LEARN ▸ Operations with Fractions and Multi-Digit Numbers
What does this painting have to do with math?
An intersection in Paris on a gray, rainy day is the subject of this atmospheric Impressionist painting. Gustave Caillebotte creates depth in this scene by using perspective and proportion in a variety of ways, including by placing large figures in the foreground and smaller ones in the distance. Imagine there is a coordinate grid on the building in the background. How might you determine the distance from the front of the building to the back by using the coordinate plane?
On the cover
Paris Street; Rainy Day, 1877
Gustave Caillebotte, French, 1848–1894
Oil on canvas
The Art Institute of Chicago, Chicago, IL, USA
Gustave Caillebotte (1848–1894). Paris Street; Rainy Day, 1877. Oil on canvas, 212.2 x 276.2 cm (83 1/2 x 108 3/4 in). Charles H. and Mary F. S. Worcester Collection (1964.336). The Art Institute of Chicago, Chicago, IL, USA. Photo Credit: The Art Institute of Chicago/Art Resource, NY
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Dividing Fractions by Using the Invert and Multiply Strategy
Lesson 11
Applications of Fraction Division
119
133
Lesson 12
Fraction Operations in a Real-World Situation
Topic D
Decimal Addition, Subtraction, and Multiplication
Lesson 13 .
Decimal Addition and Subtraction
Lesson 14
Patterns in Multiplying Decimals
Lesson 15
Decimal Multiplication
Lesson 16
Applications of Decimal Operations
Topic E 221
Division of Multi-Digit Numbers
Lesson 17 223
Partial Quotients
Lesson 18
The Standard Division Algorithm
Lesson 19
Expressing Quotients as Decimals
Lesson 20
Real-World Division Problems
Topic F
Decimal Division
Lesson 21
Dividing a Decimal by a Whole Number
Lesson 22
Dividing a Decimal by a Decimal Greater Than 1
TOPIC A Factors, Multiples, and Divisibility
Waste-Free Sharing
Here. Please make some cupcakes for everybody to share. Sure! How many people? I’m not exactly sure. enough so that everyone gets an equal share.
How many did you make?!
58,140, of course! That way, they can be evenly divided into 17, 18, 19, or 20!
LATER....
Why not just make 20 cupcakes, and worst- case scenario, there are a few leftovers?
Leftovers? But that’s so wasteful!
If you’re expecting 17 to 20 people for a party, then you have two choices. Sure, you can just make 20 cupcakes. That way, each person gets one, and depending on how many people come, you’ll have anywhere from 0 to 3 cupcakes left over. Or—bear with me—you can choose the smallest number that is a multiple of 17, 18, 19, and 20, which happens to be 58,140.
Thus, if there are 17 people, each person gets 3,420 cupcakes.
If there are 18 people, each person gets 3,230 cupcakes.
If there are 19 people, each person gets 3,060 cupcakes.
And if there are 20 people, each person gets 2,907 cupcakes.
No matter what, you’ll have no leftovers—just a whole lot of cupcakes for each and every person!
Grade 6, Module 2, Topic A, Lesson 1
Factors and Multiples
Factor Find Rules
• Decide which player will go first.
• Player 1 chooses a number from the board and marks the number with an X. Player 1 earns that number of points.
• Player 2 identifies all possible factors of the number Player 1 chose, other than that number itself, and marks each factor with an X. Player 2 earns as many points as the sum of those factors.
• The chosen number and the identified factors are removed from play and can no longer be selected for the remainder of the game.
• The game continues until there are no more numbers left to choose.
• The player with the greatest number of points wins.
1. Play the Factor Find game.
Player 1’s score:
Player 2’s score:
2. Play the Factor Find game again, letting the other player go first.
Player 1’s score:
Player 2’s score:
Tiling with Squares
3. Find the side length of a square tile, other than 1 foot, that tiles a 30 ft × 24 ft rectangle completely. Draw the square tiles on the grid to show your thinking.
4. Find the side length of a square tile, other than 1 foot, that tiles a 36 ft × 48 ft rectangle completely.
5. Find the side length of a square tile, other than 1 foot, that tiles a 5 ft × 4 ft rectangle completely. Draw the square tiles on the grid to show your thinking.
Tiling with Rectangles
6. Using 3 in × 2 in rectangular tiles, create a square on the 24 in × 18 in grid. There should be no gaps between the rectangular tiles. What is the side length in inches of the square you created?
7. Use the 24 in × 18 in grid to determine the side length in inches of the smallest square you can create with 6 in × 4 in rectangular tiles.
Grade 6, Module 2, Topic D, Lesson 14
Tiling with Squares
Problem 3: 30 × 24 Grid
Tiling with Squares
Problem 4: 36 × 48 Grid
Tiling with Squares
18 × 12 Grid
Tiling with Rectangles
Problems 6 and 7 Grid
Name Date
1. What is the side length in feet of the largest square tile that can tile a 6-foot by 9-foot rectangle completely? 9ft
2. What is the side length in inches of the smallest square that can be created with 6-inch by 9-inch rectangular tiles?
Name Date Factors and Multiples
In this lesson, we
• found common factors of two numbers.
• found common multiples of two numbers.
Examples
1. The dimensions of a rectangular room are 18 feet by 12 feet.
18ft
12ft
Two whole numbers that multiply to make a given number are called a factor pair of that number. A number is a factor of the given number if it is a whole number in a factor pair.
For example, 5 and 2 are factors of 10 because 5 × 2 = 10.
A multiple of a given number is the product when the given number is multiplied by another number. For example, 30 is a multiple of 10 because 10 × 3 = 30
When answering questions like parts (a) and (b), consider sketching the squares on the rectangle.
a. What is the side length in feet of the largest square tile that can tile the room completely? 18ft
12ft
The side length of the largest square tile that can tile the room completely is 6 feet because 6 is the largest factor that is common to both dimensions of the room, 18 feet and 12 feet.
The side length of the largest square tile that can tile the room completely is 6 feet.
b. What is the side length in feet of another square tile that can tile the room completely?
The side length of another square tile that can tile the room completely is 3 feet.
2. A rectangular tile measures 15 inches by 6 inches.
6in6in6in6in6in 15in 15in
Other common factors of 18 and 12 are 1, 2, and 3. This means that in this case, the side lengths of square tiles that can tile this room completely can be 1 foot, 2 feet, or 3 feet.
a. What is the side length in inches of the smallest square that can be created with rectangular tiles of this size? Draw a diagram to support your answer. 30in 30in
All of the possible side lengths of a square that can be created must be multiples of both 15 and 6 because the square is made from 15-inch by 6-inch tiles. The side length of the smallest square that can be created with tiles of this size is 30 inches because 30 is the smallest number that is a common multiple of 15 and 6
The side length of the smallest square that can be created is 30 inches.
b. What is the side length in inches of another square that can be created with rectangular tiles of this size?
Sample: The side length of another square that can be created is 60 inches.
Another possible side length of a square that can be created is 60 inches because 60 is a multiple of both 15 and 6. To make this square, 4 rows of 10 tiles are needed.
Name Date
PRACTICE
1. A rectangle measures 12 inches by 20 inches. What size squares can tile the rectangle completely? Choose all that apply. 12in 20in
A. 12 inches by 12 inches
B. 6 inches by 6 inches
C. 5 inches by 5 inches
D. 4 inches by 4 inches
E. 2 inches by 2 inches
For problems 2–5, list all the factors of the number.
For problems 6–8, list all the common factors of the given numbers.
6. 12 and 30
7. 12 and 25
8. 24 and 45
For problems 9–12, list six multiples of the number.
9. 3
6
8
12
For problems 13–15, list two common multiples for the given numbers.
13. 8 and 12
14. 6 and 12
15. 12 and 10
16. A hallway measures 3 feet by 9 feet. 3ft 9ft
a. What is the side length in feet of the largest square tile that can tile the hallway completely?
b. What is the side length in feet of another square tile that can tile the hallway completely?
17. A rectangle measures 14 centimeters by 25 centimeters. Sana says that the largest square that can tile the rectangle completely measures 1 centimeter by 1 centimeter. Do you agree with Sana? Explain your reasoning.
18. What size square can you create with rectangles that each measure 12 inches by 6 inches? Choose all that apply.
A. 12 inches by 12 inches
B. 24 inches by 24 inches
C. 36 inches by 36 inches
D. 6 inches by 6 inches
E. 2 inches by 2 inches
19. To make a piece of art, Ryan uses rectangular tiles that each measure 2 inches by 7 inches. What is the side length in inches of the smallest square Ryan can create with these rectangular tiles?
20. A rectangle measures 10 meters by 6 meters.
a. What is the side length in meters of the smallest square that can be created with rectangles of this size?
b. What is the side length in meters of another square that can be created with rectangles of this size?
c. Toby says the largest square that can be created with rectangles of this size cannot be determined. Do you agree with Toby? Explain.
2in 7in
Remember
For problems 21–24, multiply.
3 × 1 7
25. A blue whale weighs about 200 tons. It eats only small sea creatures called krill. The blue whale eats 2% of its body weight each day. How many tons of krill does the blue whale eat each day? Explain your thinking.
26. Match the expressions to the correct power of 10
× 10 × 10
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Divisibility
1. Make a conjecture about the rule for each diagram. Numbers inside the circle follow the rule. Numbers outside the circle do not follow the rule.
A
B
C
Conjecture:
Divisibility by 3
Conjecture:
Conjecture:
2. Group the blocks to determine whether 432 is divisible by 3. Draw and label your work to show your thinking.
Diagram
Diagram
Diagram
Divisibility by 6
3. Make a conjecture about the rule for each diagram. Numbers inside the circle follow the rule. Numbers outside the circle do not follow the rule.
D
E
F
Conjecture:
Conjecture:
Divisibility by Other Numbers
Conjecture:
4. Determine whether the number 272 is divisible by each of the following numbers. Explain your reasoning.
Diagram
Diagram
Diagram
a. 2
b. 3
c. 4
5. The Ancient One has been counting since the dawn of time. He has just reached the number 1,000,000,000,000,010.
What is the next number the Ancient One will count that is
a. divisible by 2?
b. divisible by 5?
c. divisible by 3?
d. divisible by 6?
d. 6
e. 8
f. 9
6. Determine whether the statements are true or false. If the statement is true, justify your reasoning. If the statement is false, provide a counterexample.
a. If a number is divisible by 9, then it is divisible by 3.
b. If a number is divisible by 3, then it is divisible by 9.
Name Date
The number 96 is divisible by which of the following numbers? Choose all that apply. Explain your reasoning for all answer choices.
Name Date
Divisibility
In this lesson, we
• determined whether a number is divisible by another number without using division.
• made conjectures based on patterns and evidence and looked for counterexamples to show that conjectures are false.
Examples
1. Is 352 divisible by 3? Explain how you know.
Sample: No. The sum of the digits of 352 is 10 because 3 + 5 + 2 = 10. Because 10 is not divisible by 3, 352 is also not divisible by 3.
2. Is 192 divisible by 6? Explain how you know.
Sample: Yes. The sum of the digits of 192 is 12 because 1 + 9 + 2 = 12. Because 12 is divisible by 3, 192 is also divisible by 3. Because 192 is even, it is divisible by 2. Because 192 is divisible by both 2 and 3, it is divisible by 6.
3. Is 418 divisible by 4? Explain how you know.
Sample: No. I know that 400 is divisible by 4. Skip-counting by 4 from 400 gives me 404, 408, 412, 416, and 420. Because 418 is between 416 and 420, it is not divisible by 4.
A number is divisible by 6 if it is divisible by both 2 and 3. If a number is not divisible by both 2 and 3, then it is not divisible by 6.
Skip-counting can be used to check the divisibility of any number. Think of a number that is a multiple of the given number. In this case, the multiple of 4 that is close to 418 is 400. From 400, skip-count by 4 to see whether 418 is also divisible by 4
4. Which of the following numbers are divisible by both 3 and 5? Choose all that apply.
A. 30
B. 25
C. 215
D. 105
E. 545
F. 510
The numbers 30, 105, and 510 are all divisible by both 3 and 5. To confirm this, add the digits in the number. If the sum is divisible by 3, then the number is divisible by 3. If the number ends in 0 or 5, then it is divisible by 5.
5. Scott says that if a number is divisible by both 3 and 6, then it is divisible by 9. Do you agree or disagree with Scott’s conjecture? If you agree, explain your reasoning. If you disagree, give a counterexample.
Sample: I disagree with Scott’s conjecture. A counterexample is the number 24, which is divisible by both 3 and 6 but is not divisible by 9
A counterexample shows that a statement is false. The number 24 is a counterexample to Scott’s statement because 24 is divisible by both 3 and 6 but is not divisible by 9
Name Date
PRACTICE 2
1. Ryan says that if a number is divisible by both 2 and 5, then it is also divisible by 10. Do you agree with Ryan? Explain.
2. Is 5,652 divisible by 3? Why?
3. Is 5,652 divisible by 6? Why?
4. The number 132 is divisible by which of the following numbers? Choose all that apply.
A. 2 B. 3 C. 4 D. 5 E. 6 F. 7
G. 8 H. 9 I. 10
5. The number 132,850 is divisible by which of the following numbers? Choose all that apply.
A. 2
B. 3
C. 4
D. 5
E. 6
F. 10
6. Consider the number 2,353.
a. Find the next number that is divisible by 2.
b. Find the next number that is divisible by 3.
c. Find the next number that is divisible by 6.
For problems 7–11, determine whether the number is divisible by 2, divisible by 3, divisible by both 2 and 3, or divisible by neither 2 nor 3.
7. 186 8. 243 9. 340
For problems 12–15, determine whether the number is a multiple of 4, a multiple of 6, or a multiple of both 4 and 6.
16. Write a 3-digit number greater than 700 that is divisible by both 2 and 3. Explain how you know this number is divisible by both 2 and 3
17. Is the number 714 divisible by 7? Explain your reasoning.
10. 451
11. 588
12. 168 13. 200 14. 300 15. 450
18. Sasha says that if a number is divisible by both 2 and 4, then the number is divisible by 8. Do you agree or disagree with Sasha’s conjecture? If you agree, explain your reasoning. If you disagree, give a counterexample.
19. Ryan says that if a number is divisible by 12, then it is divisible by both 2 and 3. Do you agree or disagree with Ryan’s conjecture? If you agree, explain your reasoning. If you disagree, give a counterexample.
Remember
For problems 20–23, multiply.
24. Riley has saved $60 to buy a new scooter. This amount is 30% of what he needs to save. What is the total amount of money Riley needs to save to buy the new scooter?
For problems 25–27, describe the change in place value in the equation shown.
25. 0.25 × 10 = 2.5
26. 0.25 × 100 = 25
27. 0.25 × 1,000 = 250
Name Date
The Greatest Common Factor
Common Factors
1. What is the side length of the largest square that can tile a 20-unit by 24-unit rectangle?
24units
2. What is the greatest common factor of 36 and 48?
20units
3. What is the greatest common factor of 36 and 55?
4. Riley has 28 pens and 14 notepads. She uses all of the pens and notepads to make identical gift bags for her teachers.
a. What is the greatest number of identical gift bags Riley can make?
b. How many pens does each gift bag have? How many notepads?
Prime Factorization and the Greatest Common Factor
5. Complete the table for the number 36
6. Complete the table for the number 48.
Product of Factor Pair
Factorization 48 = ×
= ×
= ×
= ×
7. What is the greatest common factor of 26 and 39?
8. What is the greatest common factor of 13 and 14?
9. A group of 18 sixth graders and 24 fifth graders play a trivia game. Each is placed on a team.
a. Each team must have the same ratio of the number of sixth graders to the number of fifth graders. What is the greatest number of teams that can play the trivia game?
b. How many sixth graders does each team have? How many fifth graders?
Name Date
1. What is the prime factorization of 24?
2. What is the prime factorization of 60?
3. What is the greatest common factor of 24 and 60? Explain.
4. What is the greatest common factor of 24 and 35? Explain.
RECAP
Name Date
The Greatest Common Factor
In this lesson, we
• found the greatest common factor of two whole numbers.
• wrote numbers as products of their prime factors.
Examples
1. Find the greatest common factor of 45 and 75 by listing the factor pairs for each number.
Terminology
The greatest common factor of two whole numbers that are not both zero is the greatest nonzero whole number that is a factor of both numbers. Greatest common factor is often abbreviated as GCF.
The prime factorization of a number is the number written as a product of prime numbers.
The greatest common factor of 45 and 75 is 15.
After listing all factor pairs for both numbers, identify factors that are on both lists. The greatest factor that is on both lists is the greatest common factor. The number 15 is the greatest common factor of 45 and 75
2. Find the greatest common factor of 48 and 60 by writing the prime factorization of each number.
48 = 2 × 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
The greatest common factor of 48 and 60 is 12.
48 = 2 × 2 × 2 × 2 × 3
60 = 2 × 2 × 3 × 5
Because 2, 2, and 3 are the prime factors common to both lists, multiply them together to find the greatest common factor, 12
Find the greatest common factor of two numbers by using prime factorization. For example:
48 = 4 × 12
48 = 2 × 2 × 3 × 4
48 = 2 × 2 × 3 × 2 × 2
Choose any factor pair for the given number.
Write each composite factor as a product of one of its factor pairs.
Continue until only prime factors remain.
3. Find the greatest common factor of 27 and 32 by writing the prime factorization of each number.
27 = 3 × 3 × 3
32 = 2 × 2 × 2 × 2 × 2
The greatest common factor of 27 and 32 is 1.
There are no prime numbers that are common factors of 27 and 32 So 1 is the greatest common factor because it is the only common factor.
4. A math class has 24 students. A social studies class has 30 students. The classes are combined and then grouped for an activity.
a. Each group must have the same ratio of the number of math students to the number of social studies students. What is the greatest number of groups that can be made?
24 = 2 × 2 × 2 × 3
30 = 2 × 3 × 5
2 × 3 = 6
The greatest number of groups that can be made is 6
Find the greatest common factor of 24 and 30 to determine the greatest number of groups that can be made with students from both classes.
b. How many math students does each group have? How many social studies students?
Each group has 4 math students and 5 social studies students.
Each group has 4 math students because 24 ÷ 6 = 4. Each group has 5 social studies students because 30 ÷ 6 = 5. Each group has a total of 9 students.
Name Date
1. Consider the numbers 45 and 30
a. Fill in the tables to show the factor pairs of 45 and 30. Factor Factor Factor Factor
PRACTICE
b. List all the common factors of 45 and 30.
c. What is the greatest common factor of 45 and 30?
d. Find the prime factorization of 45.
e. Find the prime factorization of 30
f. Explain how to use the prime factorizations to find the greatest common factor of 45 and 30.
For problems 2–6, find the greatest common factor of the pair of numbers.
2. 35 and 75
3. 12 and 48
4. 28 and 55
5. 45 and 75
6. 56 and 84
7. Toby ties ribbons on gifts. He has a red ribbon that is 44 inches long. He has a blue ribbon that is 33 inches long. He wants to cut both ribbons into pieces that are all equal in length and as long as possible. How long should Toby cut each piece of ribbon?
8. A group of 20 seventh graders and 24 sixth graders plays in a kickball tournament. Each student is placed on a team.
a. Each team must have the same ratio of the number of seventh graders to the number of sixth graders. What is the greatest number of teams that can be made?
b. How many seventh graders does each team have? How many sixth graders?
9. A hotel manager has 63 chocolates and 42 mints. She uses all of the chocolates and mints to make identical welcome bags for hotel guests.
a. What is the greatest number of identical welcome bags the hotel manager can make?
b. How many chocolates does each welcome bag have? How many mints?
10. Write two numbers that are both greater than 5 and have a greatest common factor of 5.
Remember
For problems 11–14, multiply.
15. Kelly has $80. She uses a 40% off coupon to buy a jacket priced at $75. After using the coupon to buy the jacket, how much money does Kelly have left?
16. Find the quotient. Draw a model to support your answer.
9,225 ÷ 5
Name Date
The Least Common Multiple
Record information gathered from the video.
1. Notes from the video:
2. Math questions from the video:
Common Multiples
3. What is the least common multiple of 3 and 5?
4. What is the least common multiple of 4 and 8?
5. What is the side length of the smallest square you can create with rectangular tiles that each measure 6 units by 10 units?
Prime Factorization and the Least Common Multiple
6. Consider the numbers 11 and 12.
a. What is the least common multiple of 11 and 12?
b. Write the prime factorizations of 11 and 12.
7. Roll the two 6-sided dice twice. If you get the same sum on the second roll, then roll again until you get a different sum.
a. List the two sums.
b. What is the least common multiple of the two numbers listed in part (a)?
Venn Diagrams
8. The Venn diagram shows the prime factors in the prime factorizations of 8 and 12.
a. Write the prime factorizations of 8 and 12.
b. What is the least common multiple of 8 and 12?
c. What is the greatest common factor of 8 and 12?
9. Each pack of hot dogs has 10 hot dogs. Each pack of hot dog buns has 8 hot dog buns.
a. A shopper wants to buy an equal number of hot dogs and hot dog buns. What is the least number of packs of hot dogs and the least number of packs of hot dog buns he must buy? How many hot dogs and how many hot dog buns will he have? Explain.
b. The lunchroom staff needs to buy an equal number of hot dogs and hot dog buns to feed 60 students. Is it possible to buy packs of hot dogs and packs of hot dog buns to have exactly 60 hot dogs and 60 hot dog buns? Explain.
c. Each pack of hot dogs costs $3.50. Each pack of hot dog buns costs $2.00. The lunchroom staff spends $48.00 when they buy an equal number of hot dogs and hot dog buns. How many packs of hot dogs and how many packs of hot dog buns do they buy?
Name Date
1. What is the least common multiple of 3 and 7?
2. What is the least common multiple of 8 and 12?
3. What is the least common multiple of 2 and 10?
Name Date
The Least Common Multiple
In this lesson, we
• found the least common multiple of two whole numbers.
Examples
1. Find the least common multiple of 3 and 12 by listing multiples.
multiples of 3: 3, 6, 9, 12, …
multiples of 12: 12, 24, 36, 48, …
The least common multiple of 3 and 12 is 12.
2. Find the least common multiple of 12 and 9 by using prime factorization.
12 = 2 × 2 × 3
9 = 3 × 3
36 = 2 × 2 × 3 × 3
Terminology
The least common multiple of two whole numbers is the smallest whole number greater than zero that is a multiple of both numbers.
Least common multiple is often abbreviated LCM.
List multiples of each number, starting with the numbers themselves. The smallest multiple in both lists is the least common multiple.
The prime factorization of 36 includes all the prime factors of 12 and 9. 36=2×2×3×3
Prime Factors of 12
The least common multiple of 12 and 9 is 36
The product of all the prime factors in the Venn diagram is the least common multiple. The product of the numbers in the intersection of the circles is the greatest common factor.
3. Toby bikes every 2 days. He jogs every 3 days. If Toby bikes and jogs on January 1, what is the next date he bikes and jogs on the same day? Show your work.
multiples of 2: 2, 4, 6, …
multiples of 3: 3, 6, …
The next date Toby bikes and jogs is January 7.
Find the least common multiple of 2 and 3 to determine how many days later Toby bikes and jogs on the same day again. The least common multiple of 2 and 3 is 6, and 6 days after January 1 is January 7.
Name Date
PRACTICE
1. Consider the numbers 2 and 3
a. Write the next 10 multiples of 2. 2, 4, , , , , , , , , ,
b. Write the next 10 multiples of 3. 3, 6, , , , , , , , , ,
c. List five common multiples of 2 and 3.
d. Explain why the number 84 is a common multiple of 2 and 3.
e. What is the least common multiple of 2 and 3?
2. Consider the numbers 12 and 10.
a. Write the prime factorization of 12.
b. Write the prime factorization of 10.
c. What is the least common multiple of 12 and 10?
For problems 3–5, find the least common multiple of the given numbers.
3. 2 and 8
4. 6 and 4
5. 5 and 7
6. Consider the numbers 9 and 12.
a. What is the least common multiple of 9 and 12?
b. What is the greatest common factor of 9 and 12?
c. What is the product of 9 and 12?
d. Divide the product of 9 and 12 by the greatest common factor of 9 and 12. What do you notice?
e. Divide the product of 9 and 12 by the least common multiple of 9 and 12. What do you notice?
f. Multiply the least common multiple of 9 and 12 by the greatest common factor of 9 and 12. What do you notice?
7. Riley and Tyler both go to the gym on June 1. During the month of June, Riley goes to the gym every 2 days and Tyler goes to the gym every 3 days. If needed, use the calendar to answer parts (a) and (b).
a. What is the next date they both go to the gym on the same day?
b. During the month of June, how many days do they both go to the gym on the same day?
8. One traffic light turns green every 3 minutes. Another traffic light turns green every 4 minutes. If both traffic lights turn green at 8:00 a.m., what is the next time they both turn green?
9. Ryan buys plates and cups for a party. There are 12 plates in a pack and 10 cups in a pack.
a. Ryan wants to have an equal number of plates and cups. What is the least number of packs of plates and the least number of packs of cups that Ryan must buy?
b. If each pack of plates costs $2.40, what is the cost per plate?
c. If each pack of cups costs $1.50, what is the cost per cup?
d. Ryan buys the least numbers of packs to have an equal number of plates and cups. What is the total amount Ryan spends?
10. Write two numbers that have a greatest common factor of 2 and a least common multiple of 30.
Remember
For problems 11–14, multiply.
15. A newspaper surveys 325 people. The newspaper says that 76% of people surveyed use social media. What is the number of people surveyed who use social media?
16. Reais are the official currency in Brazil. One of these reais is called a real. When traveling to Brazil, Lacy exchanges her 40 US dollars for 120 Brazilian reais. What is the exchange rate between US dollars and Brazilian reais? Choose all that apply.
A. 3 dollars per real
B. 3 reais per dollar C.
Name Date
The Euclidean Algorithm (Optional)
1. Consider the numbers 60 and 100.
a. What is the greatest common factor of 60 and 100?
b. What is the least common multiple of 60 and 100?
Another Square Puzzle
2. Consider the rectangle that measures 100 units by 60 units. Remove the largest square. Then remove the largest possible square from the remaining rectangle. Continue to remove the largest possible square from each remaining rectangle until only a square remains. What is the side length of the square that remains?
3. Consider the rectangle.
240units
180units
a. Remove the largest square. Continue to remove the largest possible square from each remaining rectangle until only a square remains. Complete the table to show the sizes of the squares removed, the sizes of the rectangles that remained, and the size of the final square that remains.
Size of the Rectangle That Remains
240 by 180
Size of the Square Removed
b. What is the side length of the square that remains? What is the greatest common factor of 240 and 180?
The Euclidean Algorithm
4. Use the Euclidean algorithm to find the greatest common factor of 396 and 540. Complete the table by writing the pairs of numbers and their differences for each of the remaining steps of the algorithm. Pair
Common Connection
5. Find the least common multiple of 180 and 240. Use the Venn diagram and the greatest common factor of 180 and 240 from problem 3.
Factors of 180 Factors of 240
6. Consider the numbers 544 and 238.
a. Use the Euclidean algorithm to find the greatest common factor of 544 and 238.
b. Find the least common multiple of 544 and 238. Use the greatest common factor of 544 and 238.
Name Date
Use the Euclidean algorithm to find the greatest common factor of 48 and 128. Draw a diagram or make a table to support your answer.
Name Date
The Euclidean Algorithm
In this lesson, we
• used the Euclidean algorithm to find the greatest common factor of two large numbers.
• used the greatest common factor to find the least common multiple of two large numbers.
Examples
1. Use the Euclidean algorithm to find the greatest common factor of 186 and 66.
a. Complete the table. What is the greatest common factor of 186 and 66?
Steps of the Euclidean Algorithm
1. List the two numbers in this order: larger and then smaller.
2. Find their difference by subtracting the smaller number from the larger number.
3. Keep the smaller of the two numbers. Discard the larger of the two numbers.
4. Use the smaller of the two numbers and the difference to form a new pair of numbers.
5. Repeat until the two numbers are the same.
and 6
and 6
The greatest common factor of 186 and 66 is 6.
b. What is the least common multiple of 186 and 66?
186 × 66 = 12,276
12,276 ÷ 6 = 2,046
The least common multiple of 186 and 66 is 2,046.
One way to find the least common multiple of two numbers is to first find the product of the two numbers. Then divide by the greatest common factor. The result is the least common multiple.
Consider organizing the prime factors of 186 and 66 in a Venn diagram. Multiply all the prime factors to find the least common multiple of 186 and 66
31 × 2 × 3 × 11 = 2,046
PRACTICE
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1. Consider the 120-unit by 80-unit rectangle.
a. What is the side length of the largest square that can be removed from the rectangle?
b. What is the size of the rectangle that remains?
c. Continue to remove the largest possible square from each remaining rectangle until only a square remains. What is the side length of the square that remains?
d. What is the greatest common factor of 120 and 80?
2. Consider the rectangle with dimensions 108 units by 168 units.
80units
a. What is the side length of the largest square that can be removed from the rectangle?
b. What are the dimensions of the rectangle that remains?
c. Continue to remove the largest possible square from each remaining rectangle until only a square remains. What is the side length of the square that remains?
d. What is the greatest common factor of 108 and 168? 120units
108units
168units
3. Use the Euclidean algorithm to find the greatest common factor of 136 and 96. Complete the table.
Pair of Numbers Difference 136 and 96 136 – 96 = 40
4. Use the Euclidean algorithm to find the greatest common factor of 264 and 360. Complete the table.
Pair of Numbers Difference 360 and 264
5. Consider the numbers 90 and 12.
a. What is the greatest common factor of 90 and 12?
b. What is the least common multiple of 90 and 12?
6. Consider the numbers 144 and 252.
a. What is the greatest common factor of 144 and 252?
b. What is the least common multiple of 144 and 252?
7. Consider the numbers 495 and 345
a. What is the greatest common factor of 495 and 345?
b. What is the least common multiple of 495 and 345?
8. Find two numbers that have a greatest common factor of 20 and a least common multiple of 2,400.
9. Jada makes a rectangular pizza that measures 21 inches by 36 inches. She wants to cut the entire pizza into equal-size squares with no pizza left over.
a. What is the side length of the largest square pieces in inches Jada can cut the pizza into?
b. How many pieces of this size can Jada cut?
10. Leo is making a quilt. He has a piece of fabric that measures 48 inches by 168 inches. Leo wants to cut the entire piece of fabric into equal-size squares with no fabric left over.
a. What is the side length of the largest square pieces in inches Leo can cut the fabric into?
b. How many pieces of this size can Leo cut?
Remember
For problems 11–14, write the fractions as mixed numbers and write the mixed numbers as fractions.
15. What is the side length of the smallest square in inches that can be created with rectangles that each measure 4 inches by 6 inches?
16. Complete the ratio table.
Dividing Fractions
Pizza Per Person
I
TOPIC B
Now, if I want every person to many people...
brought you here to help with my fraction problem, not to gorge yourself!
Hey, the more I eat, the easier your math gets! Win-win!
Why do so many fraction problems involve food?
Perhaps because so much of our food involves fractions!
Humans are sharing creatures, and food is something that we humans have always shared. Every culture has its own customs and traditions around food. The only universal is that we all recognize the importance of making sure everyone is well fed.
Then again … sometimes it can be tasty to take a little more than your allotted fraction. Besides, what the cartoon character says is true. It takes a little thinking to determine how many people 3 pizzas can feed, with each person eating 1 4 of a pizza. But if you eat all the pizza yourself, it’s quite clear how many other people you can feed: none!
Grade 6, Module 2, Topic B, Lesson 6
Name Date
Dividing a Whole Number by a Fraction
Pouring Water
1. How many pours of a 3 5 -cup measuring cup fill a 3-cup container? Draw a tape diagram to support your thinking.
2. How many pours of a 3 4 -cup measuring cup fill a 4-cup container? Draw a tape diagram to support your thinking.
3. 6 pours of what size cup fill a 4-cup container exactly to the top? Show your work.
Tape Diagrams
4. Mr. Evans buys 6 large sandwiches for a class party. Each student eats exactly 3 5 of a sandwich. The students eat all of the sandwiches. What number of students ate the sandwiches?
5. Miss Song wants to make lemonade with 8 pounds of lemons. She needs 3 4 pounds of lemons to make 1 liter of lemonade. How many liters of lemonade can Miss Song make?
6. Miss Baker buys 6 pounds of blueberries. She wants to put the blueberries in bags. Each bag holds 1 1 3 pounds of blueberries. How many bags can Miss Baker fill with blueberries?
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A group of friends shares 4 sandwiches. Each friend gets 2 5 of a sandwich. How many friends share the sandwiches? Show how you know.
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Dividing a Whole Number by a Fraction
In this lesson, we
• divided a whole number by a fraction by using a tape diagram.
• divided a whole number by a unit fraction.
• reasoned about the quotient of a whole number and a non–unit fraction.
Examples
1. Consider 4 ÷ 3 _ 5 .
a. Draw a tape diagram to represent 4 ÷ 3
Think of 4 ÷ 3 5 as the question, How many groups of 3 5 are in 4?
Draw a tape diagram with 4 units. Partition each unit into fifths. Then count the number of groups of 3 5 in 4 units.
b. Use any method to evaluate 4 ÷ 3 5 . Show how you know. Because 4 ÷ 1 5 = 20, I know that 4 ÷ 3
There are 6 groups of 3 5 in 4 whole units, plus an additional 2 3 of a group. The quotient is 6 2 3 .
2. A cereal box has 13 ounces of cereal. One serving of cereal is 1 1 4 ounces. How many servings of cereal does the cereal box have? Show how you know.
The cereal box has 10 2 5 servings of cereal.
Because 13 ÷ 1 4 = 52, I know that 13 ÷ 5 4 is 52 ÷ 5, or 10 2 5 .
The mixed number 10 2 5 is equivalent to the fraction 52 5
There are 52 groups of 1 4 in 13.
Because 5 4 is 5 times as much as 1 4 , there are 1 5 as many groups of 5 4 in 13 as there are groups of 1 4 in 13.
So there are 52 5 groups of 5 4 in 13.
When dividing, write mixed numbers as fractions greater than 1 1 1 4 = 4 4 + 1 4 = 5 4
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1. Consider 3 ÷ 3 4
a. Draw a tape diagram that represents 3 ÷ 3 4 .
b. Use the tape diagram from part (a) to evaluate 3 ÷ 3 4
2. Consider 5 ÷ 2 3 .
a. Draw a tape diagram that represents 5 ÷ 2 3 .
b. Use the tape diagram from part (a) to evaluate 5 ÷ 2 3
PRACTICE
3. Consider 6 ÷ 3 5 .
a. What is 6 ÷ 1 5 ?
b. Use your answer from part (a) to evaluate 6 ÷ 3 5 . Show how you know.
For problems 4–7, divide.
8. An office ceiling is 9 feet high. File boxes are each 3 4 feet tall. What is the greatest number of these file boxes that can be stacked in the office? Show how you know.
9. A deli worker cuts 11 pounds of cheese into servings that each weigh 2 3 pounds. How many servings of cheese does the deli worker cut? Show how you know.
Remember
For problems 10–12, multiply.
13. Determine the greatest common factor of 36 and 56. Show how you know.
14. Solve by using the standard algorithm.
24,165 − 9,059
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Dividing a Fraction by a Whole Number
Dividing a Fraction by a Whole Number
1. Kayla fills a glass with water. She pours the water into a pitcher that holds 1 5 gallon. If 4 glasses of water fill the pitcher, what amount of water in gallons does Kayla’s glass hold?
Making Lasagna
2. Kelly uses 3 1 2 cups of fresh basil to make 4 pans of lasagna.
a. Kelly uses the same amount of basil in each pan of lasagna. What amount of basil in cups does Kelly use in each pan of lasagna? Show how you know.
b. Check your solution to part (a).
3. Six friends share 2 3 of a pan of lasagna equally. What fraction of the pan of lasagna does each friend get? Show how you know.
Writing a Story Problem
4. Write a story problem that can be represented by 4 1 3 ÷ 5.
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Jada has 3 4 liters of bubble mix. She pours an equal amount of bubble mix into 5 bottles. What amount of bubble mix in liters is in each bottle? Show how you know.
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Dividing a Fraction by a Whole Number
In this lesson, we
• divided a fraction by a whole number.
• divided a mixed number by a whole number.
Examples
1. Kelly has 3 _ 4 pounds of granola. She divides the granola among 4 people so that each person gets the same amount.
a. Write a division expression to represent this situation.
b. Draw a tape diagram to model the division expression from part (a).
c. What amount of granola in pounds does each person get?
4 ÷ 4 = 3 16
Each person gets 3 16 pounds of granola.
Start with a tape diagram that represents 3 4 . Divide the tape diagram horizontally into 4 equal parts to represent 3 4 ÷ 4 Dividing by 4 produces the same result as multiplying by 1 4 . The darker section of the tape diagram shows 3 4 ÷ 4 , or 3 4 × 1 4 .
To check the answer, use the related unknown factor equation. Because 3 16 × 4 = 3 4 , the quotient 3 16 is correct.
2. A science teacher has a bag of 7 1 2 pounds of gravel. She wants to put an equal amount of gravel in 4 fish tanks. How many pounds of gravel can the teacher put in each fish tank? Show your work.
When dividing, write mixed numbers as fractions greater than 1
7 1 2 = 14 2 + 1 2 = 15 2
The teacher can put 1 7 _ 8 pounds of gravel in each fish tank.
Dividing 15 2 by 4 produces the same result as multiplying 15 2 by 1 4
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1. Consider 3 8 ÷ 4
a. Draw a tape diagram to represent 3 8 .
b. Use the tape diagram from part (a) to evaluate 3 8 ÷ 4.
PRACTICE
2. Which questions can be answered by evaluating 2 3 ÷ 5? Choose all that apply.
A. What is 2 _ 3 of 5?
B. 2 3 is 5 times what number?
C. How many groups of 5 are in 2 3 ?
D. How many groups of 2 3 are in 5?
E. What is 2 3 × 1 5 ?
3. Consider 5 6 ÷ 2.
a. Write 5 6 ÷ 2 as an unknown factor equation.
b. Evaluate 5 6 ÷ 2.
c. Check your answer from part (b).
For problems 4 and 5, divide.
4. 5 7 ÷2
6. Name a division expression that is greater than 2 3 ÷ 5. Explain your thinking.
7. Name a division expression that is less than 3 4 ÷ 7. Explain your thinking.
8. Tara pours 5 8 quarts of batter into 10 cupcake tins. She pours an equal amount into each tin. What amount of batter in quarts does Tara pour into each cupcake tin?
9. Leo has a rope that is 4 5 meters long. He cuts the rope into 3 pieces of equal length. What is the length of each piece of rope in meters?
10. If 7 loads of stone weigh 2 3 tons, what is the weight of 1 load of stone in tons?
11. Consider 2 5 ÷ 4
a. What is the quotient?
b. Write a word problem that is represented by 2 5 ÷ 4
Remember
For problems 12–15, add or subtract. Write your answer in the same form as the problem.
12. One third plus one third
13. 1 fifth plus 3 fifths
14. Five ninths minus two ninths
15. 8 tenths minus 4 tenths
16. Find the least common multiple of 10 and 12.
17. Find the least common multiple of 6 and 10.
18. Multiply.
3,529 × 46
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Dividing Fractions by Making Common Denominators
1. Three friends use wire to make bracelets. How many bracelets can each friend make? Show how you know.
a. Eddie needs 11 12 feet of wire for each bracelet he makes. He has a piece of wire that is 33 12 feet long. How many bracelets can Eddie make?
b. Noah needs 2 3 feet of wire for each bracelet he makes. He has a piece of wire that is 5 1 3 feet long. How many bracelets can Noah make?
c. Julie needs 1 1 4 feet of wire for each bracelet she makes. She has a piece of wire that is 7 1 2 feet long. How many bracelets can Julie make?
Dividing Fractions with a Common Unit
2. Calculate each quotient.
a. 8 ÷ 2
b. 8 ones ÷ 2 ones
c. 8 tens ÷ 2 tens
d. 8 thousands ÷ 2 thousands
e. 8 tenths ÷ 2 tenths
f. 8 thirds ÷ 2 thirds
3. Consider 8 9 ÷ 2 9 .
a. Draw a tape diagram to represent 8 9 ÷ 2 9
b. What is 8 9 ÷ 2 9 ?
4. Consider 8 9 ÷ 3 9 .
a. Draw a tape diagram to represent 8 9 ÷ 3 9 .
b. What is 8 9 ÷ 3 9 ?
5. Yuna says that 5 8 ÷ 3 8 is less than 1. Is Yuna correct? Explain.
For problems 6–8, divide. Check each solution.
Making Common Denominators to Divide Fractions
9. Consider 3 5 ÷ 2 3 .
a. Is the quotient greater than 1 or less than 1? Explain.
b. How can we rewrite 3 5 ÷ 2 3 so the fractions have a common denominator?
c. Draw a tape diagram to model 3 5 ÷ 2 3 .
d. Evaluate 3 5 ÷ 2 3 .
e. Show that the answer to part (d) is correct by writing a related multiplication equation.
10. Divide. Show your work.
a. 5 6 ÷ 2 3
b. 4 5 ÷ 3 7
c. 3 1 4 ÷ 2 5
Using Number Lines to Divide Fractions
11. Write a division equation that could be represented by the diagram shown.
12. Consider
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Divide. Show your work.
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Dividing Fractions by Making Common Denominators
In this lesson, we
• used unit language and diagrams to understand how to divide fractions with common denominators.
• divided a fraction by a fraction by using common denominators.
• divided a mixed number by a fraction by using common denominators.
Examples
1. Kayla and Lacy each write an expression that is equivalent to 15 fifths ÷ 3 fifths. Kayla writes 15 tenths ÷ 3 tenths. Lacy writes 1,500 300 . Who is correct? Explain.
Both Kayla and Lacy are correct. The expressions 15 fifths ÷ 3 fifths, 15 tenths ÷ 3 tenths, and 1,500 300 are all equal to 5. In each expression, 15 of one unit is divided by 3 of the same unit.
2. Consider the diagram.
The fraction 1,500 300 can be thought of as 15 hundreds ÷ 3 hundreds Because 15 and 3 have like units, the quotient is 5
a. Write a division expression that is represented by the diagram.
Sample: 5 12 ÷ 1 3
There are 5 of 12 units shaded. The shaded units represent the dividend. Four of the shaded units are labeled as 1 3 , which represents the divisor.
b. Use the diagram to evaluate the division expression from part (a).
1 1 4
Each group of 1 3 is represented with 4 units on the tape diagram because 1 3 is equivalent to 4 12 . There is 1 whole group of 1 3 in 5 12 , and there is 1 remaining unit. The 1 remaining unit is 1 4 of a group of 1 3 . So the quotient is 1 1 4
3. Evaluate 1
Create common denominators.
When dividing, write mixed numbers as fractions greater than 1.
Then divide the numerators. If the dividend and divisor have common denominators, then the quotient of the fractions is the quotient of their numerators.
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PRACTICE
1. How are the following expressions similar? How are they different? Explain.
2. Tara and Adesh each write an expression that is equivalent to 12 tenths ÷ 4 tenths. Tara writes 12 thirds ÷ 4 thirds. Adesh writes 1,200 ____ 400 . Who is correct? Explain.
For problems 3 and 4, divide.
5. Consider the diagram.
a. Write a division expression that is represented by the diagram.
b. Use the diagram to evaluate the division expression from part (a).
6. Which expression has the same value as 5 7 ÷ 3 4 ?
A. 15 7 ÷ 15 4
B. 5 28 ÷ 3 28
C. 20 28 ÷ 21 28
D. 9 11 ÷ 10 11
For problems 7–10, divide.
11. Choose one problem from problems 7–10 and check your solution.
Remember
For problems 12–15, add or subtract.
1 4 + 1 4
16. A book is 5 8 inches thick. How many copies of the book fit on a shelf that is 20 inches long?
17. Which is greater, 5% of 90 or 90% of 5? Explain.
How Much Sugar?
But... the recipe...
Just pour in the whole bag. You can never have
The thing about baking is that you have to be precise.
Cooking on a stove is more forgiving. You can throw in a little seasoning, or forget the onion, or pour in an extra glug of oil … and it’ll all work out fine (if perhaps a little different from last time).
But baking in the oven? You’d better have your chemistry just right. Leave out one ingredient and your cake may come out as brittle as a brick, as rubbery as a hot dog, or as sludgy as mud—or, in the worst case, all three at once.
That’s why bakers know their fractions. The difference between 2 3 and 3 4 may be the difference between perfection and defeat.
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Dividing Fractions by Using Tape Diagrams
Dividing a Fraction by a Unit Fraction
1. For each division expression,
• write the division expression as an unknown factor equation,
• write the interpretation of the unknown factor equation,
• draw a tape diagram to represent the unknown factor equation,
• determine the value of one unit in the tape diagram, and
• calculate the quotient.
2.
Dividing a Fraction by a Non–Unit Fraction
3. Complete the table.
Division Situations
For problems 4–6, answer the question and draw a diagram to justify your thinking.
4. Leo pours 1 4 gallon of lemonade into a jug. The lemonade fills 1 8 of the jug. How many gallons of lemonade must Leo pour to fill the whole jug?
5. A faucet fills a bucket at a rate of 1 5 gallon per minute. How many minutes does it take to fill a bucket that holds 7 8 gallons? Write your answer as a mixed number.
6. Ryan picks blueberries and freezes 3 4 of them. He freezes 2 1 2 pounds of blueberries. How many pounds of blueberries did Ryan pick? Write your answer as a mixed number.
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Consider 2 3 ÷ 4 5
a. Write 2 3 ÷ 4 5 as an unknown factor equation.
b. Draw a tape diagram that represents the unknown factor equation from part (a).
c. What is the value of one unit in the tape diagram from part (b)?
d. What is 2 3 ÷ 4 5 ?
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Dividing Fractions by Using Tape Diagrams
In this lesson, we
• related division of a fraction by a fraction to an unknown factor equation.
• used a tape diagram to divide a fraction by a fraction.
Examples
1. Consider 2 3 ÷ 3 5 .
a. Write 2 3 ÷ 3 5 as an unknown factor equation. 2 3 = 3 5 × ?
Think of this unknown factor equation as the question “ 2 3 is 3 5 of what number?”
b. Draw a tape diagram that represents the unknown factor equation from part (a).
Draw a tape diagram with 5 units. Because 2 3 is 3 5 of an unknown number, 3 of the 5 units, or 3
are labeled 2 3
One unit of the tape diagram represents
Because there are 5 units in the whole tape diagram and each unit represents
2. Leo pours 3 4 liters of liquid into a container. The liquid fills 1 4 of the container. How many liters of liquid must Leo pour to fill the whole container? Draw a tape diagram to justify your solution.
The tape diagram represents the container, and 3 4 liters fills 1 4 of the container.
Leo must pour 3 liters of liquid to fill the whole container.
Because each unit represents 3 4 liters, the whole tape diagram represents 3 liters.
3 4 × 4 = 12 4 = 3
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PRACTICE
1. Julie and Sasha evaluate 4 5 ÷ 1 2 .
• Julie thinks of 4 5 ÷ 1 2 as the question, “How many groups of 1 2 are in 4 5 ?”
• Sasha thinks of 4 5 ÷ 1 2 as the question, “ 4 5 is 1 2 of what number?”
Who is correct?
A. Only Julie is correct.
B. Only Sasha is correct.
C. Both Julie and Sasha are correct.
D. Neither Julie nor Sasha is correct.
For problems 2–4, consider the tape diagram. Write a division expression and its related unknown factor equation. Then calculate the quotient.
For problems 5–10, divide. Show your work.
11. A physical education teacher divides 3 4 of a field into sections for games. Each section is 1 8 of the whole field. How many sections does the teacher make? Show your work.
12. It takes 5 6 gallons of water to fill 1 3 of a bucket. How many gallons of water fill the whole bucket? Draw a diagram to justify your solution.
Remember
For problems 13–16, write the fraction as a mixed number, or write the mixed number as a fraction. 13. 4 3 14. 14 6 15. 2 2 3 16. 3 3 4
17. Eddie shares 4 pies with his friends. Each friend gets exactly 2 5 of a pie. How many friends does Eddie share his pies with? Show your work.
18. Which of the following statements is true?
A. The value of 6 in 620 is 100 times the value of 6 in 62.
B. The value of 6 in 62 is 10 times the value of 6 in 6.2.
C. The value of 6 in 6.2 is 1 100 the value of 6 in 0.62
D. The value of 6 in 0.62 is 1 10 the value of 6 in 0.062.
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Dividing Fractions by Using the Invert and Multiply Strategy
The Invert and Multiply Strategy
1. Consider 2 3 ÷ 3 4 .
a. Calculate the quotient by using a tape diagram.
b. Calculate the quotient by using the invert and multiply strategy.
2. Consider 5 ÷ 6 7 . Which number sentences are true? Choose all that apply.
A. 5 ÷ 6 7 = 5 × 6 ÷ 7
B. 5 ÷ 6 7 = 5 × 7 ÷ 6
C. 5 ÷ 6 7 = 5 × 7 × 1 6
D. 5 ÷ 6 7 = 5 × 6 7
E. 5 ÷ 6 7 = 5 × 7 6
Fraction Division Error Analysis
3. Riley explains to her friend that 5 6 ÷ 3 4 = 15 24 because 5 6 × 3 = 15 6 and 15 6 × 1 4 = 15 24 . Explain any errors Riley made.
4. Leo says that 2 5 ÷ 7 9 = 35 18 because
2 ×
. Explain any errors Leo made.
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For problems 1–3, divide. Show your work.
5 8 ÷ 3 5
1 2 3 ÷ 4 7
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Dividing Fractions by Using the Invert and Multiply Strategy
In this lesson, we
• divided a fraction by a fraction by using the invert and multiply strategy.
Examples
For problems 1–3, divide. Show your work.
Terminology
A reciprocal is the number obtained by inverting a fraction. The reciprocal of 3
3 . The reciprocal of 5 2 is 2 5
To divide a fraction by a fraction, multiply the dividend by the reciprocal of the divisor. Write 10 as the fraction 10 1 . Its reciprocal is 1 10
Write a mixed number divisor as a fraction greater than 1 before determining its reciprocal.
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1. Consider 3 5 ÷ 2 3
a. Draw a tape diagram of 3 5 ÷ 2 3 . Use it to calculate the quotient.
PRACTICE
b. Confirm your solution to part (a) by using the invert and multiply strategy.
For problems 2–4, determine the unknown number that makes the number sentence true.
5. Consider 2 3 ÷ 4 7 . Which number sentences are true? Choose all that apply. A. 2 _ 3 ÷ 4 _ 7 = 2 _ 3 × 7 ÷ 4 B. 2 _ 3 ÷ 4 _ 7 = 2 _ 3 × 4 ÷ 7
For problems 6–11, divide.
12. Choose one problem from problems 6–11. Explain which strategy you used in that problem and why.
21. Julie has 2 1 2 bags of granola. A recipe calls for 1 4 bag of granola to make one snack. How many snacks can Julie make?
22. Match each expression to the correct product.
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1. Ryan’s tabletop measures 4 1 2 feet by 1 1 4 feet. He wants to cover the tabletop with copies of the colored tile without cutting any tiles. Can Ryan cover the tabletop? If so, how many tiles does he need? If not, why?
Interpreting Multiplication and Division Situations
2. Match each situation with the expression it represents. Justify your thinking.
a. If 2 3 quarts of water fill 3 4 of a container, how many containers will 1 quart of water fill?
b. If 2 3 quarts of water fill 3 4 of a container, how many quarts of water are in 1 container?
c. If Jada pours 2 3 of 3 4 quarts of water out of a container, how much water does Jada pour out?
3. Kayla has 8 cups of oats. Her recipe for one batch of scones calls for 1 2 3 cups of oats. How many batches of scones can Kayla make?
4. A kitchen faucet flows at a rate of 7 8 gallons per minute. At this rate, how many minutes does it take to fill a jug that holds 2 1 4 gallons?
5. A runner plans to run 7 1 2 miles. He runs only 2 1 4 miles. What fraction of his planned distance does the runner run?
6. When 6 1 2 tiles are placed end to end, the total length of the tiles is 4 feet 4 inches. What is the length of each tile in feet?
7. One serving of yogurt is 3 4 cups. Tyler eats 1 cup of yogurt. How many servings of yogurt does Tyler eat?
8. Lacy is 3 1 2 feet tall. Her big brother Leo is 5 1 4 feet tall. Leo is how many times as tall as Lacy?
9. The orange scissors are 8 1 4 inches long. The pink scissors are 2 3 as long as the orange scissors. How long are the pink scissors in inches?
Pink Scissors
Orange Scissors
10. Write a word problem that could be solved by using the expression
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Lisa has a rope that is 2 1 4 yards long. Blake has a rope that is 5 1 2 yards long. Blake’s rope is how many times as long as Lisa’s rope?
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In this lesson, we • divided fractions and mixed numbers to solve real-world problems.
Examples
1. Noah pours 8 gallons of water into a tank. The water fills 2 5 of the tank.
a. How many gallons must Noah pour to fill the whole tank? ?
This tape diagram models the question, “8 is 2 5 of what number?”
Noah must pour 20 gallons of water to fill the whole tank.
Invert the divisor, 2 5 , and multiply by the dividend, 8 1
b. When the tank is full, how many 1 4 -gallon bottles can be filled from the tank? 20 ÷ 1 4 = 20 × 4 = 80
Eighty 1 4 -gallon bottles can be filled from the tank.
Think of this situation as the question, “How many groups of 1 4 are in 20?”
2. Adesh places square tiles end to end along the edge of a table that is 24 3 4 inches long. Each tile is 5 8 inches long.
a. How many tiles can Adesh place along the edge of the table?
Adesh can place 39 tiles along the edge of the table.
Consider using common denominators to solve this division problem.
99 4 × 2 2 = 198 8
Then divide the numerators.
b. Adesh says, “I rounded my solution up to 40. I can place 40 tiles along the edge of the table.” Is Adesh correct? Explain.
No. Adesh is not correct. There is only enough room to place 39 3 5 , or 39, tiles along the edge of the table.
When rounding, consider the situation. The quotient 39 3 5 is closer to 40 than it is to 39. However, in this case, another whole tile will not fit. So the answer must be 39
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1. Riley pours 12 gallons of water into a tank. The water fills 3 4 of the tank.
a. How many gallons can the whole tank hold?
PRACTICE
b. Riley fills the whole tank with water. He fills 1 2 -gallon bottles from the tank. How many of these bottles can Riley fill?
2. Scott finishes 1 6 of the levels on a video game in 3 1 2 months. Which expression can be used to determine the number of months it will take Scott to finish all of the levels at this rate?
A. 3 1 2 × 1 6
B. 3 1 2 ÷ 1 6
C. 1 6 ÷ 3 1 2
D. 6 ÷ 3 1 2
3. One batch of pancakes needs 1 1 4 cups of pancake mix. One box has 9 cups of pancake mix. How many batches of pancakes can be made with one box of pancake mix?
4. Sana eats 2 1 2 servings of cereal. She eats 1 2 3 cups of cereal. How many cups of cereal are in one serving?
5. Eddie stacks boxes in his garage. Each box is 1 3 4 feet tall. The distance from the garage’s floor to its ceiling is 8 1 2 feet.
a. What is the greatest number of boxes Eddie can stack in his garage?
b. Explain how you decided how to round your answer in part (a).
6. At cross country practice, Yuna runs 2 1 4 miles. Ryan runs 5 1 2 miles. How many times as far as Ryan does Yuna run?
7. Write a word problem that can be solved by using the expression 3 4 ÷ 1 3 .
8. Tyler grows a tomato that weighs 3 4 pounds. Lisa grows a tomato that weighs 1 1 5 pounds. What fraction of Lisa’s tomato’s weight is Tyler’s tomato’s weight?
Remember For problems 9–12, add or subtract.
13. Sasha uses 3 _ 5 of her peaches in a fruit salad. If she uses 3 _ 4 pounds of peaches, how many pounds of peaches does Sasha have at first?
14. Match each product to the correct expression.
Grade 6, Module 2, Topic C, Lesson 12
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Fraction Operations in a Real-World Situation
Building a Race Car
1. The blocks of wood used to make wood box race cars weigh 3 2 5 ounces. Some amount of wood must be removed from the block of wood to create the desired shape. Choose a car and determine its weight.
Note: Complete one row of the table.
2. The weight of one set of 5 wheels is 1 ounce. The weight of one set of 3 axles is 3 50 ounces.
a. What is the weight of each wheel?
b. What is the weight of each axle?
c. What is the total weight of your car with 4 wheels and 2 axles?
3. 4 1 5 ounces of paint can cover 10 cars.
a. What color is your car?
b. What is the total weight of your car with paint?
4. The weight of one decal sticker is 1 25 ounce. Suppose you place 3 decal stickers on your car. What is the total weight of your car with these decal stickers?
5. The total weight of your car should be as close as possible to 6 ounces without weighing more than 6 ounces. The weight of each weight is 1 2 ounce. How many weights should you add to your car?
6. What is the total weight of your car after adding the weights?
Racing the Car
7. In trial runs, car A traveled 36 1 2 feet in 2 3 4 seconds. Car B traveled 27 feet in 2 1 4 seconds. Which car traveled faster?
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Leo has 2 3 cups of flour. This amount is 4 5 of the amount of flour he needs to make 1 batch of dinner rolls.
How many cups of flour does Leo need to make 2 batches of dinner rolls?
RECAP
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Fraction Operations in a Real-World Situation
In this lesson, we
• solved real-world problems by adding, subtracting, multiplying, and dividing fractions and mixed numbers.
Examples
1. Kelly’s math textbook weighs 3 1 2 pounds. Riley’s history textbook weighs 2 1 4 pounds. Find the number that makes each statement true.
a. Together, Kelly’s and Riley’s textbooks weigh pounds.
b. Kelly’s textbook weighs
Find the combined weight of Kelly’s and Riley’s textbooks by adding.
than Riley’s textbook.
Find how many pounds heavier Kelly’s textbook is than Riley’s textbook by subtracting.
c. Kelly’s textbook
First, think of an unknown factor equation. What number times the weight of Riley’s textbook is the weight of Kelly’s textbook? Then, evaluate the related division expression.
2. Noah has a pitcher of 1 3 4 liters of lemonade. Noah pours an equal amount of lemonade into 4 cups. Afterward, there is 1 3 liter of lemonade remaining in the pitcher. How many liters of lemonade are in each cup?
First, subtract 1 3 from 1 3 4 to determine how many liters of lemonade Noah pours into the 4 cups. Then, divide the difference by 4 to determine how many liters of lemonade are in each cup.
There are 17 48 liters of lemonade in each cup.
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PRACTICE
1. Two middle school classes work together to paint a mural in their school. Match each situation with the expression that could be used to answer the question.
Mr. Perez’s class paints 1 5 of the entire mural, which is 2 3 of the section they must paint. What portion of the entire mural is their section?
Mrs. Chan’s class must paint 1 5 of the mural. The class paints 2 3 of their section. What portion of the entire mural does Mrs. Chan’s class paint?
Mrs. Chan’s class uses 2 3 of a gallon of white paint. Mr. Perez’s class uses 1 5 of the gallon of white paint. How much more paint does Mrs. Chan’s class use than Mr. Perez’s class?
Mrs. Chan’s class uses 2 3 of a gallon of white paint. Mr. Perez’s class uses 1 5 of the gallon of white paint. What portion of the gallon of white paint do the classes use altogether?
a. Together, Leo’s and Sana’s backpacks weigh pounds.
b. Leo’s backpack weighs pounds more than Sana’s backpack.
c. Leo’s backpack weighs times as much as Sana’s backpack.
3. Tara spends 2 3 of her money on a video game. She spends half of her remaining money on lunch. If her lunch costs $10, how much money did Tara have at first?
4. Mr. Perez gets money for his birthday. He spends 1 5 of the money on new shoes. He gives the remaining money to 3 different charities. He gives the same amount to each charity.
a. What fraction of his money does Mr. Perez give to each charity?
b. If Mr. Perez gives $60 to each charity, how much money did he get for his birthday?
5. Leo uses three bags of flour to make cookies. The amount of flour in each bag is the following:
• 2 3 cups of flour
• 1 1 2 cups of flour
• 1 1 3 cups of flour
Each batch of cookies needs 1 3 4 cups of flour. How many batches of cookies can Leo make?
6. Tyler has a pitcher of 1 3 4 liters of lemonade. Tyler pours an equal amount of lemonade into 3 cups. Afterward, there is 1 3 liter of lemonade remaining in the pitcher. How many liters of lemonade are in each cup?
Remember
For problems 7–10, add or subtract.
For problems 11 and 12, divide.
13. Kayla has 3 red ribbons. Each ribbon is 1 2 5 meters long. What is the total length of all Kayla’s ribbons in meters?
Decimal Addition, Subtraction, and Multiplication
World’s Greatest Carrot?
Are you sure? That
plus 9 is 16... put the decimal back in... so your total is $16.
You must be one
I thought so! What a relief!
$1,600.
When doing arithmetic, you may feel tempted to “ignore” the decimal point. Beware this temptation! Ignoring the decimal point is like ignoring the difference between 7 and 70, or between 16 and 1,600.
Our system of numbers is all about “place value.” That is to say: value depends on place. To know what each digit is worth, you need to see where it is written. Writing a 7 right after the decimal point and writing a 7 right before the decimal point mean two different things.
The decimal point is a landmark, a reference point, helping determine those values.
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Decimal Addition and Subtraction
Decimal Point Puzzle
1. Choose two or three decimals less than 1,000. Decimals can extend to the thousandths place. Make sure decimals have at least one number other than 0 to the right of the decimal point.
a. List the chosen decimals.
b. Use the standard algorithm to compute the sum of the decimals from part (a).
Place Value in Addition and Subtraction
2. Subtract.
3. Consider the list of numbers. Which one of the numbers is the sum of two other numbers in the list?
1 2 , 1 331 1,000 , 0.701, 1.218, 63 100
4. A baseball pitcher throws a baseball. The baseball reaches home plate in about 0.458 seconds. A softball pitcher throws a softball. The softball reaches home plate in about 0.419 seconds. How many fewer seconds does it take for the softball to reach home plate than the baseball?
For problems 5 and 6, write the unknown digits so that the sum or difference is correct.
5. + 5.3475 10.0000
6. 2.4 1.031
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684.49 + 731.287
2. Subtract.
745.406 − 61.97
1. Add.
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Decimal Addition and Subtraction
In this lesson, we
• added decimals by using the standard algorithm.
• subtracted decimals by using the standard algorithm.
Examples
1. Consider 3.458 + 32.63.
a. Estimate the sum. Show how you estimated.
3 + 33 = 36
b. Find the sum.
2. Find the difference.
− 15.47
3. T-shirts cost $8.99 each. A package of socks costs $5.50 Toby buys 2 T-shirts and 1 package of socks. He pays with $40.00. How much change should Toby get back?
8.99 + 8.99 + 5.50 = 23.48
40 − 23.48 = 16.52
Toby should get back $16.52 in change.
Round to the nearest whole numbers to estimate the sum.
When adding decimals,
Compare the sum with the estimate to determine whether the sum is reasonable.
When subtracting decimals, be sure to subtract like units.
Subtract the total cost of the items from the $40.00 Toby pays with to find how much change he should get back.
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1. Estimate the sum. Show how you estimated.
15.236 + 24.901
2. Estimate the difference. Show how you estimated.
91.887 − 61.4
PRACTICE
3. Insert a decimal point into each addend to make the number sentence true. 154 + 31 + 8 = 5.44
For problems 4–7, determine whether the number sentence is true or false. If false, calculate the correct answer.
12. A restaurant sells tacos for $0.99 each and drinks for $1.19 each. Kelly orders 2 tacos and 1 drink. She pays with a $5.00 bill. How much change should Kelly get back? Show your work.
13. The price of frozen yogurt is based on the weight of the yogurt. The table shows the prices for different weights of yogurt.
Riley’s yogurt weighs 8.641 ounces. Sasha’s yogurt weighs 5.77 ounces. To pay a lower price, should Riley and Sasha pay for their yogurt together or separately? Justify your answer.
14. Write the unknown digits so that the sum is correct. + 0.204 1.000
15. Write the unknown digits so that the difference is correct. 8 2.157
Remember
For problems 16–19, add or subtract.
16. 2 3 + 2 5
2 9 + 2 5
20. Lisa runs 3 1 2 miles. Noah runs 5 1 2 miles. How many times as far as Lisa does Noah run?
21. Which of the following situations can be represented by 1 4 ÷ 3? Choose all that apply.
A. The amount of syrup on each plate if 1 4 cup of syrup is split among 3 plates of pancakes
B. The amount Yuna swims if she swims 1 4 mile 3 times
C. The amount of time spent watching 1 4 of a 3-hour movie
D. The amount of oatmeal needed for a recipe if 1 4 pound is 3 times as much as Eddie needs
Grade 6, Module 2, Topic D, Lesson 14
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Patterns in Multiplying Decimals
1. Choose two rectangles from the set of four to complete parts (a)–(c).
a. Write a multiplication number sentence that represents the area of the first rectangle you chose.
b. Write a multiplication number sentence that represents the area of the second rectangle you chose.
c. Consider the factors that represent the lengths and widths of the rectangles in the multiplication number sentences you wrote. Use these factors to describe how the rectangles are related.
2. Choose two rectangles from the set of four to complete parts (a)–(c).
a. Write a multiplication number sentence that represents the area of the first rectangle you chose.
b. Write a multiplication number sentence that represents the area of the second rectangle you chose.
c. Consider the factors that represent the lengths and widths of the rectangles in the multiplication number sentences you wrote. Use these factors to describe how the rectangles are related.
Area Model
3. Use the 10 × 10 grid to complete the problem. Each side length of the grid represents 1 unit.
a. On the grid, shade a rectangle with an area of 0.42 square units.
b. Write a multiplication number sentence that represents the area of your rectangle in part (a).
4. Use the 10 × 10 grid to complete the problem. Each side length of the grid represents 1 unit.
a. Shade a rectangle on the grid.
b. Write a multiplication number sentence by using only decimals to describe the area of your rectangle in part (a).
c. Write a multiplication number sentence by using only fractions to describe the area of your rectangle in part (b).
Which Is Greatest?
For problems 5–7, order the expressions from least to greatest.
1. Which expressions are equal to 0.15? Choose all that apply.
A. 5 × 3 10
B. 5 × 0.03
C. 5 10 × 3 10
D. 0.05 × 3
E. 0.05 × 0.03
2. Fill in the blanks with the factor that makes each equation true. Use the given equation. 24 × 314 = 7,536
a. 240 × = 7,536
b. 2.4 × = 7,536
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In this lesson, we • recognized patterns in factors when multiplying whole numbers and decimals.
Examples
1. Fill in the blanks with the factor that makes each equation true.
a. 0.036 × 100 = 3.6
b. 0.36 × 10 = 3.6
c. 3.6 × 1 = 3.6
d. 36 × 0.1 = 3.6
e. 360 × 0.01 = 3.6
Notice how the place value of each digit in the factors changes when multiplied by powers of 10. Each equation results in the same product.
2. Fill in the blanks with the factor that makes each equation true. Use the given equation. 4.3 × 6.04 = 25.972
a. 0.43 × 60.4 = 25.972
b. 0.43 × 6.04 = 2.5972
c. 43 × 0.604 = 25.972
d. 4.3 × 0.604 = 2.5972
When one factor is divided by a power of 10 and the other factor is multiplied by the same power of 10, the product stays the same.
When one factor is divided by a power of 10 and the other factor stays the same, the product is divided by the same power of 10
3. Which expressions are equal to 0.48? Choose all that apply.
A. 8 × 6 10
B. 8 × 0.06
C. 8 10 × 6 10
D. 0.08 × 0.06
E. 0.08 × 6
8 groups of 6 hundredths is equal to 48 hundredths.
8 tenths multiplied by 6 tenths is equal to 48 hundredths.
6 groups of 8 hundredths is equal to 48 hundredths.
4. Which expressions are equivalent to 0.4 × 0.8? Choose all that apply.
A. 4 × 0.01 × 8 × 0.01
B. 4 × 0.1 × 8 × 0.1
C. 4 × 1 1,000 × 8 × 1 100
D. 4 × 0.001 × 8 × 0.1
E. 4 × 10 × 8 × 0.001
F. 4 × 8 × 1 10 × 1 10
G. 4 10 × 8 10
4 × 0.1 = 0.4 and 8 × 0.1 = 0.8
Therefore, this expression is equivalent to 0.4 × 0.8
(4 × 10) × (8 × 0.001) = 40 × 0.008. Multiplying the factor 40 by 0.01 and the factor 0.008 by 100 does not change the product. Therefore, this expression is equivalent to 0.4 × 0.8
The expression is equivalent to (4 × 1 10 ) × (8 × 1 10 ). 4 × 1 10 = 4 10 and 8 × 1 10 = 8 10 Therefore, this expression is equivalent to 0.4 × 0.8 4 10 is equivalent to 0.4 and 8 10 is equivalent to 0.8. Therefore, this expression is equivalent to 0.4 × 0.8
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1. Order the expressions from least to greatest.
A. 5 × 0.22
B. 5 × 0.201
C. 5 × 0.21
D. 5 × 0.2
2. Order the expressions from least to greatest.
A. 18 × 303 1,000
B. 18 × 3 10
C. 18 × 39 100
D. 18 × 37 100
3. Order the expressions from least to greatest.
A. 350 × 0.03
B. 35 × 3.1
C. 3.5 × 0.32
D. 0.35 × 0.33
PRACTICE
4. Fill in the blank to make each equation true. Use each number from the Factor column once.
5. Fill in the blanks with the factor that makes each equation true.
a. 0.024 × = 2.4
b. 0.24 × = 2.4
c. 2.4 × = 2.4
d. 24 × = 2.4
e. 240 × = 2.4
6. Fill in the blanks with the factor that makes each equation true. Use the given equation.
3.7 × 5.02 = 18.574
a. 0.37 × = 18.574
b. 0.37 × = 1.8574
c. × 0.502 = 18.574
d. × 0.502 = 1.8574
7. Match each expression in column A to an expression in column B that has the same value.
Column A Column B
1 __ 10 of 5 1,000
0.1 × 0.05
1 100 of 5 10 0.01 × 5
1 10 of 5 10 1 × 0.5
1 10 of 5 0.01 × 0.05
8. Which expressions are equal to 0.24? Choose all that apply.
A. 8 × 3 10
B. 8 × 0.03
C. 8 10 × 3 10
D. 0.08 × 0.03
E. 0.08 × 3
9. Which expressions are equivalent to 0.7 × 0.3? Choose all that apply.
A. 7 × 0.01 × 3 × 0.01
B. 7 × 0.1 × 3 × 0.1
C. 7 × 1 1,000 × 3 × 1 100
D. 7 × 0.001 × 3 × 0.1
E. 7 × 10 × 3 × 0.001
F. 7 × 3 × 1 10 × 1 10
G. 7 10 × 3 10
10. Lacy says that 54 × 2.37 is less than 54 because 54 is multiplied by a decimal. Do you agree or disagree? Explain.
11. Scott says that 5.4 × 0.237 is less than 5.4. Eddie says that 5.4 × 0.237 is greater than 5.4. Do you agree with Scott, or do you agree with Eddie? Why?
12. Fill in the blanks with the product that makes each equation true. Use the given equation.
21 × 13 = 273
a. 21 × 1.3 =
b. 210 × 13 =
c. 21 × 1,300 =
d. 2.1 × 1.3 =
e. 2.1 × 0.13 =
f. 0.21 × 0.13 =
Remember
For problems 13–16, add or subtract.
17. Ryan runs 3 4 miles. This distance is 2 3 of his daily running goal. What is Ryan’s total running goal in miles for 3 days?
18. Which situations can be represented by 6 ÷ 1 _ 2 ? Choose all that apply.
A. The number of hours a plane has been in the air if it has flown 1 _ 2 of its 6-hour flight
B. The number of 1 2 -pound burgers that can be made with 6 pounds of meat
C. The number of pieces of chocolate if 6 chocolate bars are split in half
D. The weight of Kelly’s bookbag if he has 6 binders that each weigh 1 2 pound
E. The total number of eggs Blake needs to make an omelet if 6 eggs is 1 2 of the total number of eggs he needs
F. The amount of trail mix in each bag if Ryan splits 1 _ 2 pound of trail mix equally among 6 bags
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Decimal Multiplication
1. Determine what information you need and what operations you can use to answer each question.
a. What is the total number of pennies used to make the artwork?
b. What is the length of the artwork in inches? What is the width of the artwork in inches?
c. What is the total value in dollars of the pennies used to make the artwork?
d. What is the total weight in grams of the pennies used to make the artwork?
e. If all the pennies used to make the artwork were stacked, how high would the stack be?
Multiplying Decimals
For problems 2 and 3, multiply. Justify your reasoning for the location of the decimal point in the product.
17.125 × 6
11.5 × 23.5
2.
3.
For problems 4–7, compute the target sum by adding the products for parts (a)–(c).
4. Target sum: 5.3508
a. 2.9 × 1.41 b. 8.06 × 0.03
5. Target sum: 20.8148 a. 7.5 × 0.86
4.4 × 3.21
1.5 × 0.68
6.02 × 0.04
6. Target sum: 12.5888 a. 5.07 × 0.04
5.5 × 0.74
3.6 × 2.31
7. Target sum: 18.5489
a. 7.8 × 1.81 b. 9.03 × 0.03
Applications of Decimal Multiplication
6.5 × 0.64
8. The width of a rectangular classroom is 29.25 feet. The length of the classroom is 31.5 feet.
a. What is the area of the classroom in square feet?
b. A company charges $0.40 per square foot to wax a floor. What is the cost to wax the classroom’s floor?
9. Miss Song wants to drink 2.2 liters of water daily. By lunchtime, she drinks 85% of her daily amount of water.
a. What is the total amount of water in liters Miss Song drinks by lunchtime?
b. How many more liters of water does Miss Song need to drink to meet her daily amount of water?
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× 3.6
Multiply.
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Decimal Multiplication
In this lesson, we • multiplied decimals by using the standard algorithm.
Examples
1. Insert a decimal point in the product to make the equation true.
18.6 × 3.07 = 57102
57.102
2. Multiply.
162.868
21.43 × 7.6
Use estimation to determine where to insert the decimal point in the product. One way to estimate is to round the factors to 20 and 3, because these numbers are simple to multiply mentally. The decimal point should be inserted to the right of 57 because 20 × 3 = 60 and 57 is close to 60.
When the standard algorithm is used to multiply decimals, the number of digits to the right of the decimal point in the product is equal to the sum of the number of digits to the right of the decimal point in each factor. Notice that 21.43 has two digits to the right of the decimal point, while 7.6 has one. Therefore, the product should have three digits to the right of the decimal point.
3. Blake earns $1,025.75 every month at his job. He earns an extra $13.50 every time he mows a lawn. Last month, Blake mowed 4 lawns.
a. How much money did Blake earn last month from mowing lawns?
13.50 × 4 = 54
Blake earned $54 last month from mowing lawns.
b. What is the total amount of money Blake earned last month?
1,025.75 + 54 = 1,079.75
Blake earned a total of $1,079.75 last month.
When adding decimals, be sure to line up addends so that like units are added.
It is not necessary to include zeros at the end of a decimal when using the standard algorithm. The expression 13.50 × 4 is equivalent to 13.5 × 4. The products 54.00, 54.0, and 54 are all equivalent.
4. Lacy wants to buy a bicycle that costs $65.50. Lacy has saved 70% of the amount she needs to buy the bicycle. How much money has Lacy saved?
65.5 × 0.7 = 45.85 Lacy has saved $45.85 Remember that 70% is
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PRACTICE
For problems 1–4, estimate the product.
1. 15 × 3.8 2. 7 × 4.22 3. 9.71 × 0.494
211.3 × 8.429
For problems 5–8, insert a decimal point in each product to make the equation true.
15. Which expression has a greater product, 8.2 × 4.19 or 8.28 × 4.09? Explain how you know.
16. A race is 6.21 miles long. Adesh runs at a constant rate of 1 mile every 9.5 minutes. At that rate, can Adesh run the race in less than 1 hour? Justify your answer.
17. A trainer at a gym earns $2,456.75 every month. She earns an extra $4.75 every time she sells a gym membership. Last month, the trainer sold 32 gym memberships. What is the total amount of money the trainer earned last month?
18. Kayla wants to buy an art set that costs $58.50. Kayla has saved 80% of the amount she needs to buy the art set. How much money has Kayla saved?
Remember
For problems 19–22, compute the sum or difference.
Find the sum. 554.99 + 951.287
23.
24. Find the difference.
25. Consider the diagram.
a. Use the diagram to match each description to its ratio.
The ratio of the number of stars to the number of triangles
of
b. Use the diagram to write a ratio that is different from those in part (a). Use ratio language to describe the ratio you write.
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Applications of Decimal Operations
Habitat 67 is an apartment building located in Montreal, Quebec, in Canada. Architect Moshe Safdie designed the unique building in 1967, and it still stands today as a historical monument. Safdie’s vision was to reinvent the apartment building by creating affordable city homes that had qualities of homes outside the city. His motto was, “For everyone a garden.”1 Habitat 67 is made up of 354 concrete blocks arranged in clusters. Its geometric design provides rooftop gardens, fresh air, privacy, and natural light to the residents of each apartment. Each apartment has access to at least one rooftop garden, many of which were built on the roof of a neighboring apartment.
Building a Habitat
Habitat Guidelines
Use 20 cubes to build a model of your own habitat. You must use all 20 cubes.
• Each cube represents a home in your habitat. The entire habitat must be in one piece.
• Your habitat must have at least 3 floor levels. The first floor level must have 2 to 8 cubes. The bottom face of each cube on the first floor level must sit flat.
• Visible side faces of cubes represent walls with windows. Visible top faces of cubes represent rooftop gardens.
• Your model should be built in the spirit of Habitat 67. Each cube must have at least 2 visible side faces so that the apartment has more windows for natural light. Each cube must have a visible top face or connect to a cube with a visible top face so that each apartment has access to a rooftop garden.
1 Andrew McIntosh and Jennifer Carter, “Habitat 67,” The Canadian Encyclopedia.
Cost
• The cost per floor level is $5,467.16.
• There is 1 rooftop garden for every visible top face. The cost per rooftop garden is $5,648.40.
• There are 2 windows for every visible side face. The cost per window is $546.42.
• A land unit is a square or an enclosed square that is visible from the top view of the building. Consider the top view of habitat A and the top view of habitat B.
Habitat AHabitat B
Habitat A has 10 land units because there are 10 squares visible from the top view of the building. Habitat B has 12 land units because it has 10 squares and 2 enclosed squares visible from the top view of the building. The cost per land unit is $4,260.90.
• The property tax is 3% of the total cost of the land units. The amount of tax is added to the total cost of the habitat.
Revenue
• An apartment represented by a cube with 4 visible side faces and a visible top face sells for $16,249.99
• An apartment represented by a cube with 4 visible side faces and no visible top face sells for $14,889.99.
• An apartment represented by a cube with 3 visible side faces and a visible top face sells for $11,249.99.
• An apartment represented by a cube with 3 visible side faces and no visible top face sells for $9,889.99.
• An apartment represented by a cube with 2 visible side faces and a visible top face sells for $7,249.99.
• An apartment represented by a cube with 2 visible side faces and no visible top face sells for $5,889.99.
1. Enter your habitat’s number of floor levels, number of rooftop gardens, number of windows, and number of land units. Use the given rates to calculate the total cost of each type of item.
$5,467.16 per floor level
$5,648.40 per rooftop garden
per window
2. Calculate the property tax on your habitat.
3. Calculate the total cost of your habitat.
4. Enter the number of each type of cube in your habitat. Use the given rates to calculate the total revenue for each type of cube.
4 Visible Side Faces, 1 Visible Top Face $16,249.99
4 Visible Side Faces, 0 Visible Top Faces $14,889.99
3 Visible Side Faces, 1 Visible Top Face $11,249.99
3 Visible Side Faces, 0 Visible Top Faces $9,889.99
2 Visible Side Faces, 1 Visible Top Face $7,249.99
2 Visible Side Faces, 0 Visible Top Faces $5,889.99
5. Calculate the total revenue for your habitat.
6. If your total cost from problem 3 is less than your total revenue from problem 5, complete part (a). Otherwise, complete part (b).
a. Subtract your total cost from your total revenue. Fill in the blank with the difference.
My habitat results in a profit of .
b. Subtract your total revenue from your total cost. Fill in the blank with the difference.
My habitat results in a loss of .
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In today’s lesson, how did you calculate the total cost, total revenue, and profit or loss of your habitat? Explain how the features of your design affected whether your habitat resulted in a profit or a loss.
RECAP
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Applications of Decimal Operations
In this lesson, we • used decimal operations to calculate cost, revenue, and profit or loss.
Examples
Yuna buys supplies to make bracelets to sell at a craft fair.
a. Yuna buys 4 boxes of colored beads that cost $13.49 each. She buys 3 boxes of glass beads that cost $12.00 each. She buys 1 package of elastic for $2.50. What is Yuna’s total cost to buy these supplies?
4 × 13.49 = 53.96
3 × 12.00 = 36
53.96 + 36 + 2.50 = 92.46
Yuna’s total cost to buy these supplies is $92.46
The total amount Yuna spends on supplies to make bracelets is the total cost.
b. On the morning of the craft fair, Yuna sells 12 bracelets for $4.50 each. Later, she sells 18 bracelets for $3.75 each. What is Yuna’s total revenue?
12 × 4.50 = 54
18 × 3.75 = 67.5
54 + 67.5 = 121.5
The total amount Yuna earns from the sale of bracelets is the total revenue.
Yuna’s total revenue is $121.50.
c. Does Yuna make a profit or experience a loss from the sale of her bracelets? Determine the profit or loss.
121.5 − 92.46 = 29.04
Yuna makes a profit of $29.04 from the sale of her bracelets.
Yuna makes a profit because the total revenue is greater than the total cost. The profit is calculated by subtracting the total cost from the total revenue.
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PRACTICE
1. Describe the different mathematical operations you used in today’s lesson to design your habitat and determine your profit or loss.
For problems 2–7, compute.
137.28 × 4 4. 463.2 − 16.41
2. 0.53 + 21
8. Lisa opens a lemonade stand.
a. To start the lemonade stand, Lisa needs to buy cups, lemons, and sugar. She buys 4 packs of cups that cost $2.89 each. Lisa buys 12 lemons that cost $0.39 each. She buys 1 bag of sugar that costs $1.25. What is Lisa’s total cost to start the lemonade stand?
b. On Lisa’s first day of selling lemonade, she sells 22 cups of lemonade for $1.00 each. On Lisa’s second day, she sells 18 cups of lemonade for $0.75 each. What is Lisa’s total revenue?
c. Does Lisa make a profit or experience a loss after selling lemonade for these two days? Determine the profit or the loss.
9. Noah makes 20 dog collars to sell at a craft fair. It costs him $29.80 to buy the supplies to make the collars. Noah sells 11 dog collars for $2.25 each. Does he make a profit or experience a loss at the craft fair? Determine the profit or loss.
Remember
For problems 10–13, add the fractions. Write the answer as a mixed number. 10. 6 4 + 5 12
11 15 + 2 3
14. Which expressions are equal to 0.18? Choose all that apply.
A. 6 × 3 10
B. 6 × 0.03
C. 6 10 × 3 10
D. 0.06 × 3
E. 0.06 × 0.03
15. Each number line is divided into equal parts. Write a fraction in each box to label the tick marks.
A Cost Divided
Okay, big numbers... but what do they mean?
COST PER YEAR (before dividing)
National Parks
Public Libraries
Military Healthcare
$3,500 billion
Whoa! Militaries and healthcare are expensive!
COST PER YEAR (after dividing)
National Parks
Public Libraries
Military
Healthcare
$9 per person
$36 per person
$2,121 per person
$10,600 per person
Which table do you find easier to understand?
When thinking about a country as big as the U.S., it can be hard to make sense of the numbers. How much, exactly, is 12 billion? Or 700 billion? Or 3.5 trillion?
But if you divide these gargantuan numbers by the population of the U.S. (about 330 million people), the numbers become more relatable. Now, they are per capita, or per person. In a sense, the U.S. spends $9 per person on its national parks (whether that particular person goes to 20 parks per year or none at all).
What do you notice about these numbers, and what do you wonder?
Grade 6, Module 2, Topic E, Lesson 17
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Partial Quotients
1. Four sixth grade teachers get a shipment of new pencils to share equally.
BoxesPacks
• Each box has 10 packs of pencils.
• Each pack has 10 pencils.
a. What is the total number of pencils in the shipment?
Pencils
b. Write an expression that represents the number of pencils each teacher receives.
c. Write a division number sentence that describes the situation. Explain.
d. Write a multiplication number sentence that describes the situation. Explain.
Sharing Pencils
2. Suppose you have 788 pencils and 4 teachers.
a. Estimate the number of pencils each teacher receives.
b. Do you think that the actual quotient is greater than or less than your estimate? Why?
c. How many pencils does each teacher receive?
3. Suppose you have 788 pencils and 5 teachers.
a. Estimate the number of pencils each teacher receives.
b. Do you think that the actual quotient is greater than or less than your estimate? Why?
c. How many pencils does each teacher receive?
÷ 8
Partial Quotients Method
5. Divide.
885 ÷ 7
6. Divide.
885 ÷ 15
7. Consider 3,184 ÷ 24.
a. Estimate the quotient. Justify your estimate.
b. Divide.
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Consider 453 ÷ 32
a. Estimate the quotient. Justify your estimate.
b. Divide.
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Partial Quotients
In this lesson, we
• estimated quotients.
• wrote quotients as mixed numbers.
• divided multi-digit whole numbers by using the partial quotients method.
Examples
1. Consider 475 ÷ 6.
a. Estimate the quotient. 80
b. Draw a diagram or use the partial quotients method to find the quotient.
Estimate the quotient by rounding the dividend and divisor to numbers that can be divided mentally.
480 ÷ 6 = 80
75 is distributed to each of the 6 groups, and 6 × 75 = 450
450 is subtracted from 475 to find that 25 is left over.
4 is distributed to each of the 6 groups, and 6 × 4 = 24.
24 is subtracted from 25 to find that 1 is left over.
Because the remainder is less than the divisor, it can be written as the fraction 1 6 , where the remainder is the numerator and the divisor is the denominator.
2. Consider 3,565 ÷ 65.
a. Estimate the quotient.
50
b. Draw a diagram or use the partial quotients method to find the quotient.
–260 –3,250 653,565 315 55 50454 54 55 65
Estimating the quotient can help to confirm whether the answer after dividing is reasonable.
3,500 ÷ 70 = 50
50 is the largest multiple of 10 that can be distributed to each of the groups of 65 because 50 × 65 = 3,250 and 60 × 65 = 3,900
4 is the largest whole number that can be distributed to each of the 65 groups because 4 × 65 = 260 and 5 × 65 = 325
The two partial quotients, 50 and 4, are added to get 54. Because the remainder of 55 is less than the divisor, it can be written as the fraction 55 65
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PRACTICE
1. Consider 590 ÷ 6
a. Estimate the quotient.
b. Draw a diagram or use the partial quotients method to find the quotient.
2. Consider 254 ÷ 21.
a. Estimate the quotient.
b. Draw a diagram or use partial quotients to find the quotient.
3. Consider 4,812 ÷ 75.
a. Estimate the quotient.
b. Draw a diagram or use partial quotients to find the quotient.
For problems 4–11, divide.
4. 180 ÷ 12
5. 760 ÷ 38
6. 282 ÷ 15 7. 714 ÷ 22
Remember
For problems 12–15, add.
8. 500 ÷ 48 9. 809 ÷ 83
10. 3,247 ÷ 15
11. 2,886 ÷ 40
16. Multiply.
17. The prime factorization of a number is 2 × 2 × 5. The prime factorization of another number is 2 × 3 × 5
a. What is the greatest common factor of these two numbers?
b. What is the least common multiple of these two numbers?
c. What are the two numbers?
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The Standard Division Algorithm
The Standard Division Algorithm
For problems 1–5, divide. As necessary, express quotients as mixed numbers.
1. 6,194 ÷ 19
2. 802 ÷ 32
3. 4,809 ÷ 4
4. 3,432 ÷ 143
5. 55,233 ÷ 51
The Euclidean Algorithm
6. Use the Euclidean algorithm to find the greatest common factor of 840 and 660.
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TICKET
Divide by using the standard division algorithm. Express the quotient as a mixed number.
8,247 ÷ 16
RECAP
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The Standard Division Algorithm
In this lesson, we
• divided multi-digit whole numbers by using the standard algorithm.
• used the Euclidean algorithm to find the greatest common factor of two numbers.
Examples
For problems 1 and 2, divide by using the standard division algorithm. As necessary, express quotients as mixed numbers.
One strategy that helps when dividing by a number such as 18 is to create a table that shows multiples of 18
Multiples of 18
1 × 18 = 18
2 × 18 = 36
3 × 18 = 54
4 × 18 = 72
Start by looking at the hundreds place of the dividend. There is no multiple of the divisor, 18, that is close to 7 but not greater than 7. Regroup 7 hundreds and combine with 5 tens to get 75 tens. Because 4 × 18 = 72, which is close to 75 but not greater than 75, record a 4 in the tens place of the quotient.
Subtract 72 tens from 75 tens to get 3 tens. Regroup the 3 tens with the 6 ones to get 36 ones. Then divide 36 by 18. Because 2 × 18 = 36, there is nothing left over, or no remainder.
Because the remainder is less than the divisor, write it as the fraction 2 38 , where the remainder is the numerator and the divisor is the denominator.
3. Use the Euclidean algorithm to find the greatest common factor of 480 and 396.
The greatest common factor of 480 and 396 is 12.
1. Write the larger number in the Dividend column and the smaller number in the Divisor column.
2. Divide and record the quotient and the remainder.
3. The previous divisor is the new dividend.
4. The remainder is the new divisor.
5. Repeat the process until you get a remainder of 0 The last divisor used is the greatest common factor.
We can use division in place of repeated subtraction in the Euclidean algorithm, a method for finding the greatest common factor of two numbers.
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PRACTICE
For problems 1–10, divide. As necessary, express quotients as mixed numbers.
1. 328 ÷ 82 2. 480 ÷ 16 3. 805 ÷ 23
1,360 ÷ 80
4,180 ÷ 8 7. 885 ÷ 14
7,843 ÷ 33
5. 2,378 ÷ 29
9. 5,420 ÷ 61
10. 2,500 ÷ 72
11. Choose two problems from problems 1–10. Then use estimation to show that your quotient is reasonable.
12. Noah makes a mistake when he divides 1,004 by 20.
a. What number should the quotient be close to? Use estimation.
b. Describe Noah’s mistake.
c. What is 1,004 ÷ 20?
Remember
For problems 13–16, add.
For problems 17–19, is the quotient greater than 17.2 or less than 17.2? Explain your thinking.
20. Which pairs of fractions and decimals are equivalent? Choose all that apply.
A. 3 10 and 0.03
B. 5 10 and 0.5
C. 2 6 10 and 0.26
D. 8 100 and 0.08
E. 90 100 and 0.9
17. 17.2 ÷ 3.3
18. 17.2 ÷ 0.33
19. 17.2 ÷ 33
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Expressing Quotients as Decimals
Quotients as Decimals
1. Divide. Express the quotient as a decimal.
2. Divide. Express the quotient as a decimal.
2,046 ÷ 16
3. Divide. Round your answer to the nearest tenth. 7 ÷ 9
4. Divide. Round your answer to the nearest hundredth.
626 ÷ 117
Division Problem Solving
5. Riley divides 11,586 by 24 as shown. She concludes that the quotient is 482.18
a. Why is Riley’s quotient of 482.18 incorrect?
b. Find the correct quotient expressed as a decimal.
6. During his Major League Baseball career, Hank Aaron played for the Milwaukee Braves and the Atlanta Braves.
• Hank Aaron’s first year playing for the Milwaukee Braves was 1954. In 1954, he got 131 hits in 468 at bats.
• Hank Aaron’s first year playing for the Atlanta Braves was 1966. In 1966, he got 168 hits in 603 at bats.
A player’s batting average is calculated by dividing the number of hits by the number of at bats. Batting averages are expressed in thousandths. In which year did Hank Aaron have a higher batting average? Divide to three places after the decimal point.
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Divide. Express the quotient as a decimal.
8,274 ÷ 16
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Expressing Quotients as Decimals
In this lesson, we
• divided multi-digit whole numbers by using the standard algorithm.
• expressed quotients as decimals.
Examples
For problems 1–3, divide.
1. 645 ÷ 6
Record a decimal point and a 0 in the tenths place. The remainder is now 30 tenths. Then, calculate 30 ÷ 6
Record a 0 in the hundredths place if there is still a remainder. Continue these steps until the remainder is 0 or until there are enough digits after the decimal point to round to a given place value.
When the dividend is less than the divisor, the ones must be regrouped as tenths. Start by recording a decimal point and a 0 in the tenths place. Then, calculate 360 ÷ 80.
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PRACTICE
1. Insert a decimal point where you think it belongs in the quotient. Justify your answer.
6,450 ÷ 16 = 403125
For problems, 2–11, divide. When necessary, round your answer to the nearest tenth.
2. 793 ÷ 8
1,943 ÷ 4 4. 5 ÷ 8
522 ÷ 12
6. 1,908 ÷ 25
8,700 ÷ 24
5,631 ÷ 377
3,900 ÷ 742
12. Sasha divides 1,449 by 18 and gets a quotient of 8.5.
a. Is Sasha’s answer reasonable? Why?
b. Consider Sasha’s work. Describe Sasha’s mistake.
13. An example of a baseball player’s batting average is 0.279. A player’s batting average is calculated by dividing the number of hits by the number of at bats. Higher batting averages are better batting averages.
a. Choose one player with a batting average that is better than the example, 0.279. Explain.
b. Choose one player with a batting average that is worse than the example, 0.279. Explain.
Remember
For problems 14–16, add.
17. Consider 453 ÷ 32.
a. Estimate the quotient. Explain how you estimated.
b. Calculate the quotient. Express any remainder as a fraction. Show your work.
18. Blake and Leo attend the same middle school. Blake lives 5 4 miles from the school. Leo lives
1 3 5 miles from the school. Which of the following correctly compares the two distances?
A. 5 4 miles < 1 3 5 miles
B. 5 4 miles > 1 3 5 miles
C. 5 4 miles = 1 3 5 miles
D. 1 3 5 miles < 5 4 miles
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Real-World Division Problems
1. Write a word problem for this equation.
40 ÷ 10 = 4
The Riddle of the $179.00
Problems 2–4 can be modeled by 179 ÷ 18. Consider whether this means that the problems all have the same answer.
2. You have $179.00 to buy books. Each book costs $18.00. What is the greatest number of books that you can buy?
3. You want to earn $179.00. You can earn $179.00 by selling books for $18.00 each. How many books should you sell to earn $179.00?
4. You have $179.00. You split that $179.00 equally among 18 people. How much money does each person get?
The Unknown
5. Kayla makes 175 ounces of lemonade for a lemonade sale. She wants to fill each cup with 8 ounces of lemonade. What is the greatest number of 8-ounce cups Kayla can fill?
6. In one day, a hotel rents 56 rooms and earns $6,678. Each room costs the same amount to rent. What is the cost per room?
7. A family takes a road trip. They drive 475 miles per day. On what day will the family have driven 1,615 miles?
8. A school buys 228 new laptops. This number is 6 times as many laptops as the school bought last year. How many laptops did the school buy last year?
9. The length of a wood board is 3 feet. The board is split into 8 equal pieces of wood. What is the length of each piece of wood in feet?
10. Toby has 493 nickels that he wants to put in rolls. Each roll has 40 nickels. What is the greatest number of rolls of nickels Toby can make?
Telling Stories
11. Write the problem your group created. Write the problem’s solution.
12. Write the problem the other group created. Write the problem’s solution.
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TICKET
A soccer stadium has 4,608 seats. Each section of the stadium has 144 seats. How many sections of seats are in the stadium? Explain.
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Real-World Division Problems
In this lesson, we
• solved real-world division problems.
• created real-world division problems.
Examples
1. A florist has 284 daisies to make 15 bouquets. The florist puts the same number of daisies in each bouquet. How many daisies does the florist put in each bouquet?
Be careful when rounding in certain situations. There aren’t enough daisies for each bouquet to have 19 because 19 × 15 = 285. So 18.9 should be rounded down to 18.
The florist puts 18 daisies in each bouquet.
2. The student council sets a goal to raise $400 by selling tins of popcorn. The council sells each tin of popcorn for $7
a. How many tins of popcorn does the student council need to sell to reach its goal?
The student council cannot sell a part of a tin of popcorn. Selling 57 tins will not be enough for the student council to reach its goal of $400, because 57 × 7 = 399. Therefore, the council needs to sell 58 tins of popcorn.
The student council needs to sell 58 tins of popcorn to reach its goal.
b. Leo says that the student council needs to sell 57 tins of popcorn. He says, “The quotient is about 57.1. I rounded down to 57, because 57.1 is closer to 57 than to 58.” Do you agree with Leo? Explain.
I do not agree with Leo. If the student council sells 57 tins of popcorn, it does not reach its goal of $400. In this case, we have to round up to 58 tins of popcorn because the council cannot sell 0.1 of a tin of popcorn.
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PRACTICE
1. A physical education class has 50 students. The teacher wants to organize teams of 8 students for a game.
a. Can 50 students be divided into teams of 8 students? Explain?
b. Suggest a number of students that can be on each team. All of the teams must have the same number of students. Explain your answer.
2. There are 368 students going on a field trip. One school bus can transport 64 students. How many buses are needed to transport all 368 students? Explain.
3. A hiker has 93 ounces of granola. He wants to make 6-ounce bags of granola. What is the greatest number of 6-ounce bags the hiker can make? Explain.
4. A school club raises $1,230 in a fundraiser. The club wants to use all of the money to buy backpacks to donate to students. One backpack costs $12. What is the greatest number of backpacks the club can buy? Explain.
5. Miss Baker prepares science stations for her students. Each station needs exactly 250 milliliters of vinegar. What is the greatest number of stations Miss Baker can prepare with 1,890 milliliters of vinegar? Explain.
6. Miss Song has 300 notecards to share among her 32 students. She wants to give the same number of notecards to each student. What is the greatest number of notecards she can give to each student? Explain.
7. Yuna has 3 packs of gum. Each pack has 5 sticks of gum. Yuna wants to give all of the gum to 4 friends and share the gum equally among the 4 friends. How many sticks of gum should each friend get? Explain.
8. Ryan’s goal is to raise $250 by selling candles. He sells each candle for $13.
a. How many candles does Ryan need to sell to reach his goal?
b. Julie says that Ryan needs to sell 19 candles. She says, “The quotient is about 19.23. I rounded down to 19, because 19.23 is closer to 19 than to 20.” Do you agree with Julie? Explain.
Remember
For problems 9–11, add.
12. Estimate the quotient. Then calculate the quotient by using the standard algorithm. Express the quotient as a mixed number.
7,226 ÷ 23
13. Use <, >, or = to make each number sentence true.
a. 0.756 0.736
b. 0.110 0.11
c. 0.0883 0.83
d. 1.03 1.035
Decimal Division
Snail Vacation
Hey, it’s a beautiful day. You want to go to the beach?
Sounds perfect! I’m in!
TOPIC F
Say, how far is this beach? About 70 miles.
And we go how fast? About 0.03 mph.
About 2,300 hours.
in 3 months!
70 divided by 0.03 is roughly 2,300. This, by itself, is just a mathematical fact.
But include some units—make that “70 miles” divided by “0.03 miles per hour,” giving us “roughly 2,300 hours”—and suddenly you’ve got a fact about animals. Specifically, why snails so rarely make beach vacations: Although a car-driving human can do it in a day, a snail can barely make the round trip in half a year!
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Dividing a Decimal by a Whole Number
1. Fill in the blanks with five different digits to make a true number sentence.
Reasoning about Decimal Division
2. Does 168.3 ÷ 3 equal 561, 56.1, 5.61, or 0.561? Explain.
3. Use division facts to fill in the blanks and make each number sentence true.
a. 144 ÷ 12 =
b. 14.4 ÷ 12 =
c. 1.44 ÷ 12 =
d. 1.44 ÷ 120 =
e. 14.4 ÷ 1,200 =
4. Without dividing, write three division expressions that are each equivalent to 42.68 ÷ 9.
5. Tyler determines that 82.4 ÷ 8 equals 10.3 because 80 ÷ 8 = 10 and 2.4 ÷ 8 = 0.3 and 10 + 0.3 = 10.3. Is Tyler’s reasoning correct? Explain.
Using the Standard Algorithm with a Decimal Dividend
For problems 6–9, estimate the quotient. Then use the standard algorithm to divide.
6. 61.65 ÷ 5
Estimate:
7. 316.2 ÷ 12
Estimate:
8. 15.17 ÷ 37
Estimate:
9. 0.294 ÷ 14
Estimate:
Name
Divide by using the standard algorithm.
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Dividing a Decimal by a Whole Number
In this lesson, we
• estimated quotients by using place value reasoning.
• divided a decimal by a multi-digit whole number by using the standard division algorithm.
Examples
For problems 1 and 2, estimate the quotient. Then use the standard algorithm to divide.
1. 582.61 ÷ 41
Estimate: 15
14.21
To estimate the quotient, first round 582.61 to 600 and 41 to 40. 60 tens ÷ 4 tens = 15
Therefore, 15 is a reasonable estimate for 582.61 ÷ 41.
Notice that the decimal point in the dividend and the quotient line up because the dividend and the quotient line up by place value.
Estimating a quotient before dividing can help confirm whether an answer is reasonable.
2. 0.645 ÷ 25
Estimate: 0.03
0.0258
3. Notebooks cost $4 each. What is the greatest number of notebooks Tyler can buy with $50.50?
Divide 50.5 by 4 to determine the greatest number of $4 notebooks Tyler can buy with $50.50
The quotient is 12.625. Tyler has enough money to buy 12 notebooks. He does not have enough money to buy 13 notebooks because 13 × 4 = 52
Tyler can buy 12 notebooks.
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PRACTICE
1. Use estimation to evaluate 8,627.5 ÷ 35
A. 2,465
B. 246.5
C. 24.65
D. 2.465
2. Without dividing, use powers of 10 to write three division expressions that are each equivalent to 35.78 ÷ 6.
3. Which expressions are equivalent to 486 ÷ 30? Choose all that apply.
A. 486 ÷ 10 ÷ 3
B. 4,860 ÷ 3,000
C. 48.6 ÷ 3
D. 4.86 ÷ 0.03
E. 0.0486 ÷ 0.003
For problems 4–7, calculate the quotient.
8. Toby shops for new shirts. The shirts cost $8 each. What is the greatest number of shirts Toby can buy with $107.60?
9. Adesh has 4.65 pounds of dried fruit. He puts the dried fruit in 5 bags. He puts an equal amount of dried fruit into each bag. How many pounds of dried fruit does Adesh put in each bag?
Remember
For problems 10–13, subtract.
14. Divide. Express the quotient as a decimal. Round to the nearest thousandth if needed.
÷ 14
15. Use <, >, or = to make each number sentence true. a.
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Dividing a Decimal by a Decimal Greater Than 1
Decimal Divisors
For problems 1–3, write the division expression as the division of two whole numbers. Then divide. 1. 6 ÷ 1.2 2. 4.5 ÷ 1.5 3. 5.25 ÷ 2.5
Dividing Decimals with the Standard Algorithm
For problems 4–7, write the decimal division expression as an equivalent expression with a whole-number divisor. Then estimate the quotient and use the standard algorithm to divide.
4. 39.2 ÷ 4.9
Equivalent division expression:
Estimate:
5. 34.84 ÷ 6.7
Equivalent division expression:
Estimate:
6. 11.61 ÷ 8.6
Equivalent division expression:
Estimate:
7. 0.3808 ÷ 1.12
Equivalent division expression:
Estimate:
Decimal Division Error Analysis
8. To calculate 3.69 ÷ 1.2, Lacy writes the division expression as 369 ÷ 12. Explain Lacy’s mistake.
9. To calculate 45.8 ÷ 4.16, Leo sets up his division work as follows. Explain Leo’s mistake.
416 4 5.8
10. Jada calculates 8.65 ÷ 2.5 as 34.6 by doing the following work. Explain Jada’s mistake.
11. Toby calculates
÷ 2.4 as
by doing the following work. Explain Toby’s mistake.
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Divide by using the standard algorithm.
RECAP
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Dividing a Decimal by a Decimal Greater Than 1
In this lesson, we
• divided a decimal by a decimal greater than 1 by using the standard algorithm.
Examples
1. Jada says that she can find 64.28 ÷ 2.5 by calculating 642.8 ÷ 25. Is Jada correct? Explain.
Yes. Jada multiplies both the dividend and the divisor of 64.28 ÷ 2.5 by 10 to create the equivalent division expression 642.8 ÷ 25
Multiplying the dividend and divisor by the same value does not change the quotient.
For problems 2–4, write the decimal division expression as an equivalent expression with a whole-number divisor. Then use the standard algorithm to divide.
2. 149.1 ÷ 3.55
Equivalent expression: 14,910 ÷ 355
Multiply both the dividend and the divisor by 100 to create an equivalent division expression with a whole-number divisor. The expression 14,910 ÷ 355 is equivalent to 149.1 ÷ 3.55.
3. 17.68 ÷ 3.4
Equivalent expression: 176.8 ÷ 34
Multiply the dividend and the divisor by the same power of 10 to create an equivalent division expression with a whole-number divisor. Notice that it is not necessary to have a whole-number dividend.
5.2
4. 826.8 ÷ 19.5
The remainder is not 0 when subtracting 390 from 468 To continue dividing, put a decimal point to the right of the 8 in the ones place in the dividend so that a 0 can be written in the tenths place.
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PRACTICE
1. Which expressions are equivalent to 634.7 ÷ 1.23? Choose all that apply.
A. 6,347 ÷ 12.3
B. 63,470 ÷ 123
C. 634.7 ÷ 12.3
D. 634.7 ÷ 123
E. 63.47 ÷ 12.3
F. 63.47 ÷ 123
2. Jada says that she can find 38.4 ÷ 1.6 by calculating 384 ÷ 16. Is Jada correct? Explain.
16. A chicken farmer collects eggs and puts them in cartons. He collects 384 eggs and puts 12 eggs in each carton. How many cartons does the farmer fill?
17. Mr. Perez plans a pizza party for his class. He wants to know how many pizzas to order for 30 students. There are 8 slices per pizza. What is the least number of pizzas he can order so that each student gets 3 slices?
A. 8 pizzas
B. 10 pizzas
C. 11 pizzas
D. 12 pizzas
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Dividing a Decimal by a Decimal Less Than 1
Decimal Division with the Standard Algorithm
For problems 1–3, write the decimal division expression as an equivalent expression with a whole-number divisor. Then estimate the quotient and use the standard algorithm to divide. Round answers to the nearest hundredth if needed.
1. 1.472 ÷ 0.32
Equivalent division expression:
Estimate:
2. 4.714 ÷ 0.021
Equivalent division expression:
Estimate:
3. 0.128 ÷ 0.15
Equivalent division expression:
Estimate:
Decimal Division Stations
Directions: Work with your partner to complete as many stations as you can in the time allotted. After you complete a station, check your solution with your teacher before moving to a new station. You may complete the stations in any order. Justify your solutions.
Station 1
Station 2
Station 3
Station 4
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TICKET
Divide by using the standard algorithm.
RECAP
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Dividing a Decimal by a Decimal Less Than 1
In this lesson, we
• divided a decimal by a decimal less than 1 by using the standard algorithm.
• solved real-world problems by dividing a decimal by a decimal.
Examples
1. Which division expressions are equivalent to 5.2 ÷ 0.045? Choose all that apply.
A. 52 ÷ 0.45
B. 252 ÷ 0.45
C. 520 ÷ 4.5
D. 0.52 ÷ 0.045
E. 5,200 ÷ 45
2. Divide.
Multiply both the dividend and the divisor by 10 to create this equivalent expression.
Multiply both the dividend and the divisor by 100 to create this equivalent expression.
Multiply both the dividend and the divisor by 1,000 to create this equivalent expression.
Multiply both the dividend and the divisor by 100 to create an equivalent expression with no decimal point in the divisor.
3. Divide. Round to the nearest hundredth if needed.
÷ 0.54
÷
= 13.6 ÷ 54
To round to the nearest hundredth, first calculate the quotient to the thousandths place.
0.251 ≈ 0.25
The decimal point in the quotient lines up with the decimal point in the dividend because the quotient and dividend are lined up by place value.
4. A road is 12.8 miles long. There is a streetlight 0.8 miles from the start of the road. There is a streetlight every 0.8 miles after that. How many streetlights are along the road?
Divide 12.8 by 0.8 to determine the number of streetlights along the 12.8-mile road.
There are 16 streetlights along the road.
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PRACTICE
1. What is 0.21 ÷ 0.07 ? Explain.
2. Which division expressions are equivalent to 2.3 ÷ 0.056? Choose all that apply.
A. 23 ÷ 0.56
B. 2.3 ÷ 0.56
C. 230 ÷ 5.6
D. 0.23 ÷ 0.056
E. 2,300 ÷ 56
For problems 3–10, divide.
3. 16.42 ÷ 0.2
1.956 ÷ 0.03
5. 59.52 ÷ 0.62 6. 7.6 ÷ 0.8
7. 27.9 ÷ 0.45
8. 10.296 ÷ 0.78
11. A running club sets up water stations every 0.8 miles in a 6.2-mile race. How many water stations does the club set up ?
12. A pharmacist has 18.5 grams of a certain medication. The pharmacist fills capsules with the medication. Each capsule must hold 0.735 grams of the medication. How many capsules can the pharmacist fill?
Remember For problems 13–16, subtract.
17. Divide by using the standard algorithm. Express the quotient as a decimal. Round to the nearest hundredth if needed.
15.535 ÷ 12
18. Which values are greater than 409.806? Choose all that apply.
A. 409.186
B. 409.81
C. 490.806
D. 409.086
E. 409.805
Grade 6, Module 2, Topic F, Lesson 24
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Living on Mars
Life on Mars
1. Consider the following information.
• The closest distance from Mars to Earth is about 34.8 million miles.
• The farthest distance from Mars to Earth is about 250 million miles.
• The speed of light is about 186,000 miles per second.
a. How many minutes will it take your message to get from Mars to Earth when the two planets are as close as possible? Round your answer to the nearest hundredth.
b. How many minutes will it take your message to get from Mars to Earth when the two planets are as far apart as possible? Round your answer to the nearest hundredth.
c. Suppose Mars is at its farthest distance from Earth and you send a message to a person on Earth. About how many minutes will it take to receive a reply if the person responds immediately?
d. Use your answer from part (c) to determine about how many messages you can send and receive in 12 hours.
For problems 2–5, consider the following situation and information.
You plan to stay on Mars and send a shuttle to Earth to gather food, water, and oxygen. It takes about 2 years for the space shuttle to go to Earth, gather supplies, and then come back to Mars.
• An astronaut breathes about 613.2 kilograms of oxygen in 2 years.
• An astronaut needs about 25.9 liters of water each week to survive.
• An astronaut eats about 1.27 pounds of food each meal.
2. How many kilograms of oxygen do you need for 1 day? Round your answer to the nearest hundredth.
3. There are 52 weeks in 1 year. What is the least amount of water in liters you need to survive on Mars for 2 years?
4. Assume you eat exactly 3 meals each day. About how many pounds of food do you need to survive on Mars for 2 years?
5. Consider the additional information.
• 1 liter of water is 1 kilogram of water.
• 1 pound is approximately 0.45 kilograms.
a. What is the total mass in kilograms of a 2-year supply of food, water, and oxygen? Round your answer to the nearest hundredth.
b. Suppose it costs about $20,000 to send 1 kilogram of mass into space. Use your answer from part (a) to determine about how much it will cost to send a 2-year supply of food, water, and oxygen for 1 person into space.
Name Date
TICKET
The International Space Station (ISS) orbits Earth at a speed of about 4.76 miles per second. Assume that it orbits at a constant speed. About how many minutes does it take the ISS to travel 1,000 miles? Round your answer to the nearest tenth.
RECAP
Name Date
Living on Mars
In this lesson, we • solved real-world problems by using decimal operations.
Examples
1. Mars has less gravity than Earth. So the height of a person’s jump on Earth is about 0.38 times the height of what that person’s jump would be on Mars. The height of Ryan’s jump on Earth is 13.5 inches. What would the height of Ryan’s jump be on Mars? Round your answer to the nearest tenth.
The height of Ryan’s jump on Mars would be about 35.5 inches.
The height of a jump on Earth is about 0.38 times the height of a jump on Mars. Therefore, 13.5 = 0.38 × ?. To find the unknown factor, divide 13.5 by 0.38.
2. One astronaut breathes about 613.2 kilograms of oxygen in 2 years.
a. What is the total amount of oxygen needed for 3 astronauts to survive on a space station for 2 years? × 3 61 3.2 1, 83 9.6
Multiply the amount of oxygen 1 astronaut breathes in 2 years by 3 to find the total amount of oxygen needed for 3 astronauts.
About 1,839.6 kilograms of oxygen are needed for 3 astronauts to survive on a space station for 2 years.
b. What amount of oxygen is needed per day for 1 astronaut to survive on the space station for 2 years?
365 × 2 = 730
Multiply the number of days in 1 year by 2 to find the number of days in 2 years.
Divide the amount of oxygen needed for 2 years by the number of days in 2 years to determine the amount of oxygen 1 astronaut needs per day.
About 0.84 kilograms of oxygen are needed per day for 1 astronaut to survive on the space station for 2 years.
c. An air recycling system on the space station produces only about 2 kilograms of oxygen per day. More oxygen is delivered to the space station. What amount of oxygen needs to be delivered to the space station for 3 astronauts to survive for 2 years?
2 × 365 × 2 = 1,460
The recycling system produces a total of 1,460 kilograms of oxygen over 2 years.
About 379.6 kilograms of oxygen need to be delivered to the space station for 3 astronauts to survive for 2 years.
Name Date
PRACTICE
1. At takeoff, a rocket has a mass of 2.3 million kilograms. After using enough fuel to reach a height of 61 kilometers, its mass is 0.131 million kilograms. How much fuel does the rocket use to reach a height of 61 kilometers?
2. The number of days it takes Earth to orbit the sun is about 0.531 times as many days as it takes Mars to orbit the sun. It takes Earth 365 days to orbit the sun. How many days does it take Mars to orbit the sun? Round your answer to the nearest day.
3. The gravitational pull on Mars is about 38% of the gravitational pull on Earth. Therefore, an object’s weight on Mars is about 38% of the object’s weight on Earth. A bag of apples weighs 3 pounds on Earth. About how many pounds would the bag of apples weigh on Mars?
4. The distance from the sun to Earth is 92.96 million miles. Light travels at about 11 million miles per minute. About how many minutes does it take for light to get from the sun to Earth? Round your answer to the nearest hundredth.
5. The closest distance from Earth to the moon is about 225 thousand miles. The closest distance from Earth to Mars is about 34 million miles. About how many times as far as the closest distance from Earth to the moon is the closest distance from Earth to Mars? Round your answer to the nearest tenth.
Remember
For problems 6–9, add or subtract.
10. Divide by using the standard algorithm. Round your answer to the nearest thousandth.
11. Use >, <, or = to make each number sentence true.
a. 1.238 1.5 b. 6.736 6 × 1 + 7 × 0.1
c. 7 ones + 8 tenths + 9 hundredths 7.890
Mixed Practice 1
1. The area of the rectangle shown is 91 square yards. ? 7 yds
a. What is the length of the rectangle?
b. What is the perimeter of the rectangle?
2. The table shows the number of cups of flour in eight different bread recipes.
a. Use the data in the table and the number line shown to create a line plot. Mark an × on the line plot for the number of cups of flour in each recipe.
Amount of Flour (cups)
b. How many recipes need 1 1 2 cups of flour?
c. How many recipes need less than 1 1 2 cups of flour?
3. A figure is made by stacking 1-inch cubes in three layers as shown. What is the volume of the figure in cubic inches?
4. 10 of the 25 cubes in a tower are yellow. Eddie states, “The ratio of the number of yellow cubes to the total number of cubes is 10 : 25.” Sana states, “The ratio of the number of yellow cubes to the total number of cubes is 25 : 10.” Which student is correct? Explain your reasoning.
5. At Food Mart, the price of a 3-pound bag of potatoes is $3.75. At Save More, the price of a 5-pound bag of potatoes is $6.50. Which store charges less per pound for potatoes?
6. Blake picks apples and then puts them in bins. Blake works at a constant rate and fills 12 bins with apples every 8 hours.
a. How many bins does Blake fill with apples in 30 hours?
b. Each bin holds 1,000 pounds of apples. How many hours does it take Blake to fill bins with 15,000 pounds of apples?
7. At a school, 180 students eat sandwiches for lunch. This number is 45% of the total number of students at the school. What is the total number of students at the school?
A. 81
B. 99
C. 327
D. 400
For problems 8–10, write a numerical expression to represent the statement.
8. Subtract 1 2 from 3 4 . Then multiply by 3.
9. Add 8 and 6. Then divide by 2.
10. Subtract 8 from the product of 7 and 4.
Mixed Practice 2
1. Complete the table.
Name in Words Drawing Point D
Line segment OE
2. Name the point that represents each ordered pair.
(2, 6) (6, 2) (2, 5) (3, 6)
3. Determine whether each statement is true or false.
a. The value of 9 in the decimal 2.19 is 1 10 the value of 9 in 49.45
b. The value of 9 in the decimal 3.95 is 1 100 the value of 9 in 96.31.
c. The value of 9 in the decimal 9.82 is 100 times the value of 9 in 8.193
4. Which expressions have a value of 105 ? Choose all that apply.
A. 10 × 5
B. 10 + 5
C. 10,000
D. 100,000
E. 10 × 10 × 10 × 10 × 10
F. 10 + 10 + 10 + 10 + 10
5. When rounded to the nearest tenth, which of the following numbers become 12.4? Choose all that apply.
A. 12.45
B. 12.35
C. 12.445
D. 12.349
E. 12.371
6. Use >, <, or = to compare the numbers.
a. 1.8 1.08
b. 3.6 3.60
c. 0.48 0.5
7. Kelly wants to make fruit punch in a 5-gallon container. The punch recipe calls for 1 2 3 gallons of pineapple juice, 1 3 4 gallons of cranberry juice, and 2 gallons of orange juice. Can Kelly’s 5-gallon container hold the total amount of the fruit punch? Explain.
8. Write a fraction on the line to make a true statement.
5 7 × 3 4 < 5 7 ×
1. 0.2 + 0.4
2. 5.21 + 3.41
2 + 1 2. 0.2 + 0.1 3. 0.2 + 0.3
4. 0.2 + 0.4 5. 0.6 + 0.2
6. 0.6 + 0.3 7. 0.4 + 0.3
8. 0.5 + 0.3
0.5 + 0.4
Number Correct:
5.76 + 4.41
5.84 + 6.41
5.85 + 6.49
3 + 1
2. 0.3 + 0.1 3. 0.3 + 0.3
4. 0.3 + 0.4
5. 0.4 + 0.2
6. 0.4 + 0.4 7. 0.5 + 0.3 8. 0.5 + 0.4
2.5 + 3.3
3.5 + 3.3
3.5 + 4.3 18. 3.6 + 4.3 19. 3.7 + 4.3
3.8 + 4.3
Number Correct: Improvement:
6.84 + 6.41
6.85 + 6.49
Circle every number that is a factor of the given number.
ACircle every number that is a factor of the given number.
Number Correct:
Circle every number that is a factor of the given number.
Number Correct: Improvement:
Circle every number that is a multiple of the given number.
Circle every number that is a multiple of the given number.
Number Correct:
Circle every number that is a multiple of the given number.
Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module.
Cover, Gustave Caillebotte (1848–1894), Paris Street; Rainy Day, 1877. Oil on canvas, 212.2 x 276.2 cm (83 1/2 x 108 3/4 in.). Charles H. and Mary F. S. Worcester Collection. (1964.336). The Art Institute of Chicago, Chicago, IL, USA Photo Credit: The Art Institute of Chicago/Art Resource, NY; All other images are the property of Great Minds.
For a complete list of credits, visit http://eurmath.link/media-credits
Acknowledgments
Agnes P. Bannigan, Erik Brandon, Joseph T. Brennan, Beth Brown, Amanda H. Carter, Mary Christensen-Cooper, David Choukalas, Cheri DeBusk, Jill Diniz, Mary Drayer, Dane Ehlert, Scott Farrar, Kelli Ferko, Levi Fletcher, Krysta Gibbs, Winnie Gilbert, Julie Grove, Marvin E. Harrell, Stefanie Hassan, Robert Hollister, Rachel Hylton, Travis Jones, Raena King, Emily Koesters, Liz Krisher, Robin Kubasiak, Connie Laughlin, Alonso Llerena, Gabrielle Mathiesen, Maureen McNamara Jones, Bruce Myers, Marya Myers, Kati O’Neill, Ben Orlin, Darion Pack, Brian Petras, DesLey V. Plaisance, Lora Podgorny, Janae Pritchett, Bonnie Sanders, Deborah Schluben, Andrew Senkowski, Erika Silva, Ashley Spencer, Hester Sofranko, Danielle Stantoznik, Tara Stewart, Heidi Strate, James Tanton, Carla Van Winkle, Jessica Vialva, Caroline Yang
Trevor Barnes, Brianna Bemel, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe
Share Your Thinking
Talking Tool
I know . . . . I did it this way because . . . . The answer is because . . . . My drawing shows . . . .
Agree or Disagree
Ask for Reasoning
I agree because . . . . That is true because . . . . I disagree because . . . . That is not true because . . . .
Do you agree or disagree with ? Why?
Why did you . . . ? Can you explain . . . ? What can we do first? How is related to ?
Say It Again
I heard you say . . . . said . . . .
Another way to say that is . . . . What does that mean?
Thinking Tool
When I solve a problem or work on a task, I ask myself
Before
Have I done something like this before?
What strategy will I use?
Do I need any tools?
During Is my strategy working?
Should I try something else?
Does this make sense?
After
What worked well?
What will I do differently next time?
At the end of each class, I ask myself
What did I learn?
What do I have a question about?
MATH IS EVERYWHERE
Do you want to compare how fast you and your friends can run?
Or estimate how many bees are in a hive?
Or calculate your batting average?
Math lies behind so many of life’s wonders, puzzles, and plans.
From ancient times to today, we have used math to construct pyramids, sail the seas, build skyscrapers—and even send spacecraft to Mars.
Fueled by your curiosity to understand the world, math will propel you down any path you choose.
Ready to get started?
Module 1
Ratios, Rates, and Percents
Module 2
Operations with Fractions and Multi-Digit Numbers
Module 3
Rational Numbers
Module 4
Expressions and One-Step Equations
Module 5
Area, Surface Area, and Volume
Module 6
Statistics
What does this painting have to do with math?
An intersection in Paris on a gray, rainy day is the subject of this atmospheric Impressionist painting. Gustave Caillebotte creates depth in this scene by using perspective and proportion in a variety of ways, including by placing large figures in the foreground and smaller ones in the distance. Imagine there is a coordinate grid on the building in the background. How might you determine the distance from the front of the building to the back by using the coordinate plane?
On the cover
Paris Street; Rainy Day, 1877
Gustave Caillebotte, French, 1848–1894 Oil on canvas
The Art Institute of Chicago, Chicago, IL, USA
Gustave Caillebotte (1848–1894). Paris Street; Rainy Day, 1877. Oil on canvas, 212.2 x 276.2 cm (83½ x 108¾ in). Charles H. and Mary F. S. Worcester Collection (1964.336). The Art Institute of Chicago, Chicago, IL, USA. Photo Credit: The Art Institute of Chicago/Art Resource, NY
