Tables and Double Number Line Diagrams

10 min

Narrative

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

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time and 1–2 minutes to discuss with their partner.

Student Task

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.

  1. Do you agree that this table represents the situation? Explain your reasoning.
    20 100
    10 50
    1 5
    3,000
  2. Complete the last row with the missing number.

Sample Response

  1. Sample response: I agree with the first three rows, but the last row would be for 3,000 seconds instead of 3,000 meters, so it wouldn't help Han answer the question.
  2. The empty cell should contain 15,000.

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.)

Standards
Addressing
  • 6.RP.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.
  • 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.

10 min

15 min