This activity prompts students to reason about equivalent ratios on a double number line and think of reasonable scenarios for these ratios. This is a review of their work in grade 6.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by partner discussion. As students discuss their answers and reasoning with their partner, select students to share during the whole-class discussion.
Student Lesson in Spanish
Activity
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Complete the double number line diagram with the missing numbers.
A double number line with 11 evenly spaced tick marks. For the top number line, a dashed box represents the title and the number 0 is on the first tick mark, 2 on the third, 7 on the eighth, and 10 on the eleventh. For the bottom number line, a dashed box represents the title and the number 0 is on the first tick mark, 1 on the third, and 5 on the eleventh.
What could each of the number lines represent? Invent a situation and label the diagram.
Make sure your labels include appropriate units of measure.
Student Response
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Building on Student Thinking
Students may struggle thinking of a scenario with a ratio. For those students, ask them if they can draw a picture that would represent that ratio and label each line accordingly.
Activity Synthesis
Display the double number line for all to see with correct values filled in. It does not matter whether the bottom line is labeled with fractions, decimals, or mixed numbers.
Invite selected students to share the situations they came up with and the units for each quantity. After each student shares, invite others to agree or disagree with the reasonableness of the diagram representing that situation. For example, is it really reasonable to say that 7 wheels make bicycles?
MLR8 Discussion Supports. Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking
Math Community
After the Warm-up, display the revisions to the class Math Community Chart that were made from student suggestions in an earlier exercise. Tell students that over the next few exercises, this chart will help the class decide on community norms—how they as a class hope to work and interact together over the year. To get ready for making those decisions, students are invited at the end of today’s lesson to share which “Doing Math” action on the chart is most important to them personally.
1.2
Activity
Standards Alignment
Building On
6.RP.A
Understand ratio concepts and use ratio reasoning to solve problems.
The purpose of this activity is to remind students of strategies for representing and comparing ratios. The taste of the mixture depends on the ratio between the amount of water and the amount of drink mix used to make the mixture. As students describe how they know which mixture would taste different, they attend to precision (MP6).
Ideally, students come into the class knowing how to draw and use diagrams or tables of equivalent ratios to analyze contexts like the one in the task. If the diagnostic assessment suggests that some students can and some students can’t, make strategic pairings of students for this task.
Monitor for students who:
Create discrete diagrams, double number line diagrams, or tables to represent the different mixtures.
Find ratios that are equivalent to the amounts given to help compare the recipes.
Calculate unit rates to help compare the recipes.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
First, show students the three drink mixtures. Some options for how to accomplish this include:
Demonstrate making the three drink mixtures yourself.
Show the video: Three glasses are filled with different amounts of water and drink mix powder. After the mixtures are stirred, two are the same color and one is different.
Display the unlabeled image of the drinks:
Tell students that two of these mixtures taste the same, and one tastes different. Ask students which mixture would taste different and why. Give students 1 minute of quiet think time and then time to share their thinking with their partner.
If possible, ask one student volunteer to take a small taste of each drink mixture and then describe to the rest of the class how the flavors compare.
Ask students:
“What does it mean to say that the drink tastes stronger?” (It has more drink mix for the same amount of water, or it has less water for the same amount of drink mix.)
“The two glasses that taste the same have different amounts of powdered drink mix in them. Why don’t the mixtures taste different? (The amount of water and amount of drink mix were both scaled by the same factor. The ratios of drink mix to water are equivalent.)
Give students 4–5 minutes of quiet work time to answer the questions. If students finish quickly, consider asking them to find the amount of drink mix per cup of water in each recipe, thus emphasizing the unit rate.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language students use to describe how the amount of water and the amount of drink mix affects the taste of the mixture. Display words and phrases such as: “more drink mix,” “more water,” “tastes stronger,” “tastes weaker,” “ratio,” “unit rate,” “per,” etc.
Activity
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Activity Synthesis
The key takeaway from this activity is that the flavor depends on the ratio of the amounts of drink mix and water in the mixture. For a given amount of water, the more drink mix you add, the stronger the mixture tastes. Likewise, for a given amount of drink mix, the more water you add, the weaker the mixture tastes.
When comparing mixtures where both the amounts of drink mix and the amounts of water are different, we can find equivalent ratios for one or more of the mixtures. This method enables us to compare the amounts of drink mix for the same amount of water or to compare the amounts of water for the same amount of drink mix. Computing a unit rate for each situation is a particular instance of this strategy.
Direct students' attention to the reference created using Collect and Display. Ask students to share how they knew which recipe was strongest. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
If students do not create diagrams, consider showing how the reasoning they describe could be represented visually. However, it is not necessary to show every type of diagram.
Discrete diagrams:
Double number line diagrams:
Tables:
water (cups)
drink mix (teaspoons)
1
2
3
water (cups)
drink mix (teaspoons)
2
1
Identify correspondences between the recipes and various diagrams. For example, ask questions like “On the double number line diagram we see the 1 to relationship at the first tick mark. Where do we see that relationship in the discrete diagrams? Where do we see it in the tables?”
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to illustrate correspondences between the number line diagrams and ratio tables for each mixture. Supports accessibility for: Visual-Spatial Processing
1.3
Activity
Standards Alignment
Building On
6.RP.A
Understand ratio concepts and use ratio reasoning to solve problems.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
The purpose of this activity is to build on students’ recent study of scale drawings to informally introduce the concept of proportional relationships.
Initially, students may describe the difference between Moons A, B, C, and D in qualitative terms, such as “D is more squished than the others.” They may also describe Moons A, B, and C as “scaled copies,” even though the curved sides make it difficult to compare corresponding side lengths and angle measurements. The activity prompts students to articulate what they mean in quantitative terms. One possibility would be talking about the height relative to the width (after defining “height” and “width” of a moon in some appropriate way).
For example, students can note that the height of the enclosing rectangle is always times its width for Moons A, B, and C, but not for D.
Alternately, they might note that the distance tip to tip is 3 times the width of the widest part of the moon for Moons A, B, and C, but not for D.
Monitor for different ways students choose to measure the figures. Given the structure of scaled copies, as students work to describe quantitatively what Moons A, B, and C have in common, they are modeling with mathematics (MP4).
In the digital version of the activity, students use an applet to compare corresponding parts in multiple images. The applet allows students to add points and measure distances. The digital version may help students measure quickly and accurately so they can focus more on the mathematical analysis.
Activity Synthesis
The goal of this discussion is to show various representations of quantities that are in a proportional relationship for Moons A, B, and C but not for Moon D. However, the term “proportional relationship” is not introduced at this time.
Invite groups to share their representations and reasoning. To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support whole-class discussion: “All _____ have _____ except _____.” and “What makes _____ different from the others is _____.” Advances: Speaking, Representing
Lesson Synthesis
Share with students, “Today we examined situations involving ratios. In each case, the ratios of the quantities were equivalent for all but one of the things, and the other thing was different in an important way.”
To emphasize ratio and rate language, consider asking students:
“In what important way were the drink mixtures the same or different?” (the amount of drink powder per cup of water)
“How could we tell using ratios that these were the same and different?” (The mixtures that tasted the same were described by equivalent ratios, and . The ratio for the mixture that tasted different was not equivalent.)
“In what important way were the moons the same and different?” (the relationship between the width and the height)
“How could we tell using numbers that these were the same and different?” (The moons that looked the same had a height to width ratio of . The moon that was different had a ratio of .)
Student Lesson Summary
When two different situations can be described by equivalent ratios, that means they are alike in some important way.
An example is a recipe. If two people make something to eat or drink, the taste will only be the same as long as the ratios of the ingredients are equivalent. For example, all of the mixtures of water and drink mix in this table taste the same, because the ratios of cups of water to scoops of drink mix are all equivalent ratios.
water (cups)
drink mix (scoops)
3
1
12
4
1.5
0.5
If a mixture were not equivalent to these, for example, if the ratio of cups of water to scoops of drink mix were , then the mixture would taste different.
Notice that the ratios of pairs of corresponding side lengths are equivalent in Figures A, B, and C. For example, the ratios of the length of the top side to the length of the left side for Figures A, B, and C are equivalent ratios. Figures A, B, and C are scaled copies of each other.
This is the important way in which they are alike.
Figures A, B, C, and D drawn on a grid. A has left side 2 and top side 3. B has left side 1 and top side 1.5. C has left side 3 and top side 4.5. D has left side 3 and top side 3.
If a figure has corresponding sides that are not in a ratio equivalent to these, like Figure D, then it’s not a scaled copy. In this unit, you will study relationships like these that can be described by a set of equivalent ratios.
Standards Alignment
Building On
6.RP.A
Understand ratio concepts and use ratio reasoning to solve problems.
Your teacher will show you three mixtures. Two taste the same, and one is different.
Which mixture would taste different? Why?
Here are the recipes that were used to make the three mixtures:
1 cup of water with teaspoon of powdered drink mix
1 cup of water with teaspoons of powdered drink mix
2 cups of water with teaspoon of powdered drink mix
Which of these recipes is for the stronger tasting mixture? Explain how you know.
Student Response
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Building on Student Thinking
Launch
Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by 3 minutes of partner discussion.
Based on student conversations, you may want to have a whole-class discussion to ensure that they see a way to measure lengths associated with the moons. Consider asking questions like:
“What does it mean to be ‘smashed down’? What measurements might you make to show that this is true?”
“Is there anything else that A, B, and C have in common that you can identify?”
“What things might we measure about these moons to be able to talk about what makes them different in a more precise way?”
Then ask students to complete the last question with their partner.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Consider having students begin by comparing just two crescent moon shapes, like A and B. Check in with students to provide feedback and encouragement after each chunk. Look for students who create a rectangle around each moon and compare the width-height ratios. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
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Here are four different crescent moon shapes.
4 moons labeled A, B, C, D drawn on a grid. A is contained in a rectangle with a length of 8 and width of 12 . B is contained in a rectangle with a length of 4 and width of 6. C is contained in a rectangle with a length of 2 and width of 3. D is contained in a rectangle with a length of 6 and width of 4.
What do Moons A, B, and C all have in common that Moon D doesn’t?
Use numbers to describe how Moons A, B, and C are different from Moon D.
Pause here so your teacher can review your work.
Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Student Response
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Building on Student Thinking
For question 2, students might attempt to find the area of each moon by counting individual square units. Suggest that they create a rectangle around each moon instead and compare the width-height ratios.
For question 3, if students are not sure how to set up these representations, providing a template may be helpful.
