Noah is depositing money in his account every week to save money. The graph shows the amount he has saved as a function of time since he opened his account.
Elena opened an account the same day as Noah. The amount of money in her account is given by the function , where is the number of weeks since the account was opened.
Who started out with more money in their account? Explain how you know.
Who is saving money at a faster rate? Explain how you know.
Scatterplot of Noah's account, horizontal, weeks since opening account, 0 to 12 by 2, vertical, amount in the account, 0 to 140 by 20. Points at 0 comma 60, 2 comma 70, 4 comma 80, 6 comma 90, 8 comma 100, 10 comma 110, 12 comma 120, and 14 comma 130.
7.2
Activity
Is It Filling Up or Draining Out?
Standards Alignment
Building On
8.EE.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
The amount of water in gallons, , in Tank A is a function of time in minutes, , and can be represented by the equation .
The amount of water in gallons, , in Tank B is a function of time in minutes, . The amount of water starts at 400 gallons and is decreasing by 5 gallons per minute.
Which tank started out with more water?
Write an equation representing the relationship between and .
One tank is filling up. The other is draining out. Which is which? How can you tell?
The amount of water in gallons, , in Tank C is a function of time in minutes, , and can be represented by the equation . Is it filling up or draining out? Can you tell just by looking at the equation?
The graph of the function for the amount of water in gallons, , in Tank D at time is shown. Is it filling up or draining out? How do you know?
7.3
Activity
A Candle and the Moon
Standards Alignment
Building On
Addressing
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
A candle is burning. It starts out 12 inches long. After 1 hour, it is 10 inches long. After 3 hours, it is 5.5 inches long.
When do you think the candle will burn out completely?
Is the height of the candle a function of time? If yes, is it a linear function? Explain your thinking.
On the first day after the new moon, 2% of the moon’s surface that we can see is illuminated. On the second day, 6% is illuminated.
Use this information to predict the days on which the moon’s surface that we can see is 50% illuminated and 100% illuminated.
7.4
Activity
Shadows
Standards Alignment
Building On
Addressing
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
When the sun was directly overhead, the stick had no shadow. After 20 minutes, the shadow was 10.5 centimeters long. After 60 minutes, it was 26 centimeters long.
Use this information to estimate how long it will be after 95 minutes.
After 95 minutes, the shadow measured 38.5 centimeters. How does this compare to your estimate?
Is the length of the shadow a function of time? If so, is it linear? Explain your reasoning.
Student Lesson Summary
Suppose a car is traveling at 30 miles per hour. The relationship between the time in hours and the distance in miles is a proportional relationship.
We can represent this relationship with an equation of the form , where distance is a function of time (since each input of time has exactly one output of distance).
Or we could write the equation instead, where time is a function of distance (since each input of distance has exactly one output of time).
These equations are examples of a mathematical model. A mathematical model is a mathematical object, like an equation, a function, or a geometric figure, that we use to represent a real-life situation. Sometimes a situation can be modeled by a linear function. We have to analyze the information we are given and use judgment about whether using a linear model is a reasonable thing to do. We must also be aware that the model may make imprecise predictions or may only be appropriate for certain ranges of values.
More generally, if we represent a linear function with an equation like , then is the initial value (which is 0 for proportional relationships), and is the rate of change of the function.
If is positive, the function is increasing.
If is negative, the function is decreasing.
If we represent a linear function in a different way, say with a graph, we can use what we know about graphs of lines to find the and values and, if needed, write an equation.
Glossary
None
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Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.