Estimate: 3 5

13 173 31 )5 ( 1 )(

13 173 13 )5 ( 3 )( 1 3 )( 1 3 )(

Is your estimate reasonable?

13 3 17

13 3 17 5

10 3 10 7

10 10 10 7 3 10 3 7

10 20 200

10 Π10 = 100

10 Π7 = 70

3 Π10 = 30

3 Π7 = 21 100

221

221

yes

120 1

10 20 200

10 3

10 10

10 7

10 7 3 10 3 7

10 Π10 = 100

10 Π7 = 70

3 Π10 = 30

3 Π7 = 21100120

1221

221

yes

Anchor Video: “Paint the Town”

Display the first problem on RDI Practice, page 42. Read the directions aloud.

Now, let’s multiply two 2-digit numbers. First, let’s estimate the answer. How can we estimate 13 × 17? (10 × 20 = 200)

MATHEMATICAL PRACTICE Reason Abstractly

Guide students in renaming the factors and applying the Distributive Property.

First we use place value to split each factor.

■ How do we rewrite 13 × 17 as the product of two expressions? (Write 13 as 10 + 3 and 17 as 10 + 7.)

■ How do we find the product? (Multiply the parts to find the partial products, and then add the products.)

Have students complete the problem as shown and check the reasonableness of their solutions.

Multiply a 2-digit by a 2-digit number.

Review the Anchor Video “Paint the Town”.

Multiply Greater Factors

Replay the Anchor Video “Paint the Town”.

The video shows how artists use multiplication and measurement to create a giant mural.

Ask targeted questions to help students connect area and measurement to multiplication.

■ How did the artists approach the job of painting the buildings? (They made a drawing on a grid to show each of the parts.)

■ In what other ways did they use grids? (They used grids to find the dimensions of the walls.)

■ How did the grids help the artists to find the dimensions of the walls? (They used the grids to show length and width so they could find the areas of the walls.)

We use multiplication to find area.

Students apply the Distributive Property and write equations to multiply two 2-digit factors.

■ The Distributive Property is the basis for the standard algorithm.

■ Learning how to apply the Distributive Property prepares students for simplifying and solving algebraic expressions.

Reason Abstractly Using properties of operations helps students to make sense of quantities and their relationships.

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Estimate:

Ramón paints a mural on the side of his school. The mural is 24 feet long and 16 feet high. How many square feet does Ramón paint?

3 5

24 163 31 )5 ( 1 )(

24 163 13 )5 ( 3 )( 1 3 )( 1 3 )(

sq. ft.

24 3 16

24 3 16 5

20 4 10 6

20 10 20 6 4 10 4 6

20 20 400

20 Π10 = 200 20 Π6 = 120

4 Π10 = 40 4 Π6 = 24

300

384

80 4

384

20 20 400

20 4

20 10

10 6

20 6 4 10 4 6

20 Π10 = 20020 Π6 = 120

4 Π10 = 40 4 Π6 = 24

300804

384

EXploRE

EXtEnD

Write an Equation

Model this problem-solving strategy with your students.

Apply the Distributive Property to solve a contexualized problem.

Have student pairs consolidate their skills as you circulate. 43

PRACTICE Students apply the Distributive Property to multiply 2-digit by 2-digit numbers.

■ How did you split the factors? ■ How did you find the partial

products?

REFLECT Students complete the refl ection on how to apply the Distributive Property.

■ How is this process different than the process of multiplying a 1-digit number by a 2-digit number?

CHALLENGE Students apply the Distributive Property to solve another real-world problem.

■ How does writing an equation help you solve the problem?

■ Do you think the area is double the area of Ramon’s mural?

Check Understanding

Point out to students that they applied the Distributive Property to solve the problem.

Point to the equation and expression.

MODEL REASONING The problem asked how many square feet Ramó n paints, so we multiplied the length and the width of the mural to find the area.

We used an expression to represent the problem. What expression did we use? (24 × 16)

Then we estimated the answer.

Point to the estimate.

What was our estimate? (400)

MODEL REASONING Next, we split the factors and wrote an equation to find the partial products. Why did we split both factors? (Both factors had more than one place value. )

How many square feet does Ramon paint? (384) Is 384 is close to the estimate of 400? (yes)

Is our answer reasonable? (yes)

Ramó n paints 384 square feet.

Understanding the Distributive Property helps us to write equations to solve multiplication problems.

Display the second problem on RDI Practice, page 42. Read the directions aloud.

We can write an equation to help us understand and solve the problem. What expression do we need to multiply to solve the problem? (24 × 16)

How can we estimate 24 × 16? (20 × 20 = 400)

How can we write an equation to represent the problem? (24 × 16 = (20 + 4) × (10 + 6))

Display the equation.

■ What is 200 + 120 + 40 + 24? (384)

Have a student complete the equation on the screen.

Look forPatterns

Make a List

Guess-and-Check

WorkBackwards

Simplify the Problem

STRATEGY BANK

BOOST

Write an Equation

Draw aModel

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Write an Equation

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Resource LinksMath 180 RDI: page 42

SAM Keyword: Distributive Property

Ram ón paints square feet.

My answer is reasonable because

.

Solve a Problem

›› Use the Distributive Property to solve the problem.

Ramó n paints a mural on the side of his school. The mural is 24 feet long and 16 feet high. How many square feet does Ramón paint?

Use the DistributiveProperty to Multiply›› estimate the product. split both factors and use the Distributive Property to multiply. Compare your answer to the estimate to determine if your answer is reasonable.

Multiply Greater Factors

13 3 17

Estimate: 3 5

13 3 17 5 ( 1 ) 3 ( 1 )

13 3 17 5 ( 3 ) 1 ( 3 )

( 3 ) 1 ( 3 )

13 3 17 5

Is your answer reasonable?

3 5

Estimate: 3 5

Boost

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STRATEGY BANK

Write an Equation

Draw a Model

Look for Patterns

Make a List

Guess-and-Check

Work Backwards

Simplify the Problem

42 MATH 180 Block 2 › BOOST

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Apply the Distributive Property ›› estimate the product. split the factors and use the Distributive Property to multiply. Check that your answer is reasonable.

Multiply Greater Factors (continued)

Solve Another Problem›› Write an equation to solve this problem using the Distributive Property.

Vicki paints a mural that has double the length and height of Ramón’s mural. Vicki’s mural is 48 feet long by 32 feet high. How many square feet does Vicki paint?

Wrap Up›› Using an example, explain how to multiply two 2-digit factors using the Distributive Property.

.

Resource LinksMath 180 RDI: page 43

SAM Keyword: Distributive Property

12 3 48

Estimate: 3 5

12 3 48 5 ( 1 ) × ( 1 )

12 3 48 5 ( 3 ) 1 ( 3 )

( 3 ) 1 ( 3 )

12 3 48 5

Is your answer reasonable?

28 3 23

Estimate: 3 5

28 3 23 5 ( 1 ) × ( 1 )

28 3 23 5 ( 3 ) 1 ( 3 )

( 3 ) 1 ( 3 )

28 3 23 5

Is your answer reasonable?

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15

610

35

22

35

15

610

Students apply their critical thinking skills to solve complex number puzzles with factors and multiples.

Make Use of Structure Students should analyze the multiplication patterns in the structure of the arithmogons to fi nd the correct missing factors.

■ Arithmogons are rich problem-solving puzzles that help students make connections and identify relationships between numbers.

■ Like crossword puzzles, arithmogons are self-corrective.

Solve Arithmogons With Multiplication

Analyze an arithmogon.

Display the first problem on RDI Practice, page 146. Read the directions aloud.

In this puzzle, the number in each square is the product of 2 missing factors in the circles on either side. This means that the missing numbers on either side of 10 must be factors of 10.

Allow students to complete the arithmogon on their own.

Have students completeRDI Practice, pages 146–147 in pairs.

ORGuide students through the Practice pages

using the following instruction.

MATHEMATICAL PRACTICE Make use of structure

Guide students to analyze the problem structure as you complete the arithmogon on screen. Point to the factors and products as you think aloud.

MODEL REASONING I see that each pair of products has a common factor, which means that they share a factor. 10 and 6 have a common factor of 2.

■ How can you tell this is true from the other numbers in the arithmogon? (10 and 15 have a common factor of 5; 6 and 15 have a common factor of 3.)

Ask students targeted questions.

■ Does the structure of the arithmogon remind you of any other puzzles? (Sudoku; Crossword puzzles)

■ How did you solve those puzzles?

Now, let’s solve another arithmogon using multiple strategies.

Identify the rule and discuss student reasoning.

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8

126

42

3

2 x 3 = 62 x 4 = 83 x 4 = 12

42

3

2 x 3 = 62 x 4 = 83 x 4 = 12

8

126

42

3

Factors of 6 V 1, 2, 3, 6Factors of 8 V 1, 2, 4, 8Factors of 12 V 1, 2, 3, 4, 6, 12

2 x 3 = 62 x 4 = 83 x 4 = 12

3

42Factors of 6 V 1, 2, 3, 62, 3, 62Factors of 8 V 1, 2, 4, 82, 4, 82Factors of 12 V 1, 2, 3, 4, 6, 122, 3, 4, 6, 122

2 x 3 = 62 x 4 = 83 x 4 = 12

EXploRE

EXtEnD

Model this problem-solving strategy with your students.

Have student pairs consolidate their skills as you circulate.

PRACTICE Students apply their skills to complete two additional arithmogons.

■ How did you begin solving the arithmogons?

■ How do you know your answers are correct?

CHALLENGE Students create their own arithmogons. Then, have students trade problems with their partners to solve.

■ How did you make sure that each pair of products shared a common factor?

REFLECT Students complete the refl ection to explain how they solved the arithmogons.

■ Is it easier to find the missing factors of a lesser product, like 6, or a product with less factors, like 7?

147

Compare multiple strategies to solve a similar problem.

Have students list the factors for each product.

This is an organized method of finding the missing numbers. To solve this problem, we can make a list of the factors of 6. (1, 2, 3, and 6)

Guide students to use the list to identify the 2 factors in the circles adjacent to 6.

Then, have a student complete the arithmogon on screen.

MODEL REASONING If one of the factors of 8 is 2, the other factor must be 4. 4 × 3 = 12. The missing factors are 2, 3, and 4.

Have students check their work by writing the 3 equations in the arithmogon.

Ask students to guess the missing factors.

You can also use the guess-and-check strategy to find the missing factors in the circles.

Encourage students to justify their reasoning.

■ Did anyone guess a wrong factor, like 5 or 7? How did you know it was wrong?

■ Is this strategy a faster way to solve the problem?

Remind students that the process of ‘guessing’ still relies on mathematical sense.

Even though you are making a guess, knowing your multiplication facts helps to choose the right numbers in the circles.

Guess-and-Check

Make a List

STRETCH

Look forPatterns

Write an Equation

Make a List

Guess-and-Check

WorkBackwards

Simplify the Problem

Draw aModel

STRATEGY BANK

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Make a List

Guess-and-Check

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8

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Your Name Partner’s Name

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Resource LinksMath 180 RDI: page 146

SAM Keywords: Factors, Multiples

Factors of 6 1, 2, 3, 6Factors of 8 1, 2, 4, 8Factors of 12 1, 2, 3, 4, 6, 12

Analyze an Arithmogon

An arithmogon (uh-rith-muh-gon) is a puzzle made up of connected number patterns.

›› Find the missing factors. In this puzzle, each square represents the product of 2 factors in the circles on either side.

Solve an Arithmogon Using Multiple Strategies

›› list the factors for each product. Then, identify the factors that belong in the circles.

›› Guess the missing factors in the circles. Then, use multiplication to check if your answer makes sense

Solve ArithmogonsWith Multiplication

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STRATEGY BANK

Write an Equation

Draw a Model

Look for Patterns

Make a List

Guess-and-Check

Work Backwards

Simplify the Problem

146 MATH 180 Block 2 › STRETCH

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SCORE

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Resource LinksMath 180 RDI: page 147

SAM Keywords: Factors, Multiples

Wrap Up›› Which strategy did you use to solve the arithmogons?

The strategy I chose to solve the arithmogons was

.

This strategy was more effective because

.

Create Your Own Arithmogon›› Write any 1- or 2-digit numbers in the circles, and multiply to find the factors. Then, trade problems with your partner to solve.

Apply Your Skills to Solve More Problems›› Complete the missing factors in the circles.

Solve Arithmogons With Multiplication (continued)

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