Unit 2, Lesson 13
Tables and Double Number Line Diagrams
Grade 6
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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.
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time and 1–2 minutes to discuss with their partner.
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.
| 20 | 100 |
| 10 | 50 |
| 1 | 5 |
| 3,000 |
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.
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.)
To Copy (from Blackline Masters)
The International Space Station Cards
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. |
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.
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 | |
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.
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.
Help us improve by sharing suggestions or reporting issues.
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.
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.