Lesson 12

Solve Problems

Warm-up: Estimation Exploration: Large Denominators (10 minutes)

Narrative

The purpose of this estimation exploration is for students to reason about the size of a complex fraction sum with large denominators. Students can see that 1 is a good estimate because one fraction is small and the other is close to 1. In the synthesis they refine this estimate to explain why the value of the sum is a little larger than 1.

Launch

• Groups of 2
• Display the expression.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

Activity

• 1 minute: partner discussion
• Record responses.

Student Facing

What is the value of the sum?

$$\frac{3}{17}+\frac{17}{19}$$

Record an estimate that is:

too low about right too high
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Student Response

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Activity Synthesis

• “How do you know that the sum is greater than 1?” ($$\frac{17}{19}$$ is $$\frac{2}{19}$$ short of a whole. Since 17ths are bigger than 19ths, adding $$\frac{3}{17}$$ makes it greater than 1.)

Activity 1: Priya’s Salad Dressing (20 minutes)

Narrative

The purpose of this activity is for students to add and subtract fractions and estimate sums and differences of fractions using the context of a recipe. Students may have different responses and reasoning for the estimation questions. In both cases, they can calculate and compare fractions but they may have different thoughts about how these differences would affect the recipe or what exactly it means for the recipe to make “about $$1\frac{1}{2}$$ cups.” In the synthesis, students discuss the reasonableness of the estimates and how to make precise calculations (MP6). When students relate their calculations to Priya's salad dressing they reason abstractly and quantitatively (MP2).

Reading: MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. “We are going to read this question 3 times.” After the 1st Read: “Tell your partner what this situation is about.” After the 2nd Read: “List the quantities. What can be counted or measured?” Reveal the question(s). After the 3rd Read: “What strategies can we use to solve this problem?”

Launch

• Groups of 2
• “What kind of ingredients do you like to put in your salad?” (lettuce, cabbage, beans, seeds, beets, tomatoes, cheese)

Activity

• 1–2 minutes: quiet think time
• 6–8 minutes: small-group work time
• Monitor for students who:
• estimate to determine that Priya’s recipe will make about $$1\frac{1}{2}$$ cups of dressing
• add $$\frac{3}{4} + \frac{1}{3}+\frac{1}{2}$$ to determine the precise amount of dressing Priya’s recipe will make

Student Facing

• $$\frac{3}{4}$$ cup olive oil
• $$\frac{1}{3}$$ cup lemon juice
• $$\frac{1}{2}$$ cup mustard
• Pinch of salt and pepper

1. Priya has $$\frac{2}{3}$$ cup of olive oil. She is going to borrow some more from her neighbor. How much olive oil does she need to borrow to have enough to make the dressing?

2. 1 tablespoon is equal to $$\frac{1}{16}$$ of a cup. Priya decides that 1 tablespoon of olive oil is close enough to what she needs to borrow from her neighbor. Do you agree with Priya? Explain or show your reasoning.

3. Priya says her recipe will make about $$1\frac{1}{2}$$ cups of dressing. Do you agree? Explain or show your reasoning.

Student Response

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Activity Synthesis

• “If Priya borrows a tablespoon of olive oil from her neighbor and uses it to make dressing, will she be putting in more or less olive oil than the recipe calls for?” ($$\frac{1}{16}$$ is smaller than $$\frac{1}{12}$$ so she will be putting in less olive oil.)
• “Do you think 1 tablespoon is close enough?”
• Poll the class.
• “How might Priya’s decision to use 1 tablespoon of olive oil change the salad dressing?” (It won’t make a difference because the difference is so small. It might taste more lemony or more mustardy because there is not as much oil. It might affect the consistency of the dressing a little.)
• Ask previously selected students to share their estimates for the amount of salad dressing in the given order.
• “Why might Priya estimate that the recipe makes $$1\frac{1}{2}$$ cups of salad dressing?” ($$\frac{3}{4}$$ is $$\frac{1}{4}$$ away from 1 and $$\frac{1}{3}$$ is close to $$\frac{1}{4}$$.)
• “Does the recipe make more or less than $$1\frac{1}{2}$$ cups? How do you know?” (More because $$\frac{1}{3}$$ is more than $$\frac{1}{4}$$.)
• “How many cups does Priya’s recipe make? How do you know?” ($$1\frac{7}{12}$$, I added $$\frac{1}{3}$$, $$\frac{3}{4}$$, and $$\frac{1}{2}$$.)

Activity 2: More Problems to Solve (15 minutes)

Narrative

The purpose of this activity is for students to solve multi-step problems involving the addition and subtraction of fractions with unlike denominators. Students work with both fractions and mixed numbers and can use strategies they have learned such as adding on to make a whole number. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

Representation: Access for Perception. Read both problems aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Conceptual Processing, Language

Launch

• Groups of 2
• “You and your partner will each choose a different problem to solve and then you will discuss your solutions.”

Activity

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion

Student Facing

1. Choose a problem to solve.

Problem A:

Jada is baking protein bars for a hike. She adds $$\frac{1}{2}$$ cup of walnuts and then decides to add another $$\frac{1}{3}$$ cup. How many cups of walnuts has she added altogether?

If the recipe requires $$1\frac{1}{3}$$ cups of walnuts, how many more cups of walnuts does Jada need to add? Explain or show your reasoning.

Problem B:

Kiran and Jada hiked $$1 \frac{1}{2}$$ miles and took a rest. Then they hiked another $$\frac{4}{10}$$ mile before stopping for lunch. How many miles have they hiked so far?

If the trail they are hiking is a total of $$2\frac{1}{2}$$ miles, how much farther do they have to hike? Explain or show your reasoning.

2. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different?
3. Revise your work if necessary.

Student Response

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Activity Synthesis

• “How were the problems the same? How were they different?” (For both problems I had to add fractions first and then subtract that total from another number. There were mixed numbers in both problems.)
• “How did you use equivalent fractions to solve these problems?” (All the fractions we worked with had different denominators so we had to find equivalent fractions with the same denominators in order to add or subtract.)

Lesson Synthesis

Lesson Synthesis

“Today we solved problems that required adding and subtracting fractions.”

“What strategy did you use to find out how much salad dressing Priya’s recipe makes?” (The denominators for the fractions are 2, 3 and 4 so I used 12 because I know that it is a multiple of 2, 3, and 4. I put the half and fourths together first since I could use 4 as a common denominator and then I used 12 to add the fourths and third.

Display: $$\frac{1}{12} - \frac{1}{16}$$

“What strategy did you use to find this difference for the olive oil?” (I knew that 48 is $$4 \times 12$$ and $$3 \times 16$$ so I used that as a common denominator. I used $$12 \times 16$$ as a common denominator.)

“How do you decide which strategy to use?” (It depends on the numbers. If I know a small common multiple of the denominators, I use that. If I don’t, I can always use the product of the denominators.)

Cool-down: Evaluate Expressions (5 minutes)

Cool-Down

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