This Warm-up gives students a chance to create tape diagrams to represent equal-size groups and division expressions in a scaffolded way. Each tape is started on a grid and pre-labeled with the known quantity. Each grid square represents 1.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time and another minute to share their diagrams with their partner.
Activity
None
Student Lesson in Spanish
We can think of the division expression as the question: “How many groups of 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.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to articulate how they represent the quantities in division questions on the blank tape diagrams. Select a few students to share their diagrams and answers. Discuss questions such as:
“In the first question, how did you know how large each part of the diagram should be?” (The length of the tape represents 10, and there are 10 grid squares, so each grid represents 1. Because the size of each group is , each part needs to have squares.)
“In the second question, we see 3 groups of 2 and an extra square of 1. How did you know that the 1 is of a group and not of a group?” (The question asks “How many groups of 2 . . . ,” so the size of each group is 2, not 7.)
“How can you tell the answers to the questions from the completed diagrams?” (We can count the number of full groups and partial groups.)
6.2
Activity
Standards Alignment
Building On
Addressing
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. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that because of is . (In general, .) 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?
In this lesson, students transition away from using concrete tools (pattern blocks and fraction strips) for reasoning about division to using a tool that is more abstract and more flexible. To represent “How many groups?” or “How many of these are in that?” questions, they draw tape diagrams on a grid. As they annotate the diagrams to represent the quantities involved and to show their reasoning, students practice communicating with precision (MP6).
Students continue to write multiplication and division equations to make sense of given situations, but here, they also think about the reasonableness of their answers. They see that the solution to a division problem can be checked using the corresponding multiplication equation. As they relate equations to situations and vice versa, students practice reasoning abstractly and quantitatively (MP2).
Some students may choose to write fractions in the form of mixed numbers, which is fine. Depending on the situation, one form may be more useful than the other. For example, in a measurement situation, a mixed number is easier to interpret, but if we need to perform further calculations with an answer, a number of the form may be easier to work with. In the course of the unit, students should become comfortable with both forms and be flexible in using them.
Launch
Arrange students in groups of 3–4. Tell students that they have used pattern blocks and fraction strips to answer questions that ask “How many groups are there?” and “How many of these are in that?” Now they will answer such questions by drawing tape diagrams.
Give students 2–3 minutes to discuss the first question and write their response. Follow that with a brief whole-class discussion. Select 1–2 students to explain how a tape diagram shows us how many s are in 1.
Then give students 8–10 minutes of quiet work time to complete the rest of the task. Ask them to discuss their diagrams only after attempting at least 2 of the 3 remaining questions. Provide access to colored pencils. Some students may find it helpful to identify whole groups and partial groups by coloring the tape diagrams.
MLR2 Collect and Display. Circulate among the students, and listen for and collect the language that students use as they explain how Andre’s tape diagram can be used to solve the equation . On a visible display, record words and phrases such as “equal parts,” “same size,” and “group of s.” Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading
Activity Synthesis
6.3
Activity
Standards Alignment
Building On
Addressing
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. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that because of is . (In general, .) 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?
In this activity, students apply the reasoning strategies they learned in earlier activities to solve division problems in context. Though the instructions do not prompt students to draw tape diagrams, students may find them to be a handy option. Students who opt to use available tools to facilitate problem solving (such as referring to fraction strips or using graph paper to set up a diagram and colored pencils to mark up parts of it) practice choosing tools strategically (MP5).
Students may decide to express each whole-number or mixed-number dividend as a fraction. For example, they may express the 6 inches as inches and then see how many s are in . As students work, monitor for the different strategies that they use.
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Keep students in the same groups. Give students 4–5 minutes of quiet work time followed by 1–2 minutes to discuss their work with their group.
Provide access to geometry toolkits. Encourage students to think about the tools and strategies at their disposal and to check their solutions using multiplication.
If time is limited, consider “jigsawing” the activity. For example, ask each group member to answer one question, making sure that both questions are answered by the group.
Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on their own progress. For example, ask students how comfortable they are in figuring out what each given quantity represents and how to write a multiplication or a division equation to represent a situation. Supports accessibility for: Organization, Conceptual Processing
Activity
None
Write a multiplication equation or a division equation for each question, and then find the answer. Explain or show your reasoning.
How many groups of pound are in pounds?
How many -inch thick books make a stack that is 6 inches tall?
Activity Synthesis
Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question by correcting errors, clarifying meaning, and adding details.
Display this first draft:
“In every 1 inch we can fit 2 books that are -inch thick, so in 6 inches we can fit or 12 books.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2 minutes to work with a partner to revise the first draft. Invite students to include a labeled diagram, if it could further clarify their revised explanation.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, and then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
Lesson Synthesis
Focus the discussion on how to create a tape diagram to represent and reason about “How many groups?” questions.
Display the question: “How many s are in 2?” Consider drawing a tape diagram on a blank grid and annotating it while students respond to questions such as:
“What would the length of the entire tape represent?” (The total amount, which is 2, or )
“How long should the tape be? Should we use one grid box for each 1 whole?” (Since we’re dealing with , it’d be helpful to show fifths and use 5 grid boxes for each 1 whole.)
“What is the size of each group?” () “How can we show the groups on the tape?” (Partition the tape into pieces of each.)
“How does the diagram help us answer the question?” (We can count the number of groups of .)
“What if the total amount cannot be partitioned equally into s? How do we deal with the extra in this case?” (We can think of it as a partial group and compare it to the size of 1 group. We can see that is half of .)
If time permits, invite students to compare and contrast the visual reasoning tools they have used so far:
“We have used pattern blocks, fraction strips, and tape diagrams to help us think about ‘How many groups?’ questions involving fractions. How are these tools alike?” (They all show groups of fractions and allow us to count the number of groups.)
“How are tape diagrams different from pattern blocks and fraction strips?” (With tape diagrams, we can specify what the length of the tape and each segment represent. With pattern blocks or fraction strips, we may need a lot of blocks or much longer strips to represent certain numbers. Also, fractions with certain denominators are not shown on the strips and cannot be shown with available pattern block shapes.)
Student Lesson Summary
One batch of a recipe calls for kg of flour. If a baker used 2 kg of flour, how many batches did she make?
We can think of the question as “How many groups of kg make 2 kg?” and represent it with the equations:
To help us make sense of the question, we
can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.
Fraction bar diagram. 10 equal parts. Each part labeled "the fraction 1 over 5." Total labeled "2 kilograms" and "question mark batches." Every two parts labeled "1 batch."
We can see that there are 5 groups of in 2. Multiplying 5 and gives or 2, so is 5.
Sometimes the number of groups or the result of dividing is not a whole number. Suppose one serving of rice is cup. How many servings are there in cups?
Here are two equations and a diagram that represent the situation:
Fraction bar diagram. 16 equal parts. 14 parts shaded. 14 parts labeled “unknown number of groups” and “3 and one half cups.” Each part labeled the fraction 1 over 4. First 3 parts labeled “1 serving.”
The diagram shows 4 full groups of , plus 2 extra s, which make of a group. So is . We can check this quotient by multiplying and .
, and , which is or .
Standards Alignment
Building On
Addressing
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. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that because of is . (In general, .) 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?
To make sense of the question “How many s are in 1?” Andre wrote equations and drew a tape diagram.
A tape diagram with three equal parts. The first two parts are shaded and are each labeled one third, total 1. A bracket is labeled 1 group of two thirds, and contains the first two parts.
In an earlier task, we used pattern blocks to help us solve the equation .
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 s are in 1?
How many s are in 3?
How many s are in 5?
Student Response
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Building on Student Thinking
Students may think that the answer to the first question can be only a whole number because in the given diagram “1 group” is shaded but the partial group is not. Remind students of their work with pattern blocks in which we saw that the answer to the question “How many rhombuses are in a trapezoid?” could involve a whole number and a fraction. Consider asking “What fraction of a group would fit in the unshaded part of the tape?”
Student Response
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Building on Student Thinking
Some students might interpret the “stack” in the last question as a group because it is commonly used as such in everyday contexts. Clarify that in this case, “a stack that is 6 inches tall” refers to the books being piled up to reach a height of 6 inches.
