How Much in Each Group? (Part 1)
10 min
Teacher Prep
Setup
Narrative
By now, students have created several diagrams based on verbal descriptions of situations. This Warm-up invites them to reason in the other direction: to interpret a representation of an equal-group situation and write a story that it could represent. In the given diagram, the number of groups and the total amount are known but the size of one group is not known. The reasoning here prepares students to think about “How much in one group?” questions and create their own representations later in the lesson.
In writing their stories, students have opportunities to communicate with precision (MP6), for instance, by including units of measurement or adjusting how “1 group” is referred to based on their chosen quantities. Ask questions to help students clarify their descriptions. Select 2–3 students who write descriptions that match the diagram but about different contexts. Invite them to share during a class discussion.
Launch
Arrange students in groups of 2. Tell students that there are two parts to this activity: writing a story with a question that can be represented by the tape diagram, and trading stories with their partner and answering each other’s question.
Give students 2–3 minutes to write their story and another minute to read and answer their partner’s question. Follow with a whole-class discussion.
Student Task
Here is a tape diagram.
- Think of a situation with a question that the diagram can represent. Describe the situation and the question.
-
Trade descriptions with your partner. Answer your partner’s question.
Sample Response
- Sample responses:
- There are 20 envelopes in 5 packs of greeting cards. How many envelopes are in 1 pack?
- A 20-yard long ribbon is cut into 5 equal pieces. How long is each piece?
- 4. (Units vary.)
Activity Synthesis (Teacher Notes)
Ask selected students to share their stories with the class. Discuss what the stories have in common. If not mentioned by students, highlight that each story involves finding the amount in one group of something, and the answer is 4 units.
Next, display the following equations for all to see. Give students a minute to think about which equations can represent the diagram (and their stories):
- 20⋅5=?
- 20÷5=?
- 20⋅?=5
- 20÷?=5
- 5⋅?=20
- 5÷20=?
Discuss why equations B, D, and E represent the situation. If time permits, also discuss why options A, C, and F do not represent the situation.
Emphasize that from both the diagram and the equations, we can tell that the value of 1 group (represented by the “?”) is 4 units.
- Diagram: 5 groups of 4 makes 20.
- Equations: Using 4 for the “?” in B, D, and E makes each equation true.
Anticipated Misconceptions
Students may struggle to answer their partner’s question because the descriptions are unclear or do not match the given expression. Encourage the listening partners to ask clarifying questions about the story or about its connections to the diagram. Urge the story-creating partners to revise their description in response to the questions.
Standards
- 6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <em>For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?</em>
- 6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <span>For example, create a story context for <span class="math">\((2/3) \div (3/4)\)</span> and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that <span class="math">\((2/3) \div (3/4) = 8/9\)</span> because <span class="math">\(3/4\)</span> of <span class="math">\(8/9\)</span> is <span class="math">\(2/3\)</span>. (In general, <span class="math">\((a/b) \div (c/d) = ad/bc\)</span>.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? </span>
15 min
10 min
Knowledge Components
All skills for this lesson
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How Much in Each Group? (Part 1)
10 min
Teacher Prep
Setup
Narrative
By now, students have created several diagrams based on verbal descriptions of situations. This Warm-up invites them to reason in the other direction: to interpret a representation of an equal-group situation and write a story that it could represent. In the given diagram, the number of groups and the total amount are known but the size of one group is not known. The reasoning here prepares students to think about “How much in one group?” questions and create their own representations later in the lesson.
In writing their stories, students have opportunities to communicate with precision (MP6), for instance, by including units of measurement or adjusting how “1 group” is referred to based on their chosen quantities. Ask questions to help students clarify their descriptions. Select 2–3 students who write descriptions that match the diagram but about different contexts. Invite them to share during a class discussion.
Launch
Arrange students in groups of 2. Tell students that there are two parts to this activity: writing a story with a question that can be represented by the tape diagram, and trading stories with their partner and answering each other’s question.
Give students 2–3 minutes to write their story and another minute to read and answer their partner’s question. Follow with a whole-class discussion.
Student Task
Here is a tape diagram.
- Think of a situation with a question that the diagram can represent. Describe the situation and the question.
-
Trade descriptions with your partner. Answer your partner’s question.
Sample Response
- Sample responses:
- There are 20 envelopes in 5 packs of greeting cards. How many envelopes are in 1 pack?
- A 20-yard long ribbon is cut into 5 equal pieces. How long is each piece?
- 4. (Units vary.)
Activity Synthesis (Teacher Notes)
Ask selected students to share their stories with the class. Discuss what the stories have in common. If not mentioned by students, highlight that each story involves finding the amount in one group of something, and the answer is 4 units.
Next, display the following equations for all to see. Give students a minute to think about which equations can represent the diagram (and their stories):
- 20⋅5=?
- 20÷5=?
- 20⋅?=5
- 20÷?=5
- 5⋅?=20
- 5÷20=?
Discuss why equations B, D, and E represent the situation. If time permits, also discuss why options A, C, and F do not represent the situation.
Emphasize that from both the diagram and the equations, we can tell that the value of 1 group (represented by the “?”) is 4 units.
- Diagram: 5 groups of 4 makes 20.
- Equations: Using 4 for the “?” in B, D, and E makes each equation true.
Anticipated Misconceptions
Students may struggle to answer their partner’s question because the descriptions are unclear or do not match the given expression. Encourage the listening partners to ask clarifying questions about the story or about its connections to the diagram. Urge the story-creating partners to revise their description in response to the questions.
Standards
- 6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <em>For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?</em>
- 6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <span>For example, create a story context for <span class="math">\((2/3) \div (3/4)\)</span> and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that <span class="math">\((2/3) \div (3/4) = 8/9\)</span> because <span class="math">\(3/4\)</span> of <span class="math">\(8/9\)</span> is <span class="math">\(2/3\)</span>. (In general, <span class="math">\((a/b) \div (c/d) = ad/bc\)</span>.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? </span>