Multiplying Rational Numbers (Part 1)

5 min

Teacher Prep
Setup
Ask students what they remember about problems involving distance, rate, and time. 1 minute of quiet work time followed by whole-class discussion.

Narrative

This activity reminds students of previous work they have done with constant speed situations, using d=rtd=rt for the relationship between distance, rate, and time. This prepares students for representing movement in opposite directions using signed numbers in following activities. 

Students may choose to use strategies such as creating a double number line or table of equivalent ratios to make sense of these problems and come up with a solution. While students are free to use these strategies, ensure that they also understand how to use d=rtd=rt to represent the relationship between distance traveled, elapsed time, and rate of travel for constant speed situations. 

Launch

Ask students what they remember about problems involving distance, rate, and time. They might offer that distances traveled and elapsed time create a set of equivalent ratios or that the elapsed time can be multiplied by the speed to give the distance traveled. Give students 1 minute of quiet work time, and follow with a whole-class discussion.

Student Task

  1. A car is traveling at a constant speed of 60 miles per hour. How far does the car travel in:

    1. 2 hours?
    2. 5 hours?
    3. xx hours?
  2. Create a representation that shows the relationship between the elapsed time and the distance traveled for this car.

Sample Response

  1.  
    1. 120 miles
    2. 300 miles
    3. 60x60x miles
  2. Sample responses:
    • d=60td = 60t
    • elapsed time (hours) distance traveled (miles)
      2 120
      5 300
      xx 60x60x
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to remind students of how the equation d=rtd=rt can be used to solve problems involving movement at a constant speed. To find the distance traveled, we can multiply the rate of travel (or speed) by the elapsed time.

Consider drawing a diagram or table to facilitate the discussion of each problem and to remind students of the strategies they used while working with proportional relationships, such as using a scale factor or calculating the constant of proportionality. When relating distance and time in a constant speed situation, the speed is the constant of proportionality.

Standards
Addressing
  • 7.RP.A·Analyze proportional relationships and use them to solve real-world and mathematical problems.
  • 7.RP.A·Analyze proportional relationships and use them to solve real-world and mathematical problems.
Building Toward
  • 7.NS.2.a·Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
  • 7.NS.A.2.a·Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as <span class="math">\((-1)(-1) = 1\)</span> and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

15 min

15 min