolate to make 4 servings of a dish. He made sure that he used the same amount of dark chocolate in each serving. How much dark chocolate did he use in each serving? 1 3 packets 1 4 packets 4 7 1 4 of a packet 1 packets 7 7
8
Gwen has 7 cups of red frosting to use on 18 cupcakes. She wants to use the same amount of frosting on each of the cupcakes. How much frosting should she use on each cupcake? 3 4 cups 7 18 cups 7
7 of a cup 18 1 1 cups 18
9 Mr. Bruce had 9 acres of land that he wants to divide equally among his 5 sons. How many acres of land should he give to each of his sons? 1 4 acres 5 1 of a acre 9
5 4 acres 5 2 2 acres 5
10 May bought an 11-pound turkey that she will serve for Thanksgiving. If she is expecting 23 guests at Thanksgiving and wants each of her guests to have the same amount of turkey, how much turkey should she serve each guest? 1 10 pounds 11 23 pounds 11
2 1 pounds 23 11 of a pound 23 COPYING IS STRICTLY PROHIBITED BY LAW
80
ADDING & SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS
MA.5.FR.2.1 Add and subtract fractions with unlike denominators, including mixed numbers and fractions greater than 1, with procedural reliability. Add and subtract fractions with unlike denominators by replacing the fractions with equivalent fractions with like denominators.
Example: 2 + 5 = ?
3
4
STEP 1: Identify the denominators of the fractions you are working with. Denominators are the bottom numbers of the fractions that represent the total number of equal parts in a whole. STEP 2: Determine the least common multiple (LCM) of the denominators. The LCM is the smallest multiple that is divisible by both denominators. This will be your common denominator for the fractions.
STEP 3: Convert each fraction to an equivalent fraction with the common denominator. Multiply the numerator and denominator of each fraction by the same factor that will make the denominator equal to the LCM. STEP 4: Perform the desired operation (addition or subtraction) on the numerators of the fractions. For addition, add the numerators together. For subtraction, subtract the second numerator from the first numerator. STEP 5: Keep the common denominator unchanged. STEP 6: If the resulting fraction is an improper fraction, simplify it by reducing it to its simplest form, if necessary. Find the greatest common divisor (GCD) of the numerator and denominator, and divide both by the GCD.
2 × 3
5 +
4
×
4 4
3 3
=
=
8 12
15 12 23 12
81
NOTE: The final answer should be written as a simplified mixed number, unless otherwise requested as an improper number.
=
1 12
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Add the fractions below. 2 + 1 = 7 6 4 12
1 + 2 = 2 3 6 3
1
1
5
3 + 1 =1 1 4 2 4
1
2 16 + 1 3 = 3 2
2 23 + 1 6 = 3 6
1
9
3 25 + 2 2 = 5 10
DIRECTIONS: Subtract the fractions below.
8 - 1 = 7 12 5 15
4 - 4 = 1 6 8 6
1
1
2 34 - 1 2 = 1 4
1
2 23 - 1 6 = 1 21
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3 - 2 = 1 5 4 10
1
1
3 34 - 1 2 = 2 4
82
DIRECTIONS: Add the fractions below. 3 + 1 = 13 8 6 24
2 + 1 = 15 7 4 28
3
17 2 26 + 2 8 = 424
6
1 3 + 3 = 4 8 1 8
5
3 68 + 1 9 = 5 12
3
14
3 13 + 2 5 = 5 15
DIRECTIONS: Subtract the fractions below.
3 - 2 = 4 5 6 15
2
1
3 46 - 1 4 = 2 6
83
6 - 4 = 2 9 7 21
7 - 5 = 1 8 6 24
1
1
3 34 - 1 2 = 2 4
1
5 13 - 1 2 = 3 56
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Find the sum. 1 2 1 6 5
7 8 + 2 3
6
1 6 1 6 3 6
2 The sum of 5 and 7 is equal to all of the fractions below except_____ . 3 4 35 12
3 10 24
3 15 36
3 20 60
3 Which of these pairs of equivalent fractions can be used to find the sum of the fractions 8 3 and 3 , respectively? 2 16 and 9 6 6
8 and 8 5 5
24 and 9 6 6
5 and 5 10 10
4 Find the difference.
9 2 ─ 4 3
1 11 12
1 7 12
11 4
11 3
5
The difference of 8 and 3 is equal to which of the following? 5 2 5 1 1 10 7 1 10
15 7 COPYING IS STRICTLY PROHIBITED BY LAW
84
DIRECTIONS: Add the fractions below. 2 + 1 = 13 8 9 36
3
6 + 4 =1 7 8 9 36
5 + 1 = 36 11 5 55
5
4 46 + 2 7 = 7218
1
3 12 + 2 5 = 6 10
5
41
4 17 + 1 6 = 542
DIRECTIONS: Subtract the fractions below.
7 - 1 = 23 10 8 40
9 - 7 = 25 8 9 72
2
1
4 23 - 2 6 = 2 3
85
3
7 - 5 = 19 6 7 42
11
6 13 - 1 5 = 4 15
3
29
7 59 - 3 4 = 336
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of these pairs of equivalent fractions can be used to find the difference of the fractions 10 and 3 , respectively? 3 4 7 and 3 4 4
20 and 6 6 6 40 and 9 12 12
10 and 10 3 3
2 Find the sum.
12 + 25 3 8
3 10 24
37 24
10 24
4
4
7 24
3 The sum of 2 4 and 2 1 is equal to all of the fractions below except_____ . 5 2 6 24 5 5 20 80 54 10
5 15 50
4
Which of these pairs of equivalent fractions can be used to find the sum of the fractions 9 8 and 15 , respectively? 12 18 and 18 27 and 30 16 16 24 24 22 and 24 20 20
44 and 50 40 40
5 Find the difference. 1 17 40 7 1 16
3
4 3 ─ 2 5 8
17 10 17 1 20
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86
1 The difference of 5 7 and 1 4 is equal to all of the fractions below except_____ . 10 5 3 27 30
3 36 40
3 18 20
33 10
2 Which of these pairs of equivalent fractions can be used to find the difference of the fractions 3 3 4 and 2 2 , respectively? 3 3 9 and 2 8 4 9 and 4 8 12 12 12 12 6 9 and 6 8 8 9 and 8 8 12 12 12 12
3 Which of these pairs of equivalent fractions can be used to find the sum of the fractions 1 5 and 6 3 3 , respectively? 4 10 9 10 9 3 and 6 4 and 4 12 12 12 12 10 9 10 9 1 and 3 3 and 3 12 12 12 12
4 Find the sum.
3 4 + 4 3
21 7
2 1 3
2 1 12
2 1 4
5 Find the difference. 11 15 14 3 87
7 6 ─ 3 5 12 5 2 1 15
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 Which of these pairs of equivalent fractions can be used to find the difference of the fractions 11 and 5 , respectively? 6 4 33 and 10 12 12
11 and 5 10 10 33 and 5 10 10
22 and 10 8 8
7 Find the sum.
24 + 12 5 7
3 12 35
34 35
3 35
4
4
7 35
8 The sum of 1 2 and 3 1 is equal to all of the fractions below except_____ . 5 3 22 11 4 4 30 30 4 11 15
4 33 45
9
Which of these pairs of equivalent fractions can be used to find the sum of the fractions 6 5 and 13 , respectively? 11 18 and 18 27 and 30 7 7 11 11 22 and 24 55 55
66 and 65 55 55
10 Find the difference. 12 21 12 1 25
4
2 4 ─ 3 3 7
13 18 8 1 19
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88
MULTIPLYING FRACTIONS BY FRACTIONS MA.5.FR.2.2 Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability. Multiplication to multiply a fraction by a fraction often requires converting mixed numbers to improper fractions.
Example: 2 1 ×
4
1 4 = ? 5
STEP 1: If you have mixed numbers, convert them to improper fractions. Multiply the whole number by the denominator, then add the numerator. The result becomes the new numerator, and the denominator remains the same. STEP 2: Simplify the fractions before multiplying to make calculations easier. Divide the numerator and denominator by their greatest common divisor.
STEP 3: Multiply the numerators of the fractions together to get the new numerator, and multiply the denominators together to get the new denominator. STEP 4: If possible, simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor. STEP 5: When multiplying a fraction greater than 1 by another fraction, follow the same steps. The resulting product will be greater than the original fractions. STEP 6: When dealing with mixed numbers, first convert them to improper fractions (Step 2). Then follow the same steps for multiplying fractions. Remember to simplify if needed.
Example:
2 1 × 1 4 = ? 4
5
2 1 = 9 4
4
1 4 = 9 5
5
Convert to an improper fraction.
NOTE: The final answer should be written as a simplified mixed number, unless otherwise requested as an improper number.
9 × 9 = 81 Multiply. 4 5 20 = 4 1 Convert to a mixed number. 20 89
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Multiply the proper fractions below. 3 × 1 = 3 8 2 4
2 × 1 = 2 20 4 5
4 × 3 = 12 20 4 5
5 × 2 = 10 18 3 6
DIRECTIONS: Multiply the mixed fractions below.
2 2 × 1 1 = 4 3
2 1 × 1 1 = 2 14
2
5
7
2 3 × 1 2 = 4 12 4
3
3
15
2 1 × 1 1 = 27 4
COPYING IS STRICTLY PROHIBITED BY LAW
5
10
90
DIRECTIONS: Multiply the proper fractions below. 9 3 × 3 = 24 6 4
15 5 × 3 = 45 9 5
6 × 7 = 42 72 8 9
9 × 3 = 27 48 8 6
DIRECTIONS: Multiply the mixed fractions below. 1
3 4 × 2 1 =9 2
2 5 × 2 1 = 6 11
5 3 5 × 2 1 = 8 32
1 8 × 11 = 1 15 9 5
5
8
91
2
4
6
3
18
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What is the product of
1 1 × 4 ? 8 3
14 24 4
13 24
1 24
4
11 24
1
2 40
2 Multiply 5 × 2 . 4 10 7 14 10 40
50 8
3 Florence has 7 cup of flour in her pantry. She uses 1 of the flour for her banana bread recipe. 8 4 How much flour does she use? 28 cups 7 cup 8 32 8 12 cup
7 12 cup
4 Frank has 1 2 pints of hot sauce in his fridge. He uses 3 of the sauce for his buffalo chicken 3 7 recipe. How much sauce does he have left? 20 pint 21
15 pint 21
35 pints 12
35 pints 9
5 Which number makes the equation below true? 9 3 × 13 4 = ________ 27 52
36 39
12 17
75 52 COPYING IS STRICTLY PROHIBITED BY LAW
92
DIRECTIONS: Multiply the proper fractions below. 11 7 77 × = 156 13 12
64 8 8 × = 150 10 15
50 10 5 × = 132 12 11
81 9 × 9 = 40 5 8
DIRECTIONS: Multiply the mixed fractions below.
2 3 × 2 1 =61
1 1 × 2 1 = 2 19
3 5 × 2 1 =97
4 × 6 1 = 34 7 7 4
5
6
93
3
2
15
12
6
6
36
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 4 Stephanie multiplies 11 × . Which of the following could not be her answer? 2 5 44 4 4 10 10 55 42 5 8
2 Harrison has 7 of a jug of bubbles on the porch. He uses 1 of his bubbles in his bubble 18 2 machine. What fraction of the jug does Harrison use? 7 7 jug jug 16 36 8 jug 20
3 jug 9
3 Luis has 3 of a pizza left over. 1 of the leftover pizza has mushrooms on it. How much 8 3 mushroom pizza is there? 4 pizza 9 pizza 11 8 3 pizza 17 pizza 24 24
4 What is 2 of 8 ? 3 9 16 27
18 24
10 12
14 9
5 Find the product of
3 2 and . 10 4
12 20
5 14
16 20
6 40
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94
1 What is the product of 2 × 5 1 ? 3 2 22 6
11 6
32 4
32 6
2 1 1 Sid cuts 8 of a cake for himself and his friends. He gives 3 of the cake to his cousin. How much cake does he have left for his other friends? 8 24
1 11 24
13 24
13 48
3 Which number makes the equation below true? 4 7 15 × 11 = ________ 28 165
28 11
24 165
28 15
4 Henry bought 1 pound of cupcake mix and 2 pound of cake mix. What is the total 4
3
weight of the cupcake and cake mix that Henry bought?
11 12
2 8
4 10
5 What is 3 of 8 ? 10 11
95
8 110
28 111
24 110
12 110 MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
2 12
6 What is the product of 8 and 1 ? 9 4 8 36 32 27
1 32 48 36
7
11 gallon of gasoline available. He uses 1 of the gasoline for his new motorized 16 3 invention. What fraction of gasoline did Nicholas use? Nicholas has
3 gasoline 16
1 gasoline 32
11 gasoline 48
11 gasoline 64
8 Jack has 4 of a pizza left over. 1 of the leftover pizza has olives on it. How much olive pizza is 9 4 there? 1 pizza 36 5 pizza 13
4 pizza 36 1 pizza 13
9 What is 7 of 3 ? 8 4 16 32
10 32
21 32
21 12
10 Find the product of 6 and 4 . 7 30 24 23 1 35
4 35 7 30
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96
PREDICTING THE RELATIVE SIZE OF THE PRODUCT MA.5.FR.2.3 When multiplying a given number by a fraction less than 1 or a fraction greater than 1, predict and explain the relative size of the product to the given number without calculating.
Make predictions about the relative size of the product without actually performing the multiplication.
STEP 1: Determine if the fraction is less than 1 or greater than 1. If the fraction is less than 1, the numerator is smaller than the denominator. For example, 1/2 or 3/4. If the fraction is greater than 1, the numerator is larger than the denominator. For example, 3/2 or 5/4.
STEP 2: Consider the effect of multiplying by a fraction less than 1. When you multiply a given number by a fraction less than 1, the product will be smaller than the given number. This is because you are taking a part of the given number, which is less than the whole.
Example:
2 × 75 3
?
75
2 × 75 3
?
75
50
75
STEP 3: Consider the effect of multiplying by a fraction greater than 1. When you multiply a given number by a fraction greater than 1, the product will be larger than the given number. This is because you are taking more than the whole, as the numerator of the fraction is larger than the denominator.
Example:
4 × 75 3
?
75
4 × 75 3
?
75
100 97
75
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
a check mark ( DIRECTIONS: Place greater than 1.
) next to all fractions that are
5 8
11 6
17 10
1 3
5 3
17 25
1 100
7 6
9 5
3 4
calculating, complete using greater than (>), DIRECTIONS: Without less than (< ), or equal to (=) symbols. 1 × 18 2
18
1 × 25 8
25
1 × 35 5
35
5 × 33 3
33
7 × 30 5
30
2 × 75 5
75
2 × 20 3
20
10 × 45 25
45
9 × 50 5
50
17 × 200 10
200
3 × 100 25
100
9 × 65 5
65
11 × 80 10
80
3 × 225 5
225
5 × 15 6
15
1 × 44 6
44
3 × 85 8
85
11 × 66 6
66
17 × 72 12
72
1 × 17 4
17
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98
Place a check mark (
DIRECTIONS: greater than 1.
) next to all fractions that are
14 6
2 4
7 9
16 8
8 2
11 23
32 20
6 7
10 5
5 4
calculating, complete using greater than (>), DIRECTIONS: Without less than (< ), or equal to (=) symbols.
99
4 × 25 2
25
2 × 45 5
45
1 × 43 5
43
8 × 30 3
30
8 × 15 9
15
35 × 144 35
144
5 × 32 5
32
84 × 56 63
56
10 × 65 15
65
12 × 70 12
70
7 × 90 7
90
20 × 325 60
325
11 × 80 10
80
3 × 88 3
88
6 × 16 6
16
1 × 22 8
22
35 × 73 19
73
4 × 78 4
78
5 × 55 9
55
1 × 125 4
125
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of the following expressions will have a product less than 24? 24 × 10 24 × 7 7 9 24 × 2 24 × 1 3 5 4
2 The product of 57 × 109 will be ____________ 109. 57 less than
greater than
equal to
twice as much as
3 In the equation below, “n” will equal ________________. 4 1 × 67 = n 3
a number greater than 67
a number less than 67
one-third of 67
67
4 Which of the following statements is true? 4 × 100 > 100 5 7 9 × 100 < 100 10
5 × 100 > 100 4 5 5 × 100 > 100
5 Alana wants to multiply 78 by a number to get an answer less than 78. Select all the answers that could be Alana’s number. 1 100 100
1 2
5 12
101 100
1 3
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100
a check mark ( DIRECTIONS: Place greater than 1.
) next to all fractions that are
45 18
12 9
9 10
3 7
19 8
45 65
18 25
9 2
26 13
4 5
calculating, complete using greater than (>), DIRECTIONS: Without less than (<), or equal to (=) symbols.
101
5 × 200 9
200
2 × 65 2
65
1 × 58 5
58
6 × 40 6
40
8 × 75 8
75
13 × 85 23
85
6 × 17 26
17
94 × 195 87
195
10 × 400 15
400
7 8
× 16
16
23 × 23 23
23
75 × 67 40
67
2 × 36 8
36
9 × 24 9
24
9 × 42 3
42
2 × 225 5
225
5 × 77 7
77
2 × 22 6
22
17 × 12 17
12
1 4
53
× 53
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1 Kyler multiplies 32 by a number, and his answer is greater than 32. Which of the following could not be his number? 108 100
6
1 5
4 4
99 7
2 A number multiplied by a fraction greater than a whole will always have a product that is: equal to the original number
one-tenth of the original number
greater than the original number
less than the original number
3 81 is multiplied by 1 . Steve thinks the answer will be equal to 81, and Bailey thinks the answer 8 will be less than 81. Who is correct? Steve is correct because 81 and 1 have the same digits, which means the product will 8 be the same as the original number. Bailey is correct because 1 is less than a whole, which means the product will be less 8 than the original number. Bailey is correct because 1 is greater than a whole, which means the product will be 8 less than the original number. Neither of them is correct because the product will actually be greater than 81.
4 Which of the following expressions will have a product greater than 157? 157 × 7 3 5
3 × 157 8
15 × 157 15
157 × 14 15
5 Which of the following makes the statement below true? 7 × ________ = a number greater than 7 16 17
7 7
1 7
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83 70 102
1 Which of the following expressions will have a product greater than 97? 8 1 97 × 8 97 × 2 97 × 3 97 × 3 2 5
2 Ms. Perez wrote the statement below on the board: 1 In the equation 5 × 22 = m, “m” will equal _____. What phrase makes this correct? a number greater than 22
a number lesser than 22
a number equal to 22
one-fourth of 22
3 Which of the following statements is true? 3 x 45 < 45 3 1 x 45 < 45 5
2 x 45 > 45 3 1 1 x 45 > 45 2
4 Anna wants to multiply a number by 65 to get an answer greater than 65. Select all of the possible answers that could be Anna’s number. 7 11 31 2 7 5 3 7
4 9
5 Kayla multiplies the number 45 by 5 and says the answer will be the same as 45. Mike says the 5 answer will be less than 45. Who is correct?
Mike is correct since 5 is less than a whole, and will give an answer less than 45. 5 Kayla is correct since 5 is same as 1 and the product will therefore be 45. 5 Mike is correct since 5 is greater than a whole and will give an answer greater than 45. 5 Neither of them is correct since the answer is a number greater than 45. 103
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6 Which of the following makes the statement below true? 6 × ______ = a number less than 6 2 4
5 3
9 7
12 3
7 The product of 4 × 117 will be ____________ 117. 13 less than
greater than
equal to
twice as much as
8 In the equation below, “k” will equal ________________. 3 × 1 × 56 = k 15 a number greater than 56
three times 56
one-fifth of 56
56
9 Product of a number and a fraction that is less than one will always be: equal to the original number
one-tenth of the original number
greater than the original number
less than the original number
10 Abadi multiplies 17 by a number, and his answer is less than 17. Which of the following could be his number? 43 68
32 8
7 3
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10 10
104
DIVIDING UNIT FRACTIONS BY WHOLE NUMBERS MA.5.FR.2.4 Extend previous understanding of division to explore the division of a unit fraction by a whole number and a whole number by a unit fraction.
Divide unit fractions by a whole number by multiplying its reciprocal.
STEP 1: Understand that a unit fraction is a fraction with a numerator of 1 (e.g., 1/2, 1/3, and 1/4). STEP 2: When dividing, express the whole number as a fraction with a denominator of 1. For example, if you want to divide 1/3 by 2, rewrite 2 as 2/1. STEP 3: Take the reciprocal of the whole number (flip it). For example, 1/3 ÷ 2/1 = 1/3 ÷ 1/2. STEP 4: Next, change the division sign to multiplication and multiply the unit fraction by the reciprocal of the whole number. For example, 1/3 × 1/2 = 1/6.
1 ÷ 2 = ? 3 1 × 1 = ? 3 2 1 × 1 = 1 3 6 2 Divide a whole number by a unit fractions by multiplying its reciprocal.
STEP 1: Express the whole number as a fraction with a denominator of 1 (e.g., 6 = 6/1). STEP 2: When dividing a whole number by a unit fraction, take the reciprocal (flip) of the unit fraction. For example, if you want to divide 6/1 by 1/2, rewrite 1/2 as 2/1. STEP 3: Change the division sign to multiplication and multiply the whole number by the reciprocal of the unit fraction. For example, 6/1 × 2/1 = 12/1.
6 ÷
1 = ? 2
6 × 2 = ? 1 1 6 × 2 = 12 1 1 1 105
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
unit fractions by a whole number by multiplying DIRECTIONS: Divide its reciprocal.
No.
Division Problem
Multiply the Reciprocal of the Unit Fraction
Answer
2 7 ÷ 8 =
7 8 = × 2 1
56 2
4 8 ÷ 6 = 3 9 ÷ 9 =
8 1 × 9 1 ×
9 3 =
48 4 81 3
1 4 ÷ 2 =
4 2 = × 1 1
8 1
2 2 ÷ 3 =
2 3 = 1 × 2
6 2
7 3 ÷ 9 =
3 9 = 1 × 7
27 7
6 4 =
a whole number by a unit fractions by multiplying DIRECTIONS: Divide its reciprocal.
No.
Division Problem
Multiply the Reciprocal of the Whole Number
Answer
2 ÷ 7 = 8
2 1 8 × 7 =
2 56
4 ÷ 8 = 6 3 ÷ 9 = 9
4 6 × 3 9 ×
1 9 =
4 48 3 81
5 ÷ 6 = 8
5 1 = 8 × 6
5 48
4 ÷ 7 = 5
4 1 = × 5 7 9 1 × 5 = 10
4 35
9 10 ÷ 5 =
1 8 =
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9 50
106
unit fractions by a whole number by multiplying DIRECTIONS: Divide its reciprocal.
No.
Division Problem
Multiply the Reciprocal of the Unit Fraction
Answer
1 6 ÷ 2 =
6 2 × 1 = 1
12 1
2 10 ÷ 4 = 3 4 ÷ 5 =
10 1 × 4 1 ×
5 3 =
40 2 20 3
2 6 ÷ 5 =
6 5 × 2 = 1
30 2
3 12 ÷ 4 =
12 4 1 × 3 =
48 3
3 7 ÷ 5 =
7 5 = 1 × 3
35 3
4 2 =
a whole number by a unit fractions by multiplying DIRECTIONS: Divide its reciprocal.
No.
Division Problem 1 ÷ 4 = 2
Answer 1 8
3 ÷ 5 = 5
2 4 × 3 5 ×
1 = 7 1 5 =
2 28 3 25
2 ÷ 2 = 5
2 1 × 2 = 5
2 10
3 ÷ 6 = 4
3 1 = 4 × 6
2 24
4 ÷ 9 = 5
4 1 5 × 9 =
4 45
2 ÷ 7 = 4
107
Multiply the Reciprocal of the Whole Number 1 1 = × 4 2
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1 1 Remi has 7 feet of string she is using to make bracelets. If each bracelet uses foot of string, 3 how many bracelets will she be able to make? 7 bracelets 21 bracelets 3 1 1 2 3 bracelets 21 of a bracelet
2 1 What is the quotient of 8 ÷ 4? 1 32
4 8
1 2
1 24
3 Which number makes the equation below true?
1 ÷ ______ = 1 2 10 20 10
1 5 5
4 1 Eli has 3 of a pizza he needs to divide equally for his 5 friends. What fraction of a pizza will each friend receive? 5 pizzas 3
1
2 3
pizzas
1 15 pizza
1 53
pizzas
5 Solve: 9 ÷
1 7
9 7 7 9
1 63 63 COPYING IS STRICTLY PROHIBITED BY LAW
108
unit fractions by a whole number by multiplying DIRECTIONS: Divide its reciprocal.
No.
Division Problem
Multiply the Reciprocal of the Unit Fraction
Answer
3 11 ÷ 6 =
11 6 = × 3 1
66 3
5 6 ÷ 7 = 8 7 ÷ 9 =
6 1 × 7 1 ×
9 8 =
42 5 63 8
7 9 ÷ 6 =
9 6 = × 1 7
54 7
5 12 ÷ 4 =
12 4 = 1 × 5
48 5
6 8 ÷ 7 =
8 7 1 × 6 =
56 6
7 5 =
a whole number by a unit fractions by multiplying DIRECTIONS: Divide its reciprocal.
No.
109
Division Problem
Multiply the Reciprocal of the Whole Number
Answer
3 ÷ 12 = 6
3 1 = × 12 6
3 72
5 ÷ 10 = 7 8 ÷ 8 = 9
5 7 × 8 9 ×
1 8 =
5 70 8 72
7 ÷ 7 = 6
7 1 = × 6 7
7 42
5 ÷ 9 = 4
5 1 = 4 × 9
5 36
6 ÷ 8 = 7
6 1 = 7 × 8
6 56
1 10 =
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Nora has 11 chocolate bars to share with her class. She divides each bar into bar 1 pieces. If 3 there are 24 kids in her class, will she have enough chocolate for each student to have a piece? No, because she will only have 1 bar. 33 Yes, because she will have 33 pieces. Yes, because she will have exactly 24 pieces. 2 No, because she will only have 3 3 pieces
2 Zane has 1 of a bowl of punch left at his party. He shares it equally in 10 cups for the party 4 guests. What fraction of the bowl of punch does each of the guests receive?
1 of a bowl 40 1 10 bowls 4
40 bowls 10 bowls 4
3 Ethan is running on a 5-mile path by his school. He stops every 1 mile to drink some water and 6 runs the whole distance of the path. How many breaks does he take for water? 30 breaks
11 breaks
5 6 of a break
1 1 5 breaks
5
45
1 45
1 36
9
4 1 Find the quotient of 3 and 15 . 3 15
5 Which number makes the equation below true?
1 18 ÷ 2 = __________ 1 32
2 18
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110
1 Finney marks the racing ground into tracks for an event. The width of the entire track is 235 feet. If he has to mark every 2 1 feet, how many tracks can he make? 2 94 tracks
84 tracks
96 tracks
100 tracks
2 4 1 What is the quotient of 15 ÷ 3 ? 4 45 4 5
2 15 3 15
3 Gwen wants to share 2 of an apple pie with her 4 friends. What share of the pie will each of 3 them get? 2 2 12 15 1 1 4 15
4 Bruce has 1 carton of milk left after breakfast. He shares the milk equally among his 3 cats. 4 What fraction of the carton of milk does each cat get? 1 6
3 4
4 3
1 12
5 4 What is 32 ÷ 9 ?
111
8 9
9 32
72
96
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6 Which number makes the equation below true ? 7 ÷ 5 = _____ 45 7 225 35 225
35 45 225 7
7 What is the quotient of 2 ÷ 4 ? 5 3 4 6 10 12
4 15 15 8
8 Emma wants to share 3 of a box of candies with her 4 siblings. What share of the candies will 5 each of them get? 1 3 5 20 4 3 5 9
9 Alex has 3 of a watermelon left after lunch. He wants to share the watermelon equally among 4 his 6 friends. What fraction of the watermelon does each friend get? 1 4
1 6
1 2
3 24
10 2 What is 36 ÷ 3 ? 54
54 3
54 2
18 COPYING IS STRICTLY PROHIBITED BY LAW
112
SOLVE MULTI-STEP REAL-WORLD PROBLEMS MA.5.AR.1.1 Solve multi-step real-world problems involving any combination of the four operations with whole numbers, including problems in which remainders must be interpreted within the context. Use a bar model to visually represent the quantities and relationships described when solving multi-step real-world problems.
STEP 1: Carefully read the real-word problem. Identify the keywords which indicate what needs to be solved and the operations to be used. STEP 2: Some common keywords used in multi-step word problems are as follows: ADDITION: in all, sum, altogether, combined, plus, increased by, more than, added to, totaled SUBTRACTION: less, fewer, left, minus, difference, take away, deduct, remaining, decreased by MULTIPLICATION: product, times, of, multiplied by, twice, double, triple, each*, per* DIVISION: quotient, divide, half, split, share, evenly, equal groups, each*, per*
* Keywords for either multiplication or division. Students must use other keywords to determine which of the two operations is required. STEP 3: Identify the known quantities using bars (rectangles) that are proportional to the values you wish to represent. Be sure to include the units of measurement. STEP 4: Continue manipulating the bars to represent each operation until the problem is solved. STEP 5: Verify the answer using the bar model you created. Example: Sarah baked 3 trays of cookies. Each tray had 24 cookies. If she gave half of the cookies to her friends, how many cookies did she have left? 72 cookies
Sarah baked 3 trays of cookies. Each tray had 24 cookies.
24 cookies
24 cookies
24 cookies
Tray 1
Tray 2
Tray 3
24 × 3 = 72 cookies
72 cookies This bar model shows that there were 72 cookies on all 3 trays.
72 cookies
If she gave half of the cookies to her friends, how many cookies did she have left?
72 ÷ 2 = 36 cookies 113
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Use bar models to solve the real-world problems below. Paul filled 4 containers with golf balls. Each container had 20 golf balls. If he gave half of the golf balls to his friends, how many golf balls does he have left?
40 golf balls
80 golf balls 20 × 4 = 80 golf balls
20 golf balls 20 golf balls 20 golf balls 20 golf balls
80 golf balls 40 golf balls
80 ÷ 2 = 40 golf balls
40 golf balls
A school is organizing a field trip for 240 students. Each bus can carry 48 students. If the school wants to ensure that every student can go on the field trip, how many buses are needed?
240 students 48 students 48 students 48 students 48 students 48 students
240 ÷ 48 = 5 buses
5 buses
240 -48 192 -48 144 -48 96 -48 48
Nadia is saving money to buy a bicycle. She has saved $40 so far. Her goal is $200. If she saves $8 every week, how many more weeks does she have to reach her goal?
20 weeks
$200 $40
$8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 $8 1
2
3
4
5
6
7
8
9
10
$200 - $40 = $160
11 12 13 weeks
14
15
16
17
18
19 20
$160 ÷ $8 = 20 weeks
A group of friends is planning a trip to the amusement park. The admission tickets cost $25 each, and they also want to buy snacks during the day. The cost of snacks is $10 per person. If there are 8 people in the group, how much money will they need in total for the tickets and snacks?
280 dollars
280 students $25 + $10
$25 + $10
$25 + $10
$25 + $10
$25 + $10
$25 + $10
$25 + $10
$25 + $10
($25 + $10) × 8 = $280 COPYING IS STRICTLY PROHIBITED BY LAW
114
DIRECTIONS: Use bar models to solve the real-world problems below. John bought 4 bags of tennis balls. Each bag had 16 tennis balls. If he gave one-fourth of the tennis balls to his tennis club, how many tennis balls does he have left?
48 tennis balls
64 tennis balls 16 tennis balls
16 tennis balls
16 tennis balls
16 tennis balls
64 ÷ 4 = 16 tennis balls
64 tennis balls 16 tennis balls
64 - 16 = 48 tennis balls
48 tennis balls
The town hall needs 1,200 bags of cement to rebuild the sidewalks. Knowing that each truck can transport 300 bags of cement, how many trucks are needed to transport all the bags of cement?
4 trucks
1,200 bags of cement 300 bags
300 bags
300 bags
300 bags
2
3
4
1
1200 ÷ 300 = 4 trucks
Amelia is saving money to buy a series of books. She has saved $20 so far. Her goal is $80. If she saves $6 every week, how many more weeks does she have to reach her goal?
10 weeks
$80 $20
$6
$6
$6
$6
$6
$6
$6
$6
$6
$6
1
2
3
4
5
6
7
8
9
10
$80 - $20 = $60 $60 ÷ $6 = 10 weeks
Brad, Alicia, and Jennifer plan to invite their parents to the theater and then to dinner for their 20th wedding anniversary. The cost for the theater ticket is $20 per person, and the restaurant is $110 for both parents. Knowing that Brad, Alicia, and Jennifer shared everything equally, how much will each pay?
70 dollars
$210
$20 $20 $20 $20 $20
$110
($20 × 5) + $110 = $210
$210 $70 115
$70
$70
$210 ÷ 3 = $70 each person
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1 Before a book release, a bookstore prepares by ordering boxes of the book in hardcover and paperback. The manager ordered 15 boxes of hardcover books. Each box contains 12 books. She also ordered 22 boxes of paperback books. Each of those boxes contains 24 books. How many more paperback books will the store have to sell than hardcover books? 180 more
528 more
348 more
448 more
2 Cecilia is planning a wedding and needs to transport the glass vases for the tables. There will be 3 vases on each table. There will be 28 tables at the event. She can fit 9 vases in each box that she packs. How many boxes will she need to get all of the vases to the wedding venue? 84 boxes
9 boxes
10 boxes
27 boxes
3 Seth is putting photos into albums. He has to sort through 4 boxes of 525 photos each. The albums he is filling will hold 200 pictures each. If he completely fills the albums as he sorts, how many albums will be totally full once he organizes all of his boxes of pictures? 10.5 full albums
10 full albums
11 full albums
2 full albums
4 Simon, Heather, and Sawyer are preparing bags of cookies for the school bake sale. Each bag must contain 7 cookies. If Simon bakes 64 cookies, Heather bakes 48, and Sawyer bakes 72, how many complete bags will they have to sell? How many cookies will they have left over? 27 bags, 2 left over
24 bags, 6 left over
28 bags, none left over
26 bags, 2 left over
5 The local animal shelter is asking for donations of bags of dog food. The community collected 23 large bags and 56 small bags of dog food for the shelter. If each of the small bags contains 10 pounds of food, and each of the large bags contains 18 pounds of food, how many pounds of food did the shelter collect altogether? 560 pounds
414 pounds
767 pounds
974 pounds COPYING IS STRICTLY PROHIBITED BY LAW
116
DIRECTIONS: Use bar models to solve the real-world problems below. To reward his students for the work well done, Alex bought 3 boxes of chocolate. Each box contains 15 chocolates. There are 3 classes, and each class has 12 students. If he gives out 1 chocolate per student, how much chocolate does he have left?
9 chocolates
45 chocolates 15 chocolates
15 chocolates
15 × 3 = 45 chocolates
15 chocolates
45 chocolates 12 students
12 students
12 students
45 - (12 × 3) = 9 chocolates
9 chocolates
A hotel organizes a cultural visit for its guests. The hotel has 76 guests, and the minibuses of the hotel can transport 20 people. How many minibuses are needed to transport all guests?
76 guests 20 guests
16 20
76 ÷ 20 = 3
20 guests
20 guests
16 guests
2
3
4
1
4 minibuses
The 16 people will ride on
another bus. Total of 4 buses.
Andrew wants to go camping with his friends, and for that, he needs to have $200. He already has $120. Every day he cuts 2 lawns for $8 per lawn. How many more days does Andrew need to cut grass to earn this amount?
5 days
$200
$200 - $120 = $80 $8 × 2 = $16
$120
$80 ÷ $16 = 5 days
$16
$16
$16
$16
$16
1
2
3
4
5
days
Sara and her four brothers want to buy a kennel for the dog they want to adopt. The kennel costs $80. They also want to buy dog food and toys that cost $100 total. Knowing that they share costs equally, how much did it cost for each?
$180 $36
117
$36
$36
36 dollars
$80 + $100 = $180 $36
$36
$180 ÷ 5 = $36
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1 Cyrus has a goal of running 100 miles in the month of August. For the first 14 days, he runs exactly 3 miles each day. How many more miles does Cyrus have left to run in August to hit his goal? 58 miles
42 miles
86 miles
83 miles
2 The local science museum is hosting a field trip day for 4 local schools. They plan to have lunch for all the students in the event hall at the same time. The schools will be sending the following School Number of Students Attending numbers of students on the field trip. If each table in the event hall can seat 18 students, how many tables will the science museum need to have ready to make sure each student has a seat for lunch? 833 tables
48 tables
Elm Elementary Sunset Elementary Pine Elementary Valley Elementary
851 tables
147 301 238 165
47 tables
3 A construction company is building houses in a new neighborhood. It takes 11 gallons of paint to paint the interior of each home. The company plans to build 37 of these homes, and the paint costs $26 per gallon. How much money will the company need to spend on paint for the interiors of these houses? $3,256
$10,842
$10,582
$364
4 Baylor and his brother and sister are staying at the beach for the summer. They decide to collect seashells all summer to put in vases to decorate the beach house. When their vacation is over, Baylor has collected 383 shells, his sister has found 291, and his brother has saved 414. If they put an equal number of shells in each of 16 vases, how many shells will be in each vase? 68 shells
1,088 shells
17,408 shells
70 shells
5 A sold-out concert had 32 rows of 65 seats for attendees. On the night of the concert, 117 people that purchased tickets did not attend because it was raining. How many seats were occupied the night of the concert? 2,080 seats
1,963 seats
214 seats
2,197 seats COPYING IS STRICTLY PROHIBITED BY LAW
118
1 A fruit seller has arranged 128 boxes of apples and 387 boxes of peaches in his stall. The apples are packed in boxes of 4 apples each, and the peaches are packed in boxes of 6 peaches each. How many more peaches are there than apples? 1,384
1,548
1,810
2,678
2 Nick, Emily, and Max are collecting toys for a charity program. Nick collects 487 toys, Emily collects 234 toys, and Max collects 440 toys. If each child is given 3 toys, how many children can they distribute the toys to? 387
345
427
486
3 John has 782 bottles of soda to be arranged on shelves. 16 bottles can be arranged on 1 shelf. How many shelves will be fully filled, and how many bottles will be left that partly fill a shelf? 36 shelves, 12 bottles left
48 shelves, 14 bottles left
48 shelves, 10 bottles left
50 shelves, 14 bottles left
4 Kevin drives 12 miles to his job and back every day. How many miles does he drive in 2 weeks if he works 5 days a week? 200 miles
220 miles
240 miles
360 miles
5 The cost of tickets for a magic show is given below. What is the total amount of money collected for the tickets if 46 adults and 14 children booked tickets for the show?
119
Ticket Type
Cost
Adults
$ 7.60
Children
$ 2.45
$353.60
$383.90
$396.48
$483.70 MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 Amani wants to paint his tree house green and yellow. The green paint costs $32.00 per gallon and the yellow paint costs $27.50 per gallon. What is the total cost of painting if he needs 2 gallons of green paint and 4 gallons of yellow paint? $150.00
$166.00
$174.00
$182.00
7 A fruit seller wants to donate fruit to a boarding school. Each resident will receive 0.5 pound of strawberries, 2 bananas, 3 apples and 2 pounds of grapes. Knowing that the boarding school has 76 children, how much will the seller donate? 48 pounds of strawberries, 152 bananas, 228 apples, and 152 pounds of grapes
38 pounds of strawberries, 152 bananas, 238 apples, and 162 pounds of grapes
38 pounds of strawberries, 182 bananas, 228 apples, and 152 pounds of grapes
38 pounds of strawberries, 152 bananas, 228 apples, and 152 pounds of grapes
8 Pedro wants to divide 134 balls among his 19 students. Knowing that each student will receive 6 balls, how much will he have left? 18 balls
20 balls
24 balls
30 balls
9 Fatima has decided to sell orange juice to buy a bicycle. On the first day, she sold 40 cups at $0.80 and 85 large cups at $1.20. How much did she earn? $134
$124
$143
$132
10 Joseph, Ted, and Sophie want to decorate and paint the garage to surprise their father. Here are the costs: painting-$80, storage furniture-$50, miscellaneous-$50. If the costs are shared equally, how much will each have to pay? $70
$80
$60
$55 COPYING IS STRICTLY PROHIBITED BY LAW
120
SOLVE REAL-WORLD PROBLEMS INVOLVING FRACTIONS MA.5.AR.1.2 Solve real-world problems involving the addition, subtraction, or multiplication of fractions, including mixed numbers and fractions greater than1. When solving real-world problems involving fractions, including mixed numbers and fractions greater than 1, first identify what needs to be solved and what operations (addition, subtraction, or multiplication) need to be carried out.
STEP 1: Carefully read the real-word problem. Identify the keywords which indicate what needs to be solved and the operations to be used. Some common keywords used in multi-step word problems are as follows: ADDITION: in all, sum, altogether, combined, plus, increased by, more than, added to, totaled SUBTRACTION: less, fewer, left, minus, difference, take away, deduct, remaining, decreased by MULTIPLICATION: product, times, of, multiplied by, twice, double, triple, each*, per* DIVISION: quotient, divide, half, split, share, evenly, equal groups, each*, per*
* Keywords for either multiplication or division. Therefore, students must use other keywords to determine which of the two operations is required. STEP 2: Convert mixed numbers to improper fractions where needed. Find the least common denominator (LCM) for all fractions in the problem with different denominators. STEP 3: If the problem involves addition or subtraction, use the common denominator to add or subtract the fractions. Add or subtract the numerators, keep the common denominator, and simplify if possible. STEP 4: If the problem involves multiplication, multiply the numerators together to get the new numerator. Multiply the denominators together to get the new denominator. STEP 5: Simplify the resulting fraction by dividing the numerator and the denominator by the greatest common factor (GCF). STEP 6: Convert the answer to a mixed number by dividing the numerator by the denominator. The quotient will be the whole number, and the remainder will be the numerator of the fraction. Write the mixed number in the form of a whole number and a proper fraction.
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DIRECTIONS: Solve each word problem below. Aisha is making a fruit salad. The recipe calls for 1 cup of chopped strawberries. She wants to
2
make 3 times the recipe. How many cups of chopped strawberries does she need altogether?
1 1 cups 2
Mei had 2 3 cups of lemonade in a pitcher. She poured out 1 1 cups of lemonade. How much
4
2
lemonade is left in the pitcher?
1 1 cups 4
A school basketball team won 2 of their games this season. If they played a total of 18 games,
3
how many games did they win?
12 games
A construction project requires 7 1 wooden boards. Mike already purchased 3 2 boards.
2
3
How many more boards does he need to purchase?
3 5 boards 6
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DIRECTIONS: Solve each word problem below.
A fruit smoothie recipe calls for 1 3 cups of strawberries. If you have only 1 cup of
2
4
strawberries, how many more cups do you need?
1 1
4
cups
Lisa is training for a marathon. On Monday, she ran 3 1 miles, and on Tuesday, she ran 2 3 miles. How many miles did Lisa run in total?
6 1
4
2
4
miles
Stacey is baking cupcakes. The recipe calls for 2
3 cups of flour. She only has 1 1 cups of 4 2
flour. How much more flour does Stacey need to complete the recipe?
1 1
4
cups
A pizza delivery driver traveled 5 3 miles to deliver the first order, then 2 1 miles to deliver
4
the second order. How many miles did the driver travel in total?
2
8 1 miles 4
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1 A recipe for a batch of pancakes calls for 3 3 cups of flour. Jennifer wants to make 4 1 batches 4 3 of pancakes. How much flour will she need? 3 1 12 4 cups 14 2 cups 3 1 16 8 cups 16 4 cups
2 During a science experiment, the granite rock weighed 8 7 grams, while the basalt rock was 10 4 4 grams heavier than the granite rock. What is the weight of the basalt rock? 5 3 4 12 10 grams 12 5 grams 13 1 grams 2
14 2 grams 5
3 2 At the start of the week, a bakery had 7 3 sacks of sugar in stock. By the end of the week, only 3 3 4 sacks of sugar were left. How many sacks of sugar did the bakery use during the week? 3 1 sacks 3 1 sacks 6 12 4 1 sacks 4 1 sacks 6 12
4 A bay seal weighed 3 7 pounds when it was born. After a month, the baby seal is 2 1 times 2 8 heavier than when it was born. How much does the baby seal weigh after a month? 6 3 pounds 4 10 3 pounds 8
9 11 pounds 16 12 1 pounds 4
5 Martin traveled a total of 18 1 miles. If he walked 6 3 miles and ran the rest, how many 6
4
miles did he run? 11 1 miles 12 12 5 miles 12
11 5 miles 12 12 7 miles 12 COPYING IS STRICTLY PROHIBITED BY LAW
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DIRECTIONS: Solve each word problem below.
Sarah walked 4
3 1 miles in the morning and 2 miles in the evening. How many miles did she 4 2
walk in total?
7 1 miles 4
Alex saved 3 1 dollars per week for 5 weeks. How much money did Alex save in total?
4
16 1
4
dollars
A construction worker is building a brick wall. In one day, he lays
3
2 4 rows of bricks. If each
row consists of 8 bricks, how many bricks did the construction worker lay in total?
22 bricks
A swimming pool is 10
1 3 feet long. If each swim lane is 2 feet wide, how many complete 2 4
swim lanes can fit in the pool?
3 119 lanes
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1 Imelda bought 3 1 pounds of red apples and 2 3 pounds of green apples at the supermarket. 6 4 How many pounds of apples did she buy in all? 5 5 pounds 5 3 pounds 8 4 5 11 pounds 6 1 pounds 12 4
2
A metal crate full of iron bars weighs 3 5 tons. After removing all the iron bars, the crate 12 1 weighs only of a ton. What was the weight of the iron bars? 4 1 3 tons 3 1 tons 2 3 3 1 tons 3 1 tons 6 12
3
Jake bought 3 full bags of cement and 1 bag that is 2 full. If each full bag of cement weighs 8 1 2 3 pounds, how many pounds of cement did Jake buy? 24 5 pounds 28 7 pounds 6 12 1 1 30 3 pounds 31 6 pounds
4 Tom bought a plant that was 2 9 feet tall. After just 1 week, the height of the plant increased 10 2 by 1 feet. What was the height of the plant after one week? 5 4 3 feet 3 11 feet 10 20 4 3 feet 3 7 feet 10 5
5
Coach Hedge filled a container with 9 1 gallons of water before the game. After the game, only 2 1 2 3 gallons of water were left in the container. How many gallons of water were consumed during the game? 6 2 gallons 6 1 gallons 3 6 7 2 gallons 7 1 gallons 3 6 COPYING IS STRICTLY PROHIBITED BY LAW
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1 An aquarium has a capacity of 12 1 liters. If the aquarium already contains 9 1 liters of water, 2 8 how much more water can be added into the aquarium? 1 3 3 8 liters 3 8 liters 5 7 3 8 liters 3 8 liters
2 A serving of beef stew calls for 2 3 cups of potatoes. How many cups of potatoes are needed to 4 make 7 1 servings of the beef stew? 2 14 3 cups 16 7 cups 8 8 1 5 18 8 cups 20 8 cups
3 Jodie uses 4 and 2 cups of detergent for each load of laundry she washes. If she needs to wash 3 5 1 loads of clothes, how many cups of detergent does she need? 6 5 8 cups 8 1 cups 9 6 8 3 cups 8 5 cups 5 6
4
To prepare tomato sauce, we need 3 cups of water for each tomato. How much water do we 2 need for 13 tomatoes? 19 1 cups 19 1 cups 2 3 19 3 cups 19 2 cups 2 3
5 To paint the classroom, we bought 5 cans of white paint, 2 cans of red paint, 1 can of blue paint,
and 2 cans of yellow paint. We used 1 of white paint, 2 of red paint, 3 of blue paint, and all 2 4 3 the yellow paint. How many cans of paint do we have left? 3 5 cans 4 3 cans 12 12 4 3 cans 4 8 cans 12 12 127
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6 A hen needs
1 pounds of feed every day. How much does a hen consume per week? 4
1 1 pounds 4 2 1 pounds 4
1 3 pounds 4 2 pounds
7 A horse needs 164.5 liters of water per week. How much does a horse consume per day? 21 5 liters 2 22 7 liters 2
23 3 liters 2 23 1 liters 2
8 A 30-centimeter plant grows 15 centimeters per week. In how many weeks will the plant reach 255 centimeters? 15 weeks
19 weeks
18 weeks
20 weeks
9 To fill a 100-liter water tank, 3 3 liter cans were used. How many cans did it take to fill the tank? 4 26 1 cans 4 26 2 cans 3
26 1 cans 3 26 2 cans 5
10 To make a pizza, we need 1 kg of flour. If we want to make 14 pizzas, how much flour do we 3 need?
4 1 kg 3 4 1 kg 2
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SOLVE REAL-WORLD PROBLEMS INVOLVING DIVISION MA.5.AR.1.3 Solve real-world problems involving division of a unit fraction by a whole number and a whole number by a unit fraction. When solving real-world problems involving a unit fraction by a whole number, it is important to multiply by the reciprocal of the divisor.
STEP 1: Carefully read and comprehend the information provided in the real-world problem. STEP 2: When dividing real-world problems involving division, express the whole number as a fraction with a denominator of 1. For example, if you want to divide 1/3 by 2, rewrite 2 as 2/1. STEP 3: Take the reciprocal of the whole number (flip it). For example, 1/3 ÷ 2/1 = 1/3 ÷ 1/2.
STEP 4: Next, change the division sign to multiplication and multiply the unit fraction by the reciprocal of the whole number. For example, 1/3 × 1/2 = 1/6.
When solving real-world problems involving a a whole number by a unit fraction, it is important to multiply by the reciprocal of the divisor.
STEP 1: To divide a whole number by a unit fraction, express the whole number as a fraction with a denominator of 1. STEP 2: Take the reciprocal of the unit fraction (flip it). For example, 5/1 ÷ 1/10 = 1/3 ÷ 10/1. STEP 3: Next, change the division sign to multiplication and multiply the unit fraction by the reciprocal of the whole number. For example, 5/1 × 10/1 = 50/1 = 50
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DIRECTIONS: Solve each word problem below.
Rachel has a rectangular pizza that is divided into 10 equal slices. If she ate 2 of the pizza, how
5
many slices did she eat?
4 slices
Maria has a rope that is 5
1 feet long. She needs to cut 3 equal pieces from the rope. 2
How long will each piece of rope be?
1 5 feet 6
Samantha baked a cake and divided it into 8 equal slices. She ate 3 slices. What fraction of the cake did Samantha eat?
3 slices of cake 8
A recipe for chocolate chip cookies calls for 2 cup of sugar. If you want to make only half
8
the recipe, how much sugar will you need?
1 cup 8
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DIRECTIONS: Solve each word problem below. A rectangular field has a perimeter of 20 3 meters. If each side of the field has the same
4
length, what is the length of each side?
5 3 meters 16
Marta is creating 3 dresses that require 9 1 yards of fabric to complete. If each dress is
2
identical, how many yards of fabric are needed for each dress?
3 1 yards 6
A recipe for a fruit smoothie requires 3 1 cups of fruit salad. If you want to make 1 of the
2
2
recipe, how many cups of fruit salad do you need?
1 3 cups 4
A group of friends went on a road trip and traveled a total distance of 150 3 miles. If they drove for 3 1 hours, what was their average speed in miles per hour?
4
2
43 1
14
131
mph
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1 A restaurant chef has four pounds of potatoes. How many servings of mashed potatoes can the chef make if each serving of mashed potatoes calls for one-eighth of a pound of potatoes? 2 servings
8 servings
16 servings
32 servings
2 A group of six friends, including Miriam, went to dine in a restaurant. While waiting for their dishes to be served, the waiter served them a quarter of a bowl of peanuts. If they split the bowl of peanuts evenly, how much peanuts did Miriam get? 1 1 of a bowl of a bowl 30 24 1 1 of a bowl of a bowl 6 4
3 1 of an hour to complete 3 each task, how many tasks did Carl complete during those 2 hours? Carl spent a total of 2 hours to complete several tasks. If he spent
2 tasks
3 tasks
6 tasks
12 tasks
4 Mr. Antonio equally divided half an acre of land among his three children. How many acres of land did each child receive? 6 acres
3 acres
2 acres
1 of an acre 6
5 Romaine prepared 20 sandwiches for a party. She cut each of the sandwiches into fourths and put the slices on a tray. How many slices of sandwiches did Romaine put on the tray?
5 slices of sandwiches
20 slices of sandwiches
40 slices of sandwiches
80 slices of sandwiches
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DIRECTIONS: Solve each word problem below. A container holds 7 3 gallons of water. If each person needs 1 gallon of water, how many
2
4
servings are there?
15 1
2
A farmer has 25 2
3
servings
acres of land where she grows sugar cane. If each sugar cane field covers
1 1 acres, how many sugar cane fields can the farmer create? 3
1
19 4 sugar cane fields
A corner store received a shipment of 18 3 pounds of apples. If they want to pack them
4 1 into bags of 1 pounds, how many bags can they fill? 2
1
12 2 bags
A construction project requires 30 1 feet of piping. If each section of piping is 2 3 feet long
2
except for the last section, how many sections of piping will the project have?
1
11 11 sections
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4
1 A bakery used 1 of a bag of chocolate chips to make 5 batches of chocolate chip cookies. If the 2 bakery used the same amount of chocolate chips in each batch, how much chocolate chips did the bakery use in each batch? 1 of a bag 2 1 of a bag 10
1 of a bag 5 1 of a bag 20
2 A car wash uses one-sixth of a quart of soap to wash each car. If the car wash has 8 quarts of soap, how many cars can they wash? 48 cars
24 cars
16 cars
12 cars
3 George needs to fill a 20-gallon aquarium with water. If George can only carry half a gallon of water each trip, how many trips will it take him to fill the aquarium? 10 trips
20 trips
40 trips
80 trips
4 A carpenter has a piece of log that is 1 of a yard long. He cuts the log into 9 equal pieces. How 4 long will each piece be? 1 of a yard 4 1 of a yard 18
1 of a yard 9 1 of a yard 36
5 Amanda served a 12-pound turkey last Thanksgiving. If she cut the turkey into one-third of a pound servings, how many servings of the turkey did she make? 4 servings
12 servings
24 servings
36 servings COPYING IS STRICTLY PROHIBITED BY LAW
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1 A dressmaker used one-third of a yard of fabric to make four skirts. The dressmaker used the same amount of fabric to make each skirt. How much fabric did the dressmaker use to make one skirt? 1 1 of a yard of a yard 3 9 1 of a yard 1 of a yard 12 18
2 A group of ten friends, including Sheila, equally shared half a bar of chocolate. Each friend got the same amount of chocolate. What fraction of a chocolate bar did Sheila get?
3
1 of a chocolate bar 2 1 of a chocolate bar 10
1 of a chocolate bar 4 1 of a chocolate bar 20
Jack bought 3 pounds of dog food for his dog. If his dog eats 1 of a pound of dog food each 10 day, how many days will the dog food that Jack bought last? 60 days
30 days
15 days
10 days
4 A painter has 6 liters of paint. If each wall requires 3 liter of paint, how many walls can the 4 painter paint? 8 walls
10 walls
12 walls
18 walls
5 A tailor has a roll of fabric that is 6 meters long. If he cuts fabric into pieces, each 1 meter long, 2 how many pieces will he get?
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6 pieces
9 pieces
12 pieces
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6 Mary is making lemonade and needs to fill a pitcher that holds 5 liters. She can only carry 1 of 5 lemonade at once. How many trips will she need to make to completely fill the pitcher? 3 trips
7 trips
25 trips
30 trips
7 Tim is organizing a party and bought a 10-pound cake. If he plans to serve slices of the cake that 1 each weigh a 2 pound each, how many slices can he serve from the cake? 5 slices
10 slices
15 slices
20 slices
8 A farmer has 12 liters of pesticide to spray on his crops. If each field requires 1 liter of 3 pesticide, how many fields can he spray? 4 fields
36 fields
12 fields
100 fields
9 A construction company has a cement mixer with a capacity of 1 1 cubic yards. If each 3 foundation requires 1 cubic yards of cement, how many foundations can they pour from 1 full 6 load of the cement mixer? 4 foundations
6 foundations
8 foundations
10 foundations
10 A carpenter has 25 feet of wood. If each shelf requires 2 1 feet of wood, how many shelves can 2 the carpenter make? 10 shelves
25 shelves
12 shelves
62.5 selves
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TRANSLATE REAL-WORLD PROBLEMS INTO NUMERICAL EXPRESIONS MA.5.AR.2.1 Translate written real-world and mathematical descriptions into numerical expressions and numerical expressions into written mathematical descriptions. Use keywords to help translate written real-world and mathematical descriptions into numerical expressions. STEP 1: Determine which mathematical operations are involved in the mathematical description. Look for keywords or phrases that indicate specific operations. STEP 2: Identify the quantities mentioned in the description and assign them variables or symbols. Use letters such as x, y, or z to represent unknown values or specific variables mentioned in the problem. STEP 3: Write the expression using appropriate mathematical symbols and notation.
KEYWORD MEANING KEYWORD MEANING KEYWORD MEANING KEYWORD MEANING KEYWORD MEANING
is Equals (=) of Multiply (x) sum Add (+) left Subtract ( ̶ ) quotient Divide (÷)
same as Equals (=) per Multiply (x) and Add (+) less than Subtract ( ̶ ) out of Divide (÷)
times Multiply (×) product of Multiply (x) total Add (+) take away Subtract ( ̶ ) quarter Divide (÷) by 4
each row Multiply (×) increased by Add (+) combined Add (+)
each shelf each ____ Multiply (×) Multiply (×) all together more than Add (+) Add (+) greater plus Add (+) Add (+) how many more decreased by difference Subtract ( ̶ ) Subtract ( ̶ ) Subtract ( ̶ ) remaining half out of Divide (÷) Divide (÷) by 2 Divide (÷)
Use keywords to help translate numerical expressions into mathematical descriptions.
STEP 1: Translate the mathematical operations in the numerical expression into written form. Use words such as "add," "subtract," "multiply," or "divide" to describe the operations being performed.
STEP 2: Describe the quantities involved using appropriate terms and units of measurement. Use words such as "number of," "amount of," or "value of" to specify the quantities involved. STEP 3: Arrange the numerical descriptions of operations and quantities in a clear and concise manner.
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DIRECTIONS: Interpret the numerical expressions. (3 + 4) × (7 + 9)
8 × (92 − 19) 8 times the quantity 92 minus 19
The quantity 3 plus 4 multiplied by the quantity 7 plus 9
18 + 63 ÷ 7
(5,702 − 682) + 47 The quantity 5,702 minus 682 plus 47
18 plus the quotient of
1 × (8 x 1 - 2) 2 Half the quantity of 8 times 1 minus 2
(225 ÷ 9) - 14
63 and 7
Subtract 14 from the quotient of 225 and 9
DIRECTIONS: Interpret the word expressions. #
WORD EXPRESSION
VARIABLE EXPRESSION
Multiply 8 and 4, then subtract 6
(8 x 4) - 6
Add 12 and 3, then multiply by 7
(12 + 3) x 7
Subtract 16 from the sum of 47 and 26
(47 + 26) - 16
6 times the quantity 25 divided by 5
6 x (25 ÷ 5)
Add 15, 28, and 121, then multiply by 23
(15 + 28 + 121) x 23
Divide 16 by 4, then add 24
(16 ÷ 4) + 24
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DIRECTIONS: Interpret the numerical expressions. (184 + 68) × (59 + 47)
9 × (2,120 − 375) 9 times the quantity
The quantity 184 plus 68 multiplied by the quantity 59 plus 47
2,120 minus 375
(6,865 − 487) + 242
457 + (48 ÷ 8)
The difference of 6,865
457 plus the quantity
and 487 plus 242
48 divided by 8
1 × (44 × 5 + 15) 2 Half the quantity 44 times 5 plus 15
(315 ÷ 7) - 8 Subtract 8 from the quotient of 315 and 7
DIRECTIONS: Interpret the word expressions.
#
139
WORD EXPRESSION
VARIABLE EXPRESSION
Multiply 12 and 6, then subtract 7
(12 x 6) - 7
Add 58 and 42, then multiply by 11
(58 + 42) x 11
Subtract 64 from the sum of 125 and 37
(125 + 37) - 64
12 times the quantity 72 divided by 9
12 x (72 ÷ 9)
Add 241, 34, and 87, then multiply by 19
(241 + 34 + 87) x 19
Divide 99 by 11, then add 54
(99 ÷ 11) + 54
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1 Which of the following expressions is the same as saying half the sum of fifteen and twentythree? (15 + 23) ÷ 2
15 + 23 ÷ 2
15 + (23 ÷ 2)
(15 ÷ 2) + 23
2 Which of the following expressions is the same as saying one-third the difference of twelve and six? 1 1 12 − (6 × 3 ) ( 3 × 12) − 6 1 1 3 × (12 − 6) 3 × 12 − 6
3 Which of the following expressions is the same as saying twice the sum of one-fourth and threeeighths? 1 3 1 3 2 × 4 + 8 (2 × 4 ) + 8 1 3 1 3 2 × (4 + 8 ) 2+( 4 + 8 )
4 Which of the following is the word form of the expression 1 × (16 ÷ 4)? 2 Half the quotient of sixteen and four Half of sixteen divided by four Product of one-half and sixteen divided by four Half of four divided by sixteen
5 Which of the following is the word form of the expression (24 ÷ 5) + 4 ? 5
Four-fifths the quotient of twenty-four and five Twenty-four divided by four-fifths more than five The quotient of twenty-four and five plus four-fifths. Twenty-four divided by the sum of five and four-fifths COPYING IS STRICTLY PROHIBITED BY LAW
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DIRECTIONS: Interpret the numerical expressions. (4,235 - 1,453) - 251
(736 × 538) + 4 The product of 736 and 538 added to 4
Subtract 251 from the difference of 4,235 and 1,453
(2,856 ÷ 42) + 57
3,426 × 23 - 1,456
The quantity 2,856 divided by 42 plus 57
Multiply 3,426 and 23 then subtract 1,456
1 × (15,645 + 8,372) 2 Half the quantity of 15,645 plus 8,372
(1,287 × 54) - 12,427 Subtract 12,427 from the product of 1,287 and 54
DIRECTIONS: Interpret the word expressions.
#
141
WORD EXPRESSION
VARIABLE EXPRESSION
Multiply 624 and 87, then subtract 5,235
(624 x 87) - 5,235
Add 292 and 224, then divide by 43
(292 + 224) ÷ 43
Subtract 125 from the product of 45 and 53
(45 x 53) - 125
46 times the quantity 36 divided by 3
46 x (36 ÷ 3)
Add 524, 86, and 62, then multiply by 24
(524 + 86 + 62) x 24
Divide 99 by 11, then add 54
(99 ÷ 11) + 54
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1 Which of the following is the word form of the expression (3 × 9) − 2 ? 3 Two-thirds less than the product of three and nine Two-thirds less than the product of three and nine The product of three, and nine less than two-thirds The product of three, and nine less two-thirds
2 Amy bought three baskets of apples. Each basket contained six red apples and four green apples. Which expression represents the total number of apples Amy bought? 3×6+4
3 × (6 + 4)
(3 × 6) + 4)
6 + (3 × 4)
3 Douglas bought eight boxes of marbles. Each box contains twenty-four marbles, six of which are green. Which expression represents the number of marbles Douglas bought that are not green? 8 × 24
(8 × 24) − 6
8 × 24 − 6
8 × (24 − 6)
4 A carpenter has a piece of wood seven feet and three-tenths of a foot long. He cuts the wood into two pieces of equal length. Which expression represents the length of each piece? 7 + (30 ÷ 2)
(7 + 30) ÷ 2
7.3 ÷ 2
(7 + 3 × 10) ÷ 2
5 Patrick bought seven packs of trading cards. Each pack contains eight trading cards, five of which are baseball cards. Which expression represents the total number of trading cards Patrick bought that are not baseball cards? 7×5
7×8−5
7×8
7 × (8 − 5)
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1 Felicia ordered twelve boxes of pens from an online store. Each box contained twenty-four pens, a third of which were black. Which expression represents the total number of black pens that Felicia ordered online? 1 × 24 12 × (24 × 1 ) 3 3 1 × 12 12 + (24 × 1 ) 3 3
2 There are thirty-six animals on Uncle Henry’s farm, a quarter of which are goats. Of all the goats on the farm, two have horns. Which expression represents the number of goats with no horns? 4−2
36 ÷ (4 − 2)
36 ÷ 4
(36 ÷ 4) − 2
3 Five boxes contain a total of sixty tokens. Each box contains the same number of tokens. If each box has ten green tokens and the rest are yellow tokens, which expression represents the number of yellow tokens in each box? 60 − 5 × 10
60 − (5 × 10)
60 ÷ 5 − 10
(60 ÷ 5) − 10
4 Which of the following is the word form of the expression 1 × ( 3 + 3)? 5 4 Product of 5 and the sum of three-fourths and three One-fifth of the sum of three-fourths and three Sum of one-fifth and three-fourths and three Product of three-fifths and three-fourths
5 Which of the following is the word form of the expression (7 ÷ 1 ) ̶ 3? 2
Quotient of seven and two less than three Quotient of seven less than three and one-half Three less than quotient of seven and one-half One-half less than quotient seven and three 143
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6 A hotel has 100 residents. 30 residents are Italian, 15 are French, 25 are German, and a third of the rest are Japanese. Which expression represents the number of Japanese residents?
7
(100 - 30 + 15 + 25) × 1 3 100 - (30 + 15 - 25) × 1 3
100 + (30 + 15 + 25) × 1 3 1 100 - (30 - 15 + 25) × 3
Which of the following is the word form of the expression (12 ÷ 2) - 5? Two over twelve minus five Twelve divided by two plus five Five less than the division of twelve over two Five less the division of twelve over two
8 Marcus collects the cherries from 6 trees in the garden. From each tree, he takes 20 cherries. He then stores them in 4 equal containers. Which expression represents the number in each container? (6 × 20) × 4
6 × (20 ÷ 4)
6 × (20 - 4)
(6 × 20) ÷ 4
9 Anne’s mom wants to make peanut butter, so she needs 1.1kg of peanuts for every 1.2kg of peanut butter. If she wants to make 4kg of peanut butter, which expression represents the amount of peanuts needed? (4 ÷ 1.2) × 1.1
4 + (1.1 ÷ 1.2)
4 × (1.1 + 1.2)
4 - (1.1 ÷ 1.2)
10 A beekeeper has 8 hives, each containing 200 bees except the last one which contains 600. Which expression represents the total number of bees?
200 + (600 × 7)
(200 + 600) × 7
600 + 7 × 200
(600 - 200) × 7
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USING THE ORDER OF OPERATIONS
MA.5.AR.2.2 Evaluate multi-step numerical expressions using order of operations.
Use the order of operations (PEMDAS) when evaluating multi-step numerical expressions.
STEP 1: Break down the given numerical expression into its individual components, including numbers, operations, parentheses, exponents, and the necessary mathematical symbols. STEP 2: Begin by evaluating any operations that are enclosed in parentheses or brackets. STEP 3: Next, evaluate any exponents present in the expression. STEP 4: Move from left to right within the expression and perform any multiplication or division operations. Calculate the products or quotients according to the order of operations.
STEP 5: Finally, move from left to right within the expression and perform any addition or subtraction operations. Calculate the sums or differences according to the order of operations. STEP 6: Be sure that the numerical expression has been simplified.
PEMDAS is a mnemonic device used to remember the order of operations in arithmetic: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). The phrase "Please Excuse My Dear Aunt Sally" is often used as an mnemonic to help remember the order of operations.
P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction
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from left to right
from left to right
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DIRECTIONS: Evaluate each numerical expression below. 3 × (2 + 4) ÷ 2 3 × 6 ÷ 2 18 ÷ 2 9
(5 + 8) × 6 (13) × 6 78
(15 ÷ 5) + (19 - 5) (15 ÷ 5) + (19 - 5) (3)
+ 17
(14)
9 - 3 × (2 + 5) ÷ 4 9 - 3 × (7) ÷ 4 9 21 ÷ 4 9 5.25 3.75
(10.5 × 3) × 4
2 + 3 × (6 ÷ 2) - 1
(31.5) × 4
2 + 3 × (3) 2 + 9 11
126
- 1 - 1 - 1 10
90 + 21 ÷ 3 + 15
4 + 2 × (8 ÷ 2)
90 + 7 + 15
4 + 2 ×
(4)
4 + 2 ×
(4)
97
+ 15 112
4 +
8 12
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146
DIRECTIONS: Evaluate each numerical expression below. 4.68 + (3 × 9) - 6
(25 × 16 + 3) x 9
4.68 + 27 - 6
(400 + 3) x 9
31.68 - 6
403 x 9
25.68
3,627
36 + (3 × 15) - 7
(137 × 2) × 2
36 + 45 - 7
274 x 2
81 - 7
548
74
(48 ÷ 6) × (45 - 5)
(35 + 10) × (166 - 80) - 325
8 x 40
45 x 86 - 325
320
3,870 - 325 3,545
147
9 × (8 - 1) + 100
16 + (2 × 5) - 17
9 x 7 + 100
16 + 10 - 17
63 + 100
26 - 17
163
9
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1 What is the value of this expression?
0.25 × (12 + 5 × 2) − 4 9
4.5
3
1.5
2 Which of these expressions has a value of 18? 16 + (4.5 × 2 − 1) ÷ 4
(16 + 4.5) × 2 − 1 ÷ 4
16 + 4.5 × (2 − 1 ÷ 4)
16 + 4.5 × 2 − (1 ÷ 4)
3 Erica needs to calculate the value of the expression 3 × 2 − (11 + 4 × 3.75). Which of the following highlights the first operation she should perform?
3 × 2 11 + 4 2 − 11 4 × 3.75
4 What is the value of this expression?
3.2 + (12 × 2.5 − 3) ÷ 9 6.2
12.8
16.6
32.4
5 Which of these expression has a value of 5? (1 + 16 × 1.5) − 4 ÷ 5 (1 + 16) × 1.5 − 4 ÷ 5 1 + 16 × (1.5 − 4 ÷ 5) 1 + (16 × 1.5 − 4) ÷ 5 COPYING IS STRICTLY PROHIBITED BY LAW
148
DIRECTIONS: Evaluate each numerical expression below. (6.6 + 1.4) ÷ 0.1 × 4
25 - 4 + 6 × 8
8 ÷ 0.1 × 0.4
25 - 4 + 48 21 + 48
80 × 4
69
20
(16.3 × 6) - 7 × 4 + 6
(43 - 2.84) + 4 × 7 + 2
97.8 - 28 + 6
40.16 + 28 + 2
69.8 + 6
70.16
75.8
1 2
÷ (7 × 1 + 34 - 19)
69 + (18 - 15 + 9) ÷ 4
1 ÷ (7 + 34 - 19) 2
69 + 12 ÷ 4
1 2
÷ 41 - 19
1 2
÷ 22
69 + 3 72
1 44
149
(18 - 8 ÷ 2) + 6 × 5
85 + 4 × (56 - 2) ÷ 6
(18 - 4) + 6 x 5
85 + 4 x 54 ÷ 6
14 + 6 x 5
85 + 216 ÷ 6
14 + 30
85 + 36
44
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1 Victor needs to calculate the value of the expression (8.85 × 4) − 3 × 2 + 5. Which of the following highlights the first operation he should perform? (8.85 × 4) 4 − 3 3 × 2 2 + 5
2 What is the value of this expression?
0.75 + (15 − 3 + 5) ÷ 4 5
6.75
13.75
14
3 Which of these expressions has a value of 25.75? (32 − 3.25) + 2 × 5 + 7
32 − (3.25 + 2 × 5) + 7
32 − 3.25 + 2 × 5 + 7
32 − 3.25 + 2 × (5 + 7)
4 Jessica needs to calculate the value of the expression 16 × 4 − (8.3 − 2 + 5.2). Which of the following highlights the first operation she should perform? 16 × 4
4 − 8.3
(8.3 − 2)
(2 + 5.2)
5 What is the value of this expression?
2 + (45 − 8 × 2) ÷ 2 3 10 15
12 1 2 15 1 6 COPYING IS STRICTLY PROHIBITED BY LAW
150
1 Which of these expressions has a value of 11? 3 (32 + 3 × 5) ÷ 4 − 4
3 32 + 3 × (5 ÷ 4) − 4
32 + 3 × 5 ÷ (4 − 3 ) 4
32 + 3 × 5 ÷ 4 − 3 4
2 Anthony needs to calculate the value of the expression 36 − (16 × 1 + 2) × 5. Which of the 2 following highlights the first operation he should perform? 1 + 2 2
16 × 1 2
36 − 16
2) × 5
3 What is the value of this expression?
3 + (17 ÷ 4 − 12 ) × 8 25 1 2 30 1 2
27 33
4 Select the expression that is not equal to 548.
(62.5 × 2 + 49 + 100) + 274
(25 × 16 + 3) × 9
(137 × 2) × 2
(274 ÷ 2 + 350) + 61
5 What is the value of this expression?
(10 + 20) × (7 + 3) + 52
151
252
325
300
352
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6 Select the expression that is not equal to 504.
485 - 65 + 37
126 × 3 + 55 + 71
72 × 3 × 2 + 72
2 × 257 - 20 ÷ 2
7 Which of the following represents the correct order of operations?
(20 + 12) + (16 × 2) ÷ 2 12
32
64
48
8 What is the value of this expression?
50 - (4 ÷ 23 ) × 8 2
500
3 5 6
552
9 Select the expression that is not equal to 25. (100 - 25) ÷ 3 = 25
(250 ÷ 10) + (3 × 9) = 25
(50 × 1.5) - 50 = 25
((30 × 5) + 25) / 7 = 25
10 What is the order of operations for the following?
(5 9 + 20) ÷ (7 1 + 3) × 52 50 2
P = Parentheses M = Multiply A = Add
P, D, M, A
P, A, S, M
P, P, D, M
P, P, A, M COPYING IS STRICTLY PROHIBITED BY LAW
E = Exponents D = Divide S = Subtract
152
TRUE OR FALSE EQUATIONS & DETERMINING THE UNKNOWN
MA.5.AR.2.3 Determine and explain whether an equation involving any of the four operations is true or false.
Use the order of operations (PEMDAS) when evaluating multi-step numerical expressions.
STEP 1: Use the order of operations (PEMDAS) to simplify each side of the equation, one side at a time. STEP 2: Perform any calculations within parentheses first, followed by exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). STEP 3: After simplifying and solving each side of the equation, compare the values obtained on both sides. STEP 4: If the equation is true, explain that the values on both sides are equal, and the equation holds. If the equation is false, explain that the values on both sides are not equal, and the equation does not hold.
PEMDAS is a mnemonic device used to remember the order of operations in arithmetic: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). The phrase "Please Excuse My Dear Aunt Sally" is often used as an analogy to help remember the order of operations.
P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction
153
from left to right
from left to right
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DIRECTIONS: Use a check mark to indicate if the equations below are true or false. No.
EQUATION
TRUE
FALSE
4 × 10 + 4 x 6 = 5 × 16 - 3 = (10.5 - 2.5) ÷ 8 18 × 2 3 8.6 × 3 = 6 × 5 - 4.2 42 - 3 × (56 ÷ 7) = 3 × (48 ÷ 6) - 6 2 + (14 ÷ 7) = 24 ÷ 8 5 + (14 - 4) × 3 = 72 ÷ 9 8 + 4 = 24 ÷ 8 6.1 × (4 + 2) = 8 × 3 - 4.4 20 ÷ 5 × 2 = 10 × 2 × 1 2 4 - 2 + 6 × 2 = (5 × 7) - (3 × 7) 2 1 21 3 + 3 × (8 ÷ 2) = 18 3 - 4 3 (7.2 + 8.8) ÷ 4 = 24 - 22 × 2 (7.5 × 3) ÷ 0.5 = 90 - 9 × 5 (8 - 4) + (3 × 3) × (6 ÷ 2) = 20 × 2 - 1 1 1 +1 1 2 × 4 = 30 × 2 ÷ 5 2 12 + 3 - 2 × 5 = 35 - 7 (9 ÷ 2 1 4 ) + (100 ÷ 10 + 1) = 15 × 3 ÷ 1 50 + 2 - 6 ÷ 3 × 5 = 2 × 45 - (7 - 5)
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154
DIRECTIONS: Use a check mark to indicate if the equations below are true or false. No.
EQUATION
TRUE
FALSE
3 × 12 + 2 × 7 = 2 × 25 - 2 = (16 - 8) ÷ 2 12 × 1 2 3.5 × 2 = 8 × 4 - 2.3 22 - 7 × (42 ÷ 6) = 4 × (35 ÷ 5) - 3 8 + (16 ÷ 2) = 20 ÷ 4 9 + (26 - 3) × 6 = 56 ÷ 8 5 + 4 = 36 ÷ 4
2.1 × (8 + 4) = 5 × 7 - 9.8 63 ÷ 7 × 2 = 6 × 6 × 1 2 5 - 3 + 8 × 4 = (6 × 4) - (9 × 2) 1 1 31 2 + 4 × (4 ÷ 2) = 16 3 - 5 4 (9.5 + 5.5) ÷ 3 = 16 - 3 × 7 (6.5 × 4) ÷ 0.4 = 7 - 12 × 6 (9 - 3) + (5 × 5) × (12 ÷ 3) = 18 × 9 - 38 2 1 +3 1 2 × 9 = 42 × 3 ÷ 2 2 11 + 5 - 6 × 2 = 28 - 24
(7 ÷ 2 1 2 ) + (226 ÷ 1 + 1) = 14 × 2 ÷ 1 40 + 4 - 10 ÷ 5 × 3 = 3 × 25 - (48 - 11)
155
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1 Which of these equations is true? (36 ÷ 3 + 9) − 5 = 3 × 6
(36 ÷ 3 + 9) − 5 = 20 ÷ 4
(36 ÷ 3 + 9) − 5 = 10 + 6
(36 ÷ 3 + 9) − 5 = 26 − 5
2 Peter evaluated the expression 72 ÷ 6 − 2 × 3. His work is shown below. 72 ÷ 6 − 2 × 3 = 72 ÷ 6 − 2 × 3 = 12 − 2 × 3 = 10 × 3 = 30
←Step 1 ←Step 2 ←Step 3
In which step did he make his first mistake? Step 1
Step 2
Step 3
Peter’s solution is correct.
3 Frank says the equation 35 ÷ 7 − 2 = 3 + 4 is true because the expression on both sides of the equal sign are equivalent to 7. Emma says it is false because the expression on the left side of the equal sign is equal to 3, while the expression on the right side of the equal sign is equal to 7. Who among them is correct? Frank
Emma
Both of them are correct.
Neither of them is correct.
4 Which of these equations is true? 3 + 3 = 17 − (3 + 4 × 2)
17 − 14 = 17 − (3 + 4 × 2)
2 × 3 = 17 − (2 + 4 × 2)
10 ÷ 5 = 17 − (3 + 4 × 2)
5 Mark evaluated the expression 16 ÷ 2 × (5 − 1). His work is shown below. 16 ÷ 2 × (5 − 1) = 16 ÷ 2 × (5 − 1) = 16 ÷ 2 × 4 = 16 ÷ 8 =2
←Step 1 ←Step 2 ←Step 3
In which step did he make his first mistake? Step 1
Step 2
Step 3
Mark’s solution is correct. COPYING IS STRICTLY PROHIBITED BY LAW
156
DIRECTIONS: Use a check mark to indicate if the equations below are true or false.
No.
EQUATION
TRUE
FALSE
4 × 12 + 2 × 3 = 3 × 25 - 2 = (24 ÷ 3) - 3 14 × 1 2 5×2=8×4-2 20 - 15 × (40 ÷ 5) = 4 × 10 - (35 × 0) 7 + (20 ÷ 2) = 16 ÷ 4 11 + (24 + 3) × 2 = 63 ÷ 7 50 + (49 ÷ 7) = 36 ÷ 4 + 49
13 × (5 × 2) = 8 × 7 - 5.5 63 ÷ 7 × 2 = 3 × (18 ÷ 3) 5 - 3 + 8 × 4 = (6 × 4) + (5 × 2) 1 - 51 31 + 4 × (2 ÷ 2) = 16 2 3 4 (12.5 + 5.5) ÷ 3 = 16 - 4 × 7 (10 × 4) ÷ 0.2 = 85 - 10 × 6 (81 ÷ 9) + (5 × 3) = 10 × 9 - 66 2 1 +3 1 2 × 2 = 24 × 3 ÷ 2 2 18 + 5 - 6 × 2 = 35 - 24
(10 ÷ 2 1 2 ) × 2 = (28 ÷ 2) - 6 75+ 4 - 10 ÷ 5 × 3 = 4 × 10 - (20 - 5)
157
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1 Jeff says the equation 40 × 2 − 2 = 94 − (6 + 5 × 2) is false because the expression on the left side of the equal sign is equal to 0, while the expression on the right side of the equal sign is equal to 78. Jane says it is false because the expression on the left side of the equal sign is equal to 78, while the expression on the right side of the equal sign is equal to 72. Who among them is correct? Jeff
Jane
Both of them are correct.
Neither of them is correct.
2 Which of these equations is true? 3.5 × (7 − 3 × 2) = 3.5 × 8
3.5 × (7 − 3 × 2) = 10 − 6
3.5 × (7 − 3 × 2) = 7 ÷ 2
3.5 × (7 − 3 × 2) = 5 + 2
3 Vivian evaluated the expression 5 + (14 ÷ 4 − 3). Her work is shown below. 5 + (14 ÷ 4 − 3)
= 5 + (14 ÷ 4 − 3) = 5 + (14 ÷ 1) = 5 + 14 = 19
←Step 1 ←Step 2 ←Step 3
In which step did she make her first mistake? Step 1
Step 2
Step 3
Vivian’s solution is correct.
4 Jessica says the equation 16.5 ÷ (11 ÷ 4 × 2) = 2 × 1.5 is false because the expression on the left side of the equal sign is equal to 12, while the expression on the right side of the equal sign is equal to 3. Angela says it is true because the expression on both sides of the equal sign are equivalent to 3. Who among them is correct? Jessica
Angela
Both of them are correct.
Neither of them is correct.
5 Which of these equations is true? 18 ÷ 4 = 1.75 × (32 ÷ 4) − 4
5.5 − 3.5 = 1.75 × (32 ÷ 4) − 4
2 × 8 = 1.75 × (32 ÷ 4) − 4
3 + 7 = 1.75 × (32 ÷ 4) − 4
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158
1 Beth evaluated the expression (8 − 6 ÷ 4) + 3.5. Her work is shown below. (8 − 6 ÷ 4) + 3.5 = (8 − 6 ÷ 4) + 3.5 = (8 − 1.5) + 3.5 = 6.5 + 3.5 = 10
←Step 1 ←Step 2 ←Step 3
In which step did she make her first mistake? Step 1
Step 2
Step 3
Beth’s solution is correct.
2 Will says the equation 5.5 × (3 + 5) ÷ 2 = 3 × 7 is false because the expression on the left side of the equal sign is equal to 30.25, while the expression on the right side of the equal sign is equal to 21. Vic says it is false because the expression on the left side of the equal sign is equal to 22, while the expression on the right side of the equal sign is equal to 21. Who among them is correct? Will
Vic
Both of them are correct.
Neither of them is correct.
3 Which of these equations is true? 16 + (9 × 3) ÷ 2 = 118 ÷ 4
16 + (9 × 3) ÷ 2 = 16 + 9
16 + (9 × 3) ÷ 2 = 10 + 4.5
16 + (9 × 3) ÷ 2 = 30 − 5.5
4 Harry says the equation 2.5 × (9 ÷ 2) + 3.75 = 4 × 5 - 25 ÷ 5 is true because the expression on the left side of the equal sign is equal to 15, while the expression on the right side of the equal sign is equal to 15. Sally says it is false because the expression on the left side of the equal sign is equal to 17, while the expression on the right side of the equal sign is equal to 15. Who among them is correct? Both of them are correct.
Sally
Harry
Neither of them is correct.
5 Which of these equations is true?
159
9 + (6 ÷ 5) × 10 - 8 = 26 ÷ 2 +20
9 + (6 ÷ 5) × 10 - 8 = 8 × 2 - 3
9 + (6 ÷ 5) × 10 - 8 = 15 × 3 - 18
9 + (6 ÷ 5) × 10 - 8 = 70 ÷ 2 - 10 × 2
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6 Johnny evaluated the expression (3 + 8 ÷ 2) × 2.5. Her work is shown below. (3 + 8 ÷ 2) × 2.5 = (3 + 8 ÷ 2) × 2.5 = (3 + 4) × 2.5 = 3 × 10 = 30
←Step 1 ←Step 2 ←Step 3 ←Step 4
In which step did he make his first mistake? Step 1
Step 2
Step 3
Step 4
7 Which of these equations is true? 7 + (2 × 3) - 16 ÷ 4 = 20 ÷ 5 + 2.5 × 2
7 + (2 × 3) - 16 ÷ 4 = 6 ÷ 4 × 3
7 + (2 × 3) - 16 ÷ 4 = 10 - 5 + 3 × 2
7 + (2 × 3) - 16 ÷ 4 = 16 + 10 ÷ 4
8 Annie says the equation 4 + (2 × 3) ÷ 4 + 2 = 24 ÷ 6 + 2 is true because the expression on the left side of the equal sign is equal to 6, while the expression on the right side of the equal sign is equal to 6. Charlotte says it is false because the expression on the left side of the equal sign is equal to 7.5, while the expression on the right side of the equal sign is equal to 6. Who among them is correct? Both of them are correct.
Neither of them is correct.
Charlotte
Annie
9 Amy evaluated the expression 6 × ( 4 - 3 ÷ 2) - 2.5. Her work is shown below. 6 × ( 4 - 3 ÷ 2) - 2.5 = 6 × ( 4 - 3 ÷ 2) - 2.5 = 6 ÷ ( 1 ÷ 2) - 2.5 = 6 ÷ 0.5 - 2.5 = 9.5
←Step 1 ←Step 2 ←Step 3 ←Step 4
In which step did she make her first mistake? Amy’s solution is correct.
Step 3
Step 2
Step 4
10 Which of these equations is true? 8 + 5 × (6 ÷ 3) - 7 = 13 ÷ 2 + 5
8 + 5 × (6 ÷ 3) - 7 = 4 ÷ 2 × 9 - 2
8 + 5 × (6 ÷ 3) - 7 = 10 + 2 × 5
8 + 5 × (6 ÷ 3) - 7 = 5.5 × 2
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WRITING EQUATIONS TO DETERMINE THE UNKNOWN MA.5.AR.2.4 Given a mathematical or real-world context, write an equation involving any of the four operations to determine the unknown whole number with the unknown in any position. Use this set of steps to solve mathematical or real-world context involving an unknown number.
STEP 1: Find the unknown number in the problem. STEP 2: Assign a letter to represent the unknown number. STEP 3: Write an equation using +, -, ×, or ÷ with the letter you chose in Step 2 to represent the unknown number. STEP 4: Solve the equation to find the value of the unknown number. STEP 5: Check your answer by putting the value you found in step 4 back into the original equation.
Example: Cassandra has twice as many tokens as Kyle. Together, they have 75 tokens. How many tokens does Kyle have? First, identify the unknown by using the first letter of Kyle’s name "K" to represent the number of tokens Kyle has. Next, translate the information given into an equation. Since Cassandra has twice as many tokens as Kyle, it can be expressed as: Cassandra 's tokens = 2 × Kyle’s tokens Together, they have 75 tokens, so we can write the equation: Cassandra 's tokens + Kyle’s tokens = 75 Substituting the expressions from above: 2K + K = 75 Continue by simplifying and solving the equation. Combine like terms: 2K + 1K = 75 3K = 75 Next, divide both sides by 3 to isolate the variable “K” to one side. K = 25
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DIRECTIONS: Write an equation for each situation and solve. Ebru has some money saved in her piggy bank. She decides to buy a toy that costs $20. After purchasing the toy, she has $35 remaining. How much money did Ebru originally have in her piggy bank?
x - $20 = $35
x = 55
A bakery sells cupcakes for $2 each. Laura bought some cupcakes and spent a total of $8. How many cupcakes did Laura buy? $8 = $2 x c
c = 4
Elizabeth went shopping and bought some clothes. She spent a total of $120. If she bought T-shirts for $20 each and a pair of jeans for $40, how many T-shirts did she buy? $120 = $20 x t + $40
t = 4
DIRECTIONS: Solve each equation. Solve for a: a - 15 = 35
Solve for m: m × 9 = 54
a = 50
m = 6
Solve for c: 2c + 5 = 13
Solve for t: 6 = t ÷ 7
c = 4
t = 42
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162
DIRECTIONS: Write an equation for each situation then solve. Alex and Ben are saving money to buy a video game. Alex has saved $15, and Ben has saved $25. They want to know how much more money they need to reach their goal of $60. $15 + $25 + x = $60
x = $20
Sam has $10 in his piggy bank. He receives $5 as an allowance each week. How many weeks will it take for him to save $50 in his piggy bank? $50 = $10 + ($5 x w)
w = 8
To celebrate their successful gardening project, Ms. Garcia's class is having a picnic. Ms. Garcia prepares a total of 135 cookies. She plans to distribute an equal number of cookies to each of the 24 students in her class. Additionally, she wants to set aside 15 cookies for the school staff. What is the greatest number of cookies that each student in Ms. Garcia's class can receive?
24 × c + 15 = 135
c = 5
DIRECTIONS: Solve each equation.
163
Solve for r: r + 27 = 42
Solve for d: 53 = 6d + 5
r = 15
d = 8
Solve for f: 3f + 7 = 16
Solve for h: 2h - 3 = 11
f = 3
h = 7
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1 Minerva has 8 dolls, and Anna has d more dolls than Minerva. If the 2 of them have a total of 20 dolls, which equation can be used to determine the value of d? 20 − 8d = 8
20 − d = 8
8 + d = 20
8 + d = 20 − 8
2 Gene bought 4 more than 3 times the number of apples that Mark bought. If Mark bought a apples and Gene bought 25 apples, which equation can be used to determine the value of a? 25 + 4 = 3a
25 = 4 + 3a
3a = 25 + 4
25 + 3a = 4
3 Robert took an 200-mile trip and used a total of 10 gallons of gas. If Robert’s car consumes g gallons of gas per mile traveled, which equation can be used to determine the value of g? 10 − 200 = g
g + 200 = 10
10 − g = 200
10 = 200g
4 The zoo had 15 sacks of peanuts in stock. The elephants at the zoo ate S sacks of peanuts each day for 4 days. If only 3 sacks of peanuts were left after 4 days, which equation cannot be used to determine the value of S? 4S − 3 = 15
15 = 3 + 4S
4S + 3 = 15
15 − 4S = 3
5 Sam needed to save $44 to buy a new toaster. He calculated that if he saved D dollars each day, then he will have saved enough money for the toaster after 12 days. All of the following equations can be used to determine the value of D except which one?
44 ÷ D = 12
12D = 44
44 ÷ 12 = D
12 + D = 44
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164
DIRECTIONS: Write an equation for each situation then solve. A movie theater charges $10 per adult ticket and $5 per child ticket. A group of adults and children attended a movie together, and the total cost for their tickets was $35. If there were 3 children in the group, how many adults were there? $35 = ($10 x a) + ($5 x 3)
a = 2
A store sells apples for $0.50 each and oranges for $0.75 each. If Karen bought a total of 10 fruits with 5 being apples and spent $5.75, how many oranges did she buy? $5.75 = ($0.50 x 5) + ($0.75 x o)
Karen bought 5 oranges
Sarah and John are selling cookies to raise money for a charity. Sarah sold twice as many cookies as John. Together, they sold a total of 60 cookies. How many cookies did Sarah and John each sell?
x + 2x = 60
x = 20 John sold 20 cookies and Sarah sold 40 cookies
DIRECTIONS: Solve each equation.
165
Solve for x: 4(x - 2) = 12
Solve for n: 2(n + 3) - 5 = 9
x = 5
n = 4
Solve for x: 3x - 2 = 10
Solve for s: s(15 - 12) = 15
x = 4
s = 5
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1 A notepad costs $3, and a pencil case costs $7. Jenny bought a pencil case and N notepads for a total of $16. All of the following equations can be used to determine the value of N except which one? 3N − 7 = 16
7 + 3N = 16
16 = 3N + 7
16 − 7 = 3N
2 William bought 5 packs of gum. Each pack contains 8 sticks of gum. After chewing g sticks of gum, William was left with 12 sticks of gum. Which equation can be used to determine the value of g? 5 × 8 = 12 − g
12 + g = 5 × 8
5 × 8 = 12g
g − 12 = 5 × 8
3 A water tank is half-full of water. After removing 15 gallons of water from the water tank, only 8 gallons of water were left. Which equation can be used to determine the value of g, the number gallons of water the water tank can hold? 8g = 15 × 8
2g − 15 = 8
8 + 15 = g
(g ÷ 2) − 15 = 8
4 A publishing company can print p books for every ream of paper. If the publishing company used a total of 80 reams of paper to print 320 books, which equation can be used to determine the value of p? 320p = 80
p = 80 × 320
320 ÷ 80 = p
80 ÷ p = 320
5 During a sale, hats cost $8 each and caps cost $3 each. Amy bought 4 hats and C caps during the sale and paid $59. Which equation cannot be used to determine the value of C? 59 − 3C = 4 × 8
3C + 59 = 4 × 8
(4 × 8) + 3C = 59
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166
1 Miss Annie needs to grade 36 papers. She can grade N papers every hour. After 3 hours of grading, she only needs to grade 6 more papers. Which equation cannot be used to determine the value of N? 36 = 3N + 6
3N − 6 = 36
6 + 3N = 36
36 − 3N = 6
2 Amanda needs to save $90 for a new book. After saving $3 per day for D days, she calculated that she still needs to save $54. Which equation cannot be used to determine the value of D? 90 − 3D = 54
3D = 90 − 54
90 - 54 = 3D
3D − 54 = 90
3 Angela can make 4 bracelets every hour, while Beth can make B bracelets every hour. After making bracelets for 3 straight hours, the 2 of them have made a total of 27 bracelets. Which equation can be used to determine the value of B? 27 = 3 + 4B
4(3 + B) = 27
27 = 4 + 3B
3(4 + B) = 27
4 Gordon can peel 10 potatoes every 2 minutes. He buys a bag containing 156 potatoes. After T minutes, there are only 6 potatoes left to peel. Which equation can be used to determine the value of T? 5T + 6 = 156
6 - 5T = 156
5T - 6 = 156
5T + 10 = 156
5 Alisson wants to earn money to buy a present for her mother. The gift costs $90. She wants to sell homemade cookies every day for 2 weeks. She sells 10 cookies per day. Which equation can be used to determine the value for the dollar amount she sells each day (X)?
167
14 + 10X = 90
10 × 14X = 90
10 - 14 X = 90
14X - 10= 90
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6 To reach her goal of doing 500 push-ups per week, Charlotte decided to do 70 push-ups the first 4 days, X the next 2 days, and 60 the last day. Which equation cannot be used to determine the value of X? 70 × 4 + 2X + 60 = 500
2X + 60 = 220
70 × 4 + 2X - 60 = 500
70 × 4 + 2X = 440
7 Jason wants to maintain the garden of their house; 10 square meters are already done. He divided the rest of the garden into 5 plots of A square meters, and the garden is 80 square meters. Which equation can be used to determine the value of A? 10 + 5A = 80
5A - 10= 80
10 - 5A = 80
80 + 5A = 10
8 To celebrate meeting their fundraising goal, Mr. Patel's class is having a pizza party. Mr. Patel orders a total of 120 slices of pizza. He plans to distribute an equal number of slices to each of the 15 students in his class. He also wants to save 10 slices to share with his family. What is the greatest number of slices that each student in Mr. Patel's class can receive? 120 - 15 - 10 = P
120 - (15 × P) = 10
10 + P = 120 - 15
15 × P + 10 = 120
9 Sam is saving money to buy a tablet for studying. He already has $40, and the tablet costs $200. According to his calculations, by receiving D dollars per week, he should be able to buy the tablet in 8 weeks. Which equation cannot be used to determine the value of D? 40 + 8D = 200
4D = 100 - 20
10 + D = 50
5 + D = 25
10 Mia’s dad wants to paint a part of the bedroom and finish it in one day. The total surface of the walls is 30 square meters, the surface to be painted is 25 square meters, and he can paint 3 square meters per hour. He finishes the job in T hours. Which equation can be used to determine the value of T? 3T - 30= 25
(30-25)T = 3
3T + 25 = 30
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168
RULES FOR DESCRIBING PATTERNS AS AN EXPRESSION MA.5.AR.3.1 Given a numerical pattern, identify and write a rule that can describe the pattern as an expression.
To find a pattern in a series of numbers, look for a consistent difference or ratio between each number. STEP 1: Study the given pattern of numbers. For example, 2, 4, 6, 8, 10. STEP 2: Figure out how the numbers in the pattern are related. Check for a consistent difference or ratio between each number. STEP 3: If the pattern has a consistent difference between each number (arithmetic sequence), subtract any number from the one before it to find the common difference. In our example, the common difference is 2 because each number is 2 more than the previous one. STEP 4: If the series has a consistent ratio between each number (geometric sequence), divide any number by the one before it to find the common ratio. In our example, the common ratio is 2 because each number is twice the previous one.
STEP 5: Write a formula using variables to represent the pattern. For an arithmetic sequence, the formula will include the common difference (d) and the first term (a). For a geometric sequence, the formula will include the common ratio (r) and the first term (a). STEP 6: For our example pattern (2, 4, 6, 8, 10), since it has a common difference of 2, we can write the formula as n = 2n, where “n” represents the nth term of the sequence. STEP 7: If needed, simplify the formula or adjust it to fit the specific starting term or index of the pattern. For example, if the pattern starts with the term “a1” instead of “a0”, you may need to adjust the expression accordingly. STEP 8: Check the formula by applying it to additional terms in the pattern to see if it correctly predicts the values. For our example, when we substitute “n” with 1, 2, 3, 4, and 5 in the formula (n) = 2n, we get the correct corresponding values of 2, 4, 6, 8, and 10, respectively. +3
Rule: Add 3 Example:
x
+3
+3
+3
+3
6
9
12
15
-
first
second
third
fourth
fifth
?
RULE: 3 + 3x, where x = 1, 2, 3, 4... Therefore , the sixth term may be calculated with x = 6 3 + 3x 3 + (3 × 6) 3 + 18 21 169
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the next 4 terms to create a pattern using the rules DIRECTIONS: Find given below.
No.
PATTERN RULE
FIRST TERM
SECOND TERM
THIRD TERM
FOURTH TERM
FIFTH TERM
Multiply by 3
1
3
9
27
81
Divide by 5
3,750
750
150
30
6
Add 4
15
19
23
27
31
Subtract by 214
2,398
2,184
1,970
1,756
1,542
Multiply by 3
2
6
18
54
162
Subtract by 8
600
592
584
576
568
Add by 128
1,500
1,628
1,756
1,884
2,012
DIRECTIONS: Write the rule of the pattern as an expression. No. THE FIRST FOUR TERMS
RULE AS AN EXPRESSION
5, 10, 15, 20
5 + 5x, where x = 0, 1, 2, 3
9, 13, 17, 21
9 + 4x, where x = 0, 1, 2, 3
40, 30, 20, 10
2, 5, 10, 17 20, 18, 16, 14
or 5x, where x= 1, 2, 3, 4 or 4x + 1, where x= 2, 3, 4, 5 50 - 10x, where x = 1, 2, 3, 4
x2 + 1 where x = 1, 2, 3, 4 22 - 2x, where x = 1, 2, 3, 4 COPYING IS STRICTLY PROHIBITED BY LAW
170
the next 4 terms to create a pattern using the rules DIRECTIONS: Find given below.
No.
PATTERN RULE
FIRST TERM
SECOND TERM
THIRD TERM
FOURTH TERM
FIFTH TERM
Multiply by 5
2
10
50
250
1,250
Divide by 3
3,888
1,296
432
144
48
Add 7
13
20
27
34
41
Subtract by 320
2,456
2,136
1,816
1,496
1,176
Multiply by 2
4
8
16
32
64
Subtract by 13
410
397
384
371
358
Add by 135
1,300
1,435
1,570
1,705
1,840
DIRECTIONS: Write the rule of the pattern as an expression. No. THE FIRST FOUR TERMS 9, 11, 13, 15 3, 5, 7, 9
RULE AS AN EXPRESSION 7 + 2x, where x = 1, 2, 3, 4
or 2x + 9, where x= 0, 1, 2, 3 3 + 2x, where x = 0, 1, 2, 3 or x + 2, where x= 1, 3, 5, 7
20, 15, 10, 5
4, 7, 12, 19
25 - 5x, where x = 1, 2, 3, 4
x2 + 3 where x = 1, 2, 3, 4
12, 8, 4, 0 16 - 4x, where x = 1, 2, 3, 4 171
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1 The numbers in the list follow a pattern.
28, 35, 42, 49, 56, 63, ...
Which expression best describes the pattern of the list? 28 − 7x, where x = 2, 3, 4, 5, 6, ...
14 + 7x, where x = 2, 3, 4, 5, 6, ...
28 + 14x, where x = 0, 1, 2, 3, 4, ...
14x, where x = 2, 3, 4, 5, 6, ...
2 The numbers in the list follow a pattern.
72, 68, 64, 60, 56, 52, ...
Which expression cannot describe the pattern of the list? 40 + 4x, where x = 8, 9, 10, 11, 12, …
80 − 4x, where x = 2, 3, 4, 5, 6, …
72 − 4x, where x = 0, 1, 2, 3, 4, …
100 − 4x, where x = 7, 8, 9, 10, 11, …
3 The numbers in the list follow a pattern.
36, 45, 54, 63, 72, 81, ...
Which expression best describes the pattern of the list? 6x, where x = 6, 7, 8, 9, 10, ...
33 + 3x, where x = 1, 2, 3, 4, 5, ...
27 + 9x, where x = 1, 2, 3, 4, 5, ...
5x + 6, where x = 6, 7, 8, 9, 10, ...
4 The numbers in the list follow a pattern.
98, 100, 102, 104, 106, 108, ...
Which expression best describes the pattern of the list? 100 − 2x, where x = 1, 2, 3, 4, 5, ...
90 + 4x, where x = 2, 3, 4, 5, 6, ...
50 + 2x, where x = 25, 26, 27, 28, 29, ...
2(49 + x), where x = 0, 1, 2, 3, 4, ...
5 The numbers in the list follow a pattern.
10, 13, 16, 19, 22, 25, ...
Which expression cannot describe the pattern of the list? 5x, where x = 2, 3, 4, 5, 6, …
3x + 1, where x = 3, 4, 5, 6, 7, …
7 + 3x, where x = 1, 2, 3, 4, 5, …
3x + 4, where x = 2, 3, 4, 5, 6, …
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172
the next 4 terms to create a pattern using the rules DIRECTIONS: Find given below.
No.
PATTERN RULE
FIRST
SECOND
THIRD
FOURTH
FIFTH
Multiply by 3
5
15
45
135
405
Divide by 4
256
64
16
4
1
Add 6
8
14
20
26
32
Subtract by 280
1,987
1,707
1,427
1,147
867
Multiply by 4
5
20
80
320
1,280
Subtract by 19
354
335
316
297
278
Add by 167
1,100
1,267
1,434
1,601
1,768
DIRECTIONS: Write the rule of the pattern as an expression. No. THE FIRST FOUR TERMS 13, 16, 19, 22 7, 12, 17, 22
RULE AS AN EXPRESSION 10 + 3x, where x = 1, 2, 3, 4
or 3x + 13, where x= 0, 1, 2, 3 7 + 5x, where x = 0, 1, 2, 3 or 5x + 2, where x= 1, 2, 3, 4
33, 30, 27, 24
173
36 - 3x, where x = 1, 2, 3, 4
9, 12, 17, 24
x2 + 8 where x = 1, 2, 3, 4
27, 20, 13, 6
34 - 7x, where x = 1, 2, 3, 4
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1 The numbers in the list follow a pattern.
9, 14, 19, 24, 29, 34, ...
Which expression best describes the pattern of the list? 4x + 5, where x = 0, 1, 2, 3, 4, ...
5x − 1, where x = 2, 3, 4, 5, 6, ...
8x + 1, where x = 1, 2, 3, 4, 5, ...
6x + 3, where x = 1, 2, 3, 4, 5, ...
2 The numbers in the list follow a pattern.
31, 34, 37, 40, 43, 46, ...
Which expression cannot describe the pattern of the list? 20 + 3x, where x = 4, 5, 6, 7, 8, ...
22 + 3x, where x = 3, 4, 5, 6, 6, ...
3x + 1, where x = 10, 11, 12, 13, 14, ...
25 + 3x, where x = 2, 3, 4, 5, 6, ...
3 The numbers in the list follow a pattern.
15, 18, 21, 24, 27, 30, ...
Which expression cannot describe the pattern of the list? 3x, where x = 5, 6, 7, 8, 9, …
6 + 3x, where x = 3, 4, 5, 6, 7, …
9 + 6x, where x = 1, 2, 3, 4, 5, …
3(1 + x), where x = 4, 5, 6, 7, 8, …
4 The numbers in the list follow a pattern.
60, 56, 52, 48, 44, 40, ...
Which expression best describes the pattern of the list? 5x, where x = 6, 7, 8, 9, 10, ...
60 − 4x, where x = 0, 1, 2, 3, 4, ...
30 + 6x, where x = 5, 6, 7, 8, 9, ...
60 − 3x, where x = 0, 1, 2, 3, 4, ...
5 The numbers in the list follow a pattern.
26, 31, 36, 41, 46, 51, ...
Which expression best describes the pattern of the list? 5x + 1, where x = 5, 6, 7, 8, 9, ...
3x + 5, where x = 7, 8, 9, 10, 11, ...
4x + 6, where x = 5, 6, 7, 8, 9, ...
8x + 2, where x = 3, 4, 5, 6, 7, ...
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174
1 The numbers in the list below follow a pattern.
66, 74, 82, 90, 98, 106, ... Which expression describes the pattern of the list? 8x + 2, where x = 8, 9, 10, 11, 12, …
10 + 8x, where x = 7, 8, 9, 10, 11, …
8x − 6, where x = 9, 10, 11, 12, 13, …
8x + 18, where x = 6, 7, 8, 9, 10, …
2 The numbers in the list below follow a pattern.
37, 42, 47, 52, 57, 62, ... Which expression best describes the pattern of the list? 6 + 5x, where x = 6, 7, 8, 9, 10, ...
5x − 2, where x = 8, 9, 10, 11, 12, ...
5x + 8, where x = 4, 5, 6, 7, 8, ...
5x + 2, where x = 7, 8, 9, 10, 11, ...
3 The numbers in the list below follow a pattern.
87, 95, 104, 111, 119, 127, ... Which expression cannot describe the pattern of the list? 8x − 1, where x = 11, 12, 13, 14, 15, ...
8x − 9, where x = 12, 13, 14, 15, 16, ...
8x − 12, where x = 14, 15, 16, 17, 18, ...
8x − 17, where x = 13, 14, 15, 16, 17, ...
4 The numbers in the list below follow a pattern.
23, 25, 27, 29, 31, 33,... Which expression cannot describe the pattern of the list? 2x + 3, where x = 10, 11, 12, 13…
2x + 4, where x = 7, 8, 9, 10, 11, …
2x − 3, where x = 9, 10, 11, 12, 13, …
4x - 12, where x = 6, 7, 8, 9, 10, …
5 The numbers in the list below follow a pattern.
77, 86, 95, 104, 113, 122, ... Which expression best describes the pattern of the list?
175
9x + 3, where x = 6, 7, 8, 9, 10, …
9x + 5, where x = 8, 9, 10, 11, 12, ...
9x - 2, where x = 10, 11, 12, 13,, ...
9x - 4, where x = 12, 13, 14, 15, ...
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6 The numbers in the list below follow a pattern.
64, 68, 72, 76, 80, 84, ... Which expression cannot describe the pattern of the list? 6x + 32, where x = 8, 9, 10, 11, 12, …
6x + 40, where x = 4, 5, 6, 7, 8, …
4x − 6, where x = 13, 14, 15, 16, 17…
4x + 32, where x = 8, 9, 10, 11, 12…
7 The numbers in the list below follow a pattern.
55, 61, 67, 73, 79, 85... Which expression best describes the pattern of the list? 6x + 5, where x = 10, 11, 12, 13, 14...
6x + 8, where x = 8, 9, 10, 11, 12, ...
6x - 5, where x = 10, 11, 12, 13, 14...
10x - 7, where x = 5, 6, 7, 8, 9, 10 ...
8 The numbers in the list below follow a pattern.
61, 68, 75, 82, 89, 96... Which expression cannot describe the pattern of the list? 7x + 2, where x = 8, 9, 10, 11, 12, 13...
7x - 2, where x = 9, 10, 11, 12, 13, 14 ...
7x + 9, where x = 7, 8, 9, 10, 11, 12...
7x - 9, where x = 13, 14, 15, 16, 17,..
9 The numbers in the list below follow a pattern.
69, 84, 99, 114, 129, 144,... Which expression describes the pattern of the list? 15x + 9, where x = 10, 11, 12, 13, 14,…
15x + 12, where x = 3, 4, 5, 6, 7, 8, …
15x - 12, where x = 11, 12, 13, 14, 15…
15x + 9, where x = 4, 5, 6, 7, 8, 9, …
10 The numbers in the list below follow a pattern.
90, 101, 112, 123, 134, 145, ... Which expression best describes the pattern of the list? 11x + 2, where x = 8, 9, 10, 11, 12, 13…
10x + 5, where x = 7, 8, 9, 10, 11, 12, ...
11x - 2, where x = 12, 13, 14, 15, 16...
10x - 5, where x = 13, 14, 15, 16, 17…
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176
RECORDING INPUTS AND OUTPUTS ON A TWO-COLUMN TABLE
MA.5.AR.3.2 Given a rule for a numerical pattern, use a two-column table to record the inputs and outputs.
Use a two-column table to record inputs and outputs for a given rule in a numerical pattern.
STEP 1: Study the pattern and its rule. For example, if the rule is "multiply each input by 2," it means that each input will be doubled to produce the corresponding output. STEP 2: Create a two-column table with one column labeled "Input" and the other labeled "Output." STEP 3: Add the initial input as the starting value on the input column. This can be any number, depending on the given rule and pattern. For example, if the pattern starts with 2, you can choose 2 as the first input. STEP 4: Apply the given rule to the initial input to determine the corresponding output. Write the input value in the "Input" column and the calculated output value in the "Output" column. STEP 5: Continue the pattern by using the output value from the previous step as the next input. Apply the rule to this new input to calculate the next output. Write these values in the appropriate columns of the table. STEP 6: Repeat the process until all of the inputs and outputs are added in to clearly see the pattern.
Example
Input (x)
1
2
3
4
5
RULE: 3x+ 2
output
5
8
11
14
17
The relationship between the input and the output is 3x+ 2. If the 1 is plugged into the equation, “3x+ 2”, we learn that “5 is the result. If the 2 is plugged into the equation, “3x+ 2”, we learn that “8 is the result. If the 3 is plugged into the equation, “3x+ 2”, we learn that “11 is the result. etc... 177
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DIRECTIONS: Use the given rule to calculate the missing outputs. RULE: 3x+ 5 Input (x)
0
1
2
3
4
5
Output
5
8
11
14
17
20
Input (x)
1
2
3
4
5
6
Output
2
6
10
14
18
22
Input (x)
0
1
2
3
4
5
Output
10
12
14
16
18
20
Input (x)
2
3
4
5
6
7
Output
1
5
9
17
21
25
Input (x)
2
3
4
5
6
7
Output
2
5
8
11
14
17
Input (x)
0
1
2
3
4
5
Output
3
5
7
9
11
13
Input (x)
0
1
2
3
4
5
Output
30
28
26
24
22
20
RULE: 4x - 2
RULE: 2x + 10
RULE: 4x - 3
RULE: 3x - 4
RULE: 2x + 3
RULE: 30 - 2x
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178
DIRECTIONS: Use the given rule to calculate the missing outputs. RULE: 2x+ 7 Input (x)
0
1
2
3
4
7
Output
7
9
11
13
15
21
Input (x)
1
2
3
4
5
8
Output
2
7
12
17
22
37
Input (x)
0
1
2
3
4
7
Output
8
11
14
17
20
29
Input (x)
1
2
3
4
5
10
Output
1
4
7
10
13
28
Input (x)
1
2
3
4
5
9
Output
3
10
17
24
31
59
Input (x)
0
1
2
3
4
7
Output
24
21
18
15
12
3
Input (x)
0
1
2
3
4
8
Output
6
10
14
18
22
38
RULE: 5x - 3
RULE: 3x + 8
RULE: 3x - 2
RULE: 7x - 4
RULE: 24 - 3x
RULE: 4x + 6
179
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1 A number pattern is defined by the expression x − 7, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
10 3
12 5
15 8
20 13
Input (x) Output
10 3
11 5
12 8
13 13
Input (x) Output
10 3
12 5
14 8
16 13
Input (x) Output
4 3
5 5
6 8
7 13
2 A number pattern is defined by the expression 9x, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
1 18
2 27
3 45
4 72
Input (x) Output
9 18
12 27
15 45
18 72
Input (x) Output
2 18
3 27
4 45
5 72
Input (x) Output
2 18
3 27
5 45
8 72
3 A number pattern is defined by the expression x ÷ 4, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
8 2
10 3
12 5
14 6
Input (x) Output
4 2
8 3
12 5
16 6
Input (x) Output
8 2
12 3
20 5
24 6
Input (x) Output
4 2
8 3
16 5
32 6
4 A number pattern is defined by the expression 6x, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
6 12
24 30
36 42
78 72
Input (x) Output
2 12
6 30
12 42
18 72
Input (x) Output
2 12
5 30
7 42
12 72
Input (x) Output
3 12
6 30
9 42
12 72
5 A number pattern is defined by the expression 3x − 1, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
3 14
5 23
7 29
10 35
Input (x) Output
5 14
8 23
10 29
12 35
Input (x) Output
2 14
5 23
7 29
10 35
Input (x) Output
3 14
5 23
8 29
10 35
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180
DIRECTIONS: Use the given rule to calculate the missing outputs. RULE: 4x+ 3 Input (x)
0
1
2
3
4
7
Output
3
7
11
15
19
24
Input (x)
1
2
3
4
5
8
Output
3
9
15
21
27
45
Input (x)
0
1
2
3
4
7
Output
6
8
10
12
14
20
Input (x)
1
2
3
4
5
7
Output
1
7
13
19
25
37
Input (x)
2
3
4
5
7
10
Output
4
9
14
19
29
44
Input (x)
0
1
2
3
5
7
Output
4
7
10
13
19
25
Input (x)
0
1
2
3
6
9
Output
50
46
42
38
26
14
RULE: 6x - 3
RULE: 2x + 6
RULE: 6x - 5
RULE: 5x - 6
RULE: 3x + 4
RULE: 50 - 4x
181
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1 A number pattern is defined by the expression 27 − (x ÷ 3), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
1 27
9 24
12 21
15 18
Input (x) Output
9 27
8 24
7 21
6 18
Input (x) Output
3 27
6 24
9 21
12 18
Input (x) Output
0 27
9 24
18 21
27 18
2 A number pattern is defined by the expression 4x − 6, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
2 2
3 6
5 14
6 18
Input (x) Output
2 2
4 6
6 14
8 18
Input (x) Output
3 2
4 6
5 14
6 18
Input (x) Output
3 2
6 6
9 14
12 18
3 A number pattern is defined by the expression 3 + (x ÷ 2), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
4 5
8 7
12 11
16 13
Input (x) Output
2 5
4 7
8 11
16 13
Input (x) Output
10 5
14 7
22 11
26 13
Input (x) Output
4 5
8 7
16 11
20 13
4 A number pattern is defined by the expression 3x + 1, where x represents any whole number. Which of these two-column tables does not represent the same number pattern? Input (x) Output
1 4
2 7
3 10
4 13
Input (x)
2
4
6
8
Output
7
13
19
25
Input (x) Output
3 10
6 19
9 28
12 37
Input (x) Output
0 3
5 8
10 13
15 18
5 A number pattern is defined by the expression (x ÷ 5) − 5, where x represents any whole number. Which of these two-column tables does not represent the same number pattern? Input (x) Output
40 3
45 4
50 5
55 6
Input (x) Output
20 1
40 60 80 2 3 4
Input (x) Output
30 1
40 3
50 5
60 7
Input (x) Output
40 3
60 7
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80 11
100 15 182
1 A number pattern is defined by the expression 2 × (x + 2), where x represents any whole number. Which of these two-column tables does not represent the same number pattern? Input (x) Output
0 4
1 6
2 8
3 10
Input (x) Output
3 10
6 16
9 22
12 28
Input (x) Output
0 0
2 8
4 12
6 16
Input (x) Output
0 4
4 12
8 20
12 28
2 A number pattern is defined by the expression 20 − (x ÷ 2), where x represents any whole number. Which of these two-column tables does not represent the same number pattern? Input (x) Output
0 20
2 19
4 18
6 17
Input (x) Output
6 14
8 12
10 10
12 8
Input (x) Output
4 18
6 17
8 16
10 15
Input (x) Output
8 16
10 15
12 14
14 13
3 A number pattern is defined by the expression 5x − 4, where x represents any whole number. Which of these two-column tables does not represent the same number pattern? Input (x) Output
0 4
4 16
8 36
10 46
Input (x) Output
2 6
4 16
6 26
8 36
Input (x) Output
3 11
6 26
9 41
12 56
Input (x) Output
1 1
3 11
5 21
7 31
4 A number pattern is defined by the expression 2 × (x + 4), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
4 16
5 18
9 26
14 30
Input (x) Output
1 10
3 14
7 22
8 24
Input (x) Output
0 8
1 12
6 20
7 22
Input (x) Output
11 30
13 34
15 38
17 42
5 A number pattern is defined by the expression 3 × (2x ÷ 4), where x represents any whole number. Which of these two-column tables does not represent the same number pattern?
183
Input (x) Output
0 0
2 3
4 5
6 8
Input (x) Output
2 3
8 12
14 21
20 30
Input (x) Output
10 15
12 18
16 24
18 27
Input (x) Output
22 33
24 36
26 39
28 42
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 A number pattern is defined by the expression 3 + (x ÷ 2), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
0 3
2 4
4 5
6 7
Input (x) Output
20 13
22 14
24 16
26 18
Input (x) Output
4 6
6 8
8 10
16 11
Input (x) Output
8 7
10 8
12 9
14 10
7 A number pattern is defined by the expression 5 + (2x ÷ 4), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
0 5
2 6
4 8
6 10
Input (x) Output
10 10
12 11
14 12
16 13
Input (x) Output
1 3
3 5
5 7
7 9
Input (x) Output
16 13
18 15
20 17
22 19
8 A number pattern is defined by the expression 2 × (x ÷ 3), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
0 0
3 2
6 4
9 6
Input (x) Output
9 6
12 8
15 10
18 13
Input (x) Output
12 7
21 14
24 16
27 18
Input (x) Output
30 20
33 22
36 24
39 28
9 A number pattern is defined by the expression 7 + (3x ÷ 6), where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
0 7
2 8
4 10
6 12
Input (x) Output
0 7
16 15
32 23
48 30
Input (x) Output
0 7
10 12
20 17
30 22
Input (x) Output
0 7
8 11
12 13
14 16
10 A number pattern is defined by the expression 2x - 3, where x represents any whole number. Which of these two-column tables represents the same number pattern? Input (x) Output
2 1
3 4
4 5
6 9
Input (x) Output
5 7
10 17
15 27
25 37
Input (x) Output
7 11
8 14
9 17
10 20
Input (x) Output
10 17
11 19
12 21
13 23
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CONVERTING METRIC UNITS MA.5.M.1.1 Solve multi-step real-world problems that involve converting measurement units to equivalent measurements within a single system of measurement.
Use tables to convert standard measurement units. Time
Length
Weight/Mass Capacity/Volume
1 ft = 12 in. 1 minute = 60 seconds 1 yd = 3 ft 1 hour = 60 minutes 1 mile = 5,280 ft 1 day = 24 hours 1 week = 7 days 1 meter (m) = 1,000 mm 1 month = 28 to 31 days 1 meter = 100 cm 1 month = 4 weeks 1 meter = 10 dm 1 year = 12 months 1 cm = 10 mm 1 year = 52 weeks 1 dm = 10 cm 1 year = 365 days 1 km = 1,000 m
CONVERSION TABLE
Examples:
TABLE 1
Multiply each km by 1,000
km
m
1 2 3 4
1,000 2,000 3,000 4,000
× 1,000 × 1,000 × 1,000 × 1,000
1 lb = 16 oz
1 L = 1,000 mL
1 ton = 2,000 lb
1 g = 1,000 mg
1 cup = 8 fluid ounces 1 pint = 2 cups
1 kg = 1,000 g
1 g = 1,000 mg 1 kg = 1,000 g
1 quart = 2 pints 1 gallon = 4 quarts
TABLE 2 Multiply each hour by 60 × 60 × 60 × 60 × 60
HOUR
MINUTES
1 2 3 4
60 120 180 240
Use tables to convert standard measurement units. 20 liters =
Example 1:
20 liters
1000 mL
×
1
1 liter
17 cm =
Example 2:
17 cm 1 185
milliliters 20 × 1,000 mL 1×1
= 20,000 mL
17 100
= 0.17 meter
meters 1m
×
=
100 cm
=
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
Convert the following mass measurements. Use the DIRECTIONS: mass section of the Conversion Table as needed. NO.
CONVERSION PROBLEM
ANSWER
4 pounds = _______ ounces
64 ounces
5 tons = _______ pounds
10,000 pounds
2.5 kilograms = _______ grams
2,500 grams
48 ounces = _______ pounds
3 pounds
4,000 pounds = _______ tons
2 tons
the following length measurements. Use the DIRECTIONS: Convert length section of the Conversion Table as needed. NO.
CONVERSION PROBLEM
ANSWER
6 feet = _______ inches
72 inches
8 yards = _______ feet
24 feet
3 meters = _______ centimeters
300 centimeters
21 feet = _______ yards
7 yards
48 inches = _______ feet
4 feet
the following time measurements. Use the DIRECTIONS: Convert time section of the Conversion Table as needed. NO.
CONVERSION PROBLEM
ANSWER
3 minutes = _______ seconds
180 seconds
360 seconds = _______ minutes
6 minutes
240 minutes = _______ hours
4 hours
5 hours = _______ minutes
300 minutes
9 days = ______ hours
216 hours
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the following mass measurements. Use the DIRECTIONS: Convert mass section of the Conversion Table as needed.
NO.
CONVERSION PROBLEM
ANSWER
6 pounds = _______ ounces
96 ounces
3 tons = _______ pounds
6,000 pounds
5.1 kilograms = _______ grams
5,100 grams
80 ounces = _______ pounds
5 pounds
8,000 pounds = _______ tons
4 tons
the following length measurements. Use the DIRECTIONS: Convert length section of the Conversion Table as needed. NO.
CONVERSION PROBLEM
ANSWER
3 feet = _______ inches
36 inches
11 yards = _______ feet
33 feet
5 meters = _______ centimeters
500 centimeters
27 feet = _______ yards
9 yards
60 inches = _______ feet
5 feet
the following time measurements. Use the DIRECTIONS: Convert time section of the Conversion Table as needed. NO.
187
CONVERSION PROBLEM
ANSWER
7 minutes = _______ seconds
420 seconds
300 seconds = _______ minutes
5 minutes
360 minutes = _______ hours
6 hours
3 hours = _______ minutes
180 minutes
7 days = ______ hours
168 hours
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 There are 12 inches in 1 foot. Kim has a piece of string 4 feet and 8 inches long. She cuts the string into 2 pieces of equal length. How many inches long is 1 piece? 10 inches
16 inches
28 inches
30 inches
2 There are 3 feet in 1 yard. Noel wants to put a fence around his vegetable garden. One side of his vegetable garden is 3 yards, 6 feet long and the other side is 2 yards, 3 feet wide. How many feet of fencing materials does Noel need? 8 feet
16 feet
24 feet
48 feet
3 There are 2 cups in 1 pint. A recipe calls for 4.5 pints of milk. Myrna only has 2 cartons of milk, each containing 3 cups. How much more milk does Myrna need? 1 1 1 cups 2 cups 2 2 3 cups
4 cups
4 There are 12 inches in 1 foot. A rectangular sandbox has a perimeter of 8 feet. If 1 side of the sandbox is 4 inches longer than the other side, what is the length of the longer side of the sandbox? 18 inches
22 inches
24 inches
26 inches
5 There are sixteen ounces in one pound. A box contains seven and a half pounds of nails. How many nails are there in the box if each nail weighs three ounces? 36 nails
40 nails
50 nails
65 nails
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the following length measurements. Use the DIRECTIONS: Convert mass section of the Conversion Table as needed.
NO.
CONVERSION PROBLEM
ANSWER
10 pounds = _______ ounces
160 ounces
9 tons = _______ pounds
18,000 pounds
8.8 kilograms = _______ grams
8,800 grams
128 ounces = _______ pounds
8 pounds
10,000 pounds = _______ tons
5 tons
the following length measurements. Use the DIRECTIONS: Convert length section of the Conversion Table as needed. NO.
CONVERSION PROBLEM
ANSWER
12 feet = _______ inches
144 inches
4 yards = _______ feet
12 feet
9 meters = _______ centimeters
900 centimeters
18 feet = _______ yards
6 yards
36 inches = _______ feet
3 feet
the following time measurements. Use the DIRECTIONS: Convert time section of the Conversion Table as needed. NO.
189
CONVERSION PROBLEM
ANSWER
13 minutes = _______ seconds
780 seconds
180 seconds = _______ minutes
3 minutes
660 minutes = _______ hours
11 hours
8 hours = _______ minutes
480 minutes
15 days = ______ hours
360 hours
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 There are 12 inches in 1 foot and 3 feet in 1 yard. Debbie had a piece of fabric 3 yards and 9 feet long. She used 1.5 yards of the fabric to make a tablecloth and used the remaining fabric to make pillowcases. If Debbie used 18 inches of fabric to make 1 pillowcase, how many pillowcases did she make in total? 9 pillowcases
10 pillowcases
12 pillowcases
15 pillowcases
2 There are 12 inches in 1 foot. If the perimeter of a square is 1 1 feet, which of the following 3 gives the length of the side of the square in inches? 2 inches
3 inches
4 inches
6 inches
3 There are 12 inches in 1 foot. Rowena used 11.25 feet of yarn to tie 15 bows. If she used the same amount of yarn to make each bow, how many inches of yarn did Rowena use to make 1 bow? 8 inches
9 inches
12 inches
15 inches
4 There are 2 pints in 1 quart, 4 quarts in 1 gallon, and 60 minutes in 1 hour. An orange squeezer machine can produce 8 pints of orange juice every 5 minutes. How many gallons of orange juice can the machine produce in 2 hours? 24 gallons
36 gallons
48 gallons
192 gallons
5 There are 60 minutes in 1 hour. A machine can make 55 widgets per minute. At this rate, how many hours will it take the machine to make 4,400 widgets? 1 of an hour 3 1 1 hours 6
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190
1 There are 60 minutes in 1 hour. A printer can print 1 picture every 2 of a minute. How many 3 pictures can the printer print if it runs continuously for 2 hours? 80 pictures
120 pictures
180 pictures
240 pictures
2 There are 60 minutes in 1 hour. If Jerry can complete 3 of his task in one hour, how many 4 minutes will it take him to fully complete the task? 45 minutes
75 minutes
80 minutes
90 minutes
3 There are 12 inches in 1 foot. George wants to use a piece of wire 8 feet long to make several frames. Each frame requires 15 inches of wire. What is the greatest possible number of frames that George can make? 5 frames
6 frames
7 frames
8 frames
4 There are 3 feet in 1 yard. A rectangular courtyard has a perimeter of 12 yards. If one side of the courtyard is 2 feet longer than the other side, what is the length of the longer side of the sandbox? 6 feet
8 feet
10 feet
12 feet
5 There are 12 inches in 1 foot and 3 feet in 1 yard. Margret had a piece of fabric 5 yards and 6 feet long. She used 2 yards of the fabric to make a table runner and used the remaining fabric to make dinner napkins. If Debbie used 15 inches of fabric to make 1 table runner, how many dinner napkins did she make in total?
191
9 dinner napkins
10 dinner napkins
11 dinner napkins
12 dinner napkins
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 There are 2 cups in 1 pint. A recipe calls for 6.5 pints of cream. Emma only has 2 cartons of cream, each containing 4 cups. How much more cream does Emma need? 2 1 cups 1 2 cups 4 3 1 cup 5 cups
7
There are 12 inches in 1 foot. If the perimeter of a square is 2 2 feet, which of the following 6 gives the length of the side of the square in inches? 1 inches
3 inches
5 inches
7 inches
8 There are 12 inches in 1 foot. Stephen used 13.75 feet of ribbon to wrap 33 gifts. If he used the same amount of ribbon to wrap each gift, how many inches of ribbon did Stephen use to wrap 1 gift? 5 inches
9 inches
11 inches
15 inches
9 There are 2 pints in 1 quart, 4 quarts in 1 gallon, and 60 minutes in 1 hour. A lemon squeezer machine can produce 12 pints of lemon juice every 5 minutes. How many gallons of lemon juice can the machine produce in 3 hours? 54 gallons
36 gallons
216 gallons
192 gallons
10 There are 60 minutes in 1 hour. A secretary can type 120 words per minute. At this rate, how many hours will it take the secretary to type 2,400 words? 1 of an hour 3
2 of an hour 3
1 1 hours 6
1 1 hours 3
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REAL-WORLD PROBLEMS INVOLVING MONEY
MA.5.M.2.1 Solve multi-step real-world problems involving money using decimal notation.
Use decimal notation to solve multi-step real-world problems involving money. STEP 1: Identify the prices, quantities, discounts, or any other monetary values mentioned in the problem. Record the decimal notation for each value. STEP 2: Determine the unknown values that need to be solved, along with any comparisons that must be made. STEP 3: Perform the necessary operations to find the answer. Example 1 John is at the supermarket and wants to buy orange juice. The first option offers a 32-ounce bottle for $2.99, while the second offer is a 16-ounce bottle for $1.50. Which option is cheaper per ounce? $2.99 ÷ 32 ounces = $0.09 per ounce
and
$1.65 ÷ 16 ounces = $0.10 per ounce
$0.09 per ounce < $0.10 per ounce Therefore, $2.99 for 32 ounces of orange juice is a better buy. Example 2 Alice, Bob, and Carol decided to contribute money to buy a gift for their friend. Bob contributed $5 more than twice the amount contributed by Alice, and Carol contributed $4 less than the amount contributed by Bob. If the price of the gift was $41, how much did Carol contribute? B = Bob, A = Alice, and C = Carol B = $5 + 2A
(Bob contributed $5 more than twice the amount contributed by Alice)
C = B - $4
(Carol contributed $4 less than the amount contributed by Bob)
If the price of the gift was $41, how much did Carol contribute?
$41 = A
+
B
+
C
$41 = 1A + $5 + 2A + $5 + 2A - $4 $41 = 5A + $6 $41 - $6 = 5A + $6 - $6 193
$35 = 5A
B = $5 + 2A
C = B - $4
$35 = 5A 5 5
B = $5 + 2 × $7
C = $19 - $4
B = $5 + $14
C = $15
$7 = A
B = $19
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
$45
64 ounces
$0.70
$0.72
$23
32 ounces
$12.50
12 feet
$1.05
$1.22
$10.99
9 feet
$100
32 gallons
$3.125
$4.17
$75
18 gallons
$63
24 hours
$0.70
$2.50
$30
12 hours
$12.99
12 slices
$1.08
$1.37
$10.96
8 slices
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$12
20 bags
$0.60
$0.48
$7.60
16 bags
$25
16 cookies
$1.56
$1.56
$50
32 cookies
$45
5 toy cars
$9.00
$8.28
$82.75
10 toy cars
$29.95
2 shirts
$14.98
$9.375
$37.50
4 shirts
$2.99
20 liters
$0.15
$0.16
$5.25
32 liters
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$24.95
5 books
$4.99
$5.00
$49.95
10 books
$1,000
12 hours
$83.33
$95.83
$575
6 hours
$64
8 yards
$8
$9
$45
5 yards
$15
10 games
$1.50
$1.06
$12.75
12 games
$7.99
24 pencils
$0.33
$0.28
$9.99
36 pencils
>, <, or =
COPYING IS STRICTLY PROHIBITED BY LAW
194
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
$16
35 ounces
$0.46
$0.58
$26
45 ounces
$9.25
11 feet
$0.84
$1.20
$10.76
9 feet
$70
21 gallons
$3.33
$2.87
$66
23 gallons
$18
20 hours
$0.9
$1.41
$24
17 hours
$9.45
6 slices
$1.58
$1.33
$10.65
8 slices
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$19
22 bags
$0.86
$1.34
$22.78
17 bags
$34
18 cookies
$1.89
$1.81
$29
16 cookies
$48
12 toy cars
$4.00
$4.00
$36
9 toy cars
$27.89
3 shirts
$9.3
$7.38
$29.50
4 shirts
$6.29
13 liters
$0.48
$0.47
$7.46
16 liters
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
195
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$57.8
7 books
$8.26
$7.6
$60.8
8 books
$500
16 hours
$31.25
$33.53
$570
17 hours
$55
12 yards
$4.58
$4.8
$48
10 yards
$52
11 games
$4.73
$4.86
$43.75
9 games
$4.47
15 pencils
$0.30
$0.17
$3.91
21 pencils
>, <, or =
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Sarah is at the store and wants to buy juice. Which option would be cheaper: buying one 32-ounce bottle for $2.56 or buying two 16-ounce bottles for $1.44 each? 16-ounce bottle is $0.09 per ounce
32-ounce bottle is $0.09 per ounce
16-ounce bottle is $0.08 per ounce
32-ounce bottle is $0.08 per ounce
2 Robert, Doug, and Kevin agreed to put their money together buy a new gaming table. Doug gave $4 more than twice the amount that Kevin gave, and Kevin gave $3 less than the amount that Robert gave. If the price of the new gaming table was $27, how much did Kevin give for the new gaming table? $5
$8
$10
$12
3 1 1 Mr. Romero left half of the money in his estate to his wife, 4 to his eldest daughter, 5 to his youngest daughter, and the remaining $100,000 to charity. How much was the total amount of money in Mr. Romero’s estate? $1,000,000
$1,500,000
$2,000,000
$4,000,000
4 Gene has $15 more than Andrew, and together they have a total of $32. How much money does Andrew have? $8.50
$23.50
$24.50
$25.50
5 Together, Frank and Alice initially had $100 when they went to the fair. Frank spent $10 on a bag of chips at the fair. At this point, he has twice as much money as Alice, who has not yet spent any of her money. How much money did Frank have initially? $30
$60
$65
$70 COPYING IS STRICTLY PROHIBITED BY LAW
196
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
$25
49 ounces
$0.51
$0.7
$28
40 ounces
$6.89
13 feet
$0.53
$0.54
$5.38
10 feet
$34
15 gallons
$2.27
$2.57
$36
14 gallons
$17
12 hours
$1.42
$0.94
$17
18 hours
$15.45
14 slices
$1.10
$0.91
$13.65
15 slices
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$24
19 bags
$1.26
$1.37
$23.3
17 bags
$17
21 cookies
$0.81
$1.19
$19
16 cookies
$53
11 toy cars
$4.82
$4.82
$106
22 toy cars
$18.5
5 shirts
$3.7
$4.57
$13.70
3 shirts
$10.69
23 liters
$0.46
$0.49
$8.26
17 liters
COST PER UNIT 3
DOLLAR AMOUNT 2
NUMBER OF UNITS 2
>, <, or =
DIRECTIONS: Compare the price points below. Use the symbols >, <, or =. NO.
197
DOLLAR AMOUNT 1
NUMBER OF UNITS 1
COST PER UNIT 1
$27.8
9 books
$3.08
$4.8
$28.8
6 books
$400
26 hours
$15.38
$21.74
$500
23 hours
$60
9 yards
$6.67
$5.33
$64
12 yards
$56
11 games
$5.09
$6.75
$54
8 games
$6.72
10 pencils
$0.67
$0.56
$7.90
14 pencils
>, <, or =
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 A mechanic normally works 8 hours per day and earns $13.50 per hour. For each hour the mechanic works in excess of 8 hours on a given day, he is paid 1.5 times his normal rate. If the mechanic works 10 hours on a given day, how much does he earn on that particular day? $136.50
$148.50
$155.50
$162.50
2 Evelyn is paid $15 per hour for the first 40 hours she works per week and $20 per hour for each hour she works in excess of the first 40 hours per week. If Evelyn earned $740 last week, how many hours did she work last week? 45 hours
46 hours
47 hours
48 hours
3 If Albert gives Bernard $17 and Bernard gives David $13, the three of them will have the same amount of money. How much more money does Albert have than Bernard at the beginning? $21
$22
$23
$25
4 Vicky has $6 more than Hank has. If Hank gives Leo $2 and Noel gives Hank $5, how much more money does Vicky have than Hank now? $12
$10
$6
$3
5 John worked 40 hours last week, including 6 hours during the weekend. John earns $18 per hour during weekdays (Monday to Friday) and 1.5 times that amount during weekends. How much did John earn last week? $752
$764
$768
$774
COPYING IS STRICTLY PROHIBITED BY LAW
198
1 Emily has $3 more than Cynthia has, but $5 less than Kim has. If Emily has $17, how much money do Cynthia and Kim have altogether? $32
$34
$36
$40
2 George went to the hardware store and spent 2 of his money to buy a new power drill, 1 to 5 3 1 buy some bolt locks, and to buy some nails. If George spent the remaining $30 to buy a new 4 hammer, how much did George spend at the hardware store? $1,200
$1,500
$1,800
$2,000
3 If Gene has four times the amount of money that he has now, he will have the exact amount of money to buy four video games and two CDs. If each video game costs $12.95 and each CD costs $8.50, how much money does Gene have now? $14.60
$15.80
$16.40
$17.20
4
Mr. Richie left half of the money in his estate to his wife, 1 to his eldest daughter, 1 to his 3 9 youngest daughter, and the remaining $300,000 to charity. How much was the total amount of money in Mr. Richie’s estate? $2,900,000
$5,400,000
$3,000,000
$4,600,000
5 Together, Billy and Laura initially had $150 when they went to the concert. Billy spent $15 on a snack at the concession stand. At this point, he has twice as much money as Laura, who has not yet spent any of her money. How much money did Billy have initially?
199
$90
$105
$100
$120 MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
6 A carpenter normally works 9 hours per day and earns $11.50 per hour. For each hour the carpenter works in excess of 9 hours on a given day, he is paid 1.7 times his normal rate. If the carpenter works 11 hours on a given day, how much does he earn on that particular day? $142.60
$149.60
$151.60
$164.60
7 Cate is paid $17 per hour for the first 30 hours she works per week and $25 per hour for each hour she works in excess of the first 30 hours per week. If Cate earned $660 last week, how many hours did she work last week? 30 hours
32 hours
35 hours
36 hours
8 If Zack gives Gary $19 and Gary gives Will $11, the three of them will have the same amount of money. How much more money does Zack have than Gary at the beginning? $23
$27
$32
$36
9 Bob has $7 more than Henry has. If Henry gives Logan $4 and Nagel gives Henry $6, how much more money does Bob have than Henry now? $11
$9
$5
$4
10 Jeremy worked 36 hours last week, including 4 hours during the weekend. Jeremy earns $15 per hour during weekdays (Monday to Friday) and 2.5 times that amount during weekends. How much did Jeremy earn last week?
$830
$630
$530
$730
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200
CLASSIFYING TRIANGLES AND QUADRILATERALS MA.5.GR.1.1 Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category. Identify polygons based on the number of sides and the number of angles. Triangles have 3 sides and 3 angles, whereas quadrilaterals have 4 sides and 4 angles.
A quadrilateral has 4 sides and 4 angles.
A triangle has 3 sides and 3 angles.
A pentagon has 5 sides and 5 angles.
A hexagon has 6 sides and 6 angles.
Classify triangles based on the size of the angle. acute acute acute acute
acute
Isosceles Triangle: 2 equal sides
obtuse right
A right triangle has 1 right angle.
acute
An obtuse triangle has 1 obtuse angle.
Scalene Triangle: 3 unequal sides
An acute triangle has 3 acute angles.
Equilateral Triangle: 4 equal sides
Classify quadrilaterals by the number of sides, types of lines, and the size of its angles.
A trapezoid has 1 pair of parallel sides. A quadrilateral has 4 sides and 4 angles.
201
A parallelogram has 2 pairs of parallel sides.
A rectangle has 4 right angles and 2 pairs of parallel sides.
A square has 4 right angles, 4 equal sides, and 2 pairs of parallel sides.
A rhombus has 4 equal sides and 2 pairs of parallel sides. Opposite angles are equal.
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
Circle the numbers for all of the quadrilaterals. (2) Write an “o” DIRECTIONS: (1) on all the obtuse angles and an “a” on all of the acute angles.
a
o
a
a
a o a
a
a
a
a
a a
o
o
o
Identify the triangles below based on the measures of the angles.
DIRECTIONS: Determine each type of triangle as right, obtuse, or acute. TRIANGLE
TYPE OF TRIANGLE
ANGLE MEASURES Number of ACUTE angles
3
Number of OBTUSE angles
0
Number of RIGHT angles
0
Number of ACUTE angles
2
Number of OBTUSE angles
0
Number of RIGHT angles
1
Number of ACUTE angles
2
Number of OBTUSE angles
1
Number of RIGHT angles
0
Acute Triangle Equilateral Triangle
Right Triangle Scalene Triangle
Obtuse Triangle Scalene Triangle
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DIRECTIONS: (1) Circle the numbers for all of the quadrilaterals. (2) Write an “o” (2) on all the obtuse angles and an “a” on all of the acute angles.
o
o
o
o o
o
o o
o a
a
o
o
o a
Identify the triangles below based on the measures of the angles.
DIRECTIONS: Determine each type of triangle as right, obtuse, or acute. TRIANGLE
TYPE OF TRIANGLE
ANGLE MEASURES Number of ACUTE angles
2
Number of OBTUSE angles
1
Number of RIGHT angles
0
Number of ACUTE angles
3
Number of OBTUSE angles
0
Number of RIGHT angles
0
Number of ACUTE angles
2
Number of OBTUSE angles
0
Number of RIGHT angles
1
Obtuse Triangle Scalene Triangle
Acute Triangle Isosceles Triangle
Right Triangle
Scalene Triangle 203
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1 An equilateral triangle is ___________ an acute triangle. Never
Always
Sometimes
Rarely
2 Select all the shapes below that can be classified as quadrilaterals. Triangle Parallelogram Square Trapezoid Hexagon
3 Anastasia and Chapman are investigating shapes. Anastasia thinks that a square is always a rectangle, and Chapman thinks that a square is always a rhombus. Who is correct? Anastasia is correct because a square has two pairs of parallel sides and four right angles, like a rectangle. Chapman is correct because a square has two pairs of parallel sides and all sides of equal length, like a rhombus. Anastasia and Chapman are both correct because a square is a special type of rectangle and a special type of rhombus. Neither is correct because a square is not a quadrilateral, but rectangles and rhombuses are.
4 Which terms below will make the statement true? An obtuse, isosceles triangle always has _______ obtuse angle(s) and _________ equal side(s). one, three
two, zero
one, two
three, two
5 Which of the following can a square not be classified as? Quadrilateral
Parallelogram
Rectangle
Trapezoid COPYING IS STRICTLY PROHIBITED BY LAW
204
if each of the statements below are TRUE or FALSE. DIRECTIONS: Decide Explain your answers. NO.
STATEMENT
DECISION
Andrew says that every right triangle has 2 acute angles.
TRUE
Galen states that every square is a parallelogram.
TRUE
EXPLANATION Answers may vary.
FALSE
Answers may vary.
FALSE
Tami declares that each angle in an equilateral triangle is 60o.
Dayana says that in a right isosceles triangle, each angle, other than the 90° angle, is equal to 45°.
Priscilla claims that a cube is made up of 6 square faces.
TRUE
Answers may vary.
FALSE
TRUE
Answers may vary.
FALSE
TRUE
Answers may vary.
FALSE
Fabiola declares that if a figure is a rhombus, then it is a regular quadrilateral.
TRUE
Gabriel claims that a rhombus is a parallelogram.
TRUE
Answers may vary.
FALSE
Answers may vary.
FALSE 205
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which shape always has two pairs of parallel sides, two pairs of sides of equal length, and four right angles? Rectangle
Trapezoid
Rhombus
Quadrilateral
Sometimes
Rarely
2 A quadrilateral is ___________ a trapezoid. Never
Always
3 Which of the following shapes can be classified as both a quadrilateral and parallelogram? Select all that apply. Triangle Trapezoid
Square Rectangle Rhombus
4 Which of the following statements is true? Scalene triangles always have three equal sides. Isosceles triangles never have any equal sides. Right triangles always have one right angle. Acute triangles always have one obtuse angle.
5 Which of the following statements is not true?
Trapezoids have exactly two pairs of parallel sides. Parallelograms can have four right angles. Quadrilaterals always have four sides. Rectangles can sometimes be rhombuses. COPYING IS STRICTLY PROHIBITED BY LAW
206
1 How many equal sides can an isosceles triangle have? no
one
two
three
2 The last question on Ronald’s math quiz is: A triangle can have two ________ angles. Which of the following is the correct answer? acute obtuse
right straight
3 Sally picks a shape that has 4 sides. She says her shape has only 1 pair of equal and parallel sides. Which shape did she pick? square
trapezoid
rectangle
rhombus
4 Which of these shapes has 2 pairs of equal and parallel sides but no right angles? trapezoid
square
rectangle
parallelogram
5 Emma wrote the following sentence on the board: A rectangle is _________ a quadrilateral. A quadrilateral is ___________ a rectangle. Which words will make the sentence correct? always, never
never, sometimes
always, sometimes
sometimes, never
6 What is the least number of acute angles that a triangle can have?
207
0
1
2
3 MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
7 Which of the following statements is true? A right triangle has all right angles. A scalene triangle has no equal sides An equilateral triangle has 2 acute angles. An isosceles triangle has 3 equal sides.
8 Will drew the triangle below. Which of the following best classifies the triangle?
isosceles, right
isosceles, acute
equilateral, acute
scalene, acute
9 Mr. Perkins notices the sign below as he crosses the street. Which statement below best describes the sign?
triangle and quadrilateral
obtuse triangle
equilateral triangle
triangle, not a quadrilateral
10 What is the least number of acute angles that an acute triangle can have? 0
1
2
3
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208
CLASSIFYING TRIANGLES AND QUADRILATERALS MA.5.GR.1.2 Identify and classify three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres. Classify three-dimensional figures into categories based on their attributes.
Right Pyramid - pyramid with a base that is a polygon and triangular faces that meet at a common vertex called the apex.
Right Prism - a prism with rectangular bases and rectangular lateral faces that are perpendicular to the bases.
Right Circular Cylinder - a cylinder with circular bases that are perpendicular to the lateral surface.
Right Circular Cone - a cone with a circular base and a vertex that is directly above the center of the base.
Sphere - a three-dimensional figure with all points equidistant from a center point.
209
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DIRECTIONS: Answer the questions for each of the three-dimensional shapes listed.
NO.
FIGURE
What is the shape of the base?
How many faces?
Square Base
5
Rectangular Base
2
Circular Base
2
Circular Base
1
No Base
0
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210
DIRECTIONS: Answer the questions for each of the three-dimensional shapes listed.
NO.
211
FIGURE
What shapes are present?
How many bases?
1 Square 4 Triangles
1
4 Rectangles 2 Squares
2
2 Circles 1 Cylinder
2
1 Circle 1 Cone
1
1 Circle
0
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Which of the following shapes contains no flat faces? Rectangular prism
Cone
Sphere
Pyramid
2 Which of the following shapes contains exactly two circular, flat faces? Pyramid
Sphere
Cylinder
Cone
3 Ryder is describing a shape to his friend. The shape has one circular flat face, a curved surface, and an apex. Which shape is Ryder describing? Cone
Rectangular prism
Pyramid
Cylinder
4 Which of the following statements is true? A pyramid has a circular base. A sphere has no flat surfaces. A rectangular prism has two curved surfaces. A cone has two apexes.
5 Winnie sorts shapes into two categories: those that have curved surfaces, and those that have flat surfaces. In which category would a cylinder fit? A cylinder would fit into the curved surfaces category.
A cylinder would fit into the flat surfaces category. A cylinder would fit into both categories. A cylinder would not fit into either category. COPYING IS STRICTLY PROHIBITED BY LAW
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DIRECTIONS: Answer the questions for each of the three-dimensional shapes listed.
NO.
FIGURE
What real life objects come to mind?
How many polygons?
Egyptian Pyramid
5
Cereal Boxes
6
Soda Cans
2
Ice Cream Cone
1
Sun
213
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0
1 How many flat surfaces does a rectangular prism have? 6
4
8
5
2 A cube is a special type of _____________. Cone
Rectangular prism
Sphere
Pyramid
3 Luke is looking for a shape that has the following attributes: 5 flat faces Rectangular base
An apex No curved surfaces or circular flat faces
Which of the following shapes is he looking for? Cone
Cylinder
Pyramid
Sphere
4 Which of the following shapes has only straight edges? Rectangular prism Cylinder Cone Sphere
5 Which of the following is an attribute that cones and pyramids share?
Both have curved edges. Both have curved surfaces. Both have exactly four faces. Both have an apex. COPYING IS STRICTLY PROHIBITED BY LAW
214
1 Which of the following shapes contain only 1 flat face? Rectangular prism
Cone
Sphere
Cylinder
2 John has a shape in his hand. The shape has 1 curved surface and 2 flat surfaces. Which of the following could be John’s shape? Rectangular prism
Sphere
Cylinder
Cone
3 How many flat surfaces does a sphere have? 0
1
2
3
4 Which of the following statements is true? A cone has two flat surfaces. A cylinder has one curved surface. A rectangular prism has no flat surfaces. A sphere has one flat face.
5 Which of the following is an attribute that cones and cylinders have? Both have an apex.
Both have curved edges. Both have straight edges. Both have two flat surfaces.
215
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6 How many edges does a rectangular prism have? 4
6
8
12
7 Sarah has a shape in her hand. The shape has no faces, but has a curved, continuous surface. Which of the following could be Sarah's shape? Cone
Sphere
Cylinder
Cube
8 How many vertices does a cube have?
4
6
8
12
9 Which figure has a square base and four triangular faces meeting at a single vertex? right circular cone right pyramid right circular cylinder rectangular prism
10 Which of the following shapes has a circular base and curved surface that tapers to a point, forming a vertex at the top?
right circular cone right pyramid right circular cylinder rectangular prism COPYING IS STRICTLY PROHIBITED BY LAW
216
FINDING THE PERIMETER AND AREA OF A RECTANGLE MA.5.GR.2.1 Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas.
Perimeter is defined as the total distance around the edge of the figure. To calculate the perimeter:
Perimeter = 2
+2w
Step 1: Identify the length of the rectangle and understand that it represents the opposite side as well. Step 2: Identify the width of the rectangle and understand that it represents the opposite side as well. Step 3: Add the length of each side to find the perimeter of the rectangle. Use your knowledge of adding fractions and decimals to complete this calculation.
1 + 3 1 + 6 1 + 3 1 = 19 2 Example 1: Perimeter = 6.2 + 3.5 + 6.2 + 3.5 = 19.4 Example 2: Perimeter = 6 5
6.2 3.5
3.5
2 1 6 5
5
1 32
1 32
6.2
1 6 5
The Area of a rectangle is calculated by using the formula length × width. To calculate the area: Step 1: Identify the length and width of the rectangle. Step 2: Multiply the length by the width to find the area of the rectangle. Use your knowledge of multiplying fractions and decimals to complete this calculation. Example 3: Area = length × width = 6 1 units × 3 units = 19 square units
3
3 6 1 3 217
2
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5
DIRECTIONS: Calculate the perimeter of the shapes below.
Perimeter = 37.8 ft
4.2 ft
14.7 ft
Perimeter = 48 25 meter
15 m
9
Perimeter = 33.2 inches
1 m 5
DIRECTIONS: Calculate the area of the shapes below.
Area = 61.74 ft2
4.2 ft
14.7 ft
Area = 138 m2
Area = 68.89 in.2
15 m
9
1 m 5
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218
DIRECTIONS: Calculate the perimeter of the shapes below.
Perimeter = 57.2 feet
6.9 ft
21.7 ft
Perimeter = 73.6 inches Perimeter = 60 meter
17 m
13 m
DIRECTIONS: Calculate the area of the shapes below.
Area = 149.73 ft2
6.9 ft
21.7 ft
Area = 221 meter2
Area = 338.56 inches2
17 m
13 m
219
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1 Sofia drew a rectangle with a length of 4.56 inches and a width of 2.17 inches. What is the perimeter of her rectangle? 9.8952 inches
6.73 inches
13.46 inches
12.26 inches
2 Find the area of the rectangle below. 5
1 feet 2
3
1 feet 3
2 18 6 square feet 2 18 6 feet
2 17 3 square feet 2 17 3 feet
3 A square has a side length of 0.9 cm. What is the area of the square? 3.6 cm²
0.81 cm²
0.18 cm²
0.81 cm
4 A farmer wants to fence in her plot of land shown below. How much fencing will be needed to fence in the plot? 32.91 miles 10.672 miles
87.164 square miles
351.215 square miles
43.582 miles
87.164 miles
5 Darius is hanging market lighting around the dining room of his restaurant. The room has a 1 length of 24 1 feet and a width of 14 4 feet. How many feet of lighting will Darius need to hang 2 around the perimeter of the room? 1 76 2 feet
1 712 2 feet
1 77 2 feet
3 38 4 feet COPYING IS STRICTLY PROHIBITED BY LAW
220
DIRECTIONS: Calculate the perimeter of the shapes below.
Perimeter = 37.4 ft
7.3 ft
11.4 ft
Perimeter = 50.4 inches Perimeter = 38 meters
14 m
5m
DIRECTIONS: Calculate the area of the shapes below.
Area = 83.22 ft2 7.3 ft
11.4 ft
Area = 70 m2
Area = 158.76 in.2
14 m
5m
221
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1 Elle and Skye are painting a wall in their clubhouse. They need to find out how much paint they will need to cover it. Elle thinks they should add up all four of the outside measurements of the wall, and Skye thinks they should multiply the length and the width of the wall. Who is correct? Elle is correct because adding the outside measurements will find the area. Skye is correct because multiplying length times width will find the area. Skye is correct because multiplying length times width will find the perimeter. Elle is correct because adding the outside measurements will find the perimeter.
2 What is the perimeter of the rectangle shown? 9 6 822 32 ft² 158 8 ft 9 822 32 ft
6 158 8 ft²
12
2 feet 8 67
1 feet 8
3 3 Marshall cut out a square piece of paper for a craft he is making. The paper has side lengths of 4 inch. What is the perimeter of Marshall’s piece of paper? 9 1 9 1 2 inches 3 inches 16 inches 16 square inches
4 What is the area of the shape below? 23.8 m²
34.84 m²
23.8 m
34.84 m
5.2 m 6.7 m
5 A rectangle is described below: •
Length: 1.32 yards
•
Width: 3.45 yards
What is the perimeter of the rectangle? 4.77 yards
4.554 square yards
9.54 square yards
9.54 yards COPYING IS STRICTLY PROHIBITED BY LAW
222
1 What is the area of the rectangle below? 5.25 ft 1.74 ft
8.154 square feet
9.135 square feet
9.234 square feet
10.348 square feet
2 1 A square has a side length of 3 5 cm. What the perimeter of the square? 12 1 centimeters 5 16 3 centimeters 5
12 4 centimeters 5 12 4 centimeters 5
3 A farmer wants to install a drip watering pipe along the sides of his rectangular vegetable garden. What is the length of pipe needed if the garden is 20 1 yards long and 14 yards wide? 2 1 1 34 yards 68 yards 2 2
69 yards
72 yards
4 Danny is painting a portrait that is 9.4 feet long and 3.6 feet wide. What is the area that he has to paint? 23.45 square feet
28.78 square feet
32.96 square feet
33.84 square feet
5 Lily and Duke are planning to buy carpet for their square study room. Lily says they should multiply the sides to get the carpet area. Duke says they should add the sides. Who is correct? Lily is correct because the area of the room is the area of the carpet.
Duke is correct because the sum of all sides of the room is the area of the carpet. Both of them are correct because they give the same answer. Both of them are incorrect.
223
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6 What is the area of the shape below? 6.4 m 6.4 m
12.8 square meters
30.97 square meters
36.16 square meters
40.96 square meters
7 Rachel is making a rectangular banner for a party. The banner has a length of 6 feet and a width of 3 feet. What is the total area of the banner? 9 square feet
12 square feet
15 square feet
18 square feet
8 Emily is making a quilt using square fabric patches. Each patch has a side length of 4 inches. If she arranges the patches in a 4x3 pattern, what is the total area covered by the patches? 48 square inches
192 square inches
194 square inches
200 square inches
9 Jason is making a rectangular photo frame. The frame has a length of 12 inches and a width of 8 inches. What is the perimeter of the frame? 20 inches
32 inches
40 inches
48 inches
10 John is planning to place a border around his rectangular artwork. The artwork measures 18 inches in length and 12 inches in width. What is the total length of the border needed?
30 inches
48 inches
60 inches
78 inches
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224
USING VOLUME AS AN ATTRIBUTE OF THREE-DIMENSIONAL FIGURES MA.5.GR.3.1 Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.
Count the number of unit cubes to determine the volume. Key:
= 15 units
3
= 1 unit
Count the unit cubes for the length, width, and height of a rectangular prism to find its volume.
Step 1: Determine the length, width, and height of the right rectangular prism. These dimensions should be whole numbers.
Step 2: Multiply the number of unit cubes in the base layer (length × width) by the number of layers (height) to find the total number of unit cubes inside the prism. This total count represents the volume of the right rectangular prism in terms of unit cubes. Step 3: Volume = Length × Width × Height
4 unit cubes
Example:
225
Volume
= 4 units × 4 units × 4 units
Volume
=
64 units3
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DIRECTIONS: Find the volume of the right rectangular prisms below.
VOLUME = 30 cubic units
VOLUME = 30 cubic units
VOLUME = 30 cubic units
DIRECTIONS: Use the cubes to find the volume.
VOLUME = 10 cubic units
VOLUME = 12 cubic units
VOLUME = 27 cubic units
VOLUME = 24 cubic units
VOLUME = 30 cubic units
VOLUME = 80 cubic units
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DIRECTIONS: Use the unit cubes to measure the volume.
VOLUME = 125 cubic units
VOLUME = 24 cubic units
VOLUME = 20 cubic units
VOLUME = 64 cubic units
VOLUME = 18 cubic units
VOLUME = 30 cubic units
VOLUME = 30 cubic units
VOLUME = 24 cubic units
VOLUME = 216 cubic units
227
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What is the volume of the rectangular prism? 256 cubic units
256 square units
256 units
256 boxes
2 Paris is filling a box with sugar cubes. She fills the bottom of the box with exactly 15 cubes. If she can fit 7 total layers in the box, what must the volume of the box be? 22 unit cubes
115 unit cubes
75 unit cubes
105 unit cubes
3 A board game company has 800 identical dice cubes that must be shipped to a toy store. An employee packs 120 into the base of a small box, leaving no gaps. If the employee can fit exactly 5 more layers on top before the box is full, will she be able to fit all 800 dice in the box? Yes, because that means she can fit 820 dice in the box. No, because that means she can only fit 700 in the box. No, because that means she can only fit 720 in the box. Yes, because that means she can fit exactly 800 dice in the box.
4 Which of the following is not a unit cube?
1 in.
1 ft
A 1 in.
1 in.
Figure A
B 1 ft
1 cm 1 ft
Figure B
C 2 cm
1 cm
Figure C
1 cm D 1 cm 1 cm
Figure D
5 Find the volume of the figure shown. 24 unit cubes
128 unit cubes
48 unit cubes
96 unit cubes COPYING IS STRICTLY PROHIBITED BY LAW
228
DIRECTIONS: Use the unit cubes to measure the volume.
VOLUME = 48 cubic units
VOLUME = 27 cubic units
VOLUME = 72 cubic units
VOLUME = 56 cubic units
VOLUME = 112 cubic units
VOLUME = 84 cubic units
VOLUME = 343 cubic units
VOLUME = 512 cubic units
VOLUME = 1,000 cubic units
229
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 What is the volume of the rectangular prism below?
44 cubic units
220 cubic units
200 units
240 boxes
2 Nate and Katie are playing with blocks. Their prisms are represented below. Both claim that their prisms have volumes of 80 cubic units. Who is correct?
Nate is correct.
Katie is correct.
Both are correct
Neither are correct.
Nate
Katie
3 What is the volume of the prism shown below?
27 unit cubes
30 unit cubes
33 unit cubes
9 unit cubes
4 For which of the following can the volume not be found by counting unit cubes?
A
B
Figure A
C
Figure B
D
Figure C
Figure D
5 Which is not a unit to express volume? Square units
Cubic units
Unit cubes
Units 33 COPYING IS STRICTLY PROHIBITED BY LAW
230
1 What is the volume of the rectangular prism below? 80 cubic units
84 cubic units
90 cubic units
94 cubic units
2 Freddy is arranging Rubik’s cubes (in the shape of a rectangular prism) in a box. The box holds 144 cubes in the bottom layer. He arranges 8 layers. What is the volume of the box? 1,152 unit cubes
2,457 unit cubes
1,512 unit cubes
2,547 unit cubes
3 What is the volume of the figure below? 49 unit cubes
50 unit cubes
98 unit cubes
99 unit cubes
4 Kim and Natalie are looking at the figures given below. Kim claims that both cubes have different volumes, but Natalie says they have the same volume. Who is correct? Kim is correct. A
B
Natalie is correct. Both are correct. Neither are correct.
5 What is the volume of the prism shown below?
231
18 unit cubes
20 unit cubes
22 unit cubes
24 unit cubes
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6 Charlie arranges blocks of a game into 11 rows of 8 blocks each. If he arranges 6 layers, what is the volume of the structure he makes? 428 unit cubes
456 unit cubes
880 unit cubes
528 unit cubes
7 What is the volume of the prism shown below ?
35 unit cubes
40 unit cubes
45 unit cubes
56 unit cubes
8 Alex is stacking cubes to form a rectangular prism. He arranges 6 layers of cubes, with each layer having 7 cubes in each row and 4 cubes in each column. What is the volume of the rectangular prism? 158 unit cubes
178 unit cubes
168 unit cubes
188 unit cubes
9 A toy company wants to ship 450 identical toy blocks. The base of the crate has been filled without spaces with 50 blocks and can fit 9 more layers. Can all 450 toy blocks fit? Yes, 500 blocks fit.
No, only 400 blocks fit.
No, only 450 blocks fit.
Yes, exactly 450 blocks fit.
10 Ava is stacking cubes to form a tower. She arranges 10 layers of cubes, with each layer having 3 cubes in each row and 9 cubes in each column. What is the volume of the tower? 270 unit cubes
250 unit cubes
260 unit cubes
280 unit cubes COPYING IS STRICTLY PROHIBITED BY LAW
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FINDING THE VOLUME OF A RIGHT RECTANGULAR PRISM
MA.5.GR.3.2 Find the volume of a right rectangular prism with whole-number side lengths using a visual model and a formula.
Calculate the volume of a rectangular solid using the formula: V = length × width × height. To find the volume of a right rectangular prism with whole-number side lengths, you can use both a visual model and a formula. The volume of a rectangular prism can be calculated using the formula: Volume = length × width × height or l × w × h.
STEP 1: Copy the formula for volume (Volume = length × width × height). STEP 2: Plug in the measurements for length, width, and height found on the rectangular prism into the equation.
STEP 3: Multiply the length, width, and height together, ensuring that you are performing the multiplication correctly.
STEP 4: The result you get after performing the multiplication is the volume of the right rectangular prism.
STEP 5: Make sure to include the correct units for volume (e.g., cubic units).
Volume
=
l
×
w
×
h
4 inches
Example
Volume
= 5 inches × 3 inches × 4 inches
Volume
=
60 inches3
5 inches 233
MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
DIRECTIONS: Use the measurements to calculate the volume of the prisms. 4 ft 4m 5 ft 3 cm
8 cm
3m 7m
8 ft
3 cm VOLUME = 72 cubic cm
VOLUME = 160 cubic feet
VOLUME = 84 cubic meters
3 ft
4 ft
8 ft
9 in.
6 ft
3 ft
3 in. 4 in.
8 ft VOLUME = 96 cubic feet
VOLUME = 144 cubic feet
3 yards 12 cm
VOLUME = 108 cubic inches
5ft
5 yards 10 yards VOLUME = 150 cubic yards
3 ft
2 cm 3cm VOLUME = 72 cubic cm
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8 ft VOLUME = 120 cubic feet
234
DIRECTIONS: Use the measurements to calculate the volume of the prisms.
VOLUME = 90 cubic feet
VOLUME = 108 cubic feet
VOLUME = 80 cubic feet
VOLUME = 72 cubic feet
VOLUME = 240 cubic yards
VOLUME = 105 cubic feet
VOLUME = 420 cubic feet
VOLUME = 108 cubic feet
VOLUME = 108 cubic feet
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1 Steven is renting a storage unit and needs it to be at least 1,200 cubic feet. Which of the following could not be dimensions of Steven’s storage unit? 10 ft × 10 ft × 12 ft
5 ft × 20 ft × 10 ft
12 ft × 10 ft × 14 ft
20 ft × 7 ft × 10 ft
2 What is the volume of the figure below? 36 cubic yards
48 cubic yards
11 cubic yards
20 cubic yards
3 yd 2 yd
6 yd
3 Select all of the following expressions that can be used to find the volume of the rectangular prism below. 7×5x2
10 × 7
35 × 2 7 in.
7×7
12 × 2
5 in.
2 in.
4 Find the volume of a rectangular prism with the following dimensions: •
Length: 18 cm 4,084 cubic cm
•
Width: 9 cm 59 cubic cm
•
Height: 32 cm 194 cubic cm
5, 184 cubic cm
5 Rose and Alexandra disagree about how to find the volume of the figure below. Rose thinks that they should multiply 5 × 3 × 17, and Alexandra thinks they should multiply 15 × 17. 17 in.
Who is correct? Rose is correct because the formula for volume is l × w × h.
5 in.
3 in.
Alexandra is correct because the formula for volume is b × h. Both of them are correct because both of their formulas will find the volume. Neither is correct because they need to add the numbers, not multiply. COPYING IS STRICTLY PROHIBITED BY LAW
236
DIRECTIONS: Use the measurements to calculate the volume of the prisms.
VOLUME = 84 cubic feet
VOLUME = 96 cubic feet
VOLUME = 64 cubic feet
4m
3m 6m VOLUME = 240 cubic feet
VOLUME = 240 cubic yards
VOLUME = 315 cubic meters
VOLUME = 108 cubic feet
VOLUME = 120 cubic yards
VOLUME = 100 cubic feet
237
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1 Eleanor wants to find the volume of the inside of her desk at school. She measures it to find that it has a length of 24 inches, a width of 16 inches, and a height of 4 inches. What is the volume of the inside of Eleanor’s desk? 44 square inches
1,536 cubic inches
44 cubic inches
1,536 square inches
2 Jax loves spending his summer swimming at the neighborhood pool and is interested in knowing the volume of water that can fit inside of it. How much water will it take to fill the pool if it has a length of 22 yards, a width of 11 yards, and a depth of 3 yards? (Hint: Depth is the height from the bottom of the pool to the top.) 726 cubic yards
626 cubic yards
36 cubic yards
308 cubic yards
3 In which of the following scenarios would it be necessary to find volume? to determine how much wallpaper is needed to cover a wall to determine how much wire fence is needed to fence in a chicken coop to determine how many eggs are in a carton to determine how much room is inside of a truck for moving
4 Find the volume of the rectangular prism below. 9m 6m 45 m
2,430 meters
2,430 cubic meters
405 cubic meters
2,240 square meters
5 Isaac’s toy box has a volume of 8 cubic feet. Which of the following could be the dimensions of his toy box? 5 ft × 2 ft × 3 ft
4 ft × 1 ft × 2 ft
8 ft × 1 ft × 2 ft
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4 ft × 2 ft × 2 ft 238
1 In which of the following scenarios would it be necessary to find the volume? to determine how much paint is needed to cover a wall to determine how much water is in the bathtub to determine how much carpet is needed to cover the floor to determine how much fencing is needed for a garden
2 Thomas has an iron safe that has a capacity of 2,500 cubic feet. Which of the following could not be the dimensions of Thomas’s safe? 25 ft × 10 ft × 10 ft
100 ft × 5 ft × 5 ft
25 ft × 20 ft × 5 ft
25 ft × 40 ft × 10 ft
3 Sally bought a new fish tank that is 28 inches long, 18 inches wide, and 12 inches high. Find the quantity of water she needs to fill the tank. 6,048 cubic inches
6,504 cubic inches
7,009 cubic inches
8,468 cubic inches
4 A shoe box has a volume of 2,100 cubic inches. Which of the following could be the dimensions of the shoe box? 30in. × 12in. × 9in.
15in. × 10in. × 7in.
20in. × 15in. × 7in.
12in. × 11in. × 6in.
5 What is the volume of the rectangular prism below? 4 cm 6 cm 11 cm
239
164 cubic centimeters
198 cubic centimeters
264 cubic centimeters
302 cubic centimeters
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6 What is the volume of the rectangular prism below? 21 yd 16 yd 19 yd
5,789 cubic yards
6,384 cubic yards
6,864 cubic yards
7,894 cubic yards
7 A shipping crate has a volume of 3,000 cubic inches. Which of the following dimensions could be the dimensions of the crate? 30 inches × 20 inches × 5 inches
30 inches × 10 inches × 5 inches
30 inches × 20 inches × 15 inches
40 inches × 8 inches × 10 inches
8 A swimming pool has the shape of a rectangular prism. The pool is 20 meters long, 10 meters wide, and 3 meters deep. Calculate the volume of water needed to fill the pool. 600 cubic meters
6,000 cubic meters
200 cubic meters
2,000 cubic meters
9 The dimensions of a rectangular prism are 8 inches, 5 inches, and 6 inches. Which of the following represents its volume? 120 cubic meters
180 cubic meters
240 cubic meters
360 cubic meters
10 A water tank in the shape of a rectangular prism measures 6 feet in length, 4 feet in width, and 3 feet in height. Calculate the volume of the water tank. 24 cubic feet
72 cubic feet
48 cubic feet
36 cubic feet COPYING IS STRICTLY PROHIBITED BY LAW
240
SOLVING REAL-WORD PROBLEMS INVOLVING VOLUME MA.5.GR.3.3 Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.
Solve real-world volume problems of right rectangular prisms using a visual model or formula.
STEP 1: Read the problem carefully to identify what is given and what needs to be solved. STEP 2: List the known values, such as the volume of the rectangular prism and any known edge lengths (length, width, height). STEP 3: Create a visual representation of the rectangular prism using the given information. STEP 4: Record the formula for the volume of a right rectangular prism. STEP 5: Set up the equation with the variable for the unknown. STEP 6: Solve the equation for the unknown. STEP 7: Check the solution.
Example A truck with various boxes of merchandise has limited space and needs to calculate the height of the boxes to ensure they fit. What is the maximum height (h) of the boxes that can be loaded into the truck? V = length (l) × width (w) × height (h) 75 cubic meters = 10 meters × 2.5 meters x h
? Volume = 75 cubic meters
75 cubic meters = 25 meters2 x h 3
1
75 cubic meters = 25 meters2 x h 25 meters2 = 25 meters2 1
1
3 cubic meters = 1h 3 cubic meters = h 241
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DIRECTIONS: Use the volume equation to solve the unknown widths. ?
5 in.
21.7 in.
1.5 ft
?
?
Volume = 360 cubic inches
Volume = 7,812 cubic in.
Volume = 4.5 cubic feet
WIDTH = 1.5 feet
WIDTH = 8 inches
40 in.
WIDTH = 9 inches
DIRECTIONS: Use the volume equation to solve the unknown lengths.
7.2 in.
14 cm
10 in.
Volume = 1,820 cubic in.
Volume = 2,646 cubic cm
Volume = 280.8 cubic in.
14 in.
LENGTH = 13 inches
?
?
LENGTH = 21 centimeters
LENGTH = 13 inches
DIRECTIONS: Use the volume equation to solve the unknown heights.
Volume = 12.5 cubic inches
?
?
HEIGHT = 6 inches
Volume = 86 cubic inches
?
Volume = 150 cubic inches
4.3 in.
HEIGHT = 1 inch COPYING IS STRICTLY PROHIBITED BY LAW
HEIGHT = 4 inches 242
DIRECTIONS: Use the measurements to calculate the unknowns.
Volume = 1.89 cubic ft
3 in.
1.5 ft
Volume = 194.25 cubic in.
7 in. ?
0.7 ft
LENGTH = 9.25 inches
?
LENGHTH = 1.8 inches
Volume = 12 cubic ft
12 in.
Volume = 864 cubic in.
?
?
2 ft
18 in.
3 ft
HEIGHT = 4 inches
HEIGHT = 2 feet
Volume = 5,107.2 cubic in.
5 ft
14 in.
Volume = 37.5 cubic ft
24 in.
? ?
3 ft
WIDTH = 2.5 feet 243
WIDTH = 15.3 inches
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1 The base of a gas container, shaped like a rectangular prism, has an area of 6 square feet. If 24 cubic feet of gas can fill half the container, how tall is the gas container? 4 feet
8 feet
10 feet
12 feet
2 A water tank, shaped like a rectangular prism, is 10 feet tall. The base of the water tank is a square. If 120 cubic feet of water can fill one-third of the tank, what is the length of one side of the base of the water tank? 2 feet
3 feet
4 feet
6 feet
3 John made a cube of ice using 64 cubic inches of water. What is the length of one side of the cube of ice that John made? 2 inches
4 inches
8 inches
It cannot be determined.
4 Jane’s water container, shaped like a rectangular prism, is 12 inches long, 6 inches wide, and 4 inches deep. She wants to fill the water container with 200 cubic inches of water. Does she have enough water to fill the water container to capacity? No
Yes
Maybe
It cannot be determined.
5 Frank has two shoe boxes. Both boxes are shaped like a rectangular prism. Shoebox A has dimensions of 8 inches by 5 inches by 4 inches, and shoebox B has dimensions of 15 inches by 3 inches by 3 inches. Which shoebox has a greater volume? Shoebox A
Shoebox B
Both shoeboxes have the same volume.
The volumes cannot be compared.
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244
DIRECTIONS: Use the measurements to calculate the volumes or the unknowns.
Volume = 384 cubic in.
? Volume = 156 cubic cm
8 in.
3.5 in. 7 in. 6 in.
5.2 cm
VOLUME = 147 cubic inches
6 in.
10 cm
?
LENGTH = 8 inches
HEIGHT = 3 cm
?
Volume = 0.625 cubic in.
2 cm
12 cm
4 cm
2.5 in.
3 cm
1 in.
VOLUME = 24 cubic cm
HEIGHT = 0.25 in.
7 cm
5 cm
VOLUME = 420 cubic cm
8 ft 3 ft
7.5 in.
Volume = 4,000 cubic ft
Volume = 990 cubic in.
?
WIDTH = 11 inches
245
Volume = 81 cubic ft
12 in.
?
?
6 ft
WIDTH = 4.5feet
LENGHTH = 40 feet
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12.5 ft
1 A water tank is 12 meters long, 10 meters wide, and 15 meters tall. An inlet pipe fills the water tank at a rate of 90 cubic meters per minute. How many minutes will it take the inlet pipe to fill the empty water tank? 10 minutes
15 minutes
20 minutes
It cannot be determined.
2 A room, shaped like a rectangular prism, is 40 meters long and 25 meters wide. The ceiling of the room is 8 meters high. One air conditioner is enough to cool 2,000 square meters. How many air conditioners are needed to cool the entire room? 2 units of air conditioner
3 units of air conditioner
4 units of air conditioner
It cannot be determined.
3 An oil tank, shaped like a cube, is 8 feet long on one side. How many containers of oil can be poured into the oil tank if each container holds 16 cubic feet of oil? 16 containers
20 containers
32 containers
64 containers
4 A box is 30 feet long, 20 feet wide, and 10 feet tall. The box is filled with cartons each having dimensions of 2 feet by 2 feet by 2 feet. How many such cartons can be put inside the box? 750 cartons
800 cartons
850 cartons
It cannot be determined.
5 A box, shaped like a rectangular prism, has a volume of 25 cubic inches. A box, shaped like a cube, has a volume that is 5 times the volume of the rectangular box. What is the length of one side of the box that is shaped like a cube? 5 inches
15 inches
25 inches
It cannot be determined.
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246
1 An ice tray, shaped like a cube, is 5 inches on one side. How many ice cubes, with a side length of 1 inch, can fit inside the ice tray? 5 ice cubes
25 ice cubes
75 ice cubes
125 ice cubes
2 A metalsmith melted a bar of iron 12 inches long, 8 inches wide, and 6 inches tall. Using the melted iron, the metalsmith made cubical iron bricks that are 2 inches on one side. How many such cubical iron bricks did the metalsmith make? 56 cubical iron bricks
64 cubical iron bricks
72 cubical iron bricks
84 cubical iron bricks
3 Jake molded and then melted three identical cubes of ice. Each cube measured 2 inches on one side. If all three cubes were melted and stored in a container, what would be the volume of the resulting water? 8 inches
16 inches
24 inches
It cannot be determined.
4 A rectangular swimming pool is 20 meters long, 8 meters wide, and 2 meters deep. If the pool is filled using a hose that delivers water at a rate of 5 cubic meters per minute, how many minutes will it take to fill the pool? 32 minutes
40 minutes
64 minutes
56 minutes
5 A storage room is 12 feet long, 8 feet wide, and 6 feet tall. If one shelf takes up 36 cubic feet of space, how many shelves can be placed in the storage room?
247
10 shelves
12 shelves
14 shelves
16 shelves
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6 An aquarium is shaped like a rectangular prism with dimensions of 40 inches long, 20 inches wide, and 18 inches tall. If each fish requires a minimum space of 20 cubic inches, how many fish can the aquarium hold at most? 72 fish
90 fish
100 fish
720 fish
7 A rectangular prism box has dimensions of 10 inches by 8 inches by 6 inches. If each small cube used to fill the box has a side length of 2 inches, how many small cubes are needed to completely fill the box? 60 cubes
180 cubes
240 cubes
360 cubes
8 A toy company packs their toy blocks in a rectangular box with dimensions of 8 inches by 6 inches by 4 inches. How many toy blocks can be packed in the box if each block has a volume of 2 cubic inches? 24 blocks
36 blocks
48 blocks
96 blocks
9 A water tank is in the shape of a cuboid with dimensions of 6 meters by 4 meters by 3 meters. If the tank is filled to its maximum capacity, what is the volume of water it can hold? 72 cubic meters
96 cubic meters
144 cubic meters
216 cubic meters
10 Ella and Sam both have gift boxes. Each gift box is shaped like a rectangular prism. Ella's gift box (Box X) has dimensions of 10 inches by 6 inches by 4 inches, while Sam's gift box (Box Y) has dimensions of 8 inches by 5 inches by 6 inches. Which gift box has a larger volume? Box X
Box Y
Both boxes have the same volume
Not enough information to determine
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248
PLOTTING AND LABELING ORDERED PAIRS MA.5.GR.4.1 Identify the origin and axes in the coordinate system. Plot and label ordered pairs in the first quadrant of the coordinate plane.
To identify the origin and axes, find the intersection of the x-axis and y-axis. STEP 1: Find the point where the x-axis and y-axis intersect. This is the origin and is labeled (0, 0). STEP 2: The x-axis is the horizontal line to the right of the origin. Positive values are to the right of the origin and negative values are to the left. STEP 3: The y-axis is the vertical line above the origin. Positive values are above the origin and negative values are below.
y-axis
Origin (0, 0) 9 8 7 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
9
x-axis
To plot an ordered pair in the first quadrant, move x units right and y units up from the origin, and label the point with its ordered pair. STEP 1: The first quadrant is the top-right section of the coordinate plane. STEP 2: To plot an ordered pair (x, y), start from the origin (0, 0), and move x units RIGHT along the x-axis and y units UP along the y-axis. STEP 3: Label the plotted point with its ordered pair.
FIRST: Move 3 units right along the x-axis. SECOND: Move 5 units up along the y-axis. THIRD: Plot the point where these two movements intersect as (3, 5).
y-axis
Example: (3, 5)
9 8 7 6 5 4 3 2 1
(3, 5) Up 5 Right 3 0
1
2
3
4
5
6
7
x-axis 249
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8
9
DIRECTIONS: Plot and label the ordered pairs below.
K
9
H
8
O
C
7
6
F
y-axis
5
M
N
4
I
J
A
3
B
2
L
G
E
1
D 0
1
2
3
4
5
6
P 7
8
9
x-axis
(2, 3)
(8, 3)
(4, 2)
(3, 3)
(5, 7)
(5, 9)
(6, 0)
(9, 3)
(9, 1)
(4, 5)
(2, 5)
(0, 4)
(6, 2)
(8, 7)
(2, 8)
(7, 0)
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250
DIRECTIONS: Plot and label the ordered pairs below.
9
L
8
B
P
7
H
O F
A
6
M
D I
y-axis
5
C
4
G
3
N K
2
E
1
0
1
J
2
3
4
5
6
7
8
x-axis
251
(1, 6)
(6, 5)
(3, 7)
(3, 1)
(7, 5)
(0, 7)
(8, 6)
(8, 2)
(1, 1)
(1, 8)
(4, 7)
(5, 6)
(5, 3)
(6, 3)
(7, 8)
(4, 8)
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9
1 What are the coordinates of the point shown on the plane below? y
(5,6)
9 8 7 6 5 4 3 2 1
(6,5) (65) (6x, 5y)
0 1 2 3 4 5 6 7 8 9
x
2 In the coordinates (3,7), what does the 7 represent? The distance right along the x-axis
The distance up along the y-axis
The distance up along the x-axis
The distance right along the y-axis
3 Which of the following shows the point (4,2)? y
y
y
y
A. 9
B. 9
C. 9
D. 9
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
0 1 2 3 4 5 6 7 8 9 x
Figure A
0 1 2 3 4 5 6 7 8 9 x
Figure B
0 1 2 3 4 5 6 7 8 9 x
Figure C
0 1 2 3 4 5 6 7 8 9 x
Figure D
4 Which set of directions properly explains how to plot the point (9,5)? Begin at the origin. Move 9 units right along the x-axis and then 5 units up from that point. Begin at the origin. Move 9 units up along the y-axis and then 5 units right from that point. Begin at (9,5). Move 9 units left to the y-axis and then 5 units down to the origin. Begin at the origin. Move 9 units right along the y-axis and then 5 units down from that point. COPYING IS STRICTLY PROHIBITED BY LAW
252
DIRECTIONS: Plot and label the ordered pairs below.
9
P
N
8
K
C
6
I
5
y-axis
D
M
7
O G
4
H
3
A
E
2
L
F
1
J
B 0
1
2
3
4
5
6
7
8
x-axis
253
(7, 2)
(0, 5)
(3, 0)
(8, 0)
(3, 6)
(4, 6)
(9, 7)
(5, 1)
(0, 2)
(6, 7)
(2, 1)
(2, 8)
(1, 4)
(9, 5)
(7, 3)
(3, 8)
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9
1 What are the coordinates and name of the point shown below? y
(1,0); the beginning
9 8 7 6 5 4 3 2 1
(0,1); the origin (0,0); the starting point (0,0); the origin
0 1 2 3 4 5 6 7 8 9
x
2 A table with coordinate pairs is shown below. Which pair is missing from the corresponding y coordinate plane? (4,7)
x-coordinate 4
y-coordinate 7
(1,8)
1
5
(7,4)
7 0
4 8
(0,8)
9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9
x
3 Which of the following is not true? The x-coordinate is listed first in an ordered pair. The y-axis runs left to right. The origin sits at point (0,0). The y-coordinate is listed second in an ordered pair.
4 What is true of the ordered pair (0,7)? The point will lie at the origin. The point will lie on the x-axis.
The point will lie on the y-axis. The point will lie 7 units to the right of the origin.
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254
1 In the coordinates (2,5), what does 2 represent? The distance right along the x-axis.
The distance up along the x-axis.
The distance right along the y-axis.
The distance up along the y-axis.
2 Which of the following is not true? The y-axis runs down to up. The y-coordinate is listed second in an ordered pair. The origin is at the point (1,1). The x-coordinate is listed first in an ordered pair.
3 What are the coordinates of the point shown in the plane below? y 9 8 7 6 5 4 3 2 1
(6,2)
(2,6) (0,6) (2,0)
0 1 2 3 4 5 6 7 8 9 x
4 Which of the following is true about the ordered pair (3,1)? The x-coordinate is 0, and the y-coordinate is 1. The x-coordinate is 3, and the y-coordinate is 0. The x-coordinate is 0, and the y-coordinate is 1. The x-coordinate is 3, and the y-coordinate is 1.
5 A table with coordinate pairs is shown below. Which pair is missing from the corresponding coordinate plane? y x-coordinate y-coordinate 1 0 2 5 4 6 6 3
9 8 7 6 5 4 3 2 1
(1,0)
(2,5) (4,6) 0 1 2 3 4 5 6 7 8 9 x
255
(6,3)
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6 Which of the following shows the point (5,6)? y
y
y
y
A. 9
B.9
C.9
D.9
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
0 1 2 3 4 5 6 7 8 9 x
0 1 2 3 4 5 6 7 8 9 x
Figure A
Figure B
0 1 2 3 4 5 6 7 8 9x
0 1 2 3 4 5 6 7 8 9 x
Figure C
Figure D
7 What are the coordinates of the point shown in the plane below? y
9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 x
(7,2)
(8,2)
(9,2)
(6,2)
8 Which of the following points is not located on the y-axis? (0,2)
(0,-5)
(0,0)
(0,8)
9 In the ordered pair (3, -7), what does the number 3 represent? The x-coordinate
The y-coordinate
The origin
The slope
10 Which of the following is not true?
The y-axis is vertical
The x-axis is horizontal
The origin is the point (0, 0)
The x-coordinate is listed second
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256
PLOTTING POINTS REPRESENTATIVE OF REAL-WORLD PROBLEMS MA.5.GR.4.2 Represent mathematical and real-world problems by plotting points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation. Interpreting the coordinate values can effectively represent mathematical and real-world problems.
STEP 1: Read the real-world problem, and identify the relevant variables and quantities involved. STEP 2: Understand what each coordinate represents, such as a measurement of time, distance, height, length, etc. STEP 3: Assign appropriate coordinates to the quantities, with the x-coordinate representing the first quantity (time, number of items, etc.) and the y-coordinate (distance, cost, etc.) representing the second value. STEP 4: Plot the points in the first quadrant of the coordinate plane based on the assigned coordinates for each situation. STEP 5: Read the coordinates (x, y) of each plotted point and interpret them in the context of the problem.
9
Example: Plot the hotel at (3, 5). 8 7
(3, 5)
y-axis
6 5 4 3
Up 5
2 1 Right 3
0
1
2
3
4
5
6
7
x-axis 257
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8
9
DIRECTIONS: Determine the coordinates for the locations listed below.
PLACE
LOCATION
2 RIGHT & 1 DOWN
3 LEFT & 2 UP
4, 6
6, 5
1, 8
7, 3
9, 2
4, 5
3, 2
5, 1
0, 4
6, 8
8, 7
3, 10
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258
DIRECTIONS: Determine the coordinates for the locations listed below.
9 8
7
6
y-axis
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
x-axis
PLACE
259
LOCATION
1 RIGHT & 4 DOWN
2 LEFT & 1 UP
5, 5
6, 1
3, 6
8, 6
9, 2
6, 7
6, 4
7, 0
4, 5
3, 7
4, 3
1, 8
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1
How tall was the tree when Edgar planted it in his backyard? 31 inches
32 inches
30 inches
33 inches
Height (in inches)
Edgar planted a tree in his backyard. He used the graph given to record the tree’s height at the end of each week.
38 37 36 35 34 33 32 31 30 0 1 2 3 4 5 6 7 8 9
Weeks
2
1 inch
2 inches
3 inches
4 inches
Height (in inches)
Using Edgar’s graph that records the tree’s height over a 9week period, how many inches did the tree grow between Week 5 and Week 7?
38 37 36 35 34 33 32 31 30 0 1 2 3 4 5 6 7 8 9
Weeks
3
At what time did Camilo start walking? 8:00 AM
7:00 AM
9:00 AM
10:00 AM
Distance (in miles)
Camilo tracked the number of miles he walked while exercising at the end of each hour using the graph given.
18 16 14 12 10 8 6 4 2 0 8a
10
12
2p
4p
2p
4p
Time
4
2 miles
6 miles
8 miles
10 miles COPYING IS STRICTLY PROHIBITED BY LAW
Distance (in miles)
Using Camilo’s graph on the right, how far had he traveled two hours after he started walking?
18 16 14 12 10 8 6 4 2 0 8a
10
12
Time 260
DIRECTIONS: Determine the coordinates for the locations listed below.
9 8
7
6
y-axis
5 HOSPITAL 4
3
2
1
0
1
2
3
4
5
6
7
8
9
x-axis
PLACE
LOCATION
5 RIGHT & 2 DOWN
2 RIGHT & 4 UP
0, 3
5, 1
2, 7
3, 5
8, 3
5, 9
2, 2
7, 0
4, 6
4, 4
9, 2
6, 8
HOSPITAL
261
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1 Lisa kept track of the number of emails she sent at the end of each day using the graph given.
Day 6
Day 7
Day 8
Day 9
Emails sent
On which day did she send the most number of emails?
45 40 35 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9
Day
2 Lisa analyzed the graph she made tracking the number of emails she sent over 9 days.
Day 1
Day 2
Day 3
Day 4
Emails sent
On which day did she send no emails?
45 40 35 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9
Day
3
One
Two
Three
Four
Emails sent
Using Lisa’s graph detailing the number of emails she sent over a 9-day period, how many days did she send more than 25 emails per day?
45 40 35 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9
Day
4 According to Lisa’s graph, how many more emails did she send on Day 6 than on Day 5?
30 emails
35 emails
40 emails
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Emails sent
20 emails
45 40 35 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9
Day
262
1
In which week did he watch the greatest number of videos? Week 2
Week 4
Week 6
Week 8
Videos watched
Arnold kept track of the number of videos he watched online each week using the graph given.
90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9
Weeks
2
How many videos did he watch during Week 2 and Week 3 combined? 30 videos
90 videos
50 videos
40 videos
Videos watched
Arnold’s graph shows the number of videos he watched online for the last 9 weeks.
90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9
Weeks
3
Seven
Eight
Five
Four
Videos watched
Using Arnold’s graph which kept track of the number of videos he watched online, how many weeks did he watch less than 50 videos?
90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9
Weeks
4
Between which two hours did Frank walked 4 miles? 10:00 AM and 11:00 AM
11:00 AM and 12:00 PM
Distance (in miles)
Frank tracked the number of miles he walked while exercising at the end of each hour using the graph given.
18 16 14 12 10 8 6 4 2 0 8a
1:00 PM and 2:00 PM 2:00 PM and 3:00 PM 263
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10
12
Time
2p
4p
5
11:00 - 2:00 PM
12:00 - 2:00 PM
1:00 - 2:00 PM
2:00 - 4:00 PM
Distance (in miles)
Using Frank’s graph on the right, at what time did Frank appear to take a break from walking on his hike?
90 80 70 12 10 8 6 4 2 0 8a
10
12
2p
4p
Time
6
30 fruits
50 fruits
60 fruits
70 fruits
Number of Fruits
Sam is tracking the number of fruits he eats each week using the graph on the right. How many fruits did he eat during Week 3?
90 80 70 60 50 40 30 20 10
0 1 2 3 4 5 6 7 8 9
Weeks
7
Day 1
Day 2
Day 3
Day 8
Number of Pushups
The graph represents the number of push-ups Ahmed did each day for a month. On which day did he do the least number of push-ups?
90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9
Days
8
90
110
100
130 COPYING IS STRICTLY PROHIBITED BY LAW
Number of Books
Mr. Perez is keeping track of the number of books his class reads each year using the graph on the right. How many books did his class read during Year 5 and Year 6 combined?
90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9
Years 264
CREATING TABLES, GRAPHS, AND LINE PLOTS
MA.5.DP.1.1 Collect and represent numerical data, including fractional and decimal values, using tables, line graphs or line plots. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). STEP 1: Determine the range of your data, which is the minimum and maximum values of your measurements. This will help you set up the axis appropriately. STEP 2: Create a horizontal axis (x-axis) for your line plot. Label it with the fractions you are working with, such as 1/8, 1/4, 3/8, 1/2, and so on. STEP 3: Divide the x-axis into equal segments based on the fractions you are dealing with. For example, if you are using 1/5, each segment would represent 1/5 of the unit. STEP 4: Give your line plot a descriptive title that explains what the plot is displaying, such as "Measurements in Fractions of a Unit." STEP 5: Title the x-axis with a clear label like "Fractions" or "Measurement Units." Add a label to the y-axis if your measurements have corresponding values on another scale. Example:
Consumed (each)
Number of Slices
Personal Pan Pizza Consumed by Students
0
STUDENTS
1
2
3
4
5
6
7
8
9
10 11 12 13 14
AMOUNT EATEN
1 5
3 5
2 5
1 5
3 5
3 5
1 5
4 5
2 5
2 5
x x x
x x x
x x x x
1 5
2 5
3 5
x x x 4 5
3 5
Personal Pan Pizza Cut in Fifths 265
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4 5
5 4 5 5
x 1
DIRECTIONS: Use the fractions below to create a tally table and line plot.
2 4
1 4
1 4
2 4
2 4
4 4
1 4
2 4
3 4
4 4
4 4
2 4
1
0
1/4
0
4 8
1 8
3 8
2 8
5 8
6 8
7 8 x x x x 1/8
2/8
3/8 4/8
4 4
1 4
3 8
5 8
2 8
1/2
3/4
4 8
4 4
x x x x x x 1
2/4
6 8
1/8
3 4
x x
x x x x
3/4
2 4
x x x x x x
1/4 1/2
4 4
4/4
1 8
3 8
3 8
1 8
1 8
x x
x x x x
x x
x
x
x
x
x
1/4
3/8
1/2
5/8
3/4
7/8
5/8 6/8
0
7/8
0
2/8
4/8
1
6/8
8/8
8/8
DIRECTIONS: Plot the set of data in the graph provided.
1
7
3
9
5
10
7
12
9
13
11
14
Time (weeks)
Weight (pounds)
1
Week
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
WEIGHT OF A NEWBORN
WEIGHT OF A NEWBORN
0
1
2
3
4
5
6
7
8
9
Weight (pounds) COPYING IS STRICTLY PROHIBITED BY LAW
10
11
12
13
14
266
DIRECTIONS: Use the fractions below to create a tally table and line plot.
1 6
5 6
1 6
6 6
2 6
5 6
5 6
2 6
2 6
1/6
1 6
4 6
3 6
2 6
1 6
1 6
4 6
3 6
x x x x x
x x x x x
x x
x x
x x x
x
1/6
2/6
3/6
4/6
5/6
6/6
1/3
1/2
2/3
2/6 3/6 4/6
2 6
5/6 6/6
0 0
1 4
1
1
3 4
2
1
1 4
1
1 4
1
2 4
1
3 4
1 4
1
x x x x x
1
1 1/4 1 2/4
1
1
3 4
1
2 4
x x x x
1
2 4
1
1 4
1
2 4
x x x
x
1 3/4
2
1 3/4 2
1
1 1/4
1 1/2 1 2/4
DIRECTIONS: Plot the set of data in the graph provided.
267
Distance
1
2
2
4
3
6
4
8
5
10
6
12
7
13
1
Time (hours)
Time
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
WALKATHON
WALKATHON
0
1
2
3
4
5
6
7
8
9
10
11
Distance (miles) MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
12
13
14
1 Jeff measured the length, in inches, of 12 grasshopper specimens during a science experiment. The table below shows the data he gathered. SPECIMEN
1
2
3
4
5
6
7
8
9
10 11 12
LENGTH (in inches) 9 1 8 2 9 2 9 1 9 1 8 2 9 1 9 2 9 1 9 2 8 2 9 1
3
3
3
3
3
3
3
3
3
3
3
3
Which of these tables matches the data displayed above? LENGTH (in inches) 2 83 1 93 2 93
LENGTH (in inches) 2 83 1 93 2 93
NUMBER OF SPECIMENS
LENGTH (in inches)
4
2 83 1 93 2 93
NUMBER OF SPECIMENS
LENGTH (in inches)
4 4
3
6 3
2 83 1 93 2 93
NUMBER OF SPECIMENS
3 3 6 NUMBER OF SPECIMENS
4
5 3
2 Alfred recorded the number of hours he spent playing video games each day for seven days. He played 1 34 hours on Day 1, 1 14 hours on Day 2, 1 12 hours on Day 3, 1 14 hours on Day 4, 1 34
hours on Day 5, 1 14 hours on Day 6, and 1 12 hours on Day 7. Which of these line graphs matches the data?
COPYING IS STRICTLY PROHIBITED BY LAW
268
DIRECTIONS: Use the fractions below to create a tally table and line plot.
2 5
5 5
4 5
3 5
2 5
4 5
5 5
2 5
2 5
1 5
2 5
1 5
2 5
3 5
4 5
x x x x
x x x
x x
2/5
3/5
4/5
5/5
2/5 x x
4/5
2 5
x x x x x x x
1/5 3/5
3 5
3 5
5/5 0
1/5
0
1 7
5 7
4 7
3 7
2 7
6 7
1
6 7
5 7
2 7
3 7
2 7
3 7
1 7
2 7
3 7
3 7
6 7
5 7
1/7
x x
x x x x
x x x x x
1/7
2/7
3/7
2/7
3/7 4/7 5/7 6/7
0
7/7
x
x x x
x x x
4/7
5/7
6/7
7/7
0
1
DIRECTIONS: Plot the set of data in the graph provided.
269
Weight Loss
1
5
2
8
3
8
4
10
5
11
6
13
7
14
0
Months (each)
WEIGHT LOSS
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
MONTHS
1
WEIGHT LOSS
1
2
3
4
5
6
7
8
9
Weight (pounds)
10
11
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12
13
14
1 The table below shows the length of some grasshopper specimens. Which of the line plots below matches the data? LENGTH (in inches) NUMBER OF SPECIMENS
● ● ● 6
7 8 Length (in inches) ● ● ● ●
6
● ● ● ● ● ● ● ● ● ● ● ●
3 1 3 64 72 74
8
1 3 84 84
3
2
4
1
5
3 ● ● ● ●
● ● ● 9
6
●
● ● ● ● 9
6
● ● ● ● ● ● ●
7 8 Length (in inches)
● ● ● ● ● ● ● ● ● ● ● ● ● 7 8 Length (in inches)
● ● ● ● ●
9
● ● ● ● ● ● ● ● ● ● ● ● ●
7 8 Length (in inches)
9
2 A vendor recorded how many pounds of potatoes she sold each day for seven days. She sold 5 14 pounds on Day 1, 4 34 pounds on Day 2, 4 12 pounds on Day 3, 5 14 pounds on Day 4, 5 12 pounds on Day 5, 5 34 pounds on Day 6, and 5 14 pounds on Day 7. Which of these line graphs matches the data?
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270
1 Kim found 12 cans of paint in her garage. The table below shows the amount of paint, in pints, in each of the cans she found. 1
CAN
2
3
4
5
6
7
8
9
10 11 12
51 51 41 51 41 41 43 41 51 41 51 41 4 4 4 4 4 4 4 4 4 4 4 4
PAINT (in pints)
Which of these tables matches the data displayed above? NUMBER OF CANS
NUMBER OF CANS
PAINT (in pints)
8
44
2
44
2
54
2
1 54
3
PAINT (in pints)
NUMBER OF CANS
PAINT (in pints)
NUMBER OF CANS
PAINT (in pints) 1
44 3
44 1
1 44 3 44 1 54
7 1
4
1
7
3
1 44 3 44 1 54
6 1
5
2 Alfred recorded the amount of rainfall each day for seven days. He recorded 6 12 inches of rainfall on Day 1, 5 14 inches on Day 2, 6 34 inches on Day 3, 6 12 inches on Day 4, 5 34 inches on Day 5, 5 12 inches on Day 6, and 5 34 inches on Day 7. Which of these line graphs matches the data?
271
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3 The table below shows the distances the students lived from the school. Which of the line plots below matches the data?
● 0
1 2
11 11 1 3 21 21 4 2 4 4 2
NUMBER OF STUDENTS
1
5
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
3
4
1
● ● ●
1 2 Distance (in miles)
● 0
DISTANCE (in miles)
● 3
0
● ● ●
1 2 Distance (in miles)
2
● ●
1 2 Distance (in miles)
● 3
● ● ● ● ● ● ● ● ● ● ● ●
0
● ● ● ● ● ● ● ● ● ● ●
3
● ● ●
1 2 Distance (in miles)
3
4 1
Ben filled six bottles with oil. He filled the first bottle with 3 4 quarts of oil, the second bottle 1
with 2 34 quarts of oil, the third bottle with 3 2 quarts of oil, the fourth bottle with 3 14 quarts of oil, the fifth bottle with 2 12 quarts of oil, and the sixth bottle with 3 quarts of oil. Which of these line graphs matches the data?
COPYING IS STRICTLY PROHIBITED BY LAW
272
MEAN, MEDIAN, MODE, AND RANGE MA.5.DP.1.2 Interpret numerical data, with whole-number values, represented with tables or line plots by determining the mean, mode, median or range. Calculating the mean (average) of a set of whole-number values. STEP 1: Add up all of the values provided in the table or line plot.
Example:
TIME (HOURS)
1
2
3
4
5
6
7
8
9
10
NUMBER OF TICKETS SOLD
5
12
7
12
16
28
22
20
15
23
FIRST: 5 + 7 + 12 + 12 + + 15 + 16 + 20 + 22 + 23 + 28 = 160 SECOND: 160 / 10 = 16 THIRD: Interpret the result as “the mean number of tickets sold over 10 hours is 16”.
Calculate the median (middle number of an ordered data set) STEP 1: Arrange the data in ascending order. STEP 2: If the data set has an odd number of values, the median is the middle value. Example: FIRST: 5, 7, 12, 12, 15, 16, 18, 20, 22, 23, 28
THIRD: 34 ÷ 2 = 17
SECOND: 16 + 18 = 34 (two numbers in the middle)
FOURTH: Interpret the result as the median is 17.
Calculate the mode (the number that appears most frequently).
Calculate the range (the number that appears most frequently).
STEP 1: Identify the value that occurs with the highest frequency.
STEP 1: The range is the difference between the highest and lowest values in the data set.
STEP 2: If multiple values share the highest frequency, the data set is multimodal, and there is no mode.
STEP 2: Subtract the smallest value from the largest value to find the range.
Example:
Example:
273
FIRST: 5, 7, 12, 12, 15, 16, 18, 20, 22, 23, 28
FIRST: Identify the highest value: 28
SECOND: 12 appears twice whereas, all other numbers appear only once.
SECOND: Identify the lowest value: 5 THIRD: Subtract 28 - 5 = 23
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DIRECTIONS: 5
3
Mean
12 10 15
8
2
6
4
8
2
7
4
2
4
2, 2, 4, 4, 4, 6, 7, 8, 8, 9 —> 4 + 6 = 10 ÷ 2 = 5 is the median
Mode
4 appears three times. No other numbers appears at that frequency. Therefore, 4 is the mode.
Range
9 - 2 = 7 is the range
50 25 50 15 15
Mean
25
10 25
25
10
10 + 10 + 15 + 15 + 25 + 25 + 25 + 25 + 50 + 50 = 150 ÷ 10 = 15
Median
10, 10, 15, 15, 25, 25, 25, 25, 50, 50 —> 25 + 25 = 50 ÷ 2 = 25 is the median
Mode
25 appears four times. No other numbers appears at that frequency. Therefore, 25 is the mode.
Range
Range
15
2 + 2 + 4 + 4 + 4 + 6 + 7 + 8 + 8 + 9 = 54 ÷ 10 = 5.4
Median
Mode
12
15 - 2 = 13 is the range
Mean
Median
12
12 appears four times. No other numbers appears at that frequency. Therefore, 12 is the mode.
Range
Mean
5
2, 3, 5, 5, 10, 12, 12, 12, 12, 15, 15 —> 12 is the median (middle number)
Mode
17
12
2 + 3 + 5 + 5 + 10 + 12 + 12 + 12 + 12 + 15 + 15 = 103 ÷ 10 = 10.3
Median
9
Calculate the mean, median, mode, and range of the data sets below.
50 - 10 = 40 is the range 5
10
4
2
21
10 16
15
3
18
2 + 3 + 4 + 5 + 10 + 10 + 15 + 16 + 17 + 18 + 21 = 121 ÷ 11 = 11 2, 3, 4, 5, 10, 10, 15, 16, 17, 18, 21 —> 10 is the median (middle number) 10 appears two times. No other numbers appears at that frequency. Therefore, 25 is the mode.
21 - 2 = 19 is the range COPYING IS STRICTLY PROHIBITED BY LAW
274
DIRECTIONS: 4
9
Mean
3
Calculate the mean, median, mode, and range of the data sets below. 8
14
Mean
45 55
64
36
18, 28, 33, 36, 45, 55, 64, 71, 77, 82 —> 45 + 55 = 100 ÷ 2 = 50 is the median
Mode
All of the numbers appear only once and therefore, there is no mode.
Range
82 - 18 = 64 is the range
10 12
Mean
3
7
13
5
1
13
8
6
1 + 3 + 5 + 6 + 7 + 8 + 10 + 12 + 13 + 13 = 78 ÷ 10 = 7.8
Median
1, 3, 5, 6, 7, 8, 10, 12, 13, 13 —> 7 + 8 = 15 ÷ 2 = 7.5 is the median
Mode
13 appears two times. No other numbers appears at that frequency. Therefore, 13 is the mode.
Range
275
82
18 + 28 + 33 + 36 + 45 + 55 + 64 + 71 + 77 + 82 = 500 ÷ 10 = 50
Median
Range
6
14 - 3 = 11 is the range
71 28 33 77 18
Mode
5
14 appears three times. No other numbers appears at that frequency. Therefore, 14 is the mode.
Range
Median
11
3, 4, 5, 6, 8, 9, 10, 11, 14, 14 —> 9 is the median (middle number)
Mode
Mean
14 14
3 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 14 + 14 = 70 ÷ 10 = 7
Median
7
10
13 - 1 = 12 is the range 9
7
17
6
19
4
13
20
12
7
4 + 6 + 7 + 7 + 7 + 9 + 12 + 13 + 17 + 19 + 20 = 121 ÷ 11 = 11 4, 6, 7, 7, 7, 9, 12, 13, 17, 19, 20 —> 9 is the median (middle number) 7 appears three times. No other numbers appears at that frequency. Therefore, 7 is the mode.
20 - 4 = 16 is the range MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 Michelle did a survey in her class. She asked the students in her class how many siblings they had and recorded the responses to her survey in the table below. Based on the results of her survey, what was the mean number of siblings per student?
NUMBER OF SIBLINGS
0
1
2
3
4
NUMBER OF STUDENTS
1
3
2
4
2
2 siblings per student
2.25 siblings per student
2.5 siblings per student
3 siblings per student
2 Michelle did a survey in her class. She asked the students in her class how many siblings they had and recorded the responses to her survey in the table below. Based on the results of her survey, what was the mode of the number of siblings of the students in her class? 2 siblings
2.25 siblings
NUMBER OF SIBLINGS
0
1
2
3
4
NUMBER OF STUDENTS
1
3
2
4
2
2.5 siblings
3 siblings
3 Michelle did a survey in her class. She asked the students in her class how many siblings they had and recorded the responses to her survey in the table below. Based on the results of her survey, what was the median number of siblings of NUMBER OF SIBLINGS 0 1 2 3 4 NUMBER OF STUDENTS 1 3 2 4 2 the students in her class? 2 siblings
2.25 siblings
2.5 siblings
3 siblings
4 Michelle did a survey in her class. She asked the students in her class how many siblings they had and recorded the responses to her survey in the table below. Based on the results of her survey, what was the range of the number of siblings of the students in her class? 1 sibling
2 siblings
NUMBER OF SIBLINGS
0
1
2
3
4
NUMBER OF STUDENTS
1
3
2
4
2
3 siblings
4 siblings
5 The organizers used the table below to keep track of the number of medals won by the participants of the contest. What is the range of the number of medals won by the participants of the contest? 8 medals
9 medals
NUMBER OF MEDALS WON
8
9
10
11
12
NUMBER OF PARTICIPANTS
6
5
4
3
2
4 medals
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12 medals 276
DIRECTIONS:
Calculate the mean, median, mode, and range of the data sets below.
23 27 22 28 23
Mean Median Mode Range
Median Mode Range
Median Mode Range
Median Mode Range 277
21 23
23 appears four times. No other numbers appears at that frequency. Therefore, 23 is the mode.
15 - 2 = 13 is the range 38
44 72
94
47
14 + 26 + 37 + 38 + 44 + 47 + 71 + 72 + 72 + 94 = 658 ÷ 10 = 65.8 14, 26, 37, 38, 44, 47, 71, 72, 72, 94 —> 44 + 47 = 91 ÷ 2 = 45.5 is the median 72 appears two times. No other numbers appears at that frequency. Therefore, 72 is the mode.
94 - 14 = 80 is the range 41
47 49
50
41
41 + 41 + 41 + 43 + 44 + 45 + 47 + 49 + 49 + 50 = 450 ÷ 10 = 45 41, 41, 41, 43, 44, 45, 47, 49, 49, 50 —> 44 + 45 = 89 ÷ 2 = 44.5 is the median 41 appears three times. No other numbers appears at that frequency. Therefore, 41 is the mode.
50 - 41 = 9 is the range
37 33 37 32 37 33
Mean
23
21, 21, 22, 23, 23, 23, 23, 25, 26, 27, 28 —> 23 is the median (middle number)
41 43 49 45 44
Mean
21 26
21 + 21 + 22 + 23 + 23 + 23 + 23 + 25 + 26 + 27 + 28 = 103 ÷ 10 = 10.3
72 26 14 71 37
Mean
25
37 33
32
37 37
32 + 32 + 33 + 33 + 33+ 10 + 15 + 16 + 17 + 18 + 21 = 385 ÷ 11 = 35 32, 32, 33, 33, 33, 37, 37, 37, 37, 37, 37 —> 37 is the median (middle number) 37 appears six times. No other numbers appears at that frequency. Therefore, 37 is the mode.
37 - 32 = 5 is the range MATH BOOTCAMP® SMART TO THE CORE | EDUCATIONAL BOOTCAMP, INC.
1 A bake shop kept track of the number of cupcakes it sold each day using the line plot given. What was the mean number of cupcakes sold by the bake shop per day?
●
● ●
● ● ● ● ●
● ●
● ● ● ●
● ●
●
●
● ●
30 31 32 33 34 35 36 37 38 39 40 Cupcakes Sold
30 cupcakes per day
32 cupcakes per day
34 cupcakes per day
36 cupcakes per day
2 A bake shop kept track of the number of cupcakes it sold each day using the line plot given. What was the median number of cupcakes the bake shop sold? 33 cupcakes
33.5 cupcakes
●
● ●
● ● ● ● ●
● ●
● ● ● ●
● ●
●
●
● ●
30 31 32 33 34 35 36 37 38 39 40 Cupcakes Sold
34 cupcakes
34.5 cupcakes
3 A bake shop kept track of the number of cupcakes it sold each day using the line plot given. What was the mode of the number of cupcakes that the bake shop sold? 30 cupcakes
32 cupcakes
●
● ●
● ● ● ● ●
● ●
● ● ● ●
● ●
●
●
● ●
30 31 32 33 34 35 36 37 38 39 40 Cupcakes Sold
34 cupcakes
36 cupcakes
4 The organizers used the table given to keep track of the number of medals won by the NUMBER OF MEDALS WON 8 9 10 participants of the contest. What is the median number of medals won by the participants of the contest? 32 medals
32.5 medals
NUMBER OF PARTICIPANTS
33 medals
6
5
11
12
3
2
4
9 medals
5 During the school’s recycling drive, the students in Ms. Anne’s class collected the following numbers of empty bottles per student: 12, 15, 10, 8, 16, 20, 14, 12, 7, 11, 9, and 10. What is the mean number of empty bottles per student that Ms. Anne’s class collected? 10 empty bottles per student
11.5 empty bottles per student
12 empty bottles per student
13 empty bottles per student
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278
1 A bakeshop kept track of the number of cupcakes it sold each day using the line plot given. What was the range of the number of cupcakes that the bakeshop sold? 6 cupcakes
8 cupcakes
●
● ●
● ● ● ● ●
● ●
● ● ● ●
● ●
●
●
● ●
30 31 32 33 34 35 36 37 38 39 40 Cupcakes Sold
9 cupcakes
10 cupcakes
2 The organizers used the table given to keep track of the number of medals won by the NUMBER OF MEDALS WON 8 9 10 participants of the contest. What is the mode of the number of medals won by the participants of the contest? 8 medals
NUMBER OF PARTICIPANTS
10 medals
6
5
11
12
3
2
4
35 medals
34 medals
3 The organizers used the table given to keep track of the number of medals won by the NUMBER OF MEDALS WON 8 9 10 participants of the contest. What is the mean number of medals won per participant?
NUMBER OF PARTICIPANTS
6
5
8 medals per participant
9 medals per participant
9.5 medals per participant
10 medals per participant
11
12
3
2
4
4 A convenience store kept track of the number of cold drinks it sold each day using the line plot given. What was the range of the number of cold drinks that the store sold? 6 drinks
8 drinks
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27 28 29 30 31 32 33 34 35 36 37 Number of cold drinks
10 drinks
12 drinks
5 A convenience store kept track of the number of cold drinks it sold each day using the line plot given. What was the median number of cold drinks that the store sold? 30 drinks 279
32 drinks
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27 28 29 30 31 32 33 34 35 36 37 Number of cold drinks
30.5 drinks
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29 drinks
6 During the school’s food drive, the students in Mr. Allen’s class collected the following numbers of soup cans: 23, 17, 11, 14, 21, 16, 10, 18, 12, 24, 6, and 20. What is the mean number of used soup cans per student that the students that Mr. Allen’s class collected? 10 pages per student
12 pages per student
14 pages per student
16 pages per student
7 The organizers used the table given to keep track of the number of trophies won by the NUMBER OF TROPHIES WON 6 7 8 participants of the contest. What is the mode of the number of trophies won by the participants of the contest? 6 trophies
7 trophies
NUMBER OF PARTICIPANTS
4
3
6
9
10
5
3
10 trophies
8 trophies
8 The organizers used the table given to keep track of the number of trophies won by the NUMBER OF TROPHIES WON 6 7 8 participants of the contest. What is the mean number of trophies won by the participants of the contest? 8 trophies
9 trophies
NUMBER OF PARTICIPANTS
4
3
6
9
10
5
3
11 trophies
10 trophies
9 The organizers used the table given to keep track of the number of trophies won by the NUMBER OF TROPHIES WON 6 7 8 participants of the contest. What is the range number of trophies won by the participants of the contest? 3 trophies
4 trophies
NUMBER OF PARTICIPANTS
4
6 trophies
3
6
9
10
5
3
7 trophies
10 The organizers used the table given to keep track of the number of trophies won by the NUMBER OF TROPHIES WON 6 7 8 participants of the contest.
What is the median number of trophies won by the participants of the contest? 6.5 trophies
7 trophies
NUMBER OF PARTICIPANTS
7.5 trophies
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4
3
6
9
10
5
3
8 trophies
280
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