1. Engage NY Math Module 3
Lesson 3: Write and interpret numerical expressions and compare expressions using a visual model.

2. Multiply by Multiples of 10
Use the commutative, associative, or distributive properties to multiply. Show all your steps.
21 x 40
213 x 30
4,213 x 20
(21 x 1) (4 x 10)
213 x (3 x 10)
4,213 x (2 x 10)
(21 x 4) (1 x 10)
(213 x 3) x 10
(4,213 x 2) x 10
84 x 10 =
200 x 3 = 600
4,000 x 2 = 8,000
840
10 x 3 = 30
200 x 2 = 400
3 x 3 = 9
10 x 2 = 20
= 639 x 10=
3 x 2 = 6
8, = 8,426 x 10 =
6,390
84,260

3. 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

4. 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 ____ = ____
1,950 x 42 ≈ 2,000 x 40 = 80,000
2,480 x 27 ≈ ____ x ____ = ____
2,480 x 27 ≈ 2,000 x 30 = 60,000
2,480 x 27 ≈ 2,500 x 30 = 75,000

5. 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) = ____
____ = ____
(5 x 3) + (4 x 3) = 27
= 27

6. 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) = ____
____ = ____
(5 x 4) + (2 x 4) = 28
= 28

7. Decompose a Factor: The Distributive Property
Solve these two math facts using the distributive property.
8 x 2 = ____
9 x 6 = ____
(5 x 2) + ( 3 x 2) = 16
= 16
(5 x 6) + (4 x 6) = 54
= 54

8. Application Problem:
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.
Robin
Gwen is 35 years old. 11
Gwen 11 11 11 2
(3 x 11) + 2 = ?

9. Concept Development From 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

10. Concept Development – Problem 1:
3 times the sum of 26 and 4. Show a tape diagram and expression.
26 + 4
3 x (26 + 4) or (26 + 4) x 3
Why are parentheses necessary around 26+4? Talk to your table.
We want 3 times as much as the total of
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

11. Concept Development – Problem 2:
6 times the difference between 60 and 51. Show a tape diagram and expression to match these words.
60 – 51
6 x (60-51) or (60-51) x 6
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.

12. Concept Development – Problem 3:
The sum of 2 twelves and 4 threes. Represent this with a tape diagram and expression.
(2 x 12) + (4 x 3) 12 12 3 3 3 3
Evaluate the expression. 36

13. Concept Development – Problem 4 From numerical expressions to word form.
Read this expression in words.
Eight times
Let me write down what I hear you saying.
8 x
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.

14. Concept Development – Problem 4 From 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
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.

15. Concept Development – Problems 5-6 From 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?

16. Concept Development – Problem 7 Comparison 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

17. Concept Development – Problem 8 Comparison 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 9 10 9 10 10 9 9
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.

18. Concept Development – Problem 9 Comparison 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 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 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.

19. Exit Ticket

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

21. HOMEWORK TASK
Assign Homework Task. Due Date: Wednesday, January 15.