# Unit 3Big Ideas

## Representing Linear Relationships

This week your student will learn how to write equations representing linear relationships. A linear relationship exists between two quantities where one quantity has a constant rate of change with respect to the other. The relationship is called linear because its graph is a line.

For example, say we are 5 mile into a hike heading toward a lake at the end of the trail. If we walk at a speed of 2.5 miles per hour, then for each hour that passes we are 2.5 miles further along the trail. After 1 hour we would be 7.5 miles from the start. After 2 hours we would be 10 miles from the start (assuming no stops). This means there is a linear relationship between miles traveled and hours walked. A graph representing this situation is a line with a slope of 2.5 and a vertical intercept of 5.

Here is a task to try with your student:

The graph shows the height in inches, , of a bamboo plant months after it has been planted.

- What is the slope of this line? What does that value mean in this context?
- At what point does the line intersect the -axis? What does that value mean in this context?

Solution:

- 3. Every month that passes, the bamboo plant grows an additional 3 inches.
- . This bamboo plant was planted when it was 12 inches tall.

## Finding Slopes

This week your student will investigate linear relationships with slopes that are not positive. Here is an example of a line with negative slope that represents the amount of money on a public transit fare card based on the number of rides you take:

The slope of the line graphed here is since This corresponds to the cost of 1 ride. The vertical intercept is 40, which means the card started out with $40 on it.

One possible equation for this line is It is important for students to understand that every pair of numbers that is a solution to the equation representing the situation is also a point on the graph representing the situation. (We can also say that every point on the graph of the situation is a solution to the equation representing the situation.)

Here is a task to try with your student:

A length of ribbon is cut into two pieces. The graph shows the length of the second piece, , for each length of the first piece, .

- How long is the original ribbon? Explain how you know.
- What is the slope of the line? What does it represent?
- List three possible pairs of lengths for the two pieces and explain what they mean.

Solution:

- 15 feet. When the second piece is 0 feet long, the first is 15 feet long, so that is the length of the ribbon.
- -1. For each length the second piece increases by, the first piece must decrease by the same length. For example, if we want the second piece to be 1 foot longer, then the first piece must be 1 foot shorter.
- Three possible pairs: , which means the second piece is 14.5 feet long so the first piece is only a half foot long. , which means each piece is 7.5 feet long, so the original ribbon was cut in half. , which means the original ribbon was not cut at all to make a second piece, so the first piece is 15 feet long.