• Evaluate student results in the heart-rate activity by checking substitutions for age and resting heart rate, computation for maximum heart rate, and whether the result is reasonable. [IS.6 - All Students]
• Suggest to students to self evaluate their results in the temperature activity by checking whether the result is reasonable based on what they already know about the range of Celsius and Fahrenheit temperatures. Point out that room temperature is approximately 20 degrees Celsius.

We use inverse functions every day; [IS.3 - Struggling Learners] as one example, if a friend gives you directions from your house to his/her house, you invert those directions to get back home. Mathematical inverses are carefully defined, but that definition isn’t necessary to explore and even understand the underlying ideas.

Start off class with an example; put the following function on the board: y = 3x + 2. Tell the class that we can use this function to find values of y, given some values of x. But what do we do if we are given the y values and need to find x instead? What we can do is solve the function for x instead, as such (demonstrate on the board):

For y = 3x + 2, since we want to solve for x, subtract 2 from both sides

For y − 2 = 3x, now we divide by 3 to solve for x

For  , now the function is set up where x is in terms of y

“Now that we’ve talked about an example of linear functions, we’ll look at a more general way to find linear functions, using the function y = 2x + 5 as an example. The first step is to swap x and y, so our function reads x = 2y + 5. Now, we just solve for y.”

For , work through a couple of examples with the class (just using linear functions). Before beginning the upcoming group activity, ask students: [IS.4 - All Students]

“Let’s review for a moment: when we need to graph a linear equation like y = 2x + 5, how can we do it?” (plotting points, finding intercepts, using slope-intercept form)

“Now let’s look at inverses in another sense.” On the board, give students directions from your school to a nearby landmark. Have them write the directions to get from the landmark back to school. (You could also use directions from the classroom to another classroom, the gymnasium, cafeteria, library, etc.) Explain that these “backwards” directions are a good example of the topic today, which is functions and their inverses.

Lead a general discussion about health and exercise: does everyone understand that the heart is a muscle? We must exercise the heart by running or doing other exercise in order to have a healthy heart muscle. Trainers know that each body is different, and one of the differences between people is their resting heart rate. Explain how to find resting heart rate.

Find your pulse on your wrist or neck (right underneath the jaw). Use two fingers to get a better feel. Have students start counting their pulse when you say so, and have them record it for 15 seconds. After 15 seconds, have them record the number of beats on a piece of paper.

Repeat this activity four or five times, having students record their resting heart rate (RHR) each time. Then, have students find the average of their resting heart rate. Have a student explain how to find the average of a group of numbers.

Tell students that RHR is usually listed as beats per minute, not beats per 15 seconds. Help students convert their average from beats per 15 seconds to beats per minute by multiplying by 4 (4 × 15 sec = 60 sec = one minute). Have students record their resting heart rate on their paper by clearly writing “RHR = 60 bpm” (or whatever their RHR is).

Tell students, “Another important heart rate measurement is your maximum heart rate. There is a formula to represent maximum heart rate (which we’ll abbreviate MHR), since it is impossible to know, just by exercising, if you have reached your maximum heart rate.” Provide students with the formula for maximum heart rate:

220 − age − 0.3RHR = MHR

Each student should calculate his/her own Maximum Heart Rate based upon the formula. [IS.5 - All Students] Have students substitute in 14 for their age and explain we’ll use that just to be a little more consistent. However, to personalize the formula, have them each use their resting heart rate. Remind them to follow order of operations and perform the calculation inside the parentheses first.

Tell students that this formula is an average for a large group of individuals. The maximum heart rate for any one individual may vary among those with the same ages and resting heart rates.

Ask for a volunteer to come up to the front of the room and share his/her MHR. In groups, students should try to guess the volunteer’s RHR. Begin a table listing the MHR and RHR for this volunteer. Do this several times until you sense students are finding patterns. Ask the class to generate a way to find someone’s RHR using the above formula if he/she tells you his/her MHR. Encourage students to think about “undoing” the operations that are performed on the RHR in order to get to MHR.

Instruct groups to write out their “system” for finding RHR, given MHR: most likely, students will write in words what they’ve done.

If a student group has used algebra to perform the calculation, ask those students to present to the class; if not, show how we can use algebra to solve the formula

MHR = 206 − 0.3RHR

for RHR. Guide students through “undoing” the addition of 206 and the multiplication by −0.3 until they obtain the formula

Explain that the two formulas are inverses, and for a better look at what that means, have students create a table for the first function, given RHR values of 30, 40, 50, and 60.

MHR = 206 − 0.3RHR

Then, have them take the MHR values they got and substitute those into the inverse function and see what they get for RHR. They should note that they get 30, 40, 50, and 60.

“What is the relationship between the two tables?” (they are opposites; they have the two rows/columns reversed; etc.)

Use examples from daily life to get ideas for inversions and inverting (for instance, following map directions in reverse), then remind students that mathematics is about precision, so we need to be precise with our language and terminology.

Remind students of the vocabulary definition of inverse function at the beginning of the lesson.

Inverse Functions: two functions that “undo” each other.

To invert: to undo.

Before proceeding, review the terms opposite, reciprocal, and inverse, and make sure students are clear on the differences (and similarities) of these terms. Explain that in less-precise language, these words might all seem to be the same, but in mathematical terms, while they are all related, each has a very different, very precise meaning.

“Now, we’ll look at another formula and talk about its inverse.”

Introduce the formula

to the class and explain that the input is a temperature in degrees Celsius and the output is a temperature in degrees Fahrenheit.

To practice with the formula, tell students that in Celsius, water boils at 100 degrees. Ask students to use the formula to determine what the boiling point of water is in Fahrenheit. (This can be repeated with the fact that water freezes at 0 degrees Celsius, although many students will know that water freezes at 32 degrees Fahrenheit without use of the formula.)

Now, ask students to use this formula to determine what Celsius temperature is equivalent to 98 degrees Fahrenheit. Students should note that it requires a fair amount of work to determine, because the formula is designed to have Celsius temperatures as an input, rather than Fahrenheit. Remind students that by finding the inverse, we essentially “flip” the inputs and outputs.

“If we want to use a formula that has Fahrenheit temperatures as inputs and Celsius temperatures as outputs, we should find the inverse of the given function.”

Use algebra and inverse operations to find the formula

Now, ask students to find a Celsius temperature equivalent to 98 degrees Fahrenheit. (Their answer should be .) Also, ask them to verify that 100 degrees Celsius and 212 degrees Fahrenheit are equivalent temperatures.

(Depending on the class and time constraints, students can also be asked to find the temperature that is the same in Celsius and Fahrenheit. The solution is −40 degrees.)

Make sure that students remember how to graph lines, and then divide the class into groups and give each group copies of graph paper with a dashed line representing y = x already marked on it (M-A2-6-2_Graph Paper.doc). First, have each group work together to find the inverse of a function. After they are sure they have the inverse, have them graph the original function and the inverse on the same graph.

Have students repeat this activity a couple of times (each time using a new sheet of graph paper with x = y marked on it). Ask students to describe, in their own words, what is the difference between y = x and x = y. Also ask them if this is the same kind of relationship as for 4 + 1 = 5 and 5 = 4 + 1.

When each group has done the activity two or three times, ask students:

“What do you notice about the graph of the original linear function and its inverse?” (They cross on the dashed line, they are reflections across the dashed line.)

Have students select a graph in their groups and find the y-coordinate for the original function when x = 2. Then, have them find the y-coordinate for the inverse function, using the y-coordinate that they just found as the new x-coordinate.

“What do you notice about the two ordered pairs?” (They are opposites; the x- and y-coordinates are reversed.)

Have students find the dashed line (y = x) on their graph paper. Point out that on this line, reversing the x- and y-coordinates has no effect (since the coordinates are the same). Explain that the fact that the two coordinates are the same for any point on the dashed line helps to explain why that line serves as the line of reflection for inverse functions.

“Remember that there are multiple representations of functions, as we discussed earlier—tables, equations, and graphs. Now, we’ve covered what inverse functions look like in each of these forms.”

Have students volunteer to explain how to find inverses if they start with any of the three forms covered in the lesson: tables, equations, and graphs. Ask students which form they prefer, reminding them that in many cases, they can switch from one form to another. For instance, if they have an equation that is graphed and they have to draw the graph of the inverse function, they can find the inverse algebraically and then graph that equation if they like, or they can create a table of values, reverse the x and y columns, and graph the new points.

Extension:

• All conversion factors in measurements are functions and can be used in a way similar to Fahrenheit/Celsius temperature representations. One quart equals 0.946333 liters, one mile equals 1.60935 kilometers, and one pound equals 453.5924 grams.

Use the equivalence between 1 quart and 0.946333 liters as a function: the number of quarts times the factor 0.946333 equals the number of liters. In other words,

Quarts × 0.946333 = Liters

0.946333 Q = L

How many liters are in 18 quarts: (0.946333) × 18 = 17.033994 liters.

Evaluate the reasonableness of your result by thinking about which unit is larger, quart or liter? Since one quart is smaller than one liter, the number of liters has to be less than the number of quarts. Since 17 < 18, that makes sense.

Using the conversion factors above, assign these representations.

• How many pounds are in 1 gram?                  [0.0022]
• How many quarts are in 18 liters?                  [19.0208]
• How many miles are in 77 kilometers?           [47.8454]