In this activity, students examine a given table of equivalent ratios to reason about a situation involving distance and time. To answer questions, students need to interpret the quantities in the situation and the values in a table that is set up to conflict with what is described. The table has no headers, which makes it less obvious that the value 3,000 is in the wrong place.
The work here prompts students to reason quantitatively and abstractly (MP2) and to attend to precision (MP6). In examining the table, students notice that labels or descriptions of the quantities are important when using a table of equivalent ratios to solve problems.
Launch
Student Lesson in Spanish
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time and 1–2 minutes to discuss with their partner.
Activity
None
Han can run 100 meters in 20 seconds. He wonders how long it would take him to run 3,000 meters at this rate. He made a table of equivalent ratios.
Do you agree that this table represents the situation? Explain your reasoning.
20
100
10
50
1
5
3,000
Complete the last row with the missing number.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Invite students to share their response to the first question. Discuss how they knew that the values in the first three rows represent the times and distances of Han’s run. Consider displaying the table and annotating it to illustrate students’ thinking, especially multiplicative reasoning.
Next, discuss how students interpreted the 3,000 in the last row.
If students saw the 3,000 as time in seconds—matching the meaning of the other values in that column—and wrote 15,000 in the blank cell, discuss what the 15,000 represents. Then, ask students: “Does this value tell Han what he wanted to know? If not, what does it tell Han?” (No. It tells Han how far he would go in 3,000 seconds.)
If students thought of the 3,000 as the distance in meters and wrote 600 in the empty cell, discuss how they arrived at the 600 and what it represents. Then, ask students to consider how the 3,000 and 600 relate to the values in the preceding rows.
Make sure students see that the values in a column are meant to represent the same quantity. While Han would run 3,000 meters in 600 seconds, the 3,000 here represents time in seconds and the missing value is the distance in meters.
Finally, ask students: “What could Han do to improve the table?” (He could label the quantity that each column represents.)
13.2
Activity
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 continue to use tables of equivalent ratios to solve problems but with decreasing scaffolding. For the first problem, there is an empty table students can use to reason about the quantities. In the second problem, no scaffolding is provided to allow students to create a table from scratch.
Monitor for the different ways students solve the problems, especially the last one. Select students who use different strategies to share during discussion later.
Launch
Arrange students in groups of 2. Give 3 minutes of quiet work time and then have students work with their partner.
Activity
None
Priya can bike 150 meters in 20 seconds. At this rate, how long would it take her to bike 3,000 meters?
Priya’s neighbor has a dirt bike that can go 360 meters in 15 seconds. At this rate, how long would it take her neighbor to go 3,000 meters? Show your reasoning.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Invite 1–2 previously selected students to display and briefly explain their solutions to each problem. Highlight the connections between different strategies, including different ways of using the same representation (such as different ways of reasoning with a table). If time permits, discuss connections across representations (such as between tables and double number line diagrams).
MLR8 Discussion Supports. For each method that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Consider selecting 1–2 students to share with the class and discussing words or phrases that might help to clarify the original statement or the revoiced statement. Advances: Listening, Speaking
13.3
Activity
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 prompts students to compare and contrast two representations of equivalent ratios. Students work collaboratively to observe similarities and differences of using a double number line diagram and using a table to express the same situation. Here are some key distinctions:
double number line
table
Distances between numbers and lengths of lines matter.
Distances and lengths do not matter because there are no lines.
The numbers on each line must be in order.
Rows of ratios can be out of order; within a column, numbers can go in any order that is convenient.
Each value of a ratio is shown on a line.
Each value of a ratio is shown in a column.
Pairs of values of a ratio are aligned vertically.
Pairs of values of a ratio appear in the same row.
Launch
To help students build some intuition about kilometers, begin by connecting it with contexts that are familiar to them. Tell students that “kilometer” is a unit used in the problem. Then ask a few guiding questions.
“Can you name two things in our town (or city) that are about 1 kilometer apart?” (Consider finding some examples of 1-kilometer distances near your school ahead of time.)
“How long do you think it would take you to walk 1 kilometer?” (Typical human walking speed is about 5 kilometers per hour, so it takes a person about 12 minutes to walk 1 kilometer.)
“What might be a typical speed limit on a highway, in kilometers per hour?” (100 kilometers per hour is a typical highway speed limit. Students might be more familiar with a speed limit such as 65 miles per hour. Since there are about 1.6 kilometers in every mile, the same speed will be expressed as a higher number in kilometers per hour than in miles per hour.)
Arrange students in groups of 2. Give one person a slip with the table and the other a slip with a double number line (shown below). Ask students to first do what they can independently, and then to obtain information from their partners to fill in all the blanks. Explain that when the blanks are filled, the two representations will show the same information.
distance traveled (km)
elapsed time (sec)
0
0
80
10
1
Activity
None
The International Space Station orbits around the Earth at a constant speed. Your teacher will give you either a double number line or a table that represents this situation. Your partner will get the other representation.
Complete the parts of your representation that you can figure out for sure.
Share information with your partner, and use the information that your partner shares to complete your representation.
What is the speed of the International Space Station?
Place the two completed representations side by side. Discuss with your partner some ways in which they are the same and some ways in which they are different.
Record at least one way that they are the same and one way they are different.
Student Response
Loading...
Building on Student Thinking
Students with the double number line representation may decide to label every tick mark instead of just the ones indicated with dotted rectangles. This is fine. Make sure they understand that the tick marks with dotted rectangles are the ones they are supposed to record in the table.
Activity Synthesis
Display completed versions of both representations for all to see. Invite students to share the ways the representations are alike and different. Consider writing some of these on the board, or this could just be a verbal discussion. Highlight the distinctions in terms of distances between numbers, order of numbers, and the vertical or horizontal orientations of the representations.
Although it is not a structural distinction, students might describe the direction in which multiplying happens as a difference between the two representations. They might say that we “multiply up or down” to find equivalent ratios in a table, and we “multiply across” to do the same on a double number line. You could draw arrows to illustrate this fact:
The vertical orientation of tables and the horizontal orientation of double number lines are conventions we decided to consistently use in these materials. Mathematically, there is nothing wrong with orienting each representation the other way. Students may encounter tables oriented horizontally in a later course. Later in this course, they will encounter number lines oriented vertically.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, to draw attention to common features, color code corresponding values in the table and double number line diagram. To highlight distinctions in the orientation of the representations and the order in which the values are listed, use arrows or other annotations. Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
Briefly revisit the two tasks, displaying the representations for all to see, and pointing out ways in which tables and double number lines are alike and different.
“Why is it important to include descriptive column headers on tables?” (Without headers, it’s unclear which quantity each column represents, whether the values we put in the cells correctly represent the situation, or whether they show equivalent ratios.)
“What are the main differences between using a double number line and using a table in solving problems that involve equivalent ratios?” (Tables can be created fairly quickly and there is flexibility in what numbers to use to help solve a problem. Double number lines may be more involved to draw, may need to be pretty long to accommodate the numbers, or may require careful thinking about the scale being used. The numbers need to be listed in order of size—from smaller to larger.)
Emphasize that because of their flexibility, tables are sometimes easier to work with.
Student Lesson Summary
On a double number line diagram, we put labels in front of each line to tell what the numbers represent. On a table, we put labels at the top of each column to tell what the numbers represent.
Here are two different ways we can represent the situation: “A snail is moving at a constant speed down a sidewalk, traveling 6 centimeters per minute.”
Both double number lines and tables can help us use multiplication to make equivalent ratios, but there is an important difference between the two representations.
On a double number line, the numbers on each line are listed in order. With a table, you can write the ratios in any order. For this reason, sometimes a table is easier to use to solve a problem.
For example, what if we wanted to know how far the snail travels in 10 minutes? Notice that 60 centimeters in 10 minutes is shown on the table, but there is not enough room for this information on the double number line.
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.
