Lesson 7Representations of Linear Relationships
Learning Goal
Let’s write equations from real situations.
Learning Targets
I can use patterns to write a linear equation to represent a situation.
I can write an equation for the relationship between the total volume in a graduated cylinder and the number of objects added to the graduated cylinder.
Lesson Terms
- linear relationship
- vertical intercept
Warm Up: Estimation: Which Holds More?
Problem 1
Which glass will hold the most water? The least?
Activity 1: Rising Water Levels
Problem 1
Move the green circle to set the starting level of water to a value you or your teacher choose.
What is the volume,
, in the cylinder after you add: 3 objects?
7 objects?
objects? Explain your reasoning.
If you wanted to make the water reach the highest mark on the cylinder, how many objects would you need?
Plot and label points that show your measurements from the experiment.
Plot and label a point that shows the depth of the water before you added any objects.
The points should fall on a line. Use the Line tool to draw this line.
Compute the slope of the line using several different triangles. Does it matter which triangle you use to compute the slope? Why or why not?
The equation of the line in the experiment has two numbers and two variables. What physical quantities do the two numbers represent? What does
represent and what does represent?
Print Version
Record data from your teacher’s demonstration in the table. (You may not need all the rows.)
number of objects
volume in ml
What is the volume,
, in the cylinder after you add objects? Explain your reasoning. If you wanted to make the water reach the highest mark on the cylinder, how many objects would you need?
Plot and label points that show your measurements from the experiment.
The points should fall on a line. Use a ruler to graph this line.
Compute the slope of the line. What does the slope mean in this situation?
What is the vertical intercept? What does vertical intercept mean in this situation?
Are you ready for more?
Problem 1
A situation is represented by the equation
Invent a story for this situation.
Graph the equation.
What do the
and the 5 represent in your situation? Where do you see the
and 5 on the graph?
Activity 2: Calculate the Slope
Problem 1
For each graph, record:
vertical change
horizontal change
slope
Describe a procedure for finding the slope between any two points on a line.
Write an expression for the slope of the line in the graph using the letters
and .
Lesson Summary
Let’s say we have a glass cylinder filled with 50 ml of water and a bunch of marbles that are 3 ml in volume. If we drop marbles into the cylinder one at a time, we can watch the height of the water increase by the same amount, 3 ml, for each one added. This constant rate of change means there is a linear relationship between the number of marbles and the height of the water. Add one marble, the water height goes up 3 ml. Add 2 marbles, the water height goes up 6 ml. Add
Reasoning this way, we can calculate that the height,
Now what if we didn’t have a description to use to figure out the slope and the vertical intercept? That’s okay so long as we can find some points on the line! For the line graphed here, two of the points on the line are
The slope of this line is the quotient of the length of the vertical side of the slope triangle and the length of the horizontal side of the slope triangle. So the slope,