Introducing Proportional Relationships with Tables

5 min

Narrative

The purpose of this Warm-up is to elicit the idea of using a table to see patterns between related quantities, which will be useful when students discuss how to use tables to analyze proportional relationships in a later activity. While students may notice and wonder many things about these images, the relationships between the rows and the relationship between the columns are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structures they might notice are:

  • Scale factors between any pair of rows (In other words, multiplying both values in one row by the same number gives the values in another row.)
  • A unit rate between the two columns (For example, multiplying any value in the first column by 12 gives the corresponding number in the second column.)
     

Launch

Arrange students in groups of 2. Display the table for all to see and read the problem stem aloud. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.

Student Task

Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases.


Table with 2 columns and 4 rows of data.
Table with 2 columns and 4 rows of data. The columns are: number of cases they order and number of rolls of paper towels. The table has the ordered pairs ( 1 comma 12), (2 comma 36), (5 comma 60) and (10 comma 120). There is an arrow pointing from row 3 to row 4 on each side with times 2 next to the arrows.

What do you notice? What do you wonder?

Sample Response

Things students may notice:

  • To go from one row to another, multiply both columns by the same number.
  • To find the number of rolls, multiply the number of cases by 12.
  • There are 12 rolls in a case.

Things students may wonder:

  • How much does a case cost?
  • How many paper towels are on a roll?
  • Why would you need 120 rolls of paper towels?

Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the relationship between the number of cases and the number of paper towels does not come up during the conversation, ask students to discuss this idea.

Standards
Building Toward
  • 7.RP.2.b·Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
  • 7.RP.A.2.b·Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

15 min

10 min

10 min