This Warm-up prompts students to carefully analyze and compare four representations of situations involving ratios. In making comparisons, students have a reason to attend to quantities carefully and use language precisely (MP6). The activity also enables the teacher to hear how students use ratio language and talk about representations of ratios learned so far before working with a new representation.
Student Lesson in Spanish
Launch
Arrange students in groups of 2–4. Display the representations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their response with their group, and then together find as many sets of three as they can.
Activity
None
Student Task Statement
Which three go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, if students make broad or vague claims (for instance, “A and B show the same thing”), ask them to clarify the terms they used (for instance, “What do you mean by “the same thing?” or “How are they the same?”).
Also, prompt students to explain the meaning of any terminology they use, such as “equivalent ratios,” “same rate,” and “per,” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
9.2
Activity
20 mins
A Huge Amount of Sparkling Orange Juice
Standards Alignment
Building On
Addressing
6.RP.A.3.a
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
In this activity, students are asked to find missing values for significantly scaled-up ratios. The work serves several purposes:
To uncover a limitation of a double number line (namely, that it is not always practical to extend it to find significantly scaled-up equivalent ratios)
To reinforce the multiplicative reasoning needed to find equivalent ratios (especially in cases when drawing diagrams or skip counting is inefficient)
To introduce a table as a way to represent equivalent ratios
Monitor for the different ways students find equivalent ratios involving large values. Here are some strategies they may use, from less practical or less flexible to more accommodating:
Extend the given double number line and then try to squeeze numbers on the extreme right end, ignoring the previously equal intervals.
Draw a new double number line diagram that is longer or where the intervals represent multiples of 4 and 5 (rather than 4 and 5).
Use multiplication (or division) and write expressions or equations to represent the given scenarios.
Select students who use different strategies to share later, including those who find the given double number line inadequate—not long enough to accommodate large numbers, requiring more marking or writing, and so on—and are consequently motivated to find a more efficient strategy.
Launch
Give students 2–3 minutes to work on the first two questions and then ask them to pause. As a class, discuss the two approaches students are likely to take: counting multiples of 4 and 5 up to 36 and 45; and multiplicative reasoning (asking “What number times 4 equals 36?”). Also discuss how a double number line as shown here might be used to support reasoning.
Reiterate the multiplicative relationship between equivalent ratios before students move on.
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, ask students: “In earlier lessons, how did you represent 1, 2, 3, or more batches of color mixtures, play clay, or radish cake?” or “How did you find the amount of each ingredient when dealing with a larger number of batches?” Supports accessibility for: Social-Emotional Functioning, Conceptual Processing
Activity
None
Student Task Statement
Noah’s recipe for one batch of sparkling orange juice uses 4 liters of orange juice and 5 liters of soda water.
Use the double number line to show how many liters of each ingredient to use for different-sized batches of sparkling orange juice.
If someone mixes 36 liters of orange juice and 45 liters of soda water, how many batches would they make?
If someone uses 400 liters of orange juice, how much soda water would they need?
If someone uses 455 liters of soda water, how much orange juice would they need?
Explain the trouble with using a double number line diagram to answer the last two questions.
Activity Synthesis
After students have a chance to share with a partner, ask previously selected students to share their reasoning for the last few questions. Sequence the discussion of students’ strategies in the order listed in the Activity Narrative. Discuss any challenges of using the double number line and merits of alternative methods students might have used.
Explain that there is a more appropriate tool—a table—that can be used to represent equivalent ratios. Display for all to see a double number line representing the sparkling juice recipe and a table of equivalent ratios. Explain that even though the table is oriented vertically and the double number line is oriented horizontally, the two representations represent the same ratios. Explain what we mean by row and column and demonstrate the use of these words. Complete the table using the values from the orange-soda ratios. Along the way, connect the two representations and compare and contrast how they show the quantities in the recipe.
A few other key insights to convey:
Just as it was important to label the double number line, it is important to label the columns of the table to indicate what the values represent.
Each row of a table shows a pair of values from a collection of equivalent ratios. Unlike a number line, distances between values do not matter.
On each line of a double number line, numbers are shown in order. In each column of a table, order is not important, that is, pairs of values can be placed in any order that is convenient. When complete, the display should look something like this:
9.3
Activity
10 mins
Batches of Nihaizu Sauce
Standards Alignment
Building On
Addressing
6.RP.A.3.a
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
This activity invites students to interact with a table in a way that discourages skip-counting. Numbers within each column are deliberately out of order. This is intended to encourage students to multiply the pairs of values from a given ratio by the same number and to emphasize that the order in which pairs of values appear is not a necessary part of the structure of a table. (Order within rows, however, is necessary.) The last question reinforces the definition of "equivalent ratios."
Students may use the given values (7 and 5) as the basis for every calculation (for instance, for every row, they think “7 times what . . . ” or “5 times what . . .”). They may also reason with values from another row (for instance, they may see 250 as rather than as ). As students work, monitor for different approaches.
Launch
Ask students if they had a favorite salad dressing or dipping sauce and, if so, whether they knew the main ingredients.
Explain that there are many dressings in Japanese cooking, and they often use vinegar as a base. The mixtures may also be used to marinate vegetables, meat, and seafood, or as dipping sauces. This activity looks at the ratios of vinegar and soy sauce in nihaizu (nee-HYE-zuu)—a sauce made only with those two ingredients—organized in a table. (The word “nihaizu” translates to “two-ingredient vinegar.”)
Explain that a tablecan be used to organize equivalent ratios. In this case, one column contains amounts of vinegar, and the other column contains corresponding amounts of soy sauce. Each row shows the amount of each ingredient in a particular batch.
Reiterate that multiplying both parts of a ratio by the same non-zero number always creates a ratio that is equivalent to the original ratio.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the last question on how they know that each row in the table shows a ratio that is equivalent to . Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Activity Synthesis
Invite one or more students who used multiplicative approaches to share their reasoning with the class. Consider displaying the table and using it to facilitate gesturing and arrow-drawing while students explain. Highlight the strategy of multiplying the 7 and 5 values by the same number.
Lesson Synthesis
In the lesson, students saw that a table can be used to represent equivalent ratios and noticed the ways in which a table and a double number line diagram represent equivalent ratios. To reinforce the key ideas from the lesson, consider asking:
“A table has rows and columns. Where are the rows and where are the columns?” (A row is a horizontal group of cells or entries. A column is a vertical group of cells or entries.)
“How does a table represent equivalent ratios?” (Every row contains a ratio that is equivalent to the ratios in other rows.)
“Compare how equivalent ratios are represented in a table and in a double number line. How are they alike or different?” (In both representations, the numbers in each equivalent ratio line up. In a double number line, they line up vertically. In a table, they line up horizontally. In a double number line, the numbers in the ratios are shown in order from left to right. In a table, the numbers aren’t necessarily listed in order.)
“How can we use a table to find equivalent ratios?” (We can multiply the numbers in a row by the same value and record the result in a new row.)
Student Lesson Summary
A table is a way to organize information. Each horizontal set of entries is called a row, and each vertical set of entries is called a column. (The table shown has 2 columns and 5 rows.) A table can be used to represent a collection of equivalent ratios.
Here is a double number line diagram and a table that both represent the situation: “The price is $2 for every 3 mangos.”
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Students may become frustrated when they “run out of number line,” but remind them of what they know about how to find ratios equivalent to (they need to multiply both 4 and 5 by the same number). Consider directing their attention to a definition of "equivalent ratios" displayed in your room or in a previous lesson, or suggesting they reexamine some of the simpler cases (for instance, the relationship between and ). Be on the lookout for students trying to tape on more paper to extend their number lines.
Activity
None
Student Task Statement
Nihaizu (nee-HYE-zuu) is a sauce used with seafood or vegetables in Japanese cooking. A recipe for nihaizu uses 7 fluid ounces of vinegar and 5 fluid ounces of soy sauce.
The table shows amounts of vinegar and soy sauce that would be in different-sized batches of the recipe.
Complete the table so that ratios represented by each row are equivalent. Explain or show your reasoning.
How do you know that each row shows a ratio that is equivalent to ? Explain your reasoning.
vinegar
(fl oz)
soy sauce
(fl oz)
7
5
28
10
3.5
250
56
Student Response
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Building on Student Thinking
Students may make patterns that do not yield equivalent ratios. For example, they may think “7 minus 2 is 5, so for the next row, 28 minus 2 is 26.” Or they may think “7 plus 21 is 28, so then 5 plus 21 is 26.” If so, consider:
Appealing to what students know about batches of recipes. “The second row represents how many batches of nihaizu?” (4, because 28 is .) “Okay, so to make 4 batches of nihaizu, how will we figure out how much soy sauce to use?” (Also multiply the 5 by 4.)
Refreshing what students learned about equivalent ratios. “We need a ratio that is equivalent to the ratio represented in row 1. So what do we need to do to the 7 and the 5?” (Multiply them by the same number.)
Students may be unsure about how to find the missing value in the row with 3.5. Encourage them to reason about it the same way they reasoned about the other rows. “We need a ratio that is equivalent to the ratio represented in row 1. So what do we need to do to the 7 and the 5?” They may have to get there by way of division. 7 divided by 2 is 3.5, so 7 times is 3.5; this means multiplying 5 by as well.
