Lesson Title: ______8.F.1_Introduction to Functions______
Course: ___________________________
Date: _____________ Teacher(s): _____________
Start/end times: 2 or 3– 50 minute class
periods______________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the
set of ordered pairs consisting of an input and the corresponding output.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Can you think of real world situations where you would
input something into a machine and get a different
output?
For example, a toaster is a machine. When bread is
input in the machine the output is toast.
In the table below list at least two machines along
with their inputs and outputs.
Input
Machine
Output
bread
toaster
toast
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.
End of Day 1:
Have students create their own function tables with either
a missing function rule or given the rule and missing
inputs and/or outputs. Have students trade papers with
each other and complete the function table created by their
peers.
End of Day 2:
Have the students in groups of four summarize in a web or
chart everything they have learned about functions over
the last two class periods. Students can expand the web
based on other student responses.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
Teacher Note: This lesson is meant to span over 2-3 50-minute class periods. There is a break to show where you
can stop and progress on day 2. There is also a note at the end of Day 1. If students need more time with the
understanding of functions, take the time using the three activities provided in #’s 5, 6, and 7 to extend the lesson for
an extra practice or review day.
1. Tell the students that another representation of a function that is sometimes used is the function machine.
Draw a simple function machine on the board (a box with part of two of the sides opened) and explain to all
students that inside the box is a function rule, a rule of correspondence that for any input there is exactly one
output. Consider extending here to show that the set of all inputs of a function are know was the domain of
the function and the set of all possible outputs is know was the range of the function.
2. Start student off with a simple rule of y  8x , place this rule inside the function machine and cover it with a
piece of paper. Ask students to give you some numbers to input into the machine. For every input, give
students an output, for example if a student gives you the number 7, you would say the output is 56.
Consider showing the input and output in a T-Chart next to the function machine so that students get used to
 students can keep track of the inputs and outputs of the function. Have
the idea of a table and so that
students guess the rule. Continue this as a whole group for a few more one-step examples; consider using
one example from each basic operation.
3. Have students break out into pairs and go on the computer to the website:
http://www.mathplayground.com/functionmachine.html. Here they are going to practice discovering a
function rule. Students can choose the level on this website. They can choose beginner or advanced and
with each level beginner or advanced there are three additional levels 1, 2, and 3. Consider assigning
students to a level or letting the students choose. Allow students to experience this function machine for
about 15-20 minutes.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: ______8.F.1_Introduction to Functions______
Course: ___________________________
Date: _____________ Teacher(s): _____________
Start/end times: 2 or 3– 50 minute class
periods______________________
4. Have two pairs join together so students are now in groups of four. Have student play in groups the “High
Card Rules” Game. Pass out to each student paper with multiple 2-column function tables or T-Charts. Each
group should have index cards with -10 to 10 or you can use playing cards red and black with Ace to 10 the
red represent negative numbers and the black cards positive numbers. Have each team start with the rule:
n  2 , then each player takes a card without looking from the pile and uses its number as the input for the
rule. The player with the greatest output gets to make the next rule for the group. Then repeat. Encourage
students to create more challenging rules including some with rational numbers and 2-step operations.
Sample function table:
 Rule: n  2

5. If students need additional practice with developing their understanding of functions and function tables
consider another day for function tables and expand on the “High Card Rules” game or using Teaching
Student Centered MATHEMATICS Grades 5-8: Van de Walle, page 272. Activity 9.4 Guess My Rule.
6. Additionally, to expand the lessons continue with the “Function Rules Pairs Check Activity Resource
Sheet.” For this activity assign partners to complete the “Function Rules Paris Check Resource Sheet.”
Have the partners share a pencil. The partner on the left completes problem #1 on the left hand side of the
paper while the partner on the right watches, checks, and assists as needed. When the problem is complete
and correct, the partners switch roles with the partner on the right completing problem #1 on the right. The
pair continues until all the problems are complete and correct.
7. Consider for extra review going back to the beginning of Day 1 and having students create their own
function machines on paper and partnering up to find the function rules, input, or output.
End Here for Day 1
8. Lesson Launch for Day 2: Tell the students: “According to a website that doctors use to help them
understand the human body, the Family Practice Notebook,
(http://www.fpnotebook.com/Endo/Exam/ArmSpn.htm), a teenager’s arm span (finger tip to finger tip to
finger tip) is equal to his/her height.”
9. Ask the students the following questions: “Do you think this is true? If not, do you think that there is a
relationship between a teenager’s arm span and his/her height? Do you think we can use the measure of a
teenager’s arm span to predict his/her height?” Allow students to have a few minutes to answer the
questions with each other and then have several students respond to each question aloud and to the whole
class.
10. After the Lesson Launch, have the students get into pairs to measure each other’s arm span and then their
height in centimeters.
11. Have the class create a class table and place it on the board in a T-Chart. Another option would be to have
the T-Chart on the document camera. Have each student come up to the T-Chart and
12. On a post-it note, have each student record an ordered pair that represents his/her measurements. The
ordered pair should have the arm span as the x-coordinate and the height as the y-coordinate.
13. Draw and create a graph on the board using the length of the board as the horizontal axis and the width of
the board as the vertical axis. Another option would be to have each student using a marker to plot the point
that represents his/her measurements on graph paper that can be displayed using the document camera.
14. Have each student place his/her note at the appropriate point on the graph drawn on the board or on graph
paper.
15. As a class take a look at the graph created using arm span vs. height of each student and discuss the graph
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: ______8.F.1_Introduction to Functions______
Course: ___________________________
Date: _____________ Teacher(s): _____________
Start/end times: 2 or 3– 50 minute class
periods______________________
with students. Ask students if arm span is a good predictor of height. Have them explain why or why not.
16. Pass out the Multiple Representations Student Resource Sheet. In the section description have the students
define a Function as “a set of ordered pairs where every x value has exactly one y value”
17. Ask students if the graph shows a function. Have students brainstorm in small groups whether or not the
graph represents a function. If there is more than one of the same x-values help students to recognize that
this particular relationship is not a function and why. Then have students share out their ideas to the whole
class. Explain to students that if you measure a lot of different people from very short to very tall, you will
likely find that there will be a predictable relationship between these two measures. We can say that arm
span is related to height. Have students discuss why this would make sense. Guide students to discover that
not everyone’s height-to-arm span ratio will be the same; however one is related to the other.
18. Have the students continue to work in small groups using the Multiple Representations Resource Sheet.
Pose the following scenario to students and have the write it in the top section of the Resource Sheet for
Numerical/Algebraic Representations. Scenario: If each lemonade recipe will serve 20 people, how many
recipes are needed to serve n people?
19. The students should continue to work in small groups. Have students use the Function Multiple
Representations Resources sheet to create table, graph the scenario, and come up with an equation.
Encourage students to model or draw pictures to help them with the representation of the function.
20. Have each student group select one of the three representations the table, the graph, or the equation and have
them place it on chart paper displayed in the class or you can have them present their representation on the
document camera. Have students explain their group’s representation and solution to the class.
21. Next change the question using the same scenario. “If it takes 3 cans of concentrate to make one lemonade
recipe, how many cans should be purchases to serve n people?”
22. The students should continue to work in small groups. Have students use the Function Multiple
Representations Resources sheet to create table, graph the scenario, and come up with an equation.
23. Have each student group select one of the three representations the table, the graph, or the equation and have
them place it on chart paper displayed in the class or you can have them present their representation on the
document camera. Have students explain their group’s representation and solution to the class.
24. Ask students, “What is different about the two questions?” “What is the difference in the equations and the
meaning of the equations?” “What is the difference in the graph of the two functions?” “What is the same
about the graph?” Consider have the questions written out beforehand and allowing students to work in
partners to answer the questions and then sharing out as a whole class.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
Students will be able to generate multiple representations of functions including graphs, tables, pictures/models, and
algebraic equations. Students will be able to explain a function the relationship between the input and output of a
function. Students will be able to determine if the example is a function when looking at a table and determining if
there are more than one of the same x-value.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Key Vocabulary: function, input, output, domain, range
Connections: Students will need prior understanding of graphing points on the coordinate plane to solve real-world
and mathematical problems from standards 5.G.1 and 5.G.2. Students will need prior understanding of evaluating
expressions at specific values of their variables from standard 6.EE.2.C.
Misconceptions: Students may have difficulties translating between the different representations of the functions.
Students may struggle when going from a table or graph to an algebraic equation.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: ______8.F.1_Introduction to Functions______
Course: ___________________________
Date: _____________ Teacher(s): _____________
Start/end times: 2 or 3– 50 minute class
periods______________________
Citations:
Scenario from: Van de Wale, J. A., & Lovin, L. H. (2006). Exploring Functions. In Teaching Student-Centered
MATHEMATICS Grades 5-8 (p. 296). Boston: Pearson.
Activity 9.4 from: Van de Walle, J. A., & Lovin, L. H. (2006). Exploring Functions. In Teaching Student-Centered
MATHEMATICS Grades 5-8 (p. 272). Boston: Pearson.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Computers with internet access
http://www.mathplayground.com/functionmachine.html
http://www.fpnotebook.com/Endo/Exam/ArmSpn.htm),
chalk board or document camera
index cards or playing cards
measuring tape
chart paper
paper with function table or T-Charts
pencil
calculator
Function Multiple Representation Resource Sheet
Function Rules Paris Check Resource Sheet
Have students choose:
1. Create a game using functions, like the game we
played during class or different.
2. Create an additional real-world scenario similar to
the one about the lemonade that we did in class.
The scenario must have two different questions
that would change the function based on the
equation.
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to
reflect on the effectiveness of this lesson?
Do the students understand functions?
Do the students know what a function is?
Can the students represent functions using a table and an equation?
How well do the students complete an input/output table for a given function?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.