This Warm-up prompts students to carefully analyze and compare fractions or expressions containing fractions. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students use to talk about the size of fractions, equivalence, mixed numbers, and addition of fractions. The reasoning also helps students to recall familiar relationships between fractions where one denominator is a factor or a multiple of the other. This awareness will be helpful later when students solve problems that involve combining quantities with different fractional parts.
Launch
Groups of 2
Display the numbers and expressions.
“Pick 3 expressions that go together. Be ready to share why they go together.”
1 minute: quiet think time
Activity
“Discuss your thinking with your partner.”
2–3 minutes: partner discussion
Share and record responses.
Which 3 go together?
A
B
C
D
Student Response
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Advancing Student Thinking
Activity Synthesis
“What are some fractions that are equivalent to the expressions in A, B, and C?” (, , , , )
“What are some ways to decide if two fractions are equivalent?” (Think about the relationship of unit fractions—for instance, the number of one-eighths that are in one-fourth, compare the fractions to a benchmark—for instance, if one fraction is greater than 1 and the other less than 1, then they’re not equivalent, or see if the numerator and the denominator of one fraction could be multiplied by the same factor to get the other fraction.)
Activity 1
Standards Alignment
Building On
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In earlier lessons, students found sums and differences of fractions (including mixed numbers) with the same denominator. In this activity, they reason about problems that involve combining or removing fractional amounts with different denominators—2, 4, and 8—in the context of stacking playing bricks. Because the denominators are familiar and are multiples or factors of one another, students can rely on what they know about the relationships of halves, fourths, and eighths to compare amounts (for example, how much greater than or less than) or to combine them.
The last question asks students to reason about the height of a tower of bricks created by combining three shorter stacks. Students may arrive at two different answers, depending on their familiarity with playing bricks and attention to precision. Some students may notice that each playing brick has studs that disappear into the bottom of another brick when stacked, so the combined height of two stacks will be less than the sum of the heights of individual stacks (MP6). Both answers are acceptable as long as they are supported.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. For example, direct students’ focus to the first part of the first problem. Invite students to write an equation to represent the problem. Then prompt them to plan a strategy, including the tools they will use (for example, number lines or fraction strips) to solve the equation. Finally, ask them to solve the equation, and then move on to the second part of the problem. Check in with students to provide feedback and encouragement after each chunk. Supports accessibility for: Organization, Memory, Attention
Launch
Groups of 2
Display the image of stacked playing bricks. Read the Task Statement (including the heights of the three stacks) together.
“The picture shows Priya’s tower. Try visualizing Kiran's and Lin’s towers in the same picture. How tall would they be?” Invite students to try sketching or describing where the top of each tower would reach.
Consider asking: “How tall was the tallest tower of playing bricks you have built?”
Activity
“Work independently on the task for a few minutes. Then share your thinking with your partner.”
5–6 minutes: independent work time
4–5 minutes: partner discussion
Monitor for students who:
Use the fact that there are 2 fourths or 4 eighths in 1 half to reason about sums or differences of the fractions.
Compare the fractional parts of the mixed number to the benchmark of or 1 (especially in the last problem).
Find differences and sums by writing equivalent fractions in fourths or eighths.
Activity Synthesis
Select students to share their responses and their reasoning.
Focus the discussion on how students found the fractional differences between the measurements. Highlight explanations about the relative sizes of halves, fourths, and eighths, and about equivalence. (See Student Responses.)
Activity 2
Standards Alignment
Building On
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Previously, students used their knowledge of equivalence to reason about the sums and differences of fractions with denominators 2, 4, or 8. In this activity, they do the same with fractions with denominators 2, 3, and 6. As before, students are not expected to write addition expressions in which the fractions are written with a common denominator (though some students may choose to do so). Instead, they rely on what they know about the relationship between and , and between and , to solve the problems. Students may choose to use visual representations to support their reasoning. When students create and compare their own representations for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).
The measurements in the task—, , and —are given in feet. Because each of them has a whole-number equivalent in inches, some students may choose to reason entirely in inches, which is a valid strategy. Ask these students to think about how they’d approach the problems if the given unit is unfamiliar or doesn’t convert handily to whole numbers in another unit.
MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to ”whether Andre can use the fraction ___ to make a stack that is ___ feet tall." Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive. Advances: Writing, Speaking, Listening
Launch
Groups of 2
Display the image of the three foam blocks.
“What do you notice? What do you wonder?”
30 seconds: quiet think time
30 seconds: partner discussion
Activity
“Take a few quiet minutes to work on the activity. Then share your thinking with your partner.”
7–8 minutes: independent work time
3–4 minutes: partner discussion
Monitor for the different reasoning strategies students use to combine different fractional parts, including use of diagrams, descriptions, and expressions or equations.
Andre is building a tower out of foam blocks. The blocks come in three different thicknesses: foot, foot, and foot.
Andre stacks 1 block of each size. Is the stack more than 1 foot tall? Explain or show how you know.
Can Andre use only the -foot and -foot blocks to make a stack that is feet tall? If you think so, show one or more ways. If not, explain why not.
Can Andre use only the -foot and -foot blocks to make a stack that is feet tall? If so, show one or more ways. If not, explain why not.
Activity Synthesis
Invite students to share their strategies for determining whether or how certain combinations of blocks would make a specified height. Record and display their reasoning.
To highlight different ways to combine different-size fractional parts, consider sketching or displaying diagrams as shown:
Making 1 foot with all blocks:
Making foot with and blocks:
Making foot with and blocks:
Lesson Synthesis
“Today we solved problems where we had to combine halves, fourths, and eighths, or remove one of those fractions from another. We also combined halves, thirds, and sixths.”
“How would you find the combined lengths of inch and inch? How would you find the difference between the two lengths?”
Consider asking students to record their response in writing, or to turn and talk to a partner after some quiet think time.
“In upcoming lessons, we’ll use some of the strategies we used today to combine tenths and hundredths.”
Standards Alignment
Building On
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Priya, Kiran, and Lin use large playing bricks to make towers. Here are the heights of their towers.
Priya: inches
Kiran: inches
Lin : inches
Show your reasoning for each question.
How much taller is Lin’s tower compared to:
Priya’s tower?
Kiran’s tower?
They are playing in a room that is 109 inches tall. Priya says that if they combine their towers to make a super tall tower, it would be too tall for the room. She says they must remove 1 brick.
Do you agree with Priya? Explain your reasoning.
Student Response
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Advancing Student Thinking
If students look at only the whole numbers when solving the problems, consider asking:
“How did you decide to compare the towers?”
“How can you use equivalent fractions to help you compare the towers?”
Student Response
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Advancing Student Thinking
4.NF.B.3.d
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.For example, express as , and add .
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.For example, express as , and add .
