Two Related Quantities, Part 2

10 min

Teacher Prep
Setup
Consider asking students to keep their materials closed. Display the statement about Lin and Jada for all to see. 1 minute to notice and wonder. A brief whole-class discussion, followed by quiet think time to complete the task.

Narrative

In this Warm-up, students reason about the relationship between distance, rate, and time to solve a problem. The purpose is to activate the idea that constant speed can be represented by a set of equivalent ratios associating distance traveled and elapsed time. In the longer activity that follows, students represent the relationship between these two quantities using a table, equations, and graphs.

As students work, monitor for different representations used (particularly tables) as well as for students who calculate each wheelchair’s speed in meters per second or each wheelchair’s pace in seconds per meter.

Launch

Arrange students in groups of 2. Give students 3–4 minutes of quiet work time and 1–2 minutes to share their thinking with a partner. Follow with a whole-class discussion. 

If needed, remind students of tools or strategies that may be appropriate for solving this problem, including double number line diagrams or tables of equivalent ratios. Consider allowing students to use calculators to ensure inclusive participation in the activity.

Student Task

A wheelchair user is considering two electric wheelchairs. Wheelchair A can travel 25 meters in 10 seconds. Wheelchair B can travel 13 meters in 5 seconds.

  1. Which wheelchair can travel at a faster rate? Be prepared to explain how you know.
  2. How many seconds will it take each wheelchair, moving at a constant speed, to travel 195 meters? Show your reasoning.

Sample Response

  1. Wheelchair B can travel faster. Sample reasoning: 
    • It can travel 26 meters in 10 seconds. Wheelchair A can travel 25 meters in the same amount of time.
    • Wheelchair A can travel 2.5 meters in 1 second. Wheelchair B can travel 2.6 meters in 1 second.
  2. 78 seconds for Wheelchair A and 75 seconds for Wheelchair B. Sample reasoning:
    Wheelchair A
    distance (meters) time (seconds)
    25 10
    1 1025\frac{10}{25} or 25\frac25
    195 195⋅25195\boldcdot\frac25 or 78
    Wheelchair B
    distance (meters) time (seconds)
    13 5
    1 513\frac{5}{13}
    195 195⋅513195\boldcdot\frac{5}{13} or 75

Activity Synthesis (Teacher Notes)

Invite students to share how they know which wheelchair can travel faster. If one of the following ways of reasoning is not mentioned by students, bring it to students’ attention:

  • Comparing the distances traveled in the same amount of time: Doubling the given distance and time for Wheelchair B shows that it can cover a longer distance in 10 seconds than Wheelchair A can. Calculating the distance that each wheelchair can travel in 1 second also shows the same result.
  • Comparing the amounts of time for travel the same distance: Wheelchair A takes more time (513\frac{5}{13} or 1026\frac{10}{26} second) to cover 1 meter than Wheelchair B does (1025\frac{10}{25} second).

Next, discuss how students found the time it would take each wheelchair to travel 195 meters. Ask students who used different reasoning strategies to share. If no student used a table of equivalent ratios, as shown in the Student Response, display a pair of blank tables and discuss how to use the table to reason about the distance each wheelchair can travel in 1 second and the time it takes to travel 1 meter.

Standards
Addressing
  • 6.EE.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <em>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.</em>
  • 6.EE.C.9·Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. <span>For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation <span class="math">\(d = 65t\)</span> to represent the relationship between distance and time.</span>
  • 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.3.b·Solve unit rate problems including those involving unit pricing and constant speed. <em>For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?</em>
  • 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.
  • 6.RP.A.3.b·Solve unit rate problems including those involving unit pricing and constant speed. <span>For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?</span>

25 min