9.1 Introducing the Sixes Multiplication Facts W the …...205 ORIGO tion 204 ORIGO Stepping Stones...

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ORIGO Stepping Stones 3 • 9.1ORIGO Stepping Stones 3 • 9.1

Introducing the Sixes Multiplication Facts9.1

Step Up 1. Look at these arrays. Complete the sentences.

a.

What do you know about this array?

How can you figure out the total number of dots?

Write a number sentence to describe the array.

What do you know about this array?

How could you use the first array to figure out the total number of dots in this array?

What other facts involving 6 could you solve using this strategy?

Write two number sentences to describe the second array.

The first array shows 5 rows of 4. That s 20 so 6 rows of 4 is 4 more. That s 24.

so

5 rows of 3 = 6 rows of 3 =

2. Writetheproductforthefivesfact.Thenusethatfacttohelpyoucomplete the sixes fact and its turnaround.

a.5 × 7 =

so

6 × 7 =

× 6 =

b.5 × 6 =

so

6 × 6 =

× =

c.5 × 8 =

so

6 × 8 =

× =

3. Use the same strategy to complete these.

a.5 × 9 =

so

6 × =

× =

b.5 × 2 =

so

6 × =

× =

c.5 × 4 =

so

6 × =

× =

Step Ahead Figure out each mystery mass.

weighs lb weighs lb

a. b.

weighs 6 lb

23 34lb lb

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Reinforcing the Sixes Multiplication Facts9.2

What multiplication fact does this whole array show?

How could you figure out the total number of dots?

Complete these facts to help you.

5 × 7 =

1 × 7 =

There are 42 dots in total because 35 + 7 is 42.

What multiplication fact does this whole array show?

What two multiplication facts could you write to help figure out the total number of dots?

What is the total number of dots?

How do you know?

× =

× =

Step Up 1. Completethefirsttwomultiplicationfactstohelpyoufigure outthetotalnumberofdots.Thencompletethesixesfact.

a.

5 × 4 =

1 × 4 =

6 × 4 =

b.

5 × 8 =

1 × 8 =

6 × 8 =

2. Complete each of these.

3. Color the besidethethinkingyoucouldusetofigureouttheproduct inthesixesfact.Thenwritetheproduct.

a.6 × 6 =

b.6 × 2 =

c.6 × 9 =

5 × 6 then add 1 × 6

5 × 7 then add 1 × 7

6 × 5 then add 1 × 5

2 × 5 then add 1 × 5

5 × 6 then add 1 × 2

5 × 2 then add 1 × 2

5 × 5 then add 1 × 5

5 × 9 then add 1 × 9

1 × 6 then add 1 × 9

Step Ahead a. Write numbers to continue the pattern.

b. Write what you notice.

6 = 1 × 5 + 1

1 2 = 2 × 5 + 2

1 8 = × +

24 = × +

30 = × +

36 = × +

42 = × +

48 = × +

54 = × +

60 = × +

a.

5 × 5

=

1 × 5

=

6 × 5

=

b.

5 ×

=

1 ×

=

6 × 7

=

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204 ORIGO Stepping Stones 3 • 9.3ORIGO Stepping Stones 3 • 9.3

Introducing the Last Multiplication Facts9.3

Step Ahead Studythisfacttrail.Eachfactcanbefiguredoutfrom10×3.

Step Up 1. Whatstrategywouldyouusetosolvethesefactsinvolving3and7?

2. Whatstrategywouldyouusetosolvethesefactsinvolving3and7?

3 × 9 9 × 7

3. Writehowyoufigureoutthesefacts.

3 × 7 7 × 3

4. Writehowyoufigureout3×3and7×7.

This multiplication chart shows the zeros and ones facts and their turnarounds.

Write all the missing tens facts and their turnarounds.

How can you use the tens facts to help figure out the fives facts?

Write all the missing fives facts and their turnarounds.

Write all the products you could find using a doubling strategy.

What facts do these show?

facts facts facts

Write the products to the sixes and nines facts and their turnarounds.

What strategy do you use to figure these out?

Loop the four last products in the chart and write the matching facts below.

× =

× =

× =

× =

What do you notice about these last facts?

What strategies could you use to figure out the products?

They all involve a 3 or a 7.

× 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2

3 0 3

4 0 4

5 0 5

6 0 6

7 0 7

8 0 8

9 0 9

10 0 10

Write your own fact trail that begins with 10 × 7.

3 × 3 9 × 3 5 × 3 6 × 33 × 3

7 × 310 × 3

8 × 3 7 × 4

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Step Ahead Write about a quick way you could calculate this total.

2. Look at the pattern in Question 1 on page 206.

a. Whatshapedidyoumakewiththedots?

b. Look at the number of dots you added each time. Write what you notice.

3. Imagine you drew two more pictures. Write the matching addition and multiplication sentences.

a.

b.

4. a. Loop the square numbers.

Exploring Square Number Patterns9.4

Look at this block pyramid.

How could you figure out the number of blocks in each layer without counting each block?

What are the dimensions of each layer?

Imagine you added a fifth layer to the bottom of the pyramid.

What would be the dimensions of this layer?

How many blocks would you need? How do you know?

b. Howdidyoufigureoutifanumberwassquare?Writeyourthinkinginwords.

5536

6430

3981

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 =

Step Up 1. Drawtwomorepicturestokeepthepatterngoing.Thencompletethe addition and multiplication sentences.

1 = ×1 1

4 = ×2 2

= ×

= ×

= ×+ + + + =

+ + + =

+ =31 4

= 11

+ + =31 5

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2. Write three multiplication facts to match each of these.

Step Up 1. Write four multiplication facts to match each of these.

Part of this multiplication chart has been covered.

What are some facts that are hidden?

Deon is thinking of a multiplication fact that has a product close to 23.

What facts could he be thinking about?

Kayla is thinking of a multiplication fact that has a product greater than 70 but less than 80.

What facts could she be thinking about?

How could you use the multiplication chart to help your thinking?

Working with All Multiplication Facts9.5

6 x 4 has a product that is close.

Facts with a product close to 39 b. Facts with a product close to 52

c. Facts with a product close to 46 d. Facts with a product close to 11

a. Facts with a product greater than 60 but less than 70

b. Facts with a product greater than 30 but less than 40

c. Facts with a product greater than 20 but less than 25

d. Facts with a product greater than 35 but less than 45

× 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

Step Ahead Figureoutthemysteryfact.Writecluesforadifferentmysteryfact.Thenexchangeyourpuzzlewithanotherstudenttofindthesolution.

MYSTERY FACT

Clue 1 If you add the digits of my product the total is 10.

Clue 2 My mystery fact is a fours fact.

Themysteryfactis:

× =

YOUR MYSTERY FACT

Clue 1 If you add the digits of my product

the total is .

Clue 2 My mystery fact is a fact.

Themysteryfactis:

× =

a.

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2. Write the dimensions and the product for each of these.

Step Up 1. For each stack, write the dimensions in the order that you would multiply.Thenwritetheproduct.

How can you figure out the number of cubes in this stack?

Write a multiplication number sentence using the dimensions of the stack.

Does it matter in what order you multiply the dimensions?

What is the easiest way to figure out the total number of blocks in the stack?

Write a number sentence to show how you would multiply the dimensions of this stack to figure out the total number of blocks.

Write another number sentence to match the dimensions of a different stack with the same number of blocks.

Exploring the Associative Property of Multiplication9.6

a.

× × =

The top layer is an array of 3 rows of 4 blocks and there are 3 layers in the stack.

3 x 4 x 3 is the same as 3 x 3 x 4

× × =

a.

× × =

b.

× × =

c.

× × =

d.

× × =

f.

× × =

e.

Step Ahead a. Use5,9,and0inanyordertowritesixdifferentmultiplicationnumbersentences.Thenwritetheproducts.

9

5 0

× ×

×

b. Write what you notice.

× × = × × =

× × = × × =

× × = × × = b.

× × =

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Step AheadA school group earns $8 for each car they wash. It takes them about 15 minutes to wash 3 cars. Howmuchmoneycouldtheyearnin1hour? $

2. Writeanumbersentenceyoucouldusetosolveeachproblem.

3. Look at the popcorn prices in Question 2. Zola spent about $25 on popcorn. Sheboughtpopcornofonlyonesize.Writemultiplicationsentencestoshowwhatpopcornshecouldhavebought.

Step Up 1. Writeanumbersentenceyoucouldusetosolveeachproblem. Thenwritetheanswer.

Amber and Maya each ate an apple with lunch on Monday and Tuesday. How many apples did they eat in total?

Imagine they ate an apple with lunch every day from Monday to Friday. How could you figure out the number of apples they ate in total?

If T represents that total, what multiplication sentence could you write to match?

T=

Noah and his friends are going bowling. Each game costs $4. They have $30 in total. How many games could they play?

Solving Word Problems Involving Multiplication9.7

a. Whatisthetotalcostof4doublescoops?

b. What is the total cost of 2 single scoopsand1triplescoop?

c. Brady sells 4 banana splits each day. How much money will he take in from bananasplitsover2days?

125g

Brady’s Frozen Yogurt

Single scoop $2

Double scoop $3

Triplescoop $5

Banana split $6

a. Whatisthetotalcostof3largepopcorn?

a. Small b. Regular c. Large d. Bucket

b. How many buckets of popcorn could youbuywith$28?

c. In two hours, a cinema recorded $36 from the sale of large popcorn. Howmanydidtheysell?

POPCORN

Popcorn

Small $3

Regular $4

Large $6

Bucket $7

Working Space

Games are $4 each so I have to think of a fours multiplication fact that has a product that is close to but less than 30.

$

$

$

$

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Step Ahead Usethefactsonpages214and215tohelpsolvetheseproblems.Write number sentences on scrap paper to show your thinking.

2. Write facts to match.

Step Up 1. Completethemultiplicationfactthatyoucouldusetofigure outthedivisionfact.Thencompletethedivisionfact.

a. b.

c. d.

How could you use multiplication to figure out 30 ÷ 6?

Introducing the Sixes and Last Division Facts9.8

12 dots in total

36 dots in total

60 dots in total

18 dots in total

Look at this array.

What is an easy way to figure out the total number of dots?

I could use division. 42 Ö 6 = ?

But it is easier to think multiplication. 6 x ? = 42

2 × = 12

12 ÷ 2 =

× 6 = 36

36 ÷ 6 =

10 × = 60

60 ÷ 10 =

× 6

= 18

18 ÷ 6 =

a.

× =

÷ =

b.

× =

÷ =

c.

× =

÷ =

d.

× =

÷ =

e.

× =

÷ =

f.

× =

÷ =

24 dots in total

21 dots in total

54 dots in total

49 dots in total

30 dots in total

9 dots in total

3. Writeamultiplicationfactanddivisionfactthatyoucoulduse tosolvethisproblem.

× =

÷ =

Ticketscost$8each.Laylapaid$48intotal.

Howmanyticketsdidshebuy?

A glass of lemonade uses one lemon.

a. Sixchildreneachsqueezedeightlemonstomake some glasses of lemonade. How many glasses of lemonadecouldtheymake?

b. Theysoldallthelemonadefor$72intotal.Iftheyshare thisamountequally,howmuchmoneywilltheygeteach?

glasses

$

42 dots in total Look at this picture.

How could you figure out the number of dots in each row?

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Step Ahead Loop an amount that you think can be equally shared amongsix.Thenuseyourcalculatortocheckyourprediction.

A group of friends have been asked to arrange 60 chairs into equal rows.

Reinforcing the Sixes and Last Division Facts9.9

Step Up 1. Coloranarraytomatchthenumbersgiven. Thencompletethefactfamilytomatch.

a.

7 × 6

=

× =

÷ =

÷ =

c.

6 × 2

=

× =

÷ =

÷ =

b.

8 × 6

=

× =

÷ =

÷ =

You could have 12 chairs in each row, or 3 rows of 20, or 6 equal rows.

How can you figure out if each of these shares is possible?

Color an array below to show another way to arrange the chairs.

What two multiplication sentences and two division sentences could you write to describe the share?

96 marbles

124 marbles

150 marbles

180 marbles

2. Completeeachfact.Thendrawlinestoconnectfactsfromthesamefamily.

18 ÷ 3 =

= 24 ÷ 4

42 ÷ 7 =

9 × 6 =

= 7 × 7

30 ÷ 5 =

54 ÷ 6 =

6 × 7 =

6 × 4 =

= 6 × 5

49 ÷ 7 =

= 18 ÷ 6

3. Write the missing number in each fact.

60 ÷ 6 =

÷ 9 = 6

÷ 6 = 6

7 = ÷ 6

21 ÷ = 3

0 ÷ 6 =

24 ÷ = 6

1 = ÷ 3

a.

e.

b.

f.

c.

g.

d.

h.

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Use the comic book prices at the top of page 218 for Questions 2 and 3.

2. Writeanequationtoshowhowyoufigureoutthetotalcostofeachpurchase.Write the products in each box to help.

Step Up1. Usethecomicbookpricesabove.Writeanequationtoshow

howyouwouldfigureoutthetotalcostforeachpurchase. Thenwritethetotal.

Look at these comic books.

Investigating Order with Multiple Operations9.10

What steps would you use to figure out the total cost?

What number sentences show your thinking?

Imagine you buy three issues of C and two issues of D .

Working left to right, what is the total cost? Using the order of multiple operations, what total do you get?

What do you notice?

If there is one type of operation in a sentence, work left to right. If there is more than one type of operation, work left to right in this order. 1. perform any operation inside

parentheses2. multiply or divide pairs of numbers 3. add or subtract pairs of numbers

These are the rules for the order of operations.

a.Buyfive C and one A

b.Buy two B and one D

3. Writeanequationtoshowhowyoufigureoutthetotalcostofeachpurchase.

a.Buy six C and six A

b.Buy three B andseven C

3 × 6 + 2 × 8 =

18 + 16

Step Ahead Carlosfiguresoutthetotalcostof3kid’smealsand2adult’smeals.Hethinks:3×8+2×10andgetsatotalof$260.

a. Thecorrecttotalis $ .

b. Describe his mistake in words.

a.

Buy 5 A and 2 B

5 × 9 + 2 × 7 =

+

b.

Buy 2 B and 3 A

+

c.

Buy 3 D and 3 C

+

A B C D

$8 an issue

$6 an issue

$7 an issue

$9 an issue

Kids $8Adults $10

Imagine you want to buy one issue of A and three issues of B .

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Step Ahead Color the besidethethinkingyoucouldusetosolve each problem.

2. Complete each of these. Follow the steps used in Question 1 on page 220.

3. Complete each equation.

Step Up 1. Calculate the part in the parentheses and write the new problem. Thenwritetheanswer.

Andre has $16 and buys one of each meal deal. How much money does he have left?

What answer will they each have?

What do you notice?

What should Jacob do to make it clear that the 4 and 7 must be added first?

16−4−7=

Maria wrote this number sentence to figure it out.

16−4+7=

Jacob wrote this number sentence.

Solving Problems Involving Multiple Operations9.11

Parentheses help make it clear what to do first or what parts of the sentence should be done together. I want to add the 4 and 7 first, so Ill write 16 - (4 + 7). The answer is 5.

Deal B $7

Deal A $4

a. 15 + (8 × 5)

15 + 40Answer is 55

b. (20 + 8) ÷ 4

÷

Answer is

c. 5×(9−2)

×

Answer is

a. (21−5)÷8

Answer is

b. 9 × (4 + 5)

Answer is

c. 56−(60÷6)

Answer is

d. 20−(40÷5)

Answer is

e. 100 + (8 × 7)

Answer is

f. 37 + (8 ÷ 2)

Answer is

a.Lela had $35. She bought 4 cups that cost $3 each. Howmuchmoneydoesshehaveleft?

35−(4×3)

(35−4)×3

4×3−35

b.6 star stickers and 10 smiley face stickers were shared equally among 8 children. Howmanystickersdideachchildget?

8 ÷ 6 + 10

(6 + 10) ÷ 8

(6+10)−8

c.Julia bought 8 t-shirts that cost $8 each.Howmuchchangedidshegetfrom$100?

(8×8)−100

100−(8+8)

100−(8×8)

6 × (7 + 3) = 6 × (9 ÷ 9) = 40−(15+16)=

36 ÷ (3 × 2) = (7−7)×6=(8 + 4) × 2 =

a.

d.

b.

e.

c.

f.

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2. Writeanumbersentencetomatcheachproblem.Thenwritetheanswer.

Step Up 1. Readthewordproblem.Thencolorthe beside the thinking youwouldusetofigureouttheanswer.

Lilly bought 3 figurines that cost $6 each. She paid with a $20 bill.

How much change should she receive?

What number sentence could you write to show your thinking?

Dixon wrote this equation to figure out the change.

What part of the equation should you do first? How do you know?

Why are parentheses not needed in Dixon’s equation?

Could you use them anyway?

Writing Equations to Match Two-Step Word Problems9.12

a.Daniel had $40. He bought 4 tickets that cost $7 each. Howmuchmoneydoeshehaveleft?

40−4×7

(40−4)×7

4×7−40

b.9 apples and 3 bananas were shared equally among 6 children. Howmanypiecesoffruitwereineachshare?

9 + 3 ÷ 6

(9 + 3) ÷ 6

(9+3)−6

c.Thecoachbought9shirtsthatcost$7each.Howmuchchangedidshegetfrom$100?

(9×7)−100

100−(9+7)

100−(9×7)

20−3×6=

You dont have to use parentheses but they can make it clearer.

Step AheadWrite a word problem that uses more than one operation. Thenexchangeproblemswithanotherstudentandhave them write a number sentence to match.

a.Larahad$25.Thensheand3friendsequallysharedaraffleprizeof$60. HowmuchmoneydoesLarahavenow?

c.Manuel earned $8 each week for 6 weeks. He then bought a game for $37. How much moneydoeshehaveleft?

e.A family pass for 4 people costs $35. How much cheaper is the family pass than paying$12foreachticket?

b.Anna had $24 in her purse. She spent $19 then withdrew $20 to pay for an $8 lunch. How muchmoneydoesshehaveleftinherpurse?

d.Zola had $18 and then earned $17 more. She bought 3 books for $5 each. How muchmoneydoesshehaveleft?

f.Teena,William,andWesleysplit$36equally.Wesley then spent $7 of his share. How muchmoneydoeshehaveleft?

g.Davidbought5ticketsthatcost$7each.Howmuchchangedidhegetfrom$50?

$6 each

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Exploring Area with Customary Units10.1 2. Count the number of square inches. Write the area.

How could you measure the amount of surface a sheet of paper covers?

You could cover the sheet of paper with tiles then count the tiles.

Area is square inches

a.


c.

The amount of surface that an item covers is called area.

Step Ahead

Area is

square inches

It covers one square inch of surface. A square inch is a unit of area.


b.

Figure out the area of the orange quadrilateral.

What is the area of the surface that the block covers?

Which of these types of tiles would you use? Why?

Use an inch ruler to measure each side of an orange pattern block. How long is each side? What shape is the block?

Step Up 1. Use orange pattern blocks to cover each rectangle with no overlaps and without leaving gaps. Count the blocks then write the area.


a.


b.

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Exploring Area with Metric Units10.2 2. a. Usethegridlinestodrawthreedifferentrectangles.

Trace around a base-10 ones block and then a tens block.

Use a centimeter ruler to measure the sides of each block. Write the measurements on your tracings.

What unit of area would you call the surface that the ones block covers?

How much area does the tens block cover?

Step Up 1. Use base-10 ones blocks to cover each rectangle with no overlaps and without leaving gaps. Write the area.

Area is sq cm

A

Area is sq cm

D

Area is sq cm

E

Area is sq cm

B

Area is sq cm

C

b. In each rectangle you drew, write the area in square centimeters (sq cm).

c. Write G inside the rectangle with the greatest area.

d. Write L inside the rectangle with the least area.

A short way to write square centimeter is sq cm.

Step Ahead

b. What do you notice?

c. Why do you think this happened?

a. Measure the area of this rectangle using these blocks and write the number of each.

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Step Ahead Calculate the area of the red shape without counting the squares one at a time. Show your thinking.

Working Space

2. For each of these, use the grid lines to draw a rectangle that matches the description. Then use multiplication to calculate the area.

This picture shows that square tiles are being used to cover a floor.

How many tiles will be needed in total?

How can you use multiplication to quickly figure it out?

What is the area of the whole floor? How do you know?

Using Multiplication to Calculate Area10.3

There are 5 rows and each row will have 4 squares. 5 x 4 = 20 so 20 tiles will be needed.

5 x 4 = 20 so the area is 20 square units.

Step Up 1. Usemultiplicationtohelpyoufigureoutthetotalarea of each large rectangle.

5

4

8 units × 3 units

Area is sq units

4 units × 7 units

Area is sq units

6 units × 9 units

Area is sq units

Area is sq units

a. b. c.

d. 7 units × 8 units

Area is sq units

e. 5 units × 6 units

Area is sq units

Area is sq units

Area is sq units

Area is sq units

Area is sq units

c.

d.

a.

b.

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What are the dimensions of each green rectangle?

What is the area of each green rectangle?

What is another way you could write the dimensions of the longest rectangle?

Does it change the area of the rectangle? How do you know?

Look at the dimensions for the other two rectangles. Could you write these another way?

What could be the dimensions of a rectangle that has an area of 8 square units? How can you figure it out?

Identifying Dimensions of Rectangles10.4

Step Up 1. Drawasmanydifferentrectanglesaspossibletomatcheacharea. Write the dimensions beside each rectangle.

×

× ×

Area is 12 sq unitsa. Area is 18 sq unitsb.

Step Ahead Draw two shapes that are not rectangles and that each have an area of 16 sq units.

2. Writetwodifferentpairsofdimensionsthatwilleachgivethesamearea.

3. a. Loop one pair of dimensions from each area in Question 2.

b. Draw a rectangle to match each pair of dimensions that you looped. Label each rectangle you draw.

a.Area is 15 sq units

×

×

b.Area is 32 sq units

×

×

c.Area is 24 sq units

×

×

d.Area is 20 sq units

×

×

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Leon's parents are buying new carpet for his bedroom. His floor is 5 yards long and 3 yards wide.

How many square yards of carpet will his parents need to buy?

Draw a picture to match the story.

Dimensions tell you about distance. One dimension is length. Another dimension is width.

What are the dimensions of the floor in Leon's bedroom?

What words would you use to describe the dimensions of a wall?

What abbreviations do you know for these measurement units?

Solving Word Problems Involving Area10.5

Step Up 1. Draw a simple picture to match each story. Then label the dimensions on your picture and calculate the area.

a. A hallway is 9 feet long and 3 feet wide. The walls are 8 feet tall.Whatistheareaofthefloor?

Area sq ft

b. A garden has 9 tomato plants in it. The garden is 2 meters by 7 meters. What is the area of the garden?

Area sq m

centimeters meters feet yards

Words used to describe dimensions include length, width, depth, height, and thickness.

2. Read the stories and answer the questions.

3. Complete the equation to match the story. Then write the area.

a.

c.

a. Emma estimates that there is enough paint to cover an area of 40 square feet. She has to paint a wall that is 8 feet high and 6 feet long?

What is the area of the wall she has to paint? sq ft

Will she have enough paint?

A bulletin board is 3 feet wide and 4 feet long. It has 8 notices on it. What is its area?

A =

Area sq ft

b. A trailer is 5 feet wide and 8 feet long. It is holding 10 large boxes. What is its area?

A =

Area sq ft

d. A solar panel costs $400 and measures 5 feet by 3 feet. What is its area?

A =

Area sq ft

A tent measures 8 feet by 7 feet. Itfits4people.Whatisitsarea?

A =

Area sq ft

b. Nina's garden is 6 meters by 3 meters. She buys a bag of fertilizer that can be spread over 50 square meters.

What is the area of the garden? sq m

Will she have enough fertilizer?

Step Ahead Write an area story to match this equation. Then write the area. A = 4 cm × 7 cm

I need to find the area of Leon's bedroom. Ill call the area A.

A = 5 x 3

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2. Draw a line to split each rectangle into two parts that are easy for you to multiply. Then calculate the area.

3. Split each rectangle into two parts that are easy for you to multiply. Then calculate the area.

Step Up 1. Write the product for each part. Then write the total.

a. b.

Think about the build-up strategy you used to figure out the sixes multiplication facts.

How would you figure out the product of 6 × 7?

How could you use the same strategy to figure out the area of a rectangle that is 14 units long and 3 units wide?

Using the Distributive Property of Multiplication to Calculate Area10.6

5 rows of 7 is 35, plus 1 more row of 7 is 42. So 6 x 7 is 42.

5

71

3

410

10

3

4

I can split 14 into 10 and 4 then multiply each part by 3.

10 x 3 is 30 4 x 3 is 12

30 plus 12 is 42

It does not matter which way the rectangle is positioned, the product is the same.

3 × 16 =

3 × 10 = 3 × 6 =

5 × 13 =

5 × 10 = 5 × 3 = Step Ahead

a.

Area sq units

a.

c.

b.

d.

b.

Area sq units

Area sq units

Area sq units

Area sq units

Area sq units

12

87

13

18

3

16

5

Cole's garden is 15 feet long and 6 feet wide. Draw and label apicturetohelpyoufigureoutthe area of his garden.

sq ft

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Step Up 1. New carpet is needed in these rooms. Calculate the area ofeachfloorplan.Thenshowhowyoufigureditout.

This is a floor plan of part of a house.

How could you figure out the total area of the family room and kitchen?

Exploring the Area of Composite Shapes10.7

I would split the plan into two rooms then add the totals together. The family room is 7 units by 4 units. The kitchen is 4 units by 3 units.

Family room

Kitchen

I would start with a larger rectangle around both rooms and subtract the squares that are not used. The larger rectangle is 7 units by 7 units. The space that is not used is 3 units by 3 units.

a.

Area sq units

b.

Area sq units

Step Ahead

Drawafloorplanwithatleast two rooms and a total area of exactly 48 square units.

2. Calculatetheareaofeachblueshape.Thenshowhowyoufigureditout.

a.

Area sq units

b.

Area sq units

3. DrawfloorplanslikethoseinQuestion2.Thenwritetheareas.

a. Less than 100 squares

Area sq units

b. More than 100 squares

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2. Calculatetheareaofeachorangeshape.Thenshowhowyoufigureditout.

Step Up 1. Use a method like those above to calculate the area ofeachgreenshape.Thenshowhowyoufigureditout.

Zoe split the shape into smaller parts in a different way.

2 × 5

2 × 8

2 × 5

Cody worked with a large whole rectangle like this.

6 × 8 less 2 × 3 less 2 × 3

Logan split the shape into smaller parts and calculated the area of each part before adding to find the total.

4 × 3 6 × 2 4 × 3

What are some different ways you could figure out the area of this shape?

Calculating the Area of Composite Shapes10.8

Area sq units Area sq units

Step Ahead Figure out the area of each shape. Show your thinking.

Orange area is sq units

Blue area is sq units

Green area is sq units

b.a.

Area sq units

c.

Area sq units

b.

Area sq units

a.

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2. Color the corners green that match a green pattern block corner. Color the corners orange that match an orange pattern block corner. Color the corners yellow that match a yellow pattern block corner.

Step Up1. Color the corners green that match a green pattern block corner.

Color the corners orange that match an orange pattern block corner. Color the corners yellow that match a yellow pattern block corner.

Look at the amount of opening between the two green sides of the shape on the left below.

Compare it to the amount of opening between the two blue sides of the shape on the right below.

Comparing Angles Using Non-Standard Units10.9

A

B

C

Step Ahead Drawahexagonwithonecornerthatfitsanorangepattern block corner.

A B

C

D

Which pair of sides has the greater amount of opening between them? How could you check?

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Step Ahead

Follow these steps to make a tool to help measure angles.

When the paper is opened out, you can place a strip on it and make the strip turn to line up along the creases.

What fraction of a full turn is the strip moving each time?

After the strip turns four times it has made one full turn around a point. This is shown by the circle in the middle.

What fraction of the circle can you see when the paper is folded again?

Measuring Angles as Fractions10.10

Tear corners off a sheet of paper. Then fold it in half.

Open and draw a curved arrow in the center on both sides.

Refold the sheet of paper.

Fold it in half again.

Step Up Use the quarter-turn tester to help you measure each corner of the shapes on page 243.

a. Colorthecornersredwhichthetesterfitsexactly.

b. Color the corners blue which are larger than the corner of the tester.

c. Color the corners green which are smaller than the corner of the tester.

This tester can now measure one-eighth of a full turn.

a. Measure the corners of the shapes above using the eighth-turn tester. Loop the corners which the new testerfitsexactly.

b. Draw a hexagon that has an eighth-turn corner.

A

B

C

D

E

What fraction of a full turn can it measure when it is folded.

This tool is called a quarter-turn tester.

Fold your quarter-turn tester like this.SA

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Which of these objects is a pyramid? How do you know?

Which of the objects above is a prism? How do you know?

Identify two different prisms in the classroom. How are they different? How are they the same?

Identifying Prisms10.11

An object that has two identical faces joined together by squares or non-square rectangles is called a prism.

Step Ahead Loop the objects that are prisms.

a. b. c. d.

e. f. g.

Step Up 1. Look at this pair of objects. Use real objects to help you answer the questions.

a. How are these objects the same?

b. Howaretheseobjectsdifferent?

2. Compare these two objects.

a. How are these objects the same?

b. Howaretheseobjectsdifferent?

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What do you know about this object?

How many vertices are there?

Use a color pencil to trace over all the edges. How many edges are there?

How many faces does the prism have? What is the prism called? How do you know?

Comparing Prisms and Pyramids10.12

Step Up1. a. Use real objects to help you complete this table. The base of each object is shaded.

Prisms

Number of faces

Number of vertices

Shape of base

Number of sides on base

Look at each object below. How many vertices, edges, and faces does each have?

When two surfaces meet, they make an edge.

When three or more edges meet, they make a vertex. When there is more than one vertex, they are called vertices.

Step Ahead Think about a pyramid that has a hexagon as its base. How many faces,vertices,andedgesdoesithave?Writehowyoufigureditout.

Faces

Vertices

Edges

b. Look at the information in the table on page 246. Write about the patterns you notice.

2. a. Complete this table.

b. Write about the patterns you notice.

Pyramids

Number of faces

Number of vertices

Shape of base

Number of sides on base

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Step Up1. Eachlargeshapeisonewhole.Lookatthefirstshape.Shadethe

sameareaonthesecondshape.Writethefractionofthesecondshapethatisshaded.

The blue shape is one whole.

What fraction of it is shaded? How do you know?

The red shape is also one whole.

What fraction of it is shaded?

What do you notice about the two fractions?

Identifying Equivalent Fractions (Area Model)11.1

What do you notice about the fraction that is shaded in each of these shapes?

Step Ahead Foreachofthese,colorpartsofthewholeshapestoshow twoequivalentfractions.Thenwritethefractions.

b.

is the same as

a.

is the same as

2. Eachlargeshapeisonewhole.Colorthefirstshapetomatchthefraction. Thencolorthesameareainthesecondshapeandwritetheequivalentfraction.

Fractions are equivalent if they cover the same amount of space in each shape.

a.

is the same as

1 3

6

b.

is the same as

1 4

c.

is the same as

1 2

d.

is the same as

1 2

3. UsetheshapesfromQuestions1and2tohelpyouwriteequivalentfractions.

a.

= 2

2 4

b.

== 8

1 4

c.

== 3

6 6

d.

==2 2

8

is the same as

a.2 4

b.

is the same as

2 3

c.

is the same as

6 8

d.

is the same as

2 2

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Exploring Equivalent Fractions (Area Model)11.2

Each of these green shapes is one whole. How has each shape been split?

What is special about the group of blue shapes?

What is another number you could use to describe them?

What fraction is shaded in total?

Each of these blue shapes is also one whole.

How has each shape been split?

What fraction is shaded in total?

=Complete this equation to show how the fractions are equivalent.

Each green and blue shape is one whole.

What would you write to describe how much is shaded in each group of shapes?

Whole numbers (like 1, 2, and 3) can be written as a fraction.

In each fraction the denominator is 1.

1 = 1 1 2 = 2

1 3 = 3 1

3. UsetheshapesfromQuestions1and2tohelpyouwriteequivalentfractions.

a.8 3

6 =

b.6 4

2 =

c. 6

5 3 =

d.14 8

4 =

e.9 3

1 =

f. 4

22 8 =

g.5 4

8 =

h. 8

10 4 =

4. Writeequivalentfractions.Usewhatyouknowaboutwholenumberstohelpyou.

a. 16 =

b. 81 =

c. 6 = 1

d. 1 = 5

Step Ahead Threepizzaswerecutintosixths.Somesliceswereeaten.

Howmuchpizzaisleftover?

6

3or

Step Up 1. Eachlargeshapeisonewhole.Foreachofthese,shadethesameareaontheblueshapesbelowandwritethematchingfraction.

a. b.5 2

5 3

2. Foreachofthese,writethefractionshownbythefirstgroupofshapes. Thencolorthesameareainthegroupofshapesbelowandwritetheequivalentfraction.

a.

b.

c.

d.

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2. Eachlargeshapeisonewhole.Writebothfractions. Thenloopthegreaterfractionineachpair.

Step Up 1. Each large shape is one whole. Color the shapes to show thefractions.Thenloopthegreaterfractionineachpair.

Using an Area Model to Compare Fractions (Same Denominators)11.3

2 3

3 3

Each hexagon is one whole.

Color one-half of the unshaded hexagon.

Write the fraction.

2 2

Which fraction is greater in each pair?

When the denominators are the same, how do you decide which fraction is greater?

1 6 or

5 6

2 3 or

1 3

4 6 or

2 6

Color parts and write fractions to keep each pattern going.

a. b.

3. Eachgreenandblueshapeisonewhole.Writethefractionshownbyeachgroupofshapes.Thenloopthegreaterfractionineachpair.

4. Writeafractiontocompleteeachsentence. UsetheshapesfromQuestions1,2,and3tohelpyou.

6

3 6

is greater than

a. 8

5 8

is greater than

b.13 6

6

is greater than

c.

Step Ahead Each large shape is one whole. Color each shape to show afractionthatmatchesthedescription.

a. greater than 3 8

less than 6 8

b. greater than 1 6

less than 3 6

c. greater than 4 8

less than 8 8

d. greater than 3 6

less than 6 6

a. b. c.c.

a.

3 4

1 4

b.

1 3

2 3

c.

5 6

1 6

6 6

5 6

4 6

2 6

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Relating and Comparing Unit Fractions (Related or Unrelated Denominators)11.4

What fraction of each strip has been shaded?

Which strip shows the greatest fraction shaded?

Which strip shows the least fraction shaded?

When you write

1 3

what does the 3 tell you?

When you write 1 5 what does the 5 tell you?

Why is

1 5

less than

1 3 ?

Which fraction is greater in each pair? How do you know?

1 8 or

1 12

1 20 or

1 50

1 2

1 2

1 4

1 4

1 4

1 4

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 8

Step Up 1. a. Coloronepartineachrowofthisfractionchart.

b. Loopthefractionthatisgreaterineachpair.

1 2

or

1 4

1 8

or

1 2

1 4

or

1 8

1

1 2

1 3

1 4

1 5

1 6

1 7

1 8

3. LookatthefractionsyouloopedinQuestions1and2.Whatdoyounotice?

4. Writeotherfractionstomakethesesentencestrue.

2. a. Coloronepartineachrowofthisfractionchart.

1 3

1 3

1 3

1 6

1 6

1 6

1 6

1 6

1 6

1 9

1 9

1 9

1 9

1 9

1 9

1 9

1 9

1 9

1

b. Loopthefractionthatisgreaterineachpair.

1 3

or

1 9

1 6

or

1 3

1 6

or

1 9

a. 1 2

is the same as

which is the same as

.

b.

1 3

is the same as

which is the same as .

Step Ahead Writethesefractionsineach sentence to make it true.

2 6

1 2

1 9

1 3

a.

is the same as

which is less than

which is more than

.

b.

is more than

which is the same as

which is more than

.

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2. Loopthefractionthatisgreaterineachpair.Usethisfractioncharttohelp.

Look at this fraction chart. What fractions could you shade in each row?

Using a Length Model to Compare Fractions (Different Denominators)11.5

1 2

1 2

1 4

1 4

1 4

1 4

1

Step Up 1. Loopthefractionthatisgreater in each pair. Usethefractioncharttohelp.

3. Eachrowofthisfractionchartshowstwowholes.

Loopthefractionthatisgreater in each pair.

Step Ahead Loop the greatestfractionineachgroup. Usethefractionchartsabovetohelpyou.

Loop the greater fraction in each of these pairs of fractions.

What helps you to decide which fraction is greater?

Which is greater: 1 3 or 2

4 ? How do you know?

I know that is less than half a row in the fraction chart. I also know that is exactly half a row. So is greater than .

1 3

1 3

2 4

2 4

1 3

or

2 4

3 4

or

2 3

1 3

or

1 4

1 3

or

1 2

3 3

or

1 2

c.1

or

1 4

d.

1 3

1 3

1 3

1

1 4

1 4

1 4

1 4

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 6

1 6

1 6

1 6

1 6

1 6

1 3

1 3

1 3

1 4

or

2 6

e. 3 4

or

2 3

f.

2 4

or

2 6

a. 2 8

or

2 3

b.

3 3

or

1 8

g. 3 6

or

3 4

h.

a. 2 6

or

2 8

b. 3 6

or

3 4

c. 4 6

or

4 8

d. 2 3

or

4 8

e. 1 8

or

1 3

f. 5 6

or

1 4

11 2

1 2

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 6

1 6

1 6

1 6

1 6

1 6

1 5

1 5

1 5

1 5

1 5

11 2

1 2

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 6

1 6

1 6

1 6

1 6

1 6

1 5

1 5

1 5

1 5

1 5

a. 12 6

or 12 8

b. 8 6

or

8 8

c. 3 6

or

3 5

d. 8 5

or 12 8

e. 7 8

or

7 5

f. 4 6

or

4 8

g. 4 5

or 1

h. 9 5

or

3 2

a. 2 6

or

2 3

or

2 8

b. 4 3

or

4 8

or

8 5

c. 5 5

or 10 6

or 15 8

11 2

1 2

1 4

1 4

1 4

1 4

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 8

1 6

1 6

1 6

1 6

1 6

1 3

1 3

1 3

1 6

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What is a reasonable estimate of the difference between the prices of these two guitars?

How could you figure out the exact difference?

Reviewing Informal Methods to Subtract11.6

Claire and Victor both split $523 into hundreds, tens, and ones and then subtracted the parts from $685. They recorded their thinking this way.

What is the same about their methods? What is different?

How could you use each method to figure out the difference between these prices?

$685 $523

$596 $325

Claire

H T O

6 8 5

− 3

6 8 2

− 2 0

6 6 2

− 0 05

6 21

Subtract the ones

Subtract the tens

Subtract the hundreds

Victor

H T O

6 8 5

− 0 05

1 8 5

− 2 0

1 6 5

− 3

6 21

Subtract the hundreds

Subtract the tens

Subtract the ones

Step Up Estimatethedifferencebetweenthetwoprices.Thenuseoneof themethodsshownonpage258tocalculatetheexactdifference.

$136$478a.

Estimate $

H T O

4 7 8

−

−

−

$431$652b.

Estimate $

H T O

−

−

−

$113$584c.

Estimate $

H T O

−

−

−

$643$758d.

Estimate $

H T O

−

−

−

$462$985e.

Estimate $

H T O

−

−

−

$147$297f.

Estimate $

H T O

−

−

−

Step Ahead Writethemissingdigitstomaketrueequations.

a. 9 6 − 4 =731b.

5 − 3 2 =156

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Step Up Estimatethedifferencebetweenthetwoprices.Thenusethestandardsubtractionalgorithmtocalculatetheexactdifference.

Evan and Dixon used different methods to calculate the difference between 578 and 263.

How are their methods the same? How are they different?

Introducing the Standard Subtraction Algorithm11.7

Dixon’s method is called the standard algorithm for subtraction.

Where else have you heard the word algorithm? How is the standard addition algorithm the same as the standard subtraction algorithm?

How do you think the subtraction algorithm works with these problems?

What is another way you could figure out some of these problems?

H T O

3 6 7

− 2 5

H T O

6 4 5

− 3

T O

8 4

− 5 3

T O

5 8

− 6

Step AheadLookattheproblemsabove.Choosetwoorthreeproblemsthatyouthinkyoucansolvewithoutusingthestandardsubtractionalgorithm.Showyourthinkingbelow.

Evan’s method

Step 3

H T O

5 7 8

− 3

5 7 5

− 6 0

5 1 5

− 0 02

1 53

Step 2

H T O

5 7 8

− 3

5 7 5

− 6 0

5 1 5

−

Step 1

H T O

5 7 8

− 3

5 7 5

−

−

Dixon’s method

Step 3

H T O

5 7 8

− 6 32

3 1 5

Step 2

H T O

5 7 8

− 6 32

1 5

Step 1

H T O

5 7 8

− 6 32

5

a.

Estimate $

T O

−

H

$769 $238b.

Estimate $

−

T O

c.

Estimate $

T O

−

H

d.

Estimate $

e.

Estimate $

T O

−

H

f.

Estimate $

T O

−

H

g.

Estimate $

T O

−

H

h.

Estimate $

T O

−

H

6 97

3 82

$32 $57 $849 $217 $68 $5

−

T O

$756 $122 $978 $6 $499 $86 $26 $859

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Chang has $92 and buys this game.

How much money will he have left over? How could you figure it out using base-10 blocks?

Working with the Standard Subtraction Algorithm (Decomposing Tens in Two-Digit Numbers)11.8

Using the standard subtraction algorithm is like using base-10 blocks.

You need to regroup when the top digit in a place-value column is less than the bottom digit in the same column.

I would show 92 using 9 tens blocks and 2 ones blocks. Then I would have to trade 1 tens block for 10 ones blocks so that I have 8 tens and 12 ones.

T O

9 2

− 3 8

T O

9 2

− 3 8

8T O

9 2

− 3 8

8 12T O

9 2

− 3 8

45

8 12

Step Up1. Changetheblockstoshowtheregrouping.Thenchange

thenumbersandusethestandardsubtractionalgorithmtocalculatethedifference.

a.

Estimate

T O

7 3

− 6 5

b.

Estimate

T O

4 6

− 1 8

c.

Estimate

T O

6 2

− 2 7

d.

Estimate

T O

9 5

− 6 8

2. Estimatethedifference. Thenusethestandardsubtractionalgorithmtocalculatetheexactdifference.

Step Ahead Writethemissingdigitsinthesestandardsubtractionalgorithmsandregroupwhennecessarysothattheanswersmakesense.

a.

−

T O

5 7

2

3 2

b.

−

T O

7

6 5

c.

−

T O

6 4

5

1 9

d.

−

T O

4

8

3 7

2

T O

7 3

− 4 6

T O

6 4

− 3 7

5 14

T O

4 5

− 2 9

T O

5 2

− 8

Step 1

Look at the digits in each place. Can you subtract each

place easily?

Step 2

You need 1 ten to help subtract the ones. Cross

out the 9 tens and write 8 tens.

Step 4

Subtract the ones then subtract

the tens. 12 ones take 8 ones. 8 tens

take 3 tens.

Step 3

Cross out the ones digit and write the

new total. 92 is now written as

8 tens and 12 ones.

$38$92

a.

c.

b.

d.

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Step Ahead

Thedifferencebetweentheirnumbersis148.

Writetwonumbersthat wouldgivethatdifference.

2. Estimatehowmuchmoneywillbeleftaftereachpurchase. Thenusethestandardsubtractionalgorithmtocalculatetheexactamount.

Step Up 1. Estimatethedifference.Thenusethestandardsubtractionalgorithmtocalculatetheexactdifference.

Working with the Standard Subtraction Algorithm (Decomposing Tens in Three-Digit Numbers)11.9

How much money will he have left if he buys the book about machines?

How could you figure out the difference using base-10 blocks?

What would you write using the standard algorithm?

If Jack buys the book about pets instead, how much money will he have left?

Use the standard algorithm to show your thinking.

$37 $9

Jack is at a bookstore and has $283.

H T O

2 8 3

− 3 7

2 4 6

7 13I need more ones so I would trade 1 ten for 10 ones. 283 is now written as 2 hundreds, 7 tens, and 13 ones.

H T

−

OI could figure out these problems in my head or use a number line but using the standard algorithm is good practice for when I subtract bigger numbers.

a.

Estimate

T O

−

H

7 33

4 9

d.

Estimate

T O

−

H

4 68

2 9

c.

Estimate

T O

−

H

7 45

4 6

b.

Estimate

T O

−

H

5 26

3 7

a.

Estimate $

T O

−

H

$35 $172b.

Estimate $

T O

−

H

c.

Estimate $

T O

−

H

d.

Estimate $

T O

−

H

e.

Estimate $

T O

−

H

f.

Estimate $

T O

−

H

g.

Estimate $

T O

−

H

h.

Estimate $

T O

−

H

$27 $145

$35 $154

$35 $161

$9 $143 $27 $175

Petaisthinkingofathree-digitnumber. Paigeisthinkingofatwo-digitnumber.

$18 $152

$8 $136

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2. Azoohasthesedifferentsnakesondisplay.

Step Up 1. Estimatethedifference.Thenusethestandardsubtractionalgorithmtocalculatetheexactdifference.

Working with the Standard Subtraction Algorithm (Decomposing Hundreds)11.10

A gardener had 235 seedlings to plant. In the first garden bed, he planted 72 seedlings. How many seedlings does he have left to plant?

How could you figure it out using base-10 blocks?

Look at the steps used in this standard subtraction algorithm. What is happening in each step?

T O

−

H

3 527 2

Step 1T O

−

H

3 527 2

Step 2

1T O

−

H

3 527 2

Step 3

131O

−

H

3 527 2

Step 4

13

6 31

a.

Estimate

T O

−

H

2 93

5 4

d.

Estimate

T O

−

H

3 75

8 6

c.

Estimate

T O

−

H

4 86

7 7

b.

Estimate

T O

−

H

1 54

3 2

Snake Length (cm)

FoxSnake 65

KingCobra 669

Bushmaster 365

Coachwhip 96

Snake Length (cm)

Anaconda 847

EasternIndigo 236

LyreSnake 74

Copperhead 81

Usethestandardsubtractionalgorithmtocalculatetheexactdifferenceinlength. Remembertoestimate,tocheckthatyouranswermakessense.

a.

KingCobra and

Coachwhip −

T OH b.

EasternIndigo and

Copperhead −

T OH

c.

Anaconda and

FoxSnake −

T OH d.

Bushmaster and

LyreSnake −

T OH

e.

KingCobra and

Copperhead −

T OH f.

Anaconda and

Coachwhip −

T OH

Step Ahead SnakeAis36cmlongerthanSnakeB.SnakeCis245cmlong.SnakeBis128cmshorterthanSnakeC.Howlongiseachsnake?

SnakeAis cm SnakeBis cm SnakeCis cm

T

I would show 235 using 2 hundreds blocks, 3 tens, and 5 ones. Then I would take away the number of seedlings he has planted.

I can take 2 ones from 5 ones but I need to trade 1 hundreds block for 10 tens blocks so that I have 13 tens.

1

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Step Up 1. Usethestandardsubtractionalgorithmtocalculatetheexactdifferencebetweeneachpairofprices.

How would you calculate the difference between each price and the amount in the wallet?

2. Usethestandardsubtractionalgorithmtocalculatehowmuchthepricehasdropped.Remembertoestimate,tocheckthatyouranswermakessense.

Exploring Subtraction Involving Zero11.11

T O

−

H

8 05

2 6

7

$126

I can't take 6 ones from 0 ones so I need to trade 1 ten for 10 ones.

$580

I can't take 8 tens from 0 tens so I need to trade 1 hundred for 10 tens.

T O

−

H

0 63

8 2

10

$306$182

T O

−

H

7 54

6 0$160

$475

I can easily take 0 ones from 5 ones. I don't need to change any digits at all.

b.

−

T OH

$608

a.

−

T OH$480

$134

a.

−

T OH b.

−

T OH

c.

−

T OH d.

−

T OH

e.

−

T OH f.

−

T OH

Step Ahead

1

1

1

2

10

240

470

58 182

978Eachbrickshowsthetotalofthetwobricksdirectlybelow.Writethemissingnumbers.Recordyourthinkingon some scrap paper.

$271

$370

$256

$147

$85

$580

$439

$409

$216

$748

$604

$138

$90

Was

Now

Was

Now

Was

Now

Was

Now

Was

Now

Was

Now

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2. Solvetheseproblems.Showyourthinking.

Think about all the different methods that you can use for subtraction.

Which method would you use for each of these problems? Why?

Consolidating Subtraction Methods11.12

What steps would you take to solve this word problem? What methods would you use?

Olivia had $590 in her bank account. If she buys a bike that costs $345 and a helmet for $23, how much money will she have left in her bank account?

320 − 315 = ? ? = 684 − 7 ? = 735 − 384

126 − 278 = ? 428 − 75 = ? 316 − ? = 43

Step Up 1. Solvetheseproblemsusinganywrittenmethodyoulike. Showyourthinking.

a.

674−358=

b.

804−72=

Step Ahead Writeastorytomatchthis.

b. Aspoolofropewas328feetlong.Apiecethatwas138feetlongwascutoff.Thenanotherpiecemeasuring76feetlongwascutoff.Howmuchropeislefton thespool?

c. Akarihas$75saved.Shesawtwoguitarsthatsheliked.Theredguitar is$167andthepurpleguitaris$135.Aguitarstrapcosts$21.IfAkariwantstheredguitarandthestrap,howmuchmoremoneydoessheneed?

a. Tylerwalked163stepsfromhishousetoschool.HisfriendGavinwalked148stepsfromhishousetothesameschool.WhatisthedifferencebetweenthedistancesTylerandGavinwalked?

steps

$

127 + 48 then subtract 136

I'll call the total cost C. C = 345 + 23

What Olivia has left is 590 Ð C.

feet

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Step Ahead The border of an analog clock is like a curved number line. What fractions can you see on an analog clock?

2. On each number line, the distance from 0 to 1 is one whole. Write the fractions shown by the green and red arrows.

Step Up1. On each number line, the distance from 0 to 1 is one whole.

Write the fraction shown by the green arrow. Then write the equivalent fraction.

On each number line, the distance from 0 to 1 is one whole.

What fraction is the green arrow pointing to? How do you know?

Identifying Equivalent Fractions (Number Line Model)12.1

This number line is also split into parts.

What fraction is the red arrow pointing to? How do you know?

=Fractions are equivalent if they are located at the same point on a number line.

What can you say about the fractions at each arrow?

Complete this equation to show how the fractions are equivalent.

0 1

0 1

a.

0 1

3. Usethenumberlinestohelpyoufigureouteachfraction.

Which fraction is closest to 1 3 ?

4


4


8


8


6


6

b.

0 1

0 1

0 1

0 1

a. b.

0 1

a.

c.

e.

b.

d.

f.

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On each number line, the distance from 0 to 1 is one whole. What fraction is the green arrow pointing to? How do you know?

What could you do to the number line to figure out the equivalent fraction in eighths?

Exploring Equivalent Fractions (Number Line Model)12.2

Complete this equation to show how the fractions are equivalent.

Draw a red arrow pointing to the equivalent fraction.

=

What fractions could you write to match where the blue arrow is pointing on this number line?

How did you figure it out? 0 31 2

I would split the distance between each whole number into eighths.

0 21

0 21

Step AheadOn this number line, the distance from 0 to 1 is one whole. Draw a line from each fraction to show where it is located on the number line.

2. On each number line, the distance from 0 to 1 is one whole. Draw a line from each fraction to its position on the number line. Then write the equivalent fractions.

0 31 2

1 5

3 5

5 5

4 5

0 1

0 41 2 3

3. Loop all the fractions that are equivalent to 2.

8 8

12 4

12 6

9 3

4 2

14 8

Step Up1. On this number line, the distance from 0 to 1 is one whole. Write

the fourths shown by the green arrows. Then write the matching fractions in eighths.

0 21

4

8

4

8

4

8

4

8

a.

e.

b.

f.

c.

g.

d.

h.

= 5

3 = 8

3 = 10

6 = 18

6

= 8

2= 5

2= 4

2= 3

2

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Step Ahead A stack of magazines is 2 inches high. Each magazine is 1 8 of an

inch thick. More of the same magazines are added to make the stack 1

2 an inch higher.

2. Usethenumberlinesonpage276tohelpyoufigureouttheanswers to these problems. Show your thinking.

The first number line has been divided into halves.

For each number line, write how the distance between each whole number has been divided.

Solving Word Problems Involving Fractions12.3

a. The library has shelves of dictionaries. Each dictionary covers 1

8 of a shelf. One of the shelves is only half full. How many dictionaries are on that shelf?

0 21

0 21

0 21

0 21 halves

Carter has some toy railroad cars that are each 1 6 of a foot long.

How many cars will be needed to make a train that is 4 3 of a foot long?

How can the number lines help you?

Step Up 1. Usethenumberlinesabovetohelpyoufigureouttheanswer to this problem.

There is a stack of blocks that is half a foot high.

Each block is 1 6 of a foot high.

How many blocks are in the stack?

b. Ashley brought two pies to a picnic. Each person ate 1

8 of a pie. At the end of the picnic only 1

4 of one pie was left over. How many people were at the picnic?

c. Sumineedsfive 1 4 cups of milk for

a recipe. She only has a 1 8 cup

measure. How many times will she needtofillthe 1

8 cup measure?

d. Jamal needs 6 8 of a liter of water

for a recipe. His measuring jug only holds 1

4 of a liter. How many times willheneedtofillthejug?

How many magazines are in the stack now?Show your thinking.

a.

How many more magazines could be added to make the stack 3 inches high? Show your thinking.

b.

0 21

The total train length is L. If L = , then first I need to know what L is in sixths. I can see that is the same as . If one car is of a foot long, then Carter will need 8 cars.

4 3 4

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Step AheadOn this number line, the distance from 0 to 1 is one whole. Write a fraction to match each description. Then draw a line from each fraction to show its location on the number line.

a.greater than

4 4

less than 7 4

c.greater than

15 4

less than 17 4

b.greater than

10 4

less than 13 4

d.greater than

17 4

less than 20 4

0 1 53 42

2. On these number lines, the distance from 0 to 1 is one whole. Loop the fraction that is greater in each pair. Use the number line to help you.

Using a Number Line Model to Compare Fractions(Same Denominators)12.4

On these number lines, the distance from 0 to 1 is one whole.

What do the marks between 0 and 1 on this number line show? How do you know?

How can you figure out which mark shows six-fourths?

Where would you label 5 4 and 7

4 on the number line? Which fraction is greater?

0 21

Where would you label 7 8 and 10

8 on the number line?

Which fraction is greater? How do you know?

What fractions could you show on this number line?

0 21

0 1 2

3. Loop the greater fraction in each pair.

0 1 32

20 6

or 18 6

16 2

or 9 2

10 4

or 12 4

Step Up1. On this number line, the distance from 0 to 1 is one whole. Draw

a line to show where each fraction is located on the number line. Then loop the fraction that is greater in each pair.

0 1 2

a. b. c. d.or

15 8

12 8

or17 8

11 8

or3 8

7 8

or6 8

9 8

e. f. g. h.or

10 6

12 6

or15 6

13 6

or2 6

11 6

or7 6

5 6

a.

c.

b.

d.

or4 4

2 4

or7 4

9 4

or3 4

5 4

or10 4

7 4

a. c.b.

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Step Ahead a. Split the distance from 0 to 1 into sixteenths and write the correct fraction below each mark.

2. On each number line, the distance from 0 to 1 is one whole. Label each green mark above the number line and each red mark below. Then write numerals and < or > to complete the number sentences.

Step Up 1. On each number line, the distance from 0 to 1 is one whole.

Using a Number Line Model to Compare Unit Fractions (Related and Unrelated Denominators)12.5

Jose wants to split the line into eighths. Ella wants to split it into sixths and Liam wants to split it into fourths.

Jose said that using eighths would mean 1 3 is the greatest fraction because

8 is greater than 6 and 4. Liam said that the greatest fraction would be 1 4 .

Who is correct? How do you know?

Three students are talking about fractions on a number line.

0 1

a. Write the correct fraction above each mark on the number line.

b. Split the distance from 0 to 1 into eighths and write the correct fraction below each mark.

0 1

c. Write the correct fraction above each mark on the number line.

d. Split the distance from 0 to 1 into sixths and write the correct fraction below each mark.

Write < or > to complete each of these.

a.

1 1

b.

1 1

c.

1 1

d.

1 1

0 1

1 8

2 8

3 8

4 8

6 8

5 8

7 8

b. Complete these equations.2 8

16=

166 8=

16=1

0 1

0 1

0 1

0 1

10

or1 6

1 4 or1

41 8 or1

81 6

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Step AheadWrite the least fraction and the greatest fraction possible using these digits. Use the number lines above and what you know about whole numbers to help you.

2. On each number line the distance from 0 to 1 is one whole. Write <, >, or = to make each sentence true.

Step Up 1. On each number line, the distance from 0 to 1 is one whole. Write <, >, or = to make each sentence true.

On each number line, the distance from 0 to 1 is one whole.

Using a Number Line Model to Compare Fractions (Different Denominators)12.6

0 1

0 1

0 1

Which fractions are equivalent to 1 2 ? How do you know?

Which fractions are less than 1 2 ? How do you know? Which fractions are greater than 1

2 ?

Use the number lines to complete these number sentences.

Which fractions are equivalent to 1?

What are some fractions that are greater than 1?

=1 2

<

1 2

>

1 2

0 31 2

e.

6 6

6 4

g.

10 4

10 6

h.

3 1

3 6

f.

12 6 2

0 21

a.

2 3

2 8

b.

5 8

5 3

c.

3 3

8 8

d.

2 32

0 42 31

0 31 2

0 631 2 4 5

a.

8 3

8 4

c.

11 4

11 3

b.

4 4 2

d.

12 4

9 3

e.

12 6

15 6

f.

9 8

9 6

g.

13 6

13 8

h.

3 6

4 8

i.

12 2

12 3

k.

8 3

8 2

l.

5 1 4

j.

3 3 2

3 1 4 6a.

least greatest

1 2 4 1b.

least greatest

8 4 3 6c.

least greatest

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2. On each number line, the distance from 0 to 1 is one whole. Complete the missing fractions so that the order is from least to greatest.

Step Up1. On each number line, the distance from 0 to 1 is one whole. Use the

number lines to help you order these fractions from least to greatest.

Amos has three different recipes to make pancakes.

Which recipe uses the greatest amount of milk?

How could you use the number lines below to help you figure it out? On each number line, the distance from 0 to 1 is one whole.

Ordering Fractions12.7RECIPE 1

cup of milk34

RECIPE 2cup of milk1

2

RECIPE 3cup of milk2

3

0 1

0 1

0 1

Write the fractions in the recipes in order from least to greatest.

0 21

b.

0 31

a.

2

0 31

a.

2

2 3

3

7 6

3

12 6

0 41

c.

2

5 2

2

13 4

4

8 2

0 1

b.

2

3 8

6

6

7 8

6

3

Step AheadUse each digit once to write fractions that are in order from least to greatest. Use the number lines above and what you know about whole numbers to help you.

6 3 1 4 4 6a.

least greatest

2 8 6 3 1 2b.

least greatest

3 1

2 3

9 4

5 3

8 6

7 4

5 6

4 4

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2. Complete each table to show equivalent fractions and whole numbers.

Which whole numbers are equivalent to these fractions? How do you know?

What do you notice about the numerators?

Analyzing Whole Numbers and Fractions12.8

Think of other fractions with 4 as a denominator.

Which of these are equivalent to whole numbers?

How can you figure it out?

0 42 31

Step Up 1. Complete each table to show the equivalent fractions and whole numbers. Use multiplication patterns to help you.

Fraction 3 3

3

3

12 3

15 3

3

3

Whole number 1 2 3 8 10

a.

Fraction 8 8

8

8

40 8

8

72 8

8

Whole number 2 3 6 10

b.

Fraction 4

4 4

4

36 4

4

40 4

4


a.

Fraction 42 6

6

12 6

6

48 6

6

6


b.

4 4

8 4

12 4

The equivalent fraction for 5 will be because there are 5 wholes made of 4 parts each.

20 4

The numerators are all products of a fours multiplication fact.

Loop the fractions that you think are equivalent to whole numbers. What do you notice?

1 2

2 2

3 2

4 2

5 2

6 2

7 2

8 2

3. Use the information in the tables in Questions 1 and 2 to help you figureouteachofthese.

a. Loop the fractions that are between 2 and 3.

b. Loop the fractions that are between 5 and 6.

c. Loop the fractions that are between 9 and 10.

8 3

7 2

17 8

26 8

10 4

16 6

20 3

32 6

14 3

43 8

23 4

35 6

37 3

40 6

78 8

32 4

22 3

19 2

Step Ahead Alisa says 3 3 is greater than 8

8 . Alex says 3 3 is less than 8

8 .

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Step Up 1. Use a centimeter ruler to measure the sides of each shape. Label each side then calculate the perimeter. Show your thinking.

Juan is building a fence to make a chicken coop. This is his plan.

What is the total length of the fence? How could you figure it out?

Exploring the Perimeter of Irregular Polygons12.9

Perimeter is another name for the total distance around a shape.It comes from two old Greek words: peri meaning “around” and metron meaning “measure”.How is calculating the perimeter different

from calculating the area of the coop?

15 ft

15 ft

8 ft 8 ftI would add the side lengths.

I would add the length and the width and double the total.

a.

Perimeter cm

2. Use a centimeter ruler to measure the sides of each shape. Label each side then calculate the perimeter.

a. b.

Perimeter cm Perimeter cm

3. This is a picture of a larger shape. Calculate its perimeter.

Perimeter m

Step Ahead This shape has been made by joining two rectangles. Calculate the length of the unknown sides.

Side A m

Side B m

B

A

20 m7 m

32 m

16 m

55 m

28 m

32 m

37 m

45 mb.

Perimeter cm

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2. These are pictures of larger shapes. Each shape has sides that are equal in length. Calculate the perimeter of each shape and write an equation to show your thinking.What do you know about squares?

How could this help you figure out the perimeter of a square?

Use a centimeter ruler to measure one side of this square.

Complete this equation to show how to calculate the perimeter.

Exploring the Perimeter of Regular Polygons12.10

+ + +

= cm

Step Up 1. For each shape, all sides are equal in length. Use a centimeter ruler to measure and calculate the perimeter of each shape. Write an equation to show your thinking.

a. =

cm

b. =

cm

=

m

a.

9 m

=

m

b.

13 m

=

mc.

=

m

d.

11 m6 m

Step Ahead Calculate the perimeter of these hexagons. Each hexagon has sides that are equal in length. Show your thinking.

Side is 14 m Perimeter is m

a.

Side is 18 m Perimeter is m

b.

Working Space

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2. Read the stories and answer the questions.

Isabelle is making edges for a rectangular garden. She wants to plant 5 rows of carrots. The dimensions of the garden are 7 feet by 6 feet.

What is the perimeter of the garden?

What picture can you draw to match the story?

Solving Word Problems Involving Perimeter12.11

Which numbers in the story helped you?

Which numbers were not important to know? Why?

Step Up 1. Draw a simple picture to match each story. Then label the dimensions on your picture and calculate the perimeter.

a. Koda is making a frame for a rectangular picture. The dimensions of the picture are 15 inches by 9 inches. What is the perimeter of the picture?

in

b. Tia walked around the outside of a rectangular playground with 3 friends. Two sides are each 16 meters and two sides are each 7 meters. How far did Tia walk?

m

When I see the dimensions for a rectangle I need to remember that opposite sides will be the same length. So if the length is 7 feet I know that two sides of the rectangle will each be 7 feet.

a. Mary’s grandma is sewing a border on a rectangular quilt. The quilt is 82 inches by 55 inches. What is the perimeter of the quilt?

in

b. The perimeter of a rectangular barnyard is 64 meters. Each long side measures 19 meters. What is the length of each short side?

m

Step Ahead The perimeter of a rectangle is 48 yards. What could be the dimensions of the rectangle? Show your thinking.

c. A corral has 8 equal sides. The fence on each side is 12 ft long and 5 ft high. What is the perimeter of the corral?

ft

d. The border of a square window has a perimeter of 64 inches. What is the length of each side?

in

Length is

yd Width is

yd

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ORIGO Stepping Stones 3 • 12.12 ORIGO Stepping Stones 3 • 12.12

Step Ahead

Draw a shape that is not a rectangle.Make the area 12 sq units.

b. Use the rectangles you drew in Question 1a on page 294 to complete this table.

What does the perimeter of a shape tell you?

What does the area of a shape tell you?

How do you calculate perimeter? How do you calculate area?

What could be the dimensions of a rectangle with an area of 12 sq cm?

Draw 3 different rectangles that each have an area of 12 sq cm and label the dimensions.

Exploring the Connection Between Perimeter and Area12.12

Rectangle Length (cm) Width (cm) Perimeter (cm) Area (sq cm)

A 16

B 16

C 16

Inside each rectangle write its perimeter. What do you notice?

Step Up 1. a. Drawthreedifferentrectanglesthateachhaveanarea of 16 sq cm. Label the rectangles A, B, and C.

2. a. Drawthreedifferentrectanglesthateachhaveanareaof20sqcm. Label the rectangles D, E, and F.

Rectangle Length (cm) Width (cm) Perimeter (cm) Area (sq cm)

D 20

E 20

F 20

b. Use the rectangles you drew above to complete this table.

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