Unit 5, Lesson 1
Describing and Graphing Situations
Algebra 1
1.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity 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 |
Student Response
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Building on Student Thinking
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”).
1.2
Activity
Instructional Routines
MLR6: Three ReadsMaterials
NoneActivity Narrative
This activity further refreshes students’ understanding of functions through contextual examples. Here students are prompted to reason graphically about the relationship between the two quantities in the situation—time in seconds and the distance of a dog from a post. Students also recall the meaning of independent and dependent variables.
Students are given descriptions of the dog’s movement while it was attached to a post and asked to sketch corresponding graphs. Along the way, students interpret each point on the graph to mean a particular point in time and a particular distance from the post.
Students later reason that the relationship between time and distance is a function because the dog can be in only one location at any given time. For instance, the dog could not be both 5 feet and 1.7 feet away from the post at the same exact time.
As students work, monitor for students that create a graph with a vertical value greater than 5 or a graph that does not go through the two given points. If many graphs show these features, discuss with the class why the former is not possible and why the latter does not match the given descriptions.
This activity uses the Three Reads math language routine to support reading comprehension and sense making.
Launch
Keep students in groups of 4.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the bulleted information, without revealing the questions.
- For the first read, read the problem aloud then ask, “What is this situation about?” (how a dog moves around a post) Listen for and clarify any questions about the context.
- After the second read, ask students to list any quantities that can be counted or measured. (the distance from the post at certain times)
- After the third read, reveal the question: “Sketch a graph that could represent the dog’s distance from the post, in feet, as a function of time, in seconds, since the owner left,” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read (start with two points representing the numerical information given and connect them so that the story makes sense).
Discuss features of a graph that could represent the dog’s movement on Day 3, and display a possible graph. An example is shown here.
Ask 2 group members to graph the dog’s movement on Day 1, and ask the other 2 members to graph the movement on Day 2.
Representation: Internalize Comprehension. Represent the same information through different modalities by using physical objects to represent abstract concepts. Use a three-dimensional prop to represent the situation and the dog’s movement. This prop might use a dowel rod for the post and a piece of string or yarn for the leash. Viewing or manipulating the prop can help students to better understand the context and quantities involved.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
NoneThree days in a row, a dog owner tied his dog’s 5-foot-long leash to a post outside a store while he ran into the store to get a drink. Each time, the owner returned within minutes.
The dog’s movement each day is described here.
- Day 1: The dog walked around the entire time while waiting for its owner.
- Day 2: The dog walked around for the first minute and then laid down until its owner returned.
- Day 3: The dog tried to follow its owner into the store but was stopped by the leash. Then, it started walking around the post in one direction. It kept walking until its leash was completely wound up around the post. The dog stayed there until its owner returned.
- Each day, the dog was 1.5 feet away from the post when the owner left.
- Each day, 60 seconds after the owner left, the dog was 4 feet from the post.
Your teacher will assign one of the days for you to analyze.
Sketch a graph that could represent the relationship between the dog’s distance from the post, in feet, and time, in seconds, since the owner left.
Day
Student Response
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Building on Student Thinking
Activity Synthesis
Select some students to share their graphs and a brief explanation of how the graphs match the descriptions.
One takeaway from this activity is that the relationship between the time since the owner left and the dog’s distance from the post is a function. Solicit as many explanations as possible of why it is. To emphasize that a function is a relationship in which one output is assigned to every input, explain that:
- The dog can be in only one spot at a time and cannot have more than one distance away from the post at any time.
- On each graph, we can see that only one value for distance corresponds to each value for time.
Ask students,
- “In this situation, what is the input and what is the output?” (It makes sense for time in seconds to be the input and distance from the post to be the output.)
- “Is it possible for the distance to be the input and the time to be the output? Why or why not?” (It is possible, but if the distance is the input, the relationship is no longer a function. This is because there could be multiple possible times at which the dog is a certain distance away from the post.)
Remind students that a quantity that is an input for a function is called an independent variable, and a quantity that is an output is called a dependent variable. In this case, time is independent and distance from the post is dependent.
1.3
Activity
Materials
NoneActivity Narrative
This activity gives students a chance to use mathematical language to describe relationships that are functions and to practice sketching a graph of a function given a description. The context is the same as in the previous activity, but the quantities are different.
Deciding on which variable is independent and which is dependent, as well as sketching a graph of the relationship, engages students in aspects of modeling (MP4). It requires students to make sense of quantities, consider how they are related, and think about what values are reasonable. Using the language of “function” to articulate the relationship between variables is an opportunity to attend to precision (MP6).
When sketching a graph for the function that defines the number of barks, students are likely to create either a discrete graph or a continuous graph. (Because the total number of barks cannot be fractional, creating a step graph would most accurately represent it as a function of time, but students are not expected to do so at this point.) Identify students who sketch each kind of graph. If time permits, select them to share their thinking during the whole-class discussion.
Launch
Arrange students in groups of 2. Tell students their task is to analyze two pairs of quantities from a familiar situation. Ask partners to each choose a different pair of quantities. Give students a few minutes of quiet work time and then time for partners to take turns sharing their functions and representations.
Tell students that the partner who is listening should listen for the following information:
- How did their partner decide which quantity is independent (the input) and which is dependent (the output)?
- How did they decide what makes a function of ?
- How does the graph represent the relationship between and ?
- On the graph, how was the scale for each axis determined?
Encourage students to notice any part of their own or their partner’s statement or graph that may not seem reasonable in the situation, then think about what might be more reasonable. (For instance, it is not reasonable for a dog to bark 1,000 times in 2 minutes.)
Leave 1–2 minutes for a whole-class discussion.
MLR8 Discussion Supports. Display sentence frames to support partner discussions: “_____ is the input or output because . . . .” “I noticed _____, so . . . .” or “To determine a scale on the graph, I . . . .”
Advances: Conversing, Representing
Advances: Conversing, Representing
Activity
NoneHere are two pairs of quantities from a situation you’ve seen in this lesson. Each pair has a relationship that can be defined as a function.
- time, in seconds, since the dog owner left and the total number of times the dog has barked
- time, in seconds, since the owner left and the total distance, in feet, that the dog has walked while waiting
Choose one pair of quantities, and express their relationship as a function.
- In that function, which variable is independent? Which one is dependent?
- Write a sentence of the form “ is a function of .”
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Sketch a possible graph of the relationship on the coordinate plane. Be sure to label and indicate a scale on each axis, and be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
For each situation, select 1–2 students who drew different graphs to display them for all to see. Ask the students to briefly explain how they decided which quantity should be the input and which should be the output and what the graph should look like.
If it is not brought up, remind students of the definition of a function and that the number of barks must be a function of time because there are many times when the total number of barks is the same (for example, in the sample response, the dog barked a total of 2 times at 10 seconds and 20 seconds).
If the dog barked consistently every second and we chose to measure time only in whole numbers of seconds, it might be possible to tell time by the number of barks and it might be reasonable for the number of seconds that have passed to be a function of the number of barks.
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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.
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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.
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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 on the function's graph tells us that 125 seconds after takeoff, the height of the plane is 400 feet.
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.