1. MULTIPLYING
WITH FRACTIONS

2. Multiplying with fractions
Mark each bar model as a thirds bar model.

3. Multiplying with fractions
Mark each bar model as a thirds bar model.
Use a colored pencil to shade 2/3.

4. Multiplying with fractions
Mark each bar model as a thirds bar model.
Use a colored pencil to shade 2/3.
Iterate step two FOUR TIMES. Use a different color each time.

5. Multiplying with fractions
Mark each bar model as a thirds bar model.
Use a colored pencil to shade 2/3.
Iterate step two 4 TIMES. Use a different color each time.
What is the total of 2/3 shaded 4 times?
Show this situation as a math equation.
Equation – A mathematical statement that two things are equal. Uses symbols.

6. Multiplying with fractions
Mark each bar model as a thirds bar model.
Use a colored pencil to shade 2/3.
Iterate step two 4 TIMES. Use a different color each time.
What is the total of 2/3 shaded 4 times?
Show this situation as a math equation.
(2/3)(4) = 8/3 or
(2/3)(4) = 2 2/3
Equation – A mathematical statement that two things are equal. Uses symbols.

7. Multiplying with fractions
6. Mark each bar model as a thirds bar model.

8. Multiplying with fractions
6. Mark each bar model as a thirds bar model.
7. Use a colored pencil to shade 2/3 of the top bar.

9. Multiplying with fractions
6. Mark each bar model as a thirds bar model.
7. Use a colored pencil to shade 2/3 of the top bar.
8. Do step number 7 one half times. Use a different color each time.

10. Multiplying with fractions
6. Mark each bar model as a thirds bar model.
7. Use a colored pencil to shade 2/3 of the top bar.
Using a different color, do step number 7 one half times on the bottom bar.
Write an equation for each of the bar models.
top bar model ________________________
bottom bar model _____________________
Equation – A mathematical statement that two things are equal. Uses symbols.

11. Multiplying with fractions
6. Mark each bar model as a thirds bar model.
7. Use a colored pencil to shade 2/3 of the top bar one time.
8. Do step number 7 one half times. Use a different color each time.
9. Write an equation for each of the bar models.
top bar model ___(2/3)(1) = 2/3_____
bottom bar model _(2/3)(1/2) = 1/3_____
Equation – A mathematical statement that two things are equal. Uses symbols.

12. Number Lines & Multiplication
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned previously.
b. Continue partitioning the segment from 1 to 2 into fourths.

13. Number Lines & Multiplication
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned yesterday.
b. Continue partitioning the segment from 1 to 2 into fourths.

14. Number Lines & Multiplication
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned yesterday.
Continue partitioning the segment from 1 to 2 into fourths.
11. Using an arrow, “jump” 3 fourths (3/4).

15. Number Lines & Multiplication
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned yesterday.
Continue partitioning the segment from 1 to 2 into fourths.
11. Use an arrow to “jump” 3 fourths (3/4).
12. Iterate step eleven so that it has happened 2 times.

16. Number Lines & Multiplication1 2
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned yesterday.
Continue partitioning the segment from 1 to 2 into fourths.
11. Use an arrow to “jump” 3 fourths (3/4).
12. Iterate that jump so that it has occurred 2 times.
13. Write an equation that describes what has been shown on the number line.

17. Number Lines & Multiplication1 2
10. Draw this number line from 0 to 2.
a. Partition the segment from 0 to 1 into fourths using the strategies you learned yesterday.
Continue partitioning the segment from 1 to 2 into fourths.
11. “Jump” 3 fourths (3/4)
12. Iterate that jump so that it has occurred 2 times.
13. Write an equation that describes what has been shown on the number line.
(3/4) (2) = 6/ or (3/4) (2) = 3/2 or (3/4) (2) = 1 ½

18. Number Lines (cont.)
Draw a new number line from 0 to 1 and label as shown above.
Use what you know about fractions to model eighths on your number line. Label your number line accurately.

19. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8

20. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8
16. Use an arrow to “jump” ¾.

21. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8
16. Use an arrow to “jump” ¾.

22. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8
Use an arrow to “jump” ¾.
Using a different color, complete a ¾ jump ½ times.

23. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8
Use an arrow to “jump” ¾.
Using a different color, complete a ¾ jump ½ times.
Write an equation stating what the number line you drew illustrates.

24. Number Lines (cont.)
0 4
2 8
2 4
6 8
4 4
0 8
4 8
8 8
Use an arrow to “jump” ¾.
Using a different color, complete a ¾ jump ½ times.
Write an equation stating what the number line you drew illustrates.
( ¾) (1/2) = 3/8

25. PULLING IT ALL TOGETHER (BAR MODEL, NUMBER LINE)
19. Complete the table.
SIZE OF BASE UNIT
NUMBER OF ITERATIONS
VISUAL MODEL
(BAR MODEL, NUMBER LINE)
RESULT
SUMMARY EQUATION
2/3 3 2 1
1/2
1/3

26. PULLING IT ALL TOGETHER (BAR MODEL, NUMBER LINE)
19. Complete the table.
SIZE OF BASE UNIT
NUMBER OF ITERATIONS
VISUAL MODEL
(BAR MODEL, NUMBER LINE)
RESULT
SUMMARY EQUATION
2/3 3 2
(2/3) (3) = 2
4/3 or 11/3
(2/3) (2) = 4/3 1
(2/3) (1) = 2/3
1/2
1/3
(2/3) (1/2) = 1/3
2/9
(2/3) (1/3) = 2/9

27. 20. Look at the following equations. a) (1/2) (3/5) = 3/10
WRITE A RULE
20. Look at the following equations.
a) (1/2) (3/5) = 3/10
b) (3/4) (2/3) = 6/12 = ½
c) (5/9) (2/7) = 10/63
d) (2) (2/5) = 4/5
It would be tedious and time consuming to have to draw a number line or bar model every time we have a multiplication problem with or without fractions. You should know your whole number multiplication facts at this point. Work with a classmate develop a rule for determining the product when one or both of the factors are fractions. Draw more examples if you need to.
Factors – the numbers that are being multiplied
Product – the answer to a multiplication problem

28. WRAP IT UP 21. Complete each of the following equations. a. (1) (5) =
b. (2) (5) =
c. (3) (7) =
d. (10) (4) =
Factors – the numbers that are being multiplied
Product – the answer to a multiplication problem
22. Compare and contrast the factors and products in the problems above.

29. WRAP IT UP 21. Complete each of the following equations.
b. (2) (5) = 10
c. (3) (7) = 21
d. (10) (4) = 40
Factors – the numbers that are being multiplied
Product – the answer to a multiplication problem
22. Compare and contrast the factors and products in the problems above.
All of the products are greater than or equal to one or both of the factors.

30. SUMMARIZING MULTIPLICATION ???
23. Bart says that in multiplication problems the answer is never smaller than the products. Use your knowledge of multiplication to support or refute his claim. Use as many words from the word bank as you can.
word bank
Numerator Denominator Product Factor
Multiply Equation Iterate/Iterated Times

31. And now…….
DIVISION

32. 10 ÷2 = 5 10 put into groups of two 10 put into two groups 5 groups
5 groups
Groups of 5

33. 10 ÷2 = 5 10 put into groups of two 10 put into two groups 5 groups
5 groups
groups of 5
24. Although the equation looks the same these two situations are different. Write a short word problem for each picture.

34. Show 6/2 =
Six broken into two groups
Six broken into groups of two 3

35. 3/2 = 1 1/2 Three split into two groups Three split into groups of two
one and a half in each group or one and a half complete groups
1 1/2

36. QUOTIENT – the answer in an equation involving division.
25. The first two that we did had whole number quotients. The last one had a mixed number quotient. How can you tell if the quotient will be a whole or mixed number? Give at least 6 examples that support your conjecture.

37. Remember it is that much of the original whole.
½ ÷ 2 =
Half divided into two groups
Half split into groups of two
Remember it is that much of the original whole.

38. 10 ÷ ½ = 20 10 groups of 1/2. What is the whole?
10 split into groups of 1/2 20

39. 26. Solve the following equations using a picture of some kind.
2 ÷ ½
1 ÷ ½
½ ÷ ½
¼ ÷ ½
e) ½ ÷ ¼
* Although sometimes nonsensical it can help to think of teams. The first number tells how many people you have and the second tells how many are needed to make a team. Your answer tells how many teams you can make.

40. 27. An algorithm is a rule or procedure to follow to get the answer to a problem. Drawing picture of all division problems can deepen or demonstrate understanding but takes a great deal of time and space. Work with a classmate to develop an algorithm that will allow you to find the quotient when fractions are involved. Provide evidence to support your algorithm.