Lesson: Functions
Developing the Concept
Now that students have been reintroduced to one-step functions involving addition and subtraction, they will focus on one- and two-step functions involving addition, subtraction, multiplication, and division.
Materials: overhead projector and blank transparency; 4 congruent triangles, 2 congruent circles, and 3 congruent squares for use on the overhead; 20 two-step functions written on index cards; Function Machine (PDF file) sheets for each pair of students (see below).
Preparation: Prepare index cards with two-step functions written on them. (You may wish to laminate the cards.) Copy Function Machine (PDF file) sheets for each pair of students.
Prerequisite Skills and Concepts: Students should be able to write and solve equations. They should be able to write the equation and create the function table for simple functions involving addition and subtraction.
- Ask: Earlier this year, we talked about functions. Who can remind the class what a function is?
Students should say that a function is a rule that describes the relationship between two variables. - Ask: Who can give me an example of a function?
Have a student give a function in equation and word form. - Say: When we worked with functions before, we looked at how they relate to patterns. In the example you just gave, the pattern would be to add
shapes to each succeeding version of the pattern. (Fill in the appropriate number, based on the student's function.) I'd like a volunteer to make a design on the overhead with these geometric shapes.
Have a volunteer come up and arrange the shapes on the overhead. Have him or her use all of the available shapes. - Say: Congratulations! You've just won the class T-shirt design contest! The school has decided to print copies of your shirt for the class and a few of the teachers. We need to figure out how many of each shape will be needed to make 30 shirts. How can we do that?
Students should say that you can count how many of each shape there are on one shirt and then multiply that number by 30. - Ask: If there are 4 triangles on one shirt, how many would be on 30 shirts?
Students should say 120. On the board, draw a table like the one shown below.Number of
T-shirtsNumber of Triangles 30 120 - Ask: The school has decided to sell your T-shirt to other students as a fundraiser. They want to print 55 shirts. How many triangles will be on 55 shirts?
Give students time to multiply; then write 55 and 220 in the table. - Ask: Your T-shirts are so popular that the PTA wants to sell them in all the schools in your town. They are going to print 350 of them. How many triangles will be on 350 shirts?
Students should say 1,400. Write 350 and 1,400 in the table. - Ask: What have we created here? (Point to the table.)
Student should answer that a function table has been created. - Ask: What is the rule for this function?
Students should say that the number of triangles is 4 times the number of T-shirts. - Ask: If we let x stand for the number of T-shirts and y stand for the number of triangles, what is the equation for this function?
Students should say “y = 4x.” Write this on the board, above the table. Also write (x) in the first column after Number of T-shirts and (y) in the second column after Number of Triangles. - Ask: How is this design different from the patterns we looked at last time?
Students should say that last time, the number of shapes changed with each new design. This time the same design is repeating over and over with no changes. - Ask: How is this function different from the ones we used to describe the patterns last time?
Students should say that this function uses multiplication rather than addition or subtraction. Have students write the equations and create function tables for the circles and squares in the design. They should use the same values for x as those used in the triangle table. Review the answers. - Ask: Now let's look at another function. Pretend you want to join an after-school art class. The class costs $6 a week, and you can sign up for as many weeks as you like. If x stands for the number of weeks and y stands for the cost of the class, what is the function of x and y?
Students should say that y is 6 times x. Write y = 6x on the board. - Say: There is also an $18 materials fee that you only have to pay once.
Add + 18 to the equation so that it reads y = 6x + 18. - Ask: How is this equation different from the first equation, y = 6x?
Students should say that in this equation you must multiply and add, not just multiply. - Say: This kind of function is called a two-step function, because you need to perform two operations. Use this equation to find the cost of taking the class for 4 weeks.
Students should say that it would cost $42 to take the class for 4 weeks. - Ask: Did you multiply or add first? (Multiply). How did you know to multiply first?
Students should say that they followed the order of operations. - Say: Let's pretend you have $78. How would you find the total number of weeks that you could take the class?
Guide students to see that you could substitute 78 for y, subtract 18 from each side of the equation, and then divide each side by 6. - Say: So, if you had $78, you could take the class for 10 weeks.
Then have students create a function table showing the weekly cost of the class for up to ten weeks.
Wrap-Up and Assessment Hints
Have students work in teams of two. Give each team a Function Machine (PDF file) sheet.
The first team takes a prepared index card with a two-step function on it. The second team picks a number. The first team applies its function to that number and gives an output number. This continues until the second team correctly identifies the function. The teams swap roles and play again. The team that correctly names the function in the fewest tries wins that round.