# Lesson 1

Proportional Relationships and Equations

### Lesson Narrative

In this lesson, students build on their work with tables and represent proportional relationships using equations of the form $$y = kx$$. The activities revisit contexts from the previous two lessons, presenting values in tables and focusing on the idea that for each table, there is a number $$k$$ so that all values in the table satisfy the equation $$y = kx$$. By expressing the regularity of repeated calculations of values in the table with the equations, students are engaging in MP8.

Teacher Notes for IM 6–8 Accelerated
The activities in this lesson refer to the contexts being seen in earlier lessons. Those lessons are not included in IM 6–8 Math Accelerated, so these instructions do not apply. Use some time in the launch of each activity to familiarize students with the contexts as needed.

### Learning Goals

Teacher Facing

• Generalize a process for finding missing values in a proportional relationship, and justify (orally) why this can be abstracted as $y=kx$, where $k$ is the constant of proportionality.
• Generate an equation of the form $y=kx$ to represent a proportional relationship in a familiar context.
• Write the constant of proportionality to complete a row in the table of a proportional relationship where the value for the first quantity is 1.

### Student Facing

Let’s write equations describing proportional relationships.

### Student Facing

• I can write an equation of the form $y=kx$ to represent a proportional relationship described by a table or a story.
• I can write the constant of proportionality as an entry in a table.

Building On

### 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