Unit 5, Lesson 1
Describing and Graphing Situations
Algebra 1
1.1
Warm-up
Bagel Shop
Instructional Routines
Materials
None
Activity Narrative
The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.
Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the best price, however, do form a function, because there is only one possible best price for each number of bagels.
Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.
Launch
Arrange students in groups of 4. Give students a few minutes of quiet time to think about the first question and then a couple of minutes to share their thinking with their group. Pause for a class discussion.
Invite students to share an explanation of how each person in the situation could be right. If possible, record and display students’ reasoning for all to see. After the reasoning behind each price is shared, direct students to the table in the second question. Ask students to write down “best price” in the header of the second column and then complete the table.
Activity
NoneYour teacher will give you instructions for completing the table.
A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
- Jada says to her friends, “Shouldn’t the total be $13.25?”
- Priya says, “I think it should be $13.00.”
- Han says, “No, I think it should be $11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
| number of bagels | |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 | |
| 13 |
Activity Synthesis
Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels) and a table showing the best price for each number of bagels bought.
| number of bagels |
shopkeeper’s price |
Jada’s price |
Priya’s price |
Han’s price |
|---|---|---|---|---|
| 1 | 1.25 | |||
| 2 | 2.50 | |||
| 3 | 3.75 | |||
| 4 | 5.00 | |||
| 5 | 6.25 | |||
| 6 | 7.50 | 6.00 | ||
| 7 | 8.75 | 7.25 | ||
| 8 | 10.00 | 8.50 | ||
| 9 | 11.25 | 9.75 | 8.00 | |
| 10 | 12.50 | 11.00 | 9.25 | |
| 11 | 13.75 | 12.25 | 10.50 | |
| 12 | 15.00 | 12.00 | 11.75 | 10.00 |
| 13 | 16.25 | 13.25 | 13.00 | 11.25 |
| number of bagels |
best price |
|---|---|
| 1 | 1.25 |
| 2 | 2.50 |
| 3 | 3.75 |
| 4 | 5.00 |
| 5 | 6.25 |
| 6 | 6.00 |
| 7 | 7.25 |
| 8 | 8.50 |
| 9 | 8.00 |
| 10 | 9.25 |
| 11 | 10.50 |
| 12 | 10.00 |
| 13 | 11.25 |
Ask students,
- “‘Number of bagels’ and ’price’ do not form a function, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
- “If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
- “If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”
After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles represent Jada’s price, red squares represent Priya’s price, and yellow triangles represent Han’s price. In the second graph, each X represents the best price for each number of bagels.
Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river”).
Student Lesson Summary
A relationship between two quantities is a function if there is exactly one output for each input. We call the input the independent variable and the output the dependent variable.
Let’s look at the relationship between the amount of time since a plane takes off, in seconds, and the plane’s height above the ground, in feet.
-
These two quantities form a function if time is the independent variable (the input) and height is the dependent variable (the output). This is because at any amount of time since takeoff, the plane could be at only one height above the ground.
For example, 50 seconds after takeoff, the plane might have a height of 180 feet. At that moment, it cannot be simultaneously 180 feet and 95 feet above the ground.
For any input, there is only one possible output, so the height of the plane is a function of the time since takeoff.
-
The two quantities do not form a function, however, if we consider height as the input and time as the output. This is because the plane can be at the same height for multiple lengths of times since takeoff.
For instance, the plane will likely be 100 feet above the ground shortly after takeoff as well as shortly before landing.
For an input, there are multiple possible outputs, so the time since takeoff is not a function of the height of the plane.
Functions can be represented in many ways—with a verbal description, a table of values, a graph, an expression or an equation, or a set of ordered pairs.
When a function is represented with a graph, each point on the graph is a specific pair of an input and output.
Here is a graph that shows the height of a plane as a function of time since takeoff.
Graph of a function, origin O. Horizontal axis from 0 to 300, by 50s, time in seconds. Vertical axis from 0 to 1500, by 250s, height in feet. Function begins at zero and goes through 50 comma 180, 250 comma 1500, and 250 comma 1500. Point 125 comma 400 is noted.
It is a function because there is one output for each input. The point
Here is a graph that shows the time since takeoff as the output and the height of the plane as the input.
This is not a function because an input of 100 feet has two possible outputs.
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