Lesson 4: How Many Groups? (Part 1)
Let’s play with blocks and diagrams to think about division with fractions.
Illustrative Math Unit 6.4, Lesson 4 (printable worksheets)
Lesson 4 Summary
The following diagram shows how to solve problems that involve equal-sized groups that also involve fractions.
Lesson 4.1 Equal-sized Groups
Write a multiplication equation and a division equation for each statement or diagram.
- Eight $5 bills are worth $40.
- There are 9 thirds in 3 ones.
Lesson 4.2 Reasoning with Pattern Blocks
Use the pattern blocks in the applet to answer the questions. (If you need help aligning the pieces, you can turn on the grid.)
- If a hexagon represents 1 whole, what fraction do each of the following shapes represent? Be prepared to show or explain your reasoning.
a. 1 triangle
b. 1 rhombus
c. 1 trapezoid
d. 4 triangles
e. 3 rhombuses
f. 2 hexagons
g. 1 hexagon and 1 trapezoid
- Here are Elena’s diagrams for 2 · ½ and 6 · ⅓ = 2. Do you think these diagrams represent the equations? Explain or show your reasoning.
- Use pattern blocks to represent each multiplication equation. Recall that a hexagon represents 1 whole.
a. 3 · ⅙ = ½
b. 2 · 3/2 = 3
- Answer the following questions. If you get stuck, use pattern blocks.
a. How many ½s are in 4?
b. How many ⅔s are in 2?
c. How many ⅕s are in 1½?
Lesson 4 Practice Problems
- A shopper buys cat food in bags of 3 lbs. Her cat eats ¾lb each week. How many weeks does one bag last?
a. Draw a diagram to represent the situation and label your diagram so it can be followed by others. Answer the question.
b. Write a multiplication or division equation to represent the situation.
c. Multiply your answer in the first question (the number of weeks) by ¾. Did you get 3 as a result? If not, revise your previous work.
- Use the diagram to answer the question: How many ⅓s are in 1⅔? The hexagon represents 1 whole. Explain or show your reasoning.
- Which question can be represented by the equation ? · ⅛ = 3?
A. How many 3s are in ⅛?
B. What is 3 groups of ⅛?
C. How many ⅛s are in 3?
d. What is ⅛ of 3?
- Write two division equations for each multiplication equation.
a. 15 · ⅖ = 6
b. 6 · 4/3 = 8
c. 16 · ⅞ = 14
- Noah and his friends are going to an amusement park. The total cost of admission for 8 students is $100, and all students share the cost equally. Noah brought $13 for his ticket. Did he bring enough money to get into the park? Explain your reasoning.
- Write a division expression with a quotient that is:
a. greater than 8 ÷ 0.0001
b. less than 8 ÷ 0.0001
c. between 8 ÷ 0.0001 and 8 ÷ 1/10
- Find each unknown number.
a. 12 is 150% of what number?
b. 5 is 50% of what number?
c. 10% of what number is 300?
d. 5% of what number is 72?
e. 20 is 80% of what number?
The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.
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