Unit 3, Lesson 7
Representations of Linear Relationships
Grade 8
7.1
Warm-up
Which container holds the most liquid? The least?
7.2
Activity
7.3
Activity
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Consider a new cylinder that is filled with 25 ml of water. Identical beads that have a volume of 0.75 ml are dropped into the cylinder one at a time.
- Write an equation that describes the volume in the cylinder as beads are added. Use for the total volume in the cylinder and for the number of beads.
- How would your original equation change if the cylinder was only filled with 12 ml of water?
- How would your original equation change if larger beads that had a volume of 1.25 ml were used?
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A situation is represented by the equation .
- Create a story for this situation.
- What does the 5 represent in your situation?
- What does the represent in your situation?
Student Lesson Summary
A glass cylinder is filled with 50 ml of water. Marbles, each with a volume of 3 ml, are dropped into the cylinder one at a time. With each marble, the water level increases in height by an amount equivalent to a volume of 3 ml. This constant rate of change means there is a linear relationship between the number of marbles and the total volume in the cylinder. If 1 marble is added, the total volume increases by 3 ml. If 2 marbles are added, the total volume increases by 6 ml. If marbles are added, the total volume goes up ml.
This means that the total volume, , for marbles is . The 3 represents the rate of change, or slope of the graph, and the 50 represents the initial amount, or vertical intercept of the graph.
Any linear relationship can be expressed in the form using just the rate of change, , and the initial amount, . For example, the equation could be used to describe a different scenario where marbles, each with a volume of 5 ml, are added to a cylinder that initially had 20 ml of water.
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