Engage NY Math Module 3

Lesson 3: Write and interpret numerical expressions and compare expressions using a visual model.

21 x 40 (21 x 1) (4 x

10) (21 x 4) (1 x

10) 84 x 10 = 840

213 x 30 213 x (3 x 10) (213 x 3) x 10 200 x 3 = 600 10 x 3 = 30 3 x 3 = 9 600 + 30 + 9 =

639 x 10= 6,390

4,213 x 20 4,213 x (2 x 10) (4,213 x 2) x 10 4,000 x 2 =

8,000 200 x 2 = 400 10 x 2 = 20 3 x 2 = 6 8,000+400+20

+6 = 8,426 x 10 =

84,260

Multiply by Multiples of 10Use the commutative, associative, or distributive properties to multiply. Show all your steps.

ESTIMATE PRODUCTS

• 421 x 18 ≈ ____ x ____ = ____ • Round 421 to the nearest hundred.• 400

• 421 x 18 ≈ 400 x ____ = ____ • Round 18 to the nearest 10• 20

• 421 x 18 ≈ 400 x 20 = ____ • What’s 400 x 20?• 8,000

• 421 x 18 ≈ 400 x 20 = 8,000

ESTIMATE PRODUCTS

• 323 x 21 ≈ ____ x ____ = ____ • In your notebooks, write the multiplication sentence rounding each

factor to arrive at a reasonable estimate of the product.

• 323 x 21 ≈ 300 x 20 = 6,000

• 1,950 x 42 ≈ ____ x ____ = ____ • In your notebooks, write the multiplication sentence rounding each

factor to arrive at a reasonable estimate of the product.

• 1,950 x 42 ≈ 2,000 x 40 = 80,000

• 2,480 x 27 ≈ ____ x ____ = ____ • In your notebooks, write the multiplication sentence rounding each

factor to arrive at a reasonable estimate of the product.

• 2,480 x 27 ≈ 2,000 x 30 = 60,000 • 2,480 x 27 ≈ 2,500 x 30 = 75,000

Decompose a Factor: The Distributive Property

• 9 x 3 = ____ Solve the multiplication sentence.• 9 x 3 = 27

• 9 x 3 = ____• (5 x 3) + ( ___ x 3) = ____

• 9 is the same as 5 and what number?• 4

• (5 x 3) + (4 x 3) = ____• 15 + ____ = ____

• 9 x 3 = 27• (5 x 3) + (4 x 3) = 27• 15 + 12 = 27

Decompose a Factor: The Distributive Property

• 7 x 4 = ____ Solve the multiplication sentence.• 7 x 4 = 28

• 7 x 4 = ____• (5 x 4) + ( ___ x 4) = ____

• 7 is the same as 5 and what number?• 2

• (5 x 4) + (2 x 4) = ____• 20 + ____ = ____

• 7 x 4 = 28• (5 x 4) + (2 x 4) = 28• 20 + 8 = 28

Decompose a Factor: The Distributive Property

• Solve these two math facts using the distributive property.• 8 x 2 = ____• 9 x 6 = ____

• 8 x 2 = ____• (5 x 2) + ( 3 x 2) = 16• 10 + 6 = 16

• 9 x 6 = ____• (5 x 6) + (4 x 6) = 54• 30 + 24 = 54

Robin is 11 years old. Her mother, Gwen, is 2 years more than 3 times Robin’s age. How old is Gwen? Use a tape diagram to solve this problem. Be sure to include a statement of solution.

Application Problem:

Robin

11

(3 x 11) + 2 = ?

Gwen

Gwen is 35 years old.

11 11 11 2

Concept DevelopmentFrom word form to numerical expressions and

diagrams.• What expression describes the total value of these 3

equal units?

5

• 3 x 5

• How about 3 times an unknown amount called A. Show a tape diagram and expression.

A• 3 x A

Concept Development – Problem 1:

26 + 4

3 x (26 + 4) or (26 + 4) x 3

3 times the sum of 26 and 4. Show a tape diagram and expression.

• Why are parentheses necessary around 26+4? Talk to your table.• We want 3 times as much as the total of 26 + 4.• If we don’t put the parentheses, it doesn’t show what we are

counting.• We are counting the total of 26 and 4 three times.

• Evaluate the expression.• 90

Concept Development – Problem 2:

60 – 51

6 x (60-51) or (60-51) x 6

6 times the difference between 60 and 51. Show a tape diagram and expression to match these words.

• We’ve recorded two different expressions for these words: • 6 x (60-51) or (60-51) x 6

• Are these expressions equal? Why or why not? Talk with your table.• Yes, they are equal. The two factors are just reversed.

• What is the name of this property?• The commutative property of multiplication

• Explain it in your own words to your partner.

Concept Development – Problem 3:

(2 x 12) + (4 x 3)

The sum of 2 twelves and 4 threes. Represent this with a tape diagram and expression.

• Evaluate the expression.• 36

12 12 3 3 3 3

Concept Development – Problem 4From numerical expressions to word form.

• 8 x (43-13).• Read this expression in words.

• Eight times 43-13.

• Let me write down what I hear you saying.• 8 x 43-13.

• It sounds like you are saying that we should multiply 8 and 43 and then subtract 13. Is that what you meant?

• Is the second expression equivalent to the first? Why or why not?• No, it’s not the same. We didn’t write any parentheses.

Without them we will get a different answer because you won’t subtract first. We are supposed to subtract 13 from 43 and then multiply by 8.

Concept Development – Problem 4From numerical expressions to word form.

• Why can’t we simply read every expression left to right and translate it?• We need to use words that tell what we should subtract first and

then multiply.• Let’s name the two factors we are multiplying. Turn and talk.

• 8 and the answer to 43-13.• We need to multiply the answer to the stuff inside the

parentheses by 8.• Since one of the factors is the answer to this part (43-13), what

could we say to make sure we are talking about the answer to this subtraction problem?

• What do we call the answer to a subtraction problem?• The difference between 43 and 13.• What is happening to the difference of 43 and 13?

• It’s being multiplied by 8.• We can say and write, “8 times the difference of 43 and 13.”

Compare these words tot the ones we said first. Do they make sure we are multiplying the right numbers together? What others ways are there to say it?• Yes, they tell us what to multiply better.

• The product of 8 and the difference between 43 and 13.• 8 times as much as the difference between 43 and 13. • The difference of 43 and 13 multiplied 8 times.

Concept Development – Problems 5-6From numerical expressions to word form.

• In your math journals, record these numerical expressions into word form.• (16 + 9) x 4

• The sum of 16 and 9 times 4• Will 16 plus 9 times 4 be equivalent?

• (20 x 3) + (5 x 3)• The sum of 20 threes and 5 threes• The sum of 3 twenties and 3 fives• The product of 20 and 3 plus the product of 5 and 3

• Will twenty times 3 plus 5 times 3 yield the same answer?

Concept Development – Problem 7Comparison of expressions in word form and numerical form.

• Let’s use <, >, or = to compare expressions.• Compare 9 x 13 and 8 thirteens. Draw a tape diagram for each

expression and compare them.13

13

• We don’t need to evaluate the solutions in order to compare them. When can just look at our diagrams. • 9 x 13 > 8 thirteens

Concept Development – Problem 8Comparison of expressions in word form and numerical form.

• Compare the sum of 10 and 9, doubled and (2x10) + (2x9). Draw a tape diagram for each expression and compare them.

10

10

• They are equal because the sum of 10 and 9, doubled is (10+9) x 2. The expression on the right is the sum of 2 tens and 2 nines. There are 2 tens and 2 nines in each bar.

9

10

10

99

9

Concept Development – Problem 9Comparison of expressions in word form and numerical form.

• Compare 30 fifteens minus 1 fifteen and 29 x 15. Draw a tape diagram for each expression and compare them.

15

• They are equal because 30 fifteens minus 1 fifteen equals 29 fifteens and 29 x 15 equals 29 fifteens. The length of the bars are equal.

15

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Exit Ticket

Display Problem Set on the board. Allow time for the students to complete the problems with tablemates.

Problem Set

Assign Homework Task. Due Date: Wednesday, January 15. 

HOMEWORK TASK