Using Diagrams to Find the Number of Groups: IM 6.4.6
[url=https://im.kendallhunt.com/MS/teachers/1/4/6/preparation.html]“Using Diagrams to Find the Number of Groups”[/url] from IM Grade 6 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 6.4.6
- IM 6.4.6 Lesson: Using Diagrams to Find the Number of Groups
- Practice 6.4.6
- IM 6.4.6 Practice: Using Diagrams to Find the Number of Groups
IM 6.4.6 Lesson: Using Diagrams to Find the Number of Groups
We can think of the division expression 10 ÷ 2 ½ as the question: "How many groups of 2 ½ are in 10?" Complete the tape diagram to represent this question. Then find the answer.
Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.
[size=150]To make sense of the question “How many [math]\frac{2}{3}[/math]s are in 1?,” Andre wrote equations and drew a tape diagram.[/size][br][br][math]?\cdot\frac{2}{3}=1[/math] [math]1\div\frac{2}{3}=?[/math] [br] 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[/img][br][br]In an earlier task, we used pattern blocks to help us solve the equation [math]1\div\frac{2}{3}=?[/math]. Explain how Andre’s tape diagram can also help us solve the equation.
Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
How many [math]\frac{3}{4}[/math]s are in 1?
Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
How many [math]\frac{2}{3}[/math]s are in 1?
Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
How many [math]\frac{3}{2}[/math]s are in 1?
Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.
How many [math]\frac{3}{8}[/math]-inch thick books make a stack that is 6 inches tall?
Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.
How many groups of [math]\frac{1}{2}[/math]pound are in [math]2\frac{3}{4}[/math]pounds?
Write a question that can be represented by the division equation [math]5\div1\frac{1}{2}=?[/math].
Find the answer to the previous question. Explain or show your reasoning.
IM 6.4.6 Practice: Using Diagrams to Find the Number of Groups
We can think of [math]3\div\frac{1}{4}[/math] as the question “How many groups of [math]\frac{1}{4}[/math] are in 3?” Draw a tape diagram to represent this question. Then find the answer.
Describe how to draw a tape diagram to represent and answer [math]3\div\frac{3}{5}=?[/math] for a friend who was absent.
How many groups of [math]\frac{1}{2}[/math] day are in 1 week?[br][br]Write a multiplication equation or a division equation to represent the question.
How many groups of ½ day are in 1 week? Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.
Diego said that the answer to the question “How many groups of [math]\frac{5}{6}[/math]are in 1?” is [math]\frac{6}{5}[/math]or [math]1\frac{1}{5}[/math]. Do you agree with him? Explain or show your reasoning.
Select [b]all[/b] the equations that can represent the question: “How many groups of [math]\frac{4}{5}[/math] are in 1?”
Calculate each percentage mentally.
What is 10% of 70?
What is 10% of 110?[br]
What is 25% of 160?
What is 25% of 48?[br]
What is 50% of 90?[br]
What is 50% of 350?[br]
What is 75% of 300?[br]
What is 75% of 48?[br]