EM2NY_G4_M1_Apply_25A_121707_Updated 10.4

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A Story of Units® Fractional Units

APPLY ▸ Place Value Concepts for Addition and Subtraction

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Place Value Concepts for Addition and Subtraction

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Place Value Concepts for Multiplication and Division

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Multiplication and Division of Multi-Digit Numbers

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Foundations for Fraction Operations

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Angle Measurements and Plane Figures

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Place Value Concepts for Decimal Fractions

Demonstrate that a digit represents 10 times the value of what it represents in the place to its right.

Write numbers to 1,000,000 in unit form and expanded form by using place value structure.

Subtract by using the standard algorithm, decomposing larger units up to 3 times.

Subtract by using the standard algorithm, decomposing larger units multiple times.

Multiplication as Multiplicative Comparison

Dear Family,

Key Term times as many

In previous grades, your student learned to compare numbers and use addition or subtraction to describe how many more or how many less. Now, your student uses their prior knowledge of multiplication and division to compare numbers and describe their relationship as times as many. Your student explores a variety of patterns and models such as blocks, tape diagrams, and money, to explain what it means to say times as many. They find an unknown quantity when two quantities are compared by writing multiplication and division equations.

dollars dimes pennies

¢

= 3 × 5

Comparison

Amy’s Tower Gabe’s Tower

The comparison phrase times as many can be adjusted to match different contexts.

Figure E Figure F

Measurement weight liquid volume capacity height length width distance

tower is 3 times as tall as Gabe’s tower. 15

times as heavy times as much times as much times as tall times as long times as wide times as far

At-Home Activity

Use Comparison Language

Help your student practice describing multiplication and division equations by using times as many language. You may find it useful to use the comparison language provided in the chart of measurements.

• Use two different-size containers that can hold water, such as a small measuring cup and a large water glass. Have your student fill the larger container by using water from the smaller container. Discuss the amount of liquid each container holds. Help your student by asking a question such as, “What can we say about how much more liquid the larger container holds?” Then say, “We can say that the larger container holds about times as much liquid as the smaller container.” Repeat with other different-size containers like a pot and a bowl, or a pitcher and a cup.

• Get two dry spaghetti noodles. Break off a small piece, about 1 inch, of the first noodle. Leave the second noodle whole. Use the piece of noodle to measure the whole noodle from one end to the other by moving the piece along the whole noodle with no gaps or overlaps. Then compare the length of the piece to the whole noodle by using times as long language. Use a comparison statement such as, “The whole noodle is about times as long as the piece of noodle.” Encourage your student to describe about how many times as long one noodle is than the other.

• Consider starting conversations with your student such as, “I noticed it took you 2 minutes to brush your teeth and 10 minutes to eat your breakfast. How many times as long did it take you to eat your breakfast than it did to brush your teeth?”

I can add to see that the total of 6 units of 8 is 48.

The tape diagram shows that 48 is 6 times as many as 8.

2. Pablo eats 7 grapes. Luke eats 3 times as many grapes as Pablo. How many grapes does Luke eat?

3 × 7 = 21

Luke eats 21 grapes.

I can draw a tape diagram to represent the problem.

I draw 1 unit of 7 for Pablo.

I draw 3 units of 7 for Luke because he eats 3  times as many grapes as Pablo.

I can multiply to find the total number of grapes Luke eats.

REMEMBER

Add or subtract.

3. 714 − 235 = 479

I can write the problem vertically.

I start by making my units ready to subtract. I can rename 1 ten as 10 ones. I add the 10 ones to 4 ones.

I also have to rename the hundreds, 1 hundred as 10 tens.

Now I am ready to subtract.

4. 252 + 388 =  640

To find 252 + 388, I can decompose each number into hundreds, tens, and ones.

252 + 388

200 50 2 300 80 8 I add like units.

200 + 300 = 500

50 + 80 = 130

2 + 8 = 10

Then I add.

500 + 130 + 10 = 640

1.

2.

Draw a tape diagram to represent the statement. Then complete the equation.

3. 21 is 3 times as many as 7

4. Adam reads 6 pages in his book. Gabe reads 4 times as many pages in his book as Adam. How many pages does Gabe read?

REMEMBER

5. Add or subtract.

a. 613 164 =

b. 497 + 213 =

Use the tape diagram to complete the statement and equations.

I need to find how many units of 7 equals 63

The value of the unknown is 9 because 63

=

. This tells me 7 is repeated 9 times. I can find the unknown with multiplication too.

Draw a tape diagram to represent the statement. Then write an equation to find the unknown and complete the statement.

2. 32 is 4 times as many as 8 . ?

I know the unknown unit is repeated 4 times to equal 32 .

I draw a unit with a question mark because I do not know what number is being repeated. Then I draw a second tape to show the single unit repeated 4 times. The total is 32 .

The value of the unknown is 8 because 32 ÷ 4 = 8 This tells me 8 is repeated 4 times.

?

8 8 8 8 32

REMEMBER 3. Which shape has an area of 10 square units? represents 1 square unit.

Circle the correct answer.

A.

Same-size square tiles are placed side by side with no gaps to measure area. The area is equal to the number of square units.

Shape B is covered with 10 same-size square tiles.

B.

Use the tape diagram to complete the statement and equations.

Draw a tape diagram to represent each statement. Then write an equation to find the unknown and complete the statement.

REMEMBER

6. Which rectangles have an area of 8 square units? represents 1 square unit. Circle the two correct answers.

A.

B.

C.

D.

The paintbrush is 4 times as heavy as the toothbrush.

I see that the paintbrush weighs 40 grams. The toothbrush weighs 10 grams.

The paintbrush weighs more than the toothbrush. I can also say that the paintbrush is heavier than the toothbrush.

I recognize the relationship between the weight of the paintbrush and the weight of the toothbrush. I can write a multiplication equation to describe the relationship.

I say “times as heavy” when weight is measured.

Use the Read–Draw–Write process to solve the problem.

2. While playing softball, Robin’s team drinks 8 liters of water, and Carla’s team drinks 24 liters of water. How many times as much water does Carla’s team drink as Robin’s team?

24 ÷ 8 = 3

Carla’s team drinks 3 times as much water as Robin’s team.

I read the problem. I read again. As I reread, I think about what I can draw.

I draw a tape diagram. I draw one tape and label it 8 to represent the 8 liters of water that Robin’s team drinks.

I draw another tape and label it 24 to represent the 24 liters of water that Carla’s team drinks.

8

8

24 ...

? times as much

I need to find how many eights are in 24. I divide 24 by 8 to get 3

REMEMBER 3. Find the area of the shape. Each represents 1 square inch.

Area: 10 square inches

I can count each square as 1 square inch. The shape is made up of 10 squares, so it has an area of 10 square inches.

Record each measurement. Then complete the statement and the equation.

The plum is 6 times as as the cherry.  g = 6 ×   g

The glue stick is 4 times as as the eraser.  cm = 4 ×   cm

Use the Read–Draw–Write process to solve each problem.

3. Deepa and David pick lemons. David picks 3 times as many lemons as Deepa. David picks 27 lemons. How many lemons does Deepa pick?

4. Amy has a yellow dog and a brown dog. The yellow dog weighs 28 pounds, and the brown dog weighs 7 pounds. How many times as heavy is the yellow dog as the brown dog?

REMEMBER

5. Find the area of the shape. Each represents 1 square inch.

Area:

Name

Date

Complete the chart to show how to make a new unit. Then complete the statement and equations.

1. dollars dimes pennies

× 10

1 dime is worth 10 times as much as 1 penny .

1 dime = 10 × 1  penny

10¢ = 10 × 1 ¢

I know that with pennies and dimes 10 of the smaller unit has the same value as 1 of the larger unit.

When I have 10 pennies, I can make 1 dime.

So 1 dime has the same value as 10 pennies.

2. Oka says that because 1 dollar is worth 10 times as much as 1 dime, 4 dollars must be worth 10 times as much as 4 dimes. Do you agree with Oka? Why?

Yes, I agree with Oka because 10 = 10 × 1 and 40 = 10 × 4

1 dollar is worth as much as 10 dimes. So 4 dollars are worth as much as 40 dimes.

REMEMBER 3. A rectangular patio has a width of 5 feet and a length of 9 feet. What is the area of the patio?

9 ft

The patio is in the shape of a rectangle. I can multiply the length and width of the patio to find its area.

5 ft

5 × 9 = 45

The area of the patio is 45 square feet.

Name

Date

Complete the chart to show how to make a new unit. Then complete the statement and equations.

1.

dollars dimes pennies ×

1 dime is worth 10 times as much as 1 .

1 dime = 10 × 1

10¢ = 10 ×  ¢

2.

dollars dimes pennies ×

1 dollar is worth 10 times as much as 1 .

1 dollar = 10 × 1

$1 = 10 × ¢

3. Mia says that because 1 dime is worth 10 times as much as 1 penny, 6 dimes must be worth 10 times as much as 6 pennies. Do you agree with Mia? Why?

4. David says that because 1 dollar is worth 10 times as much as 1 dime, 5 dollars must be worth 5 times as much as 5 dimes. Do you agree with David? Why?

REMEMBER

5. A rectangular garden has a length of 7 feet and a width of 6 feet. What is the area of the garden?

FAMILY MATH

Place Value and Comparison within 1,000,000

Dear Family,

In previous grades, your student learned about place value for numbers up to 10,000. Using that knowledge, your student explores counting with large sums of money as a context for understanding large numbers. They learn to read, write, and compare numbers up to 1,000,000. They also connect recent learning about times as much to realize that a digit represents 10 times the value of the same digit in the place to its right. A strong sense of place value understanding helps your student add, subtract, multiply, and divide with large numbers later this year.

Key Terms

A place value chart organizes numbers and shows the relationships between place value units.

56,348

50,000 + 6,000 + 300 + 40 + 8

fifty-six thousand, three hundred forty-eight

56 thousands 3 hundreds 4 tens 8 ones

Writing numbers in various forms, such as in standard form, expanded form, word form, and unit form, enables flexible thinking.

thousands hundreds tens ones

× 10

10 times as much as 3 tens is 3 hundreds.

10 × 30 = 300

3 hundreds is 10 times as much as 3 tens.

300 = 10 × 30

At-Home Activities

Comparing Money

Encourage your student to practice multiplying and dividing by 10 by using pennies, dimes, and dollars. Ask your student how many cents are in a penny (1¢), a dime (10¢), and a dollar (100¢). Talk about how many pennies equal the value of a dime and how many dimes equal the value of a dollar. Ask your student to say a multiplication sentence for each relationship such as, “10 times 1 cent is 10 cents, and 10 times 10 cents is 100 cents.” Then ask questions about larger amounts.

• “How many cents is 7 dimes worth?” (70 cents)

• “What is 10 times as much as 7 dimes?” (7 dollars or 70 dimes)

• “70 cents is 10 times as much as how many cents?” (7 cents)

• “7 dollars is worth 700 cents. How many cents is 10 times as much as 7 dollars?” (7,000 cents)

Comparing Large Numbers

Write two large numbers, such as 38,720 and 36,954. Ask your student to say which number is greater and which number is less and explain how they know. Encourage your student to draw a place value chart to help them. For an added challenge, ask your student to write a number greater than one of the numbers, write a number less than the other number, and write a number whose value is between both numbers.

1  hundred thousand = 10 ten thousands

Ten thousand is a place value unit composed of 10 thousands.

Hundred thousand is a place value unit composed of 10 ten thousands.

I know I can bundle 10 of a smaller unit to make 1 of the next larger unit.

There are 10 ten thousands. I look at a place value chart.

I find ten thousands. The next larger unit is hundred thousands.

I can bundle and rename 10 ten thousands as 1 hundred thousand.

Use the place value disks to help you complete the equation.

Million is a place value unit composed of 10 hundred thousands.

I know I can bundle 10 of a smaller unit to make 1 of the next larger unit.

There are 10 hundred thousands. I look at a place value chart.

I find hundred thousands. The next larger unit is millions.

I can bundle and rename 10 hundred thousands as 1 million.

1  million = 10 hundred thousands

REMEMBER

3. Shade the rectangle to break it into 2 smaller rectangles. Then complete the equations to find the total area of the large rectangle. Each square represents 1 square unit.

Area: 65 square units

I can use the break apart and distribute strategy to find the area.

I decompose 13 into 10 and 3.

I shade a 5 by 10 rectangle and label the side lengths of the shaded and unshaded parts of the large rectangle. 10 3 5

I find the area by multiplying each part of 13 by 5 and then adding the products.

1 = 10 ten thousands

1 = 10 hundred thousands

REMEMBER

5. Shade the rectangle to break it into 2 smaller rectangles. Then complete the equations to find the total area of the large rectangle. Each square represents 1 square unit.

Name Date

Use the place value chart to complete the statements and equations.

1. thousands hundreds tens ones ×10

10 times as much as 2 tens is 2 hundreds

10 × 20 = 200

2 hundreds is 10 times as much as 2 tens .

200 = 10 × 20

I need to find 10 times as much as 2 tens.

When I multiply by 10, the units shift one place to the left on the place value chart.

Now there are 2 units in the hundreds place. So 10 times as much as 2 tens is 2 hundreds.

When I divide by 10, the units shift one place to the right on the place value chart.

Now there are 3 units in the hundreds place. So 3,000 divided by 10 is equal to 300

Use the place value chart to complete the equation.

2. thousands ten thousands hundreds tens ones

10 3,000 ÷ 10 = 300

REMEMBER

3. Find the area of the shape. Show your strategy.

7 in

3 in

2 in

3 in

2 in

Area: 33 square inches

6 in

6 × 7 = 42

3 × 3 = 9

42 9 = 33

I can find the areas of the large and small rectangles. Then I can subtract the area of the smaller rectangle from the area of the larger rectangle.

To find the area of the smaller rectangle, I need to know the lengths of the sides. The opposite sides of rectangles have the same lengths.

I know the large rectangle is 7 inches by 6 inches.

I know the small rectangle is 3 inches by 3 inches.

6 in

2 in

7 in

3 in

3 in

2 in

Name Date

Use the place value chart to complete the statements and equations.

1. thousands hundreds tens ones ×10

2. thousands hundreds tens ones ×10

10 times as much as 6 tens is 6 .

10 × 60 =

6 hundreds is 10 times as much as 6 .

600 = 10 ×

Use the place value chart to complete the equation.

3.

10 times as much as 4 hundreds is 4 .

10 × 400 =

4 thousands is 10 times as much as 4

4,000 = 10 ×

÷ 10 thousands ten thousands hundreds tens ones

900 ÷ 10 =

4.

thousands ten thousands hundreds tens ones

÷ 10

7,000 ÷ 10 =

REMEMBER

5. Find the area of the shape. Show your strategy.

3 in

3 in

Area:

4 in

9 in

Count the number of place value disks in each column of the chart.

Write the number at the bottom of each column.

Then fill in the blanks to write the unit form of the number represented in the chart.

I count the number of disks in each column. There is 1 millions disk, so I write 1 at the bottom of the first column.

I write the other numbers of disks at the bottom of each column.

I express the number shown in the chart in unit form.

Unit form helps me know how many of each place value unit are in a number. It helps me use the correct names for each digit.

Fill in the blanks to write the number in expanded form in two ways.

I can write a number in expanded form by using addition. I write the value of each digit.

There is 1 hundred thousand. I write 100,000

There are 5 thousands. I write 5,000

There are 3 hundreds. I write 300

There are 4 tens. I write 40.

I do not need to include 0 ten thousands or 0 ones because they do not change the value of the number.

I can also write a number in expanded form by using multiplication and addition.

The value of each digit is represented by using multiplication. The parentheses help to show each place value unit.

I think about the place value units and complete each expression.

REMEMBER

a. How many feet of fencing outline the border of the patio? 40 feet

b. How many square feet of brick cover the patio? 57 square feet

To find the perimeter, or the border, of the patio, I add all the side lengths.

Area is the amount of flat space the patio takes up. To find the area, I think of the patio as two separate rectangles. I draw a line to show the rectangles. I multiply each rectangle’s length by its width. Then I add the two products.

3. Casey is building a brick patio with a fence around it. Use the picture to answer questions about the patio.

Count the number of place value disks in each column of the chart.

Write the number at the bottom of each column.

Then fill in the blanks to write the unit form of the number represented in the chart.

Use the numbers on the place value chart to complete the expanded form.

Expanded form: + + + +

Fill in the blanks to write the number in expanded form in two ways.

REMEMBER

4. Miss Diaz is making a garden with a fence around it. Use the picture to answer questions about the garden.

a. How many feet of fencing are needed to go around the garden? feet

b. How many square feet of ground does the garden have? square feet

1. 3141592

3,141,592

I know that the place value chart is organized into groups of three place value units. Those groups are called periods. A comma separates the digits between each period.

I start to the right of the digit in the ones place and count three places to the left.

I write a comma between 1 thousand and 5 hundreds.

Then I count three more places to the left of the first comma.

I write a comma between 3 millions and 1 hundred thousand.

2.

I can use a place value chart.

I see there are 4 ten thousands, 2 thousands, 6 tens, and 7 ones.

40,000 + 2,000 + 60 + 7 = 42,067

REMEMBER 4. Liz collects leaves in her backyard. The tally chart shows the number of leaves she collects in each color. Use the data in the tally chart to draw a scaled picture graph.

Liz’s Leaf Collection

Color Number of Leaves Green Yellow Red Brown Orange

In a scaled picture graph, each symbol represents more than 1 object or observation.

I make a table from the data in the tally chart.

Green Yellow Red Brown Orange 2 4 6 2 8

I create a scale. I think it is reasonable for a circle to represent 2 leaves because each number is a multiple of 2

Color

I give the graph a title. I label the left side of my graph with the colors of the leaves. I write the key I will use below the graph. Green

Yellow Red Brown Orange

Liz’s Leaf Collection

Each represents 2 leaves.

Now I can draw circles to represent the data.

I draw circles to represent the data. As I draw circles for each color, I count by twos until I represent the correct number of leaves because each circle represents 2 leaves.

Express the following numbers in standard form by using commas.

1. 9560

3. 308223

2. 36805

4. 34651047

Fill in the blank to make a true number sentence.

5. 10,000 + 5,000 + 300 + 8 =

6. 400,000 + + 900 + 20 + 6 = 430,926

7. 42 thousands + 8 hundreds + 4 ones = Complete the table.

REMEMBER

11. Mr. Davis surveys his students about their pets. The tally chart shows the results of the survey. Use the data in the table to draw a scaled picture graph.

Students’ Pets

Pet Number of Students

Dog

Cat

Fish

Rabbit

Horse

. represents Each

each number with digits on the place value chart. Then circle the greater number.

I write the digits of each number in the correct places in the place value chart.

I see that each number has a digit in the hundred thousands place. Both numbers have 5 hundred thousands.

I look at the next smallest place value unit, the ten thousands.

52 ten thousands is less than 53 ten thousands, so 521,645 is less than 534,904

Use >, =, or < to compare the numbers.

2. 5,622   <  5 thousands 8 hundreds 7 tens 5 ones

To compare 5,622 and 5 thousands 8 hundreds 7 tens 5 ones, I first write them in the same form. I can think about both numbers in unit form.

5 thousands 6 hundreds 2 tens 2 ones

5 thousands 8 hundreds 7 tens 5 ones

Both numbers have 5 thousands. The first number has 6  hundreds, and the second number has 8 hundreds.

6  hundreds < 8  hundreds, so 5,622 < 5  thousands 8 hundreds 7 tens 5 ones

Arrange the numbers from least to greatest.

I can write the numbers in a place value chart and compare the digits.

8,240 has the smallest number of thousands, 8 thousands, so it is the smallest number.

11,554 has the largest number of thousands, 11 thousands, so it is the largest number.

9,232 and 9,338 both have 9 thousands, so I compare the hundreds place.

9,232 has 2 hundreds, and 9,338 has 3 hundreds. So, 9,232 < 9,338

From least to greatest, the numbers are 8,240, 9,232 , 9,338, and 11,554

REMEMBER 4. Measure the lengths of the lollipops to the nearest quarter inch. Record the data in the table. Create a line plot to represent your data.

I measure each lollipop. I line up the end of my ruler with the end of the lollipop. I measure to the nearest quarter inch.

I organize my data in the table. I look at the data in the table to help me decide the scale on the line plot.

I write a title for the line plot, and I write a label to show that the numbers represent lengths in inches. The smallest measurement is 2 inches, and the largest measurement is 33 4 inches. I make intervals of 1 4 inch on the line plot.

I make an X above the line plot for each piece of data.

Represent each number with digits on the place value chart. Then circle the greater number.

Use >, =, or < to compare the numbers.

Arrange the numbers from least to greatest.

REMEMBER

11. Cut out the ruler. Measure the lengths of the crackers to the nearest quarter inch. Record the data in the table. Create a line plot to represent your data.

Lengths of Crackers

Rounding Multi-Digit Whole Numbers

Dear Family,

Your student is learning to round numbers to the nearest thousand, ten thousand, and hundred thousand. First, they name numbers in unit form based on the place value to which they are rounding. Then they use the vertical number line to show their understanding. Labeling the number line with two benchmark numbers and the number that is halfway between the two benchmark numbers can help your student to identify the closest benchmark. Your student also decides when it may be helpful to round to the nearest or next benchmark. They understand that some situations require an estimate greater than the actual amount, such as when estimating a cost.

190,000 = 19 ten thousands

186,045

185,000 = 18 ten thousands 5 thousands

180,000 = 18 ten thousands

186,045 ≈ 190,000

186,045 rounded to the nearest ten thousand on the vertical number line.

Liz has $70. She wants to buy a book bag that costs $34, a book that costs $19, and a calculator that costs $24

Liz rounds to the nearest ten.

Liz rounds to the next ten.

30 + 20 + 20 = 70 40 + 20 + 30 = 90

To make sure she has enough money, Liz decides to use the estimate that rounds to the next ten instead of the nearest ten.

At-Home Activities

Rounding Numbers

Get six pieces of paper. Label the first piece with the first digit from your phone number, the second piece with the second digit from your phone number, and so on until all pieces of paper have been used. Mix up the papers and place facedown in a row. Then have your student turn over all 6 pieces of paper to form a 6-digit number. Invite your student to help you round the number to the nearest thousand, ten thousand, and hundred thousand. Have them explain their reasoning as they round. Repeat with a different 6-digit number.

Estimate Costs

Practice using estimation by planning an imaginary shopping trip. Let your student choose the reason for the trip and the items to purchase. For example, your student may decide they are shopping for a party or a gift. Use a store flyer or a store’s online website and ask your student to help you think about how much money is needed to buy some items. Decide on a budget, such as $100. Make a shopping list and label items with whole-dollar amounts. Begin with 2 items, rounding to the nearest ten dollars. Ask questions to discuss the estimates.

• “What is the total cost when we estimate by rounding each price to the nearest ten dollars?”

• “Do we have enough money to buy the items if we use that estimate?”

• “Is there another way to estimate the total to make sure we have enough money?”

Name Date

1. Represent 2,137 on the place value chart to match the given unit form.

2 thousands 13 tens 7 ones

thousands tens ones hundreds

I draw to represent 2 in the thousands column, 13 in the tens column, and 7 in the ones column.

2 thousands 13 tens 7 ones is the same as 2,137.

2. Rename 1,584 in different ways.

1

I can use a place value chart to rename 1,584 in different ways.

I can rename 1 thousand as 10 hundreds. So 1,584 can be renamed as 15 hundreds 8 tens 4 ones.

I can use place value to rename 1,584 in other ways too.

Write the answer for each question.

3. How many thousands are in the thousands place in 24,308? 4 thousands

4. How many thousands are in 24,308? 24 thousands

I write 24,308 in a place value chart. There are 4 thousands in the thousands place.

thousands tens ones hundreds

I can rename 24,308 as 24 thousands, 3 hundreds, 0 tens, and 8 ones.

thousands tens ones hundreds

So there are 24 thousands in 24,308

REMEMBER

5. Fill in the blank to make the equation true. 42 ÷  7 = 6

6. Complete the equations to find 42 ÷ 6.

42 ÷ 6 = (30 ÷ 6) + ( 12 ÷ 6)

= ( 5 ) + ( 2 ) = 7

I can use a number bond to break apart 42 into smaller parts. Then I can divide by 6 by using facts I know.

42 ÷ 6 = ( 30 ÷ 6) + (12 ÷ 6)

30 12

I divide each part by 6 and add the quotients to find 42 ÷ 6 .

I can use a related multiplication equation with an unknown factor.

× 6 = 42

I skip-count by 6 to get to the total, 42

6, 12 , 18, 24, 30, 36, 42

I skip-counted by 6 seven times. So, 7 × 6 = 42 and 42 ÷ 7 = 6.

Name Date

1. Represent 1,524 on each place value chart to match the given unit form.

a. 1 thousand 5 hundreds 2 tens 4 ones thousands tens ones hundreds

b. 15 hundreds 2 tens 4 ones thousands tens ones hundreds

2. Rename 6,219 in different ways. thousands hundreds ten ones hundreds ten ones tens ones ones

Write the answer for each question.

3. How many thousands are in the thousands place in 49,225? thousands

4. How many thousands are in 49,225? thousands

REMEMBER

5. Fill in the blanks to make the equations true.

30 ÷ = 5

× 9 = 45

6. Complete the equations to find 54 ÷ 6

54 ÷ 6 = (30 ÷ 6) + ( ÷ 6) = ( ) +

Draw or cross out disks on the chart to match the statement. Complete the statement.

1 ten thousand less than 537,521 is 527,521 .

I see 3 ten thousands disks.

To find 1 ten thousand less than 537,521 , I can cross out 1 ten thousands disk.

Now I see 2 ten thousands disks.

So 537,521 10,000 = 527,521

Complete the statement and equation.

2. 1,000 more than 22,175 is 23,175

22,175 + 1,000 = 23,175

There are 22 thousands in 22,175 1 thousand more than 22 thousands is 23 thousands.

22 thousands + 1 thousand = 23 thousands

So 1,000 more than 22,175 is 23,175

Use the rule to complete the number pattern.

3. Rule: Subtract 10,000

There are 8 ten thousands in 83,504. I subtract 1  ten thousand to find the next number in the pattern.

8 ten thousands 1 ten thousand = 7 ten thousands

The next number in the pattern is 73,504 I repeat this rule to complete the pattern.

REMEMBER

4. Ivan plots a point on a number line to show what time he starts fishing. Each interval represents 5 minutes.

a. What time does Ivan start fishing?

4 : 15 p.m.

b. Ivan catches a fish at 4 : 40 p.m.

Plot and label this time on the number line.

I know that each interval represents 5 minutes. I start at 4 : 00 p.m. I skip-count by fives until I arrive at the point. Ivan starts fishing at 4 : 15 p.m.

I skip-count by fives until I reach 40. I plot a point at the tick mark that represents 40. Then I label the time on the number line.

5 10 20 25 30 35 40

15 start

Draw or cross out disks on the chart to match the statement. Complete the statement.

hundred thousand less than 464,314 is

Complete each statement and equation.

2. 10,000 more than 74,528 is . 74,528 + 10,000 =

Use the rule to complete the number pattern.

4. Rule: Add 100,000 205,699

5. Rule: Subtract 1,000

42,891

REMEMBER

6. Eva plots a point on a number line to show what time math class begins. Each interval represents 5 minutes.

11:00 a.m. 12:00 p.m.

a. What time does math class begin?

b. Eva has lunch at 11 : 55 a.m. Plot and label this time on the number line.

Name Date

Round to the nearest thousand. Draw a number line to show your thinking.

1. 205,787 ≈ 206,000

206,000 = 206 thousands

205,787

205,500 = 205 thousands 5 hundreds

The symbol ≈ means about.

205,787 ≈ 206,000

205,787 is about 206,000

205,000 = 205 thousands

There are 205 thousands in 205,787 1 more thousand is 206 thousands. I draw and label the bottom and top tick marks.

205 thousands 5 hundreds is halfway between 205 thousands and 206 thousands. I draw and label the tick mark that represents halfway between 205,000 and 206,000.

205,787 is greater than 205,500. I plot 205,787 between 205,500 and 206,000.

205,787 is greater than 205,500, which means it is more than halfway between 205,000 and 206,000. That tells me that 205,787 is closer to 206,000 than to 205,000

2. There are 14,366 people at an amusement park. To the nearest thousand, about how many people are at the amusement park?

14,366 ≈ 14,000

There are about 14,000 people at the amusement park.

I can draw a number line to round 14,366 to the nearest thousand.

14,366 is less than halfway between 14,000 and 15,000 That means 14,366 is closer to 14,000 than to 15,000.

REMEMBER 3. Carla reads from 6 : 02 p.m. to 6 : 14 p.m. How many minutes does Carla read?

Carla reads for 12 minutes.

I think about a time that is easy for me to count on to.

I choose 6 : 10 p.m. because 10 is a benchmark number.

I use the arrow way to count on from 6 : 02 to 6 : 10.

There are 8 minutes from 6 : 02 to 6 : 10.

Then I count on from 6 : 10 to 6 : 14. There are 4 minutes from 6 : 10 to 6 : 14. 6:02 + 8 6:10 + 4 6:14

I add 8 and 4 to get 12

Round to the nearest thousand. Show your thinking on the number line.

1. 3,800 ≈

2. 18,155 ≈

Round to the nearest thousand. Draw a number line to show your thinking.

3. 47,706 ≈

4. 104,621 ≈

5. There are 17,129 books in a library. To the nearest thousand, about how many books are in the library?

REMEMBER

6. Amy watches a video for science class from 1 : 13 p.m. to 1 : 24 p.m. How many minutes does Amy spend watching the video?

Name Date

Round to the nearest ten thousand. Show your thinking on the number line.

1. 84,211 ≈ 80,000

90,000 = 9 ten thousands

85,000 = 8 ten thousands 5 thousands

84,211

80,000 = 8 ten thousands

There are 8 ten thousands in 84,211 1 more ten thousand is 9 ten thousands. I label the bottom and top tick marks.

8 ten thousands 5 thousands is halfway between 8 ten thousands and 9 ten thousands. I label the tick mark that represents halfway between the ten thousands.

84,211 is less than 85,000. I plot 84,211 between 80,000 and 85,000

Because 84,211 is less than 85,000, it is less than halfway between 80,000 and 90,000. That tells me that 84,211 is closer to 80,000 than to 90,000

2. Rocky Mountain National Park covers an area of 265,769 acres. To the nearest hundred thousand, about how many acres does the park cover?

265,769 ≈ 300,000

The park covers about 300,000 acres.

I can draw a number line to round 265,769 to the nearest hundred thousand.

265,769 is more than halfway between 200,000 and 300,000. That means 265,769 is closer to 300,000 than to 200,000

REMEMBER

3.

Each side of her backyard is 9 yards.

I need to find the length of each side of Eva’s backyard.

I know that a pentagon has 5 sides. I also know that each side of Eva’s backyard has the same length. So Eva walks 45 yards around 5 sides of equal length. I can divide to determine the length of each side.

45 ÷ 5 = 9

Eva walks the perimeter of her backyard. Each side of her backyard has the same length. Her backyard is in the shape of a regular pentagon. She walks a total of 45 yards. What is the length of each side of her backyard?

Name Date

Round to the nearest ten thousand. Show your thinking on the number line.

1. 32,144 ≈

2. 508,223 ≈

Round to the nearest hundred thousand. Use the number line to show your thinking.

3. 521,309 ≈

4. 791,547 ≈

5. 612,338 people visited the Westfield Zoo last year. About how many people visited the zoo? Round to the nearest ten thousand.

REMEMBER

6. Mr. Lopez designs a dog kennel in the shape of a regular octagon. He uses 56 feet of fencing around the perimeter of the kennel. Each side of the kennel has the same length. What is the length of each side of the kennel?

Name Date

Round the number to the given place. Show your thinking on the number line.

1. 38,357

a. Nearest thousand

39,000 = 39 thousands

b. Nearest ten thousand

40,000 = 4 ten thousands

38,357

38,500 = 38 thousands 5 hundreds

38,357

38,000 = 38 thousands

38,357 ≈ 38,000

38,357 is less than 38,500, which means it is less than halfway between 38,000 and 39,000

So 38,357 is closer to 38,000 than to 39,000

35,000 = 3 ten thousands 5 thousands

30,000 = 3 ten thousands

38,357 ≈ 40,000

38,357 is greater than 35,000, which means it is more than halfway between 30,000 and 40,000

So 38,357 is closer to 40,000 than to 30,000

Round the number to the given place.

2. 783,455

Nearest

783,455 is greater than 750,000, which means it is more than halfway between 700,000 and 800,000. So it is closer to 800,000 than to 700,000

To the nearest hundred thousand, 783,455 ≈ 800,000

I can use similar thinking to round to the nearest ten thousand and nearest thousand.

REMEMBER

3. Draw and shade 2 different rectangles that each have an area of 36 square units. Show how to find the area and perimeter of each rectangle.

Equation to find area: 3 × 12 = 36

Area: 36 square units

Equation to find perimeter: (2 × 3) + (2 × 12) = 6 + 24 = 30

Perimeter: 30 units

Equation to find area: 4 × 9 = 36

Area: 36 square units

Equation to find perimeter: (2 × 4) + (2 × 9) = 8 + 18 = 26

Perimeter: 26 units

I can write expressions to represent the side lengths for rectangles with an area of 36 square units.

The expressions are 1 × 36, 2 × 18, 3 × 12 , 4 × 9, and 6 × 6

For the first rectangle, I choose to draw a rectangle with side lengths of 3 units and 12 units.

I multiply the width times the length to find the area. Opposite side lengths of rectangles are equal, so I multiply the width by 2 and the length by 2 . Then I add the products to find the perimeter. The perimeter is 30 units.

For the second rectangle, I choose to draw a rectangle with side lengths of 4 units and 9 units.

The rectangles have the same number of square units; they are just arranged differently. So the rectangles have the same area, but their perimeters are different.

Name Date

Round the number to the given place. Show your thinking on the number line.

a. Nearest thousand

77,129 ≈

Round the numbers to the given place.

Nearest hundred thousand

Nearest ten thousand

Nearest thousand

Nearest hundred thousand

Nearest ten thousand

Nearest thousand

b. Nearest ten thousand 77,129 ≈

3. 388,205

Nearest hundred thousand

Nearest ten thousand

Nearest thousand

5. 526,325

Nearest hundred thousand

Nearest ten thousand

Nearest thousand

1. 77,129

2. 81,266

4. 58,862

REMEMBER

6. Draw and shade 2 different rectangles that each have an area of 20 square units. Show how to find the area and perimeter of each rectangle.

Equation to find area:

Area: square units

Equation to find perimeter:

Perimeter: units

Equation to find area:

Area: square units

Equation to find perimeter:

Perimeter: units

Name Date

1. Jayla earns 5,615 points in Level 3 of a video game. Jayla and Ray estimate the number of points by rounding the number.

Jayla rounds to the nearest thousand, and Ray rounds to the nearest hundred.

Is Jayla’s estimate or Ray’s estimate more accurate? Explain.

Jayla rounds 5,615 to 6,000. Ray rounds 5,615 to 5,600. Jayla’s estimate is 385 more than 5,615. Ray’s estimate is more accurate because it is only 15 less than 5,615.

I draw a number line to round 5,615 to the nearest thousand.

5,615 is greater than 5,500, which means it is more than halfway between 5,000 and 6,000.

So it is closer to 6,000 than to 5,000.

To the nearest thousand, 5,615 ≈

Next, I draw a number line to round 5,615 to the nearest hundred.

5,615 is less than 5,650, which means it is less than halfway between 5,600 and 5,700

So it is closer to 5,600 than to 5,700

To the nearest hundred, 5,615 ≈ 5,600.

I compare the estimates to the actual number of points.

Jayla’s estimate of 6,000 is 385 points more than the actual number.

Ray’s estimate of 5,600 is 15 points less than the actual number.

Because Ray’s estimate is closer to the actual number, his estimate is more accurate.

2. Zara wants to save $250 to buy a guitar. She saves $68 the first week and $77 the second week. Should Zara round to the nearest hundred or to the nearest ten to estimate how much she still needs to save? Explain. Round

Zara should round to the nearest ten so that her estimates are closer to the actual amounts. If she rounds to the nearest hundred, she will get $200 for an estimate of her total savings. $200 is much higher than the actual amount she saved, so she might think she only needs to save about $50 more when she really needs to save about $100 more.

Rounded to the nearest hundred, Zara’s savings are $

and

100.

So she would estimate that she has saved about $200 so far.

Rounded to the nearest ten, Zara’s savings are $

So she would estimate that she has saved about $150 so far.

The actual amount saved is $68 + $77 = $145 Rounding to the nearest ten results in estimates that are closer to the actual values than rounding to the nearest hundred. So Zara should round to the nearest ten.

REMEMBER

3. Gabe draws a quadrilateral that has 2 pairs of parallel sides, 4 sides of equal length, and 4 right angles. Which shape does Gabe draw? Circle all the shapes that can name Gabe’s quadrilateral.

A. Square

B. Rectangle

C. Trapezoid

D. Parallelogram

E. Rhombus

I know each shape listed is a quadrilateral. I can think about other attributes of each shape.

square rectangle

• 2 pairs of parallel sides

• 4 sides of equal length

• 4 right angles

• 2 pairs of parallel sides

• Opposite sides of equal length

• 4 right angles

trapezoid

• At least 1 pair of parallel sides

parallelogram rhombus

• 2 pairs of parallel sides

• Opposite sides of equal length

• 2 pairs of parallel sides

• 4 sides of equal length

Gabe’s quadrilateral has 2 pairs of parallel sides. A square, rectangle, parallelogram, and a rhombus always have 2 pairs of parallel sides. A trapezoid has at least 1 pair of parallel sides, which means it can have 1 or more pairs of parallel sides.

Gabe’s quadrilateral has 4 sides of equal length. A square and a rhombus always have 4 sides of equal length. A trapezoid, parallelogram, and rectangle can have 4 sides equal in length.

Gabe’s quadrilateral also has 4 right angles. A square and a rectangle always have 4 right angles. A trapezoid, parallelogram, and rhombus can have 4 right angles.

Gabe’s quadrilateral is a square. Rectangle, trapezoid, parallelogram, and rhombus also name the shape because all of them can be a square.

square rectangle

• 2 pairs of parallel sides

• 4 sides of equal length

• 4 right angles

• 2 pairs of parallel sides

• Opposite sides of equal length

• 4 right angles

trapezoid

• At least 1 pair of parallel sides

parallelogram

• 2 pairs of parallel sides

• Opposite sides of equal length

rhombus

• 2 pairs of parallel sides

• 4 sides of equal length

That means that square, rectangle, trapezoid, parallelogram, and rhombus can all name Gabe’s quadrilateral.

4. Ivan measures a string that is 5 times as long as Gabe’s string. Ivan’s string is 35 inches long. How long is Gabe’s string?

35 ÷ 5 = 7

Gabe’s string is 7 inches long.

I read the problem. I read again. As I reread, I think about what I can draw. I draw a tape diagram with 1 unit to represent the length of Gabe’s string. I label it with a question mark because it is the unknown.

I draw another tape with 5 units to represent that Ivan’s string is 5 times as long as Gabe’s string. Ivan’s string is 35 inches long, so I label the 5 units with a total of 35

I need to find the length of Gabe’s string. I see that I can divide 35 by 5 to find the size of 1 unit.

Name Date

1. A principal needs to order 1,437 T-shirts for students at her school. She rounds to the nearest hundred to estimate the number of T-shirts to order. Will there be enough T-shirts for all the students? Explain.

2. Ivan’s family drives 2,213 miles on a vacation. Ivan rounds this number to the nearest thousand. His sister rounds to the nearest hundred. Whose estimate is more accurate? Explain.

3. Liz’s class will win a prize if they sell 500 magnets. They sell 62 magnets in the first month and 93 in the second month. Liz wants to estimate the number of magnets they still need to sell. Should she round to the nearest hundred or to the nearest ten? Explain.

REMEMBER

4. Liz draws a quadrilateral that has 2 pairs of parallel sides, 4 sides of equal length, and no right angles. Which shape does Liz draw? Circle all the shapes that can name Liz’s quadrilateral.

A. Square

B. Rectangle

C. Rhombus

D. Trapezoid

E. Parallelogram

Use the Read–Draw–Write process to solve the problem.

5. Carla reads 2 times as many pages as Casey. Carla reads 18 pages. How many pages does Casey read?

Multi-Digit Whole Number Addition and Subtraction

Dear Family,

Your student is learning to add and subtract numbers within 1,000,000 by using the standard algorithm. They begin by using concrete place value disks and drawings on the place value chart. The place value chart helps your student make sense of when they must rename units when adding or subtracting. Your student uses rounding to estimate to check whether their answer is reasonable. They also solve word problems by using the Read–Draw–Write process to practice adding and subtracting. They use tape diagrams to represent and make sense of the problems and write equations with a letter for the unknown.

Rounding and estimating before adding shows that the answer of 236,089 is reasonable.

Town A has 13,546 residents.

Town B has 2,243 more residents than Town A.

Town C has 1,230 fewer residents than Town B. What is the total number of residents in all three towns?

At-Home Activity

Large Number Fun

Explore adding and subtracting with large numbers related to topics that your student finds interesting. Encourage rounding, estimating, determining reasonableness, and checking subtraction problems with addition. Consider using the following example topics.

• Gather information such as the number of average visitors per year at your student’s favorite amusement parks or other attractions. You can also create your own information such as Adventure Park had 745,691 visitors this year and Discovery Park had 667,345 visitors. Ask your student questions such as, “How many more people visited Adventure Park than Discovery Park this year?”

• Gather information about the weights of large animals. For example, a crocodile weighs 1,098 pounds, an elephant weighs 10,648 pounds, a shark weighs 2,562 pounds, and a bear weighs 1,332 pounds. Ask your student questions such as, “What is the combined weight of the shark and bear?”

Add by using the standard algorithm. 1.

7

I add the ones. 7  ones + 2  ones = 9  ones. I write a 9 in the ones column below the line.

Then I add the tens. 4  tens + 9  tens = 13  tens. I rename 13 tens as 1 hundred 3 tens. I write a 1 on the line in the hundreds column and a 3 in the tens column below the line.

Next, I add the hundreds. 8  hundreds + 7  hundreds + 1  hundred = 16  hundreds. I rename 16 hundreds as 1 thousand 6 hundreds. I write a 1 on the line in the thousands column and a 6 in the hundreds column below the line.

Last, I add the thousands. 3 thousands + 2  thousands + 1  thousand = 6 thousands. I write a 6 in the thousands column below the line.

First, I write the numbers in vertical form. I line up the numbers by place value units.

Then I add like place value units. I rename a sum when there are 10 or more of a place value unit.

I see there is only one number in the hundred thousands place to add, so I write the 4 in the hundred thousands column below the line.

Use the Read–Draw–Write process to solve the problem.

3. Smithfield has a population of 65,225. Flatwood has a population of 28,183.

What is the total population of the two cities?

65,225 + 28,183 = 93,408

The total population of the two cities is 93,408.

I read the problem. I read again. As I reread, I think about what I can draw. I draw and label parts in the tape diagram to represent the populations of Smithfield and Flatwood.

I do not know the total population of the two cities. I can use the letter p to represent the total population.

65,225 p

28,183

I can write an equation to represent the problem, 65,225 + 28,183 = p I can add by using the standard algorithm.

REMEMBER 4. Use the picture to write a word problem that can be represented by the expression 2 × 7.

Carla picks 2 rows of carrots. Each row has 7 carrots. How many carrots does Carla pick?

I look at the picture.

I see 2 rows of carrots. Each row has 7 carrots.

The expression 2 × 7 represents how to find the total number of carrots in the picture.

I can find how many carrots there are in all. I write a word problem that asks about the total number of carrots.

Use the Read–Draw–Write process to solve the problem.

7. A basketball team scored 5,104 points last season and 5,568 points this season.

How many total points did the team score in the two years?

REMEMBER

8. Use the picture to write a word problem that can be represented by the expression 4 × 7

Name Date

Use the Read–Draw–Write process to solve the problem.

1. The Appalachian Trail is 2,193 miles long. The Pacific Crest Trail is 460 miles longer than the Appalachian Trail. The Continental Divide Trail is 3,100 miles long. What is the total length of the three trails? Is your answer reasonable? Explain.

2,200 + (2,200 + 500) + 3,100 = 8,000

2,193 + (2,193 + 460) + 3,100 = 7,946

The total length of the three trails is 7,946 miles.

Yes, my answer is reasonable. I estimated the answer by rounding each number to the nearest hundred before adding. My estimate was 8,000 miles. When I added the actual amounts, my answer was 7,946 miles. I know that 7,946 is very close to 8,000.

I read the problem. I read again.

As I reread, I think about what I can draw.

I draw a tape diagram to represent the problem. I know the Appalachian Trail is 2,193 miles long. I know the Pacific Crest Trail is 460 miles longer than the Appalachian Trail. I know the Continental Divide Trail is 3,100 miles long. I do not know the total length of all three trails. I can use the letter d to represent the total length of all three trails.

Appalachian Trail

Pacific Crest Trail

Continental Divide Trail

2,193

2,193 460

d

3,100

I am trying to find the total length of the three trails. I can write an equation to represent the problem, 2,193 + (2,193 + 460 ) + 3,100 = d.

Before I solve the problem, I can estimate the answer by rounding each number to the nearest hundred. 2,200 + (2,200 + 500 ) + 3,100 = 8,000. My estimate is 8,000 miles.

When I add the actual amounts, my answer is 7,946 miles.

7,946 is very close to 8,000, so my answer is reasonable.

REMEMBER 2. Write the pattern and complete the table.

Pattern: Multiply the input by 9

Input 2 3 5 7 8 10

Output 18 27 45 63 72 90

I look for a pattern in the table.

When the input number is 2 , the output number is 18. I know that 2 × 9 = 18.

I think the pattern is multiply by 9. I can check other input and output numbers to see if I am correct.

I know that 5 × 9 = 45.

I know that 8 × 9 = 72

The pattern is multiply by 9

I can multiply each input number by 9 to complete the table.

3. Miss Diaz paints 1 2 of a wall.

Partition and shade the rectangle to show how much of the wall Miss Diaz paints.

I know Miss Diaz paints 1 2 of a wall. The fractional unit is halves, so I draw 1 line to partition the rectangle into 2 equal parts.

I shade 1 part, or 1 half, to show how much of the wall Miss Diaz paints.

Name Date

Use the Read–Draw–Write process to solve each problem.

1. An apple orchard has Honeycrisp trees and McIntosh trees. The orchard has 1,603 Honeycrisp trees. It has 327 more McIntosh trees than Honeycrisp trees.

a. About how many apple trees does the orchard have?

b. Exactly how many apple trees does the orchard have?

c. Is your answer reasonable? Compare your estimate from part (a) to your answer from part (b). Explain your reasoning.

2. A restaurant made $5,987 on Friday. It made $1,527 more on Saturday than on Friday. On Sunday, the restaurant made $2,641. How much money did the restaurant make in the three days? Is your answer reasonable? Explain.

REMEMBER

3. Write the pattern and complete the table.

4. Mr. Endo decorates 1 8 of a bulletin board with blue paper.

Partition and shade the rectangle to show how much of the bulletin board is decorated with blue paper.

I look at the numbers in each place value to see whether I am ready to subtract.

I start with the ones place. I am ready to subtract the ones because 3 ones is more than 1 one.

I am also ready to subtract the tens because 8 tens is more than 4 tens.

I am not ready to subtract the hundreds because 2 hundreds is not more than 7 hundreds. I need to rename 1 thousand as 10 hundreds. Now there are 12 hundreds because 2 hundreds + 10 hundreds = 12  hundreds

I am ready to subtract the hundreds because 12 hundreds is more than 7 hundreds.

I am also ready to subtract the thousands because 4 thousands is more than 2 thousands. Now I am ready to subtract like place value units.

3 ones - 1 one = 2 ones

8 tens - 4 tens = 4 tens

12 hundreds - 7 hundreds = 5 hundreds

4 thousands - 2 thousands = 2 thousands

I record the subtraction below each place value unit.

Subtract by using the standard algorithm.

REMEMBER 2. Amy eats 1 4 of a small cake. Pablo eats 1 8 of a large cake.

Pablo’s Cake

Amy’s Cake

a. Shade the circles to represent the amount of cake that Amy eats and the amount of cake Pablo eats.

b. Pablo says that his piece of cake is bigger than Amy’s so that means that 1 8 is always greater than 1 4 . Do you agree with Pablo? Why?

No, I do not agree with Pablo because the cakes are not the same size. You have to start with the same-size wholes when comparing fractions.

I shade 1 equal part in the circle that represents Amy’s cake to show 1 4 I shade 1 equal part in the circle that represents Pablo’s cake to show 1 8

I see that Pablo’s piece of cake is bigger. However, I know that when I compare fractions, the wholes must be the same size.

3. Write an equation to represent that 32 is 4 times as much as 8.

32 = 4 × 8

I think about the statement 32 is 4 times as much as 8

I can draw a tape diagram to represent 1 unit of 8 Then I draw 4 units of 8 because the statement is 4 times as much as 8

I know the total of 4 units of 8 is 32 , so I label the total 32

8 8 8 8

8

32

I write an equation to represent the tape diagram.

Subtract by using the standard algorithm.

Use the Read–Draw–Write process to solve the problem.

7. A florist sells 17,926 flowers. 7,583 of these flowers are roses. The rest of the flowers are lilies. How many lilies are sold?

REMEMBER

8. Ivan eats 1 3 of a small pie. Oka eats 1 6 of a large pie. Oka’s Pie

Ivan’s Pie

a. Shade the circles to represent the amount of pie that Ivan eats and the amount of pie Oka eats.

b. Oka says that her piece of pie is bigger than Ivan’s so that means that 1 6 is always greater than 1 3 . Do you agree with Oka? Why?

9. Write an equation to represent that 27 is 3 times as much as 9.

I look at the numbers in each place value to see whether I am ready to subtract.

I am ready to subtract the ones because 3 ones is more than 1 one.

I am not ready to subtract the tens because 0 tens is not more than 9 tens. I need to rename 1 hundred as 10 tens. Now there are 10 tens because 0  tens + 10  tens = 10  tens

I am ready to subtract the tens because 10 tens is more than 9 tens.

I notice that I am also not ready to subtract the hundreds. I need to rename 1 thousand as 10 hundreds. Now there are 10 hundreds because 0  hundreds + 10  hundreds = 10  hundreds.

I am ready to subtract the hundreds because 10 hundreds is more than 4 hundreds.

I am also ready to subtract the thousands and ten thousands.

Now I am ready to subtract like place value units.

3  ones − 1  one = 2  ones

10  tens − 9  tens = 1  ten

10  hundreds − 4  hundreds = 6  hundreds

6  thousands − 0  thousands = 6  thousands

4  ten thousands − 2  ten thousands = 2  ten thousands

I record the subtraction below each place value unit.

2. Use the Read–Draw–Write process to solve the problem.

Mr. Davis is a pilot. He flew a total distance of 14,028 miles. He flew from New York City to London and then from London to Sydney. The distance from New York City to London is 3,470 miles.

How far did Mr. Davis fly from London to Sydney?

14,028 − 3,470 = 10,558

Mr. Davis flew 10,558 miles from London to Sydney.

I read the problem. I read again. As I reread, I think about what I can draw. I draw a tape diagram and label 14,028 to represent the total distance Mr. Davis flew. I partition a part and label 3,470 to represent the distance from New York City to London. I do not know the distance from London to Sydney. I can use the letter d to represent this amount.

3,470 d

14,028

I can write an equation to represent the problem. 14,028 − 3,470 = d

I rewrite the problem in vertical form and subtract.

REMEMBER

3. Partition the rectangle into 2 equal parts. Shade 1 part.

What fraction of the rectangle is shaded?

1

2 I draw 1 line to partition the rectangle into 2 equal parts. Then I shade 1 part.

There are 2 equal parts, which means the fractional unit is halves. One half is shaded, so I can write the fraction 1 2

4. Use the equation to complete parts (a) and (b).

18 = 3 × 6

a. Draw 2 tape diagrams to represent the equation.

I can think about 18 = 3 × 6 as 3 times as much as 6 or 6 times as much as 3

To represent 3 times as much as 6, I draw 1 unit of 6. Then I draw 3 units of 6 because 18 is 3 times as much as 6

I label the product 18.

b. Complete both statements. 18 is 3 times as much as 6. 18 is 6 times as much as 3.

To represent 6 times as much as 3, I draw 1 unit of 3. Then I draw 6 units of 3 because 18 is 6 times as much as 3.

I label the product 18

Subtract by using the standard algorithm.

Use the Read–Draw–Write process to solve each problem.

4. Pablo has 4,620 football and baseball cards in total. He has 1,486 football cards. How many baseball cards does Pablo have?

5. The sum of two numbers is 17,426. One number is 8,731. What is the other number?

REMEMBER

6. Partition the rectangle into 4 equal parts. Shade 1 part.

What fraction of the rectangle is shaded?

7. Use the equation to complete parts (a) and (b).

28 = 7 × 4

a. Draw 2 tape diagrams to represent the equation.

b. Complete both statements.

28 is times as much as 7.

28 is times as much as 4.

1. 650,000 31,387

I can rewrite the problem in vertical form. Then I get ready to subtract by renaming place value units when I do not have enough to subtract.

I rename 650,000 as 6 hundred thousands, 4 ten thousands, 9 thousands, 9 hundreds, 9 tens, and 10 ones.

Now I am ready to subtract like place value units.

10  ones 7  ones = 3  ones

9  tens 8  tens = 1  ten

9  hundreds 3  hundreds = 6  hundreds

9  thousands 1  thousand = 8  thousands

4  ten thousands 3  ten thousands = 1  ten thousand

6  hundred thousands 0  hundred thousands = 6  hundred thousands

I record the subtraction below each place value unit.

Subtract by using the standard algorithm.

Use the Read–Draw–Write process to solve the problem.

2. An Olympic-size swimming pool can hold 660,430 gallons of water. It currently has 418,795 gallons of water in it. How many more gallons of water are needed to fill the pool?

660,430 418,795 = 241,635

Carla needs 241,635 more gallons of water to fill the pool.

I read the problem. I read again. As I reread, I think about what I can draw. I draw a tape diagram and label the total as 660,430 to represent the amount of water the pool can hold. I partition a part and label it as 418,795 to represent the amount of water the pool currently has in it. I do not know how many more gallons of water are needed to fill the pool. I can use the letter g to represent this amount. 660,430

418,795 g

I can write an equation to represent the problem.

660,430 – 418,795 = g

I rewrite the problem in vertical form and subtract.

g = 241,635

REMEMBER 3.

Favorite Pets

Each represents 8 students.

a. How many fourth grade students voted?

88 students

I know each circle represents 8 students by looking at the key. That means that each half-circle represents 4 students.

I see that 28 students voted for cat, 36 students voted for dog, 12 students voted for fish, 4 students voted for hamster, and 8 students voted for lizard.

I can add to find the total.

28 + 36 + 12 + 4 + 8 = 88

Fourth grade students voted for their favorite pet. The scaled picture graph represents the data.

Type of Pet

Dog Fish Cat

Lizard Hamster

b. How many more students voted for cat than for hamster and fish combined?

12 more students

I add the number of students who chose hamster and the number of students who chose fish to find the combined total. Then I subtract the combined total from the number of students who chose cat.

28 - 16 = 12

c. How many fewer students voted for lizard than for dog?

28 fewer students

I see that 36 students voted for dog and 8 students voted for lizard.

I can subtract to find the difference.

36 - 8 = 28

Subtract by using the standard algorithm.

3. 900,000 886,071

Use the Read–Draw–Write process to solve each problem.

4. A bagel shop sells 176,663 blueberry bagels and 182,149 plain bagels. How many more plain bagels than blueberry bagels does the shop sell?

5. Liz has driven her car 71,435 miles. She needs to have her car serviced at 100,000 miles. How many more miles can Liz drive before her car needs to be serviced?

REMEMBER

6. Fourth grade students go to the aquarium. The scaled picture graph shows the number of fish of each color they see.

Fish Observed at the Aquarium

a. How many fish did the students see altogether?

b. How many fewer black fish did the students see than orange fish?

c. How many more yellow fish did the students see than blue fish?

Name Date

Use the Read–Draw–Write process to solve each problem.

1. A music festival had 22,485 attendees on Friday. It had 28,327 attendees on Saturday. After Sunday, the music festival had a total of 72,558 attendees.

a. Estimate the number of attendees the music festival had on Sunday. Round each value to the nearest thousand.

22,000 + 28,000 = 50,000

73,000 50,000 = 23,000

The music festival had about 23,000 attendees on Sunday.

b. Find the number of attendees the music festival had on Sunday.

22,485 + 28,327 = 50,812

72,558 50,812 = 21,746

The music festival had 21,746 attendees on Sunday.

c. Is your answer reasonable? Use your estimate from part (a) to explain.

Yes, my answer of 21,746 attendees is reasonable. 21,746 attendees rounded to the nearest thousand is 22,000, which is close to my estimate of 23,000.

I read the problem. I read again. As I reread, I think about what I can draw.

I draw a tape diagram and partition it into three parts, one part for each day of the festival. I label the total as 72,558 because that is the total number of attendees. I label one part as 22,485 to represent Friday’s attendees. I label another part as 28,327 to represent Saturday’s attendees.

I do not know the number of attendees on Sunday. I can use the letter a to represent this amount.

22,485 28,327

72,558

a

I need to find the number of attendees on Sunday. My tape diagram helps me see that I need to add the numbers of attendees on Friday and Saturday. Then I can subtract that total from 72,558 to find the number of attendees on Sunday.

I round each value to the nearest thousand.

22,485 ≈ 22,000

28,327 ≈ 28,000

72,558 ≈ 73,000

My estimate for a is 23,000.

The actual answer for a is 21,746.

The actual amount is close to my estimate of 23,000 attendees.

2. Aberdeen has a population of 293,097. It has 59,812 more people than Monroe. What is the total population of the two cities? Make an estimate before finding the answer. Is your answer reasonable? Explain.

290,000 60,000 = 230,000

290,000 + 230,000 = 520,000

293,097 59,812 = 233,285

293,097 + 233,285 = 526,382

The total population of both cities is 526,382. The answer is reasonable because my estimate is 520,000, which is close to 526,382.

I read the problem. I read again.

As I reread, I think about what I can draw.

I draw a tape diagram. I draw and label one tape as 293,097 to represent the population of Aberdeen. I draw a shorter tape to represent the population of Monroe. I label the difference between the two cities as 59,812

I need to find the total population of the two cities.

My tape diagram helps me see that I need to subtract 59,812 from 293,097 to find the population of Monroe. Then I can add to find the total population of the 2 cities.

I round each value to the nearest ten thousand. 293,097 ≈ 290,000 59,812 ≈ 60,000

My estimate for p is 520,000.

The actual answer for p is 526,382

The actual total population is 526,382 people, which is close to my estimate of 520,000

Aberdeen Monroe ? p

I do not know the population of Monroe. I also do not know the total population of the two cities. I use the letter p to represent this amount. 293,097 59,812

REMEMBER 3. 5,572

The value of the underlined 5 is 10 times as much as the value of the circled 5

I can use a place value chart to compare the values of the digits.

thousands ten thousands tens ones hundreds

× 10

The underlined 5 is in the thousands place. It has a value of 5,000 The circled 5 is in the hundreds place. It has a value of 500. 5 thousands is 10 times as much as 5 hundreds.

5,000 = 10 × 500

Name Date

Use the Read–Draw–Write process to solve each problem.

1. A book company sold 342,380 copies of a book in the first week. The company sold 361,419 copies in the second week. After 3 weeks, the book company had sold 923,468 copies of the book.

a. Estimate the number of copies of the book the company sold in the third week. Round each value to the nearest ten thousand.

b. Find the number of copies of the book the company sold in the third week.

c. Is your answer reasonable? Use your estimate from part (a) to explain.

2. A soccer stadium has 93,810 seats. It has 28,654 more seats than a baseball stadium. How many total seats do the baseball and soccer stadiums have? Make an estimate before finding the answer. Is your answer reasonable? Explain.

REMEMBER 3. 2, 9 9 3

The value of the underlined 9 is times as much as the value of the circled 9.

Name Date

Use the Read–Draw–Write process to solve the problem.

1. A book company has 12,532 mystery books.

It has 1,982 fewer fantasy books than mystery books. It has 1,432 more science fiction books than mystery books.

How many total mystery, fantasy, and science fiction books does the book company have?

12,532 1,982 = 10,550

12,532 + 1,432 = 13,964

12,532 + 10,550 + 13,964 = 37,046

The book company has 37,046 total mystery, fantasy, and science fiction books.

I read the problem. I read again. As I reread, I think about what I can draw.

I draw a tape diagram. I draw and label one tape as 12,532 to represent the mystery books.

I draw a second tape that is shorter than the mystery books to represent the fantasy books and label the difference 1,982

I draw a third tape that is longer than the mystery books to represent the science fiction books and label the difference 1,432

I need to find the total number of books the book company has. I can use the letter c to represent the unknown total number of books.

First, I subtract 1,982 from 12,532 to find the number of fantasy books. It is 10,550

Next, I add 12,532 and 1,432 to find the number of science fiction books. It is 13,964

Last, I add the number of mystery, fantasy, and science fiction books together to find c

REMEMBER 2. Compare the numbers by using >, =, or <. Explain how you know.

700,000 + 20,000 + 1,000 + 300 + 4  >  718,529

I write the first number in standard form, 721,304. Both numbers have 7 hundred thousands. The first number has 2 in the ten thousands place. The second number has 1 in the ten thousands place. I know that 2 ten thousands is greater than 1 ten thousand, so 721,304 > 718,529.

I write the first number in standard form, 721,304

I can use a place value chart to record 721,304 and 718,529

ten thousands hundred thousands tens ones hundreds

Both numbers have 7 hundred thousands, so I look at the unit to the right, ten thousands.

721,304 has a 2 in the ten thousands place.

718,529 has a 1 in the ten thousands place.

2 ten thousands, or 20,000, is greater than 1 ten thousand, or 10,000 So 721,304 > 718,529

Name Date

Use the Read–Draw–Write process to solve each problem.

1. Mia is an entomologist. She records the number of ants, beetles, and termites in part of a forest. She records 12,861 ants.

She records 4,859 fewer beetles than ants.

She records 5,589 fewer termites than beetles.

How many total insects does Mia record?

2. The Maplewood School District has 6,705 first grade students and 6,441 second grade students.

The school district has 1,304 more third graders than first graders.

It has 1,577 more fourth graders than second graders.

What is the total number of first through fourth graders in the district?

REMEMBER

3. Compare the numbers by using >, =, or <. Explain how you know.

FAMILY MATH Metric Measurement Conversion Tables

Dear Family,

Your student is learning to convert from larger metric units to smaller metric units. They work with units that measure length (centimeters, meters, and kilometers), mass (grams and kilograms), and liquid volume (milliliters and liters). Measurements are sometimes given as mixed units, such as 1 kilometer 300 meters. Mixed units have more than 1 unit. For example, the number 1,300 can be written with mixed units as 1 thousand 3 hundreds. Your student uses what they know about whole number place value units to rename mixed measurement units. Then they add and subtract measurements to solve word problems.

1 kilometer is 1,000 times as long as 1 meter.

1 km = 1,000 × 1 m

1 kilometer = 1,000 meters

Metric units have a similar relationship to place value units.

2,250 mL + 1 L 750 mL 1 L 750 mL = 1,750 mL

The relationship between 1 kilometer and 1,000 meters is used to convert 2, 3, and 4 kilometers to meters. 2,250 mL + 1 L 750 mL

1 L 750 mL = 1,750 mL

250

, 0 00

Mrs. Smith has 4,000 mL.

Mrs. Smith has 4,000 mL.

mL 1 L 750 mL

1 L 750 mL is a mixed measurement unit. Converting to milliliters before adding is one strategy for adding mixed measurement units.

2,250 mL 1 L 750 mL

At-Home Activities

Metric Units

Use real-world examples to practice converting metric units. For example, think of a familiar route, such as the distance from home to school. Find the distance of that route in whole number kilometers. If you don’t know the distance in kilometers then estimate. 1 kilometer is about the same as 6 10 miles. Then have your student convert kilometers to meters and centimeters.

How Much Taller?

Help your student practice subtracting mixed units by measuring the heights of members of your family. Use a meter stick or a tape measure with metric units to measure in meters and centimeters. Another option is to use an online conversion tool to convert known heights in feet and inches into meters and centimeters. Then have your student use subtraction to see how much taller one family member is than another.

Name

Complete the conversion table.

1. Kilometers Meters

9 9,000

Date

A kilometer (km) is a unit for measuring distance or length.

Convert.

Convert means to express a measurement in terms of a different, related measurement unit.

I know there are 1,000 meters in 1 kilometer. (1 km) 1 , 00 0 m 10 0 m × 1 , 000

10 m 1 m

I know that 1 kilometer is 1,000 times as long as 1 meter.

I can multiply the number of kilometers by 1,000 to find the equivalent number of meters.

9 × 1,000 m = 9,000 m

There are 9,000  meters in 9 kilometers.

I look at the measurement that is given, 17 km 86 m. This is a mixed unit because it has more than 1 unit. It has units of kilometers and meters.

I can multiply the number of kilometers by 1,000 to convert to meters.

17 × 1,000 m = 17,000 m

17 kilometers is 17,000 meters.

86 more meters is 17,086 meters.

17,000 m + 86 m = 17,086 m

2. 17,086  m = 17 km 86 m

3. 9 m 63 cm − 5 m 76 cm = 387 cm

I can rename both measurements as centimeters because when they are written in mixed units, there are not enough centimeters in the first number to subtract 76 cm. To convert meters to centimeters, I multiply the number of meters by 100.

Use the Read–Draw–Write process to solve the problem.

4. Ray kicks a field goal that is 16 meters 45 centimeters long. Casey kicks a field goal that is 21 meters 67 centimeters long. What is the total length of both field goals?

16 m 45 cm + 21 m 67 cm = 3,812 cm

The total length of both field goals is 3,812 centimeters.

I read the problem. I read again. As I reread, I think about what I can draw.

I draw a tape diagram. I partition the tape diagram into two parts. One part represents the length Ray kicks. The other part represents the length Casey kicks. I make Casey’s part longer because he kicked farther. I label each part.

I use the letter d to represent the total length, which is unknown. I see that I can add the two lengths together to find the total length of both field goals.

The lengths are in mixed units. I rename both measurements as centimeters and then add.

So d = 3,812

REMEMBER 5. Measure the length of each pencil to the nearest quarter inch. Then use the data to complete the line plot.

I use a ruler to measure the length of each pencil to the nearest quarter inch.

I line up one end of the pencil with the edge of the ruler because it represents 0 inches. The ruler shows tick marks by fourths, or quarter inches.

The other end of the pencil lines up with the mark for 1 1 2 inches on the ruler.

The length of this pencil is 1 1 2 inches. I draw an X above 1 1 2 on the line plot.

I measure the rest of the pencils. For each length, I make an X above the same number on the line plot.

There are 8 pencils, so there are 8 Xs on my line plot.

Name Date

Complete the conversion tables.

1. Meters Centimeters

4 7

Convert.

3.  cm = 235 m

4. 8 km 205 m =  m

Add or subtract.

5. 6 m 42 cm 18 cm =

6. 8 m 72 cm + 2 m 43 cm =

Use the Read–Draw–Write process to solve the problem.

7. There are two rock walls to choose from at a park. Ray climbs the wall that is 4 meters 26 centimeters tall. Ivan climbs the wall that is 335 centimeters tall. How much taller is Ray’s wall than Ivan’s wall?

2. Kilometers Meters

REMEMBER

8. Cut out the ruler. Measure the length of each fish to the nearest quarter inch. Then use the data to complete the line plot.

Fish Lengths

1. Kilograms Grams

6 6,000

I know 1 kilogram is 1,000 times as heavy as 1 gram.

I can multiply the number of kilograms by 1,000 to find the equivalent number of grams.

6 × 1,000 g = 6,000 g

2. Liters Milliliters

2 2,000

I know 1 liter is 1,000 times as much as 1 milliliter.

I can multiply the number of liters by 1,000 to find the equivalent number of milliliters.

2 × 1,000 mL = 2,000  mL

There are 2,000 milliliters in 2 liters. Convert.

There are 6,000 grams in 6 kilograms.

3. 63,213  g = 63 kg 213 g

I know that 1 kilogram is equal to 1,000 grams.

I can multiply the number of kilograms by 1,000 to convert to grams.

63 × 1,000 g = 63,000 g

213 more grams is 63,213 grams.

63,000 g + 213 g = 63,213 g

Subtract.

4. 9 L 421 mL − 3 L 410 mL = 6 L 11 mL

I can subtract like units because there are enough milliliters in the first number to subtract 410 milliliters.

9  L − 3  L = 6  L 421  mL − 410  mL = 11  mL

The answer is 6  L 11  mL.

Use the Read–Draw–Write process to solve the problem.

5. Mia has a brown horse and a white horse. The brown horse eats 47 kg 456 g of hay each week. The white horse eats 33 kg 743 g of hay each week. How much total hay do the horses eat each week?

47 kg 456 g = 47,456 g

33 kg 743 g = 33,743 g

47,456 g + 33,743 g = 81,199 g

The horses eat a total of 81,199 grams of hay each week.

I read the problem. I read again. As I reread, I think about what I can draw.

I draw a tape diagram. I partition the tape diagram into two parts. One part is longer than the other and represents the amount of hay the brown horse eats. The other part represents the amount of hay the white horse eats. I label each part.

I use the letter h to represent the total amount both horses eat, which is what I need to find.

I see that I can add the two amounts of hay together. The weights are in mixed units. I can rename both weights as grams and then add.

47  kg 456  g = 47,456  g

33  kg 743  g = 33,743  g

47 kg 456 g 33 kg 743 g

So h = 81,199.

6. Partition the interval from 0 to 1 into halves. Plot the fraction on the number line.

To partition the interval from 0 to 1 into halves, I know I need 2 equal parts. I draw 1 tick mark evenly spaced between 0 and 1 . Now the interval from 0 to 1 is partitioned into 2 equal parts, or halves.

I count by halves from 0 halves to 1 half. I plot 1 2 on the number line.

Use the Read–Draw–Write process to solve the problem.

7. Gabe has a page of 40 stickers in rows.

There are 5 stickers in each row.

How many rows of stickers are there?

40 ÷ 5 = 8

There are 8 rows of stickers.

I read the problem. I read again.

As I reread, I think about what I can draw.

I draw a tape diagram and label 40 to represent the total number of stickers. I know there are 5 stickers in each row. I draw one part and label it 5. I do not know how many rows of 5 stickers there are. I label the unknown. 5

40

? rows

I see that I can divide to find the number of rows.

Name Date Complete the conversion tables.

1. Kilograms Grams

4 17

2. Liters Milliliters

6 13

Convert

3. 51 kg 128 g = g

Add or subtract.

5. 9 L 421 mL

3 L 410 mL =

Use the Read–Draw–Write process to solve the problem.

4. 8 L 16 mL = mL

6. 14 kg 641 g + 3 kg 407 g =

7. A baseball team drinks 18 L 932 mL of sports drink at their game. A football team drinks 26 L 498 mL of sports drink at their game. How much more sports drink does the football team drink than the baseball team?

REMEMBER

8. Partition the interval from 0 to 1 into thirds. Plot the fraction on the number line.

Use the Read–Draw–Write process to solve the problem. Use a letter to represent the unknown.

9. Carla plants 30 tulips in equal rows. She plants 6 tulips in each row.

How many rows of tulips does Carla plant?

Grade 4 Mathematics Reference Sheet

Metric Conversions

1 kilogram = 1,000 grams

Customary Conversions

1 yard = 3 feet

liter = 1,000 milliliters

1 pint = 2 cups

1 quart = 2 pints

1 gallon = 4 quarts

Time Conversions

1 day = 24 hours

1 week = 7 days

1 year = 365 days

Acknowledgments

Kelly Alsup, Lisa Babcock, Adam Baker, Reshma P. Bell, Joseph T. Brennan, Dawn Burns, Leah Childers, Mary Christensen-Cooper, Jill Diniz, Christina Ducoing, Janice Fan, Scott Farrar, Gail Fiddyment, Ryan Galloway, Krysta Gibbs, Torrie K. Guzzetta, Kimberly Hager, Eddie Hampton, Andrea Hart, Rachel Hylton, Travis Jones, Emily Koesters, Liz Krisher, Courtney Lowe, Bobbe Maier, Ben McCarty, Ashley Meyer, Bruce Myers, Marya Myers, Victoria Peacock, Maximilian Peiler-Burrows, Marlene Pineda, DesLey V. Plaisance, Elizabeth Re, Meri Robie-Craven, Amanda Roose, Jade Sanders, Deborah Schluben, Colleen Sheeron-Laurie, Jessica Sims, Danielle Stantoznik, Tara Stewart, Mary Swanson, James Tanton, Julia Tessler, Jillian Utley, Saffron VanGalder, Rafael Velez, Philippa Walker, Jackie Wolford, Jim Wright, Jill Zintsmaster

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

Credits

For a complete list of credits, visit http://eurmath.link/media-credits

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?

ISBN 979-8-89012-170-7

Module 1

Place Value Concepts for Addition and Subtraction

Module 2

Place Value Concepts for Multiplication and Division

Module 3

Multiplication and Division of Multi-Digit Numbers

Module 4

Foundations for Fraction

Operations

Module 5

Angle Measurements and Plane Figures

Module 6

Place Value Concepts for Decimal Fractions

What does this painting have to do with math?

American abstract painter Frank Stella used a compass to make brightly colored curved shapes in this painting. Each square in this grid includes an arc that is part of a design of semicircles that look like rainbows. When Stella placed these rainbow patterns together, they formed circles. What fraction of a circle is shown in each square?

On the cover

Tahkt-I-Sulayman Variation II, 1969

Frank Stella, American, born 1936 Acrylic on canvas

Minneapolis Institute of Art, Minneapolis, MN, USA

Frank Stella (b. 1936), Tahkt-I-Sulayman Variation II, 1969, acrylic on canvas. Minneapolis Institute of Art, MN. Gift of Bruce B. Dayton/ Bridgeman Images. © 2020 Frank Stella/Artists Rights Society (ARS), New York

9 798890 121707