Lesson 5

Two Equations for Each Relationship

Lesson Narrative

In previous lessons students saw that a proportional relationships can be viewed in two ways, depending on which quantity you regard as being proportional to the other. In this lesson they write equations for these two ways, and they see why the two constants of proportionality associated with each way are reciprocals of each other. 

The activities in this lesson use familiar contexts, but not identical situations from previous lessons: measurement conversions and water flowing at a constant rate. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and generalize from repeated calculations to arrive at an equation (MP8). After students write or use an equation, they interpret their answers in the context of the situation (MP2).


Learning Goals

Teacher Facing

  • Use the word “reciprocal” to explain (orally and in writing) that there are two related constants of proportionality for a proportional relationship.
  • Write two equations that represent the same proportional relationship, i.e., $y=kx$ and $x=(\frac{1}{k})y$, and explain (orally) what each equation means.

Student Facing

Let’s investigate the equations that represent proportional relationships.

Learning Targets

Student Facing

  • I can find two constants of proportionality for a proportional relationship.
  • I can write two equations representing a proportional relationship described by a table or story.

CCSS Standards

Building On

Addressing

Glossary Entries

  • constant of proportionality

    In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

    In this example, the constant of proportionality is 3, because \(2 \boldcdot 3 = 6\), \(3 \boldcdot 3 = 9\), and \(5 \boldcdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.

    number of oranges number of apples
    2 6
    3 9
    5 15
  • proportional relationship

    In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.

    For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.

    We can write this relationship as \(p = 4s\). This equation shows that \(s\) is proportional to \(p\).

    \(s\) \(p\)
    2 8
    3 12
    5 20
    10 40

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