standard_description: Find inverse functions.
problem_activity: Activity
In previous activities with linear relationships, when
\(x\)
increases the
\(y\)
value increases as well; adding objects to a cylinder
increases
the water level and adding money to a bank account
increases
the balance. The slope of the lines that represent these relationships were positive. In this activity, students see negative slopes for the first time.
In this activity, students answer questions about a public transportation fare card context. After computing the amount left on the card after 0, 1, and 2 rides, they express regularity in repeated reasoning (MP8) to represent the amount remaining on the card after
\(x\)
rides. They are told that the slope of this line is -2.5, and are prompted to explain why a negative value makes sense.
While the language is not introduced in the task statement, the value of
\(x\)
for which the money on the card is 0 is called the
\(x\)
-intercept or horizontal intercept. Unlike the
\(y\)
-intercept, which can be seen in the equation
\(y = 40 - 2.5x\)
, the
\(x\)
intercept has to be calculated: it is the value of
\(x\)
for which
\(0 = 40 - 2.5x\)
.
Digital
Launch
If your students are unlikely to be familiar with public transportation, you may need to give them some quick information about how a fare card works. If possible, prepare some photos related to purchasing and using a fare card. (Some example images are provided, here.)
Explain to students that someone who wants to ride the bus or subway in a city often uses a card like this. The rider pays money which a computer system associates with the card. Every time the rider wants to ride, they swipe the card, and the cost of the ride is subtracted in the computer system from the balance on the card. Eventually, the amount available on the card runs out, and the rider must spend more money to increase the amount available on the card.
Arrange students in groups of 2 and give them 5 minutes of quiet work time. Provide access to rulers. After they have discussed their responses with a partner, discuss why a negative value for the slope makes sense in the context and ways to visually tell whether a line has a positive or negative slope.
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Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Noah put
$
40 on his fare card. Every time he rides public transportation,
$
2.50 is subtracted from the amount available on his card.
How much money, in dollars, is available on his card after he takes
0 rides?
1 ride?
2 rides?
\(x\)
rides?
Graph the relationship between amount of money on the card and number of rides.
How many rides can Noah take before the card runs out of money? Where do you see this number of rides on your graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
If your students are unlikely to be familiar with public transportation, you may need to give them some quick information about how a fare card works. If possible, prepare some photos related to purchasing and using a fare card. (Some example images are provided, here.)
Explain to students that someone who wants to ride the bus or subway in a city often uses a card like this. The rider pays money which a computer system associates with the card. Every time the rider wants to ride, they swipe the card, and the cost of the ride is subtracted in the computer system from the balance on the card. Eventually, the amount available on the card runs out, and the rider must spend more money to increase the amount available on the card.
Arrange students in groups of 2 and give them 5 minutes of quiet work time. Provide access to rulers. After they have discussed their responses with a partner, discuss why a negative value for the slope makes sense in the context and ways to visually tell whether a line has a positive or negative slope.
[IMG]
[IMG]
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Noah put
$
40 on his fare card. Every time he rides public transportation,
$
2.50 is subtracted from the amount available on his card.
How much money, in dollars, is available on his card after he takes
0 rides?
1 ride?
2 rides?
\(x\)
rides?
Graph the relationship between amount of money on the card and number of rides.
[IMG]
How many rides can Noah take before the card runs out of money? Where do you see this number of rides on your graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask students, "Why does it make sense to say the slope of this graph is
\(\text-2.5\)
rather than 2.5?" We can say that the rate of change of the amount on the card is
\(\text-2.5\)
dollars per ride (for each ride, the amount on the card
decreases
by 2.5). We can also say the slope of the graph representing the relationship is
\(\text-2.5\)
(when
\(x\)
increases by 1,
\(y\)
decreases by 2.5). Ensure that students can articulate some version of “For every ride, the amount on the card
decreased
by
$
2.50, and this is why it makes sense that the slope is
\(\text-2.5\)
.”
Ask students, "If we let
\(y\)
represent the amount of money on the card, in dollars, and
\(x\)
the number of rides, the linear relationship in this activity has the equation
\(y = 40 - 2.5x\)
. When will there be no money left on the card?" (After 16 rides, the card has no money left, so when
\(x = 16\)
,
\(y = 0\)
.) Explain that just like the point
\((0,40)\)
on the graph is called the vertical intercept, the point
\((16,0)\)
is called the horizontal intercept. In this situation, the vertical intercept tells us how much money Noah put on the card and the horizontal intercept tells us how many rides Noah can take before the card has no more money on it.
Some students may connect the plotted points with a line, or only draw the line and not each point. If this comes up, acknowledge that it’s a common practice in mathematical modeling to connect points showing a line to make a relationship easier to see. Ask if a point on the line such as
\((1.5, 36.25)\)
makes sense in this situation. We understand that the line helps us to understand the situation, but we also recognize that only integer values for the number of rides make sense in this situation.
Display once again lines
\(v\)
and
\(t\)
from the warm-up. Explain that even though they both have slope triangles similar to a triangle with vertical length of 1 and horizontal length of 3, they don’t have the same slope. Ask students to articulate which line has a slope of
\(\frac13\)
and which line has a slope of
\(\frac {\text{-}1}{3}\)
, and why. Validate students’ use of informal language to describe the differences in their own words. For example, they might say that if you put a pencil on a line and move it along the line from left to right, line
\(t\)
goes “uphill” but line
\(v\)
goes “downhill.”
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Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to support students to explain their reasoning for the question, “Why does it make sense to say the slope of this graph is
\(\text-2.5\)
rather than 2.5?” Give students time to meet with 2–3 partners, to share and get feedback on their responses. Encourage listeners to ask their partner clarifying questions such as, “Can you demonstrate your thinking by using the graph?” and “How does this negative value make sense for this situation?” Allow students to write a revised draft that reflects ideas and language from their shares. This will help students refine their thinking around mean through conversing with their partners.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


aligns: no