Authentic Lessons for 21st Century Learning

Shiver Me Functions!

Function Notation

Brittany VanCleave, Amber Stokes, Michell Eike, Mary Braggs | Published: October 21st, 2022 by K20 Center

  • Grade Level Grade Level 8th, 9th
  • Subject Subject Mathematics
  • Course Course Algebra 1
  • Time Frame Time Frame 90-105 minutes
  • Duration More 2-3 class periods


This lesson will break down the components of a function and the relationship it has to real-world contexts. Students will recall their knowledge of inputs and outputs with a function machine. Students will simplify functions and then evaluate the functions. Students will practice simplifying polynomials through a digital breakout room.

Essential Question(s)

What can you represent with function notation?



Students activate prior knowledge and show what they know about function notation.


Students recall linear equations and use pattern recognition to find inputs, outputs, and the equation.


Students eliminate misconceptions about function notation.

Extend 1

Students complete a digital breakout focused on simplifying polynomials and compete to see who can unlock the treasure map first.

Extend 2

Students apply what they have learned to evaluate functions algebraically and graphically.


Students demonstrate their understanding of the meaning of function notation within the context of a real-world scenario.


  • Lesson Slides (attached)

  • What’s My Function? handout (attached; one per pair of students; printed front only)

  • Card Matching cards (attached; one set per pairs of students; printed front only)

  • Treasure Map Hunt handout (attached; one per pair of students; printed front only)

  • Treasure Map Hunt Guide document (attached; for teacher use)

  • Using Function Notation handout (attached; one per pair of students; printed front only)

  • Student devices with internet access

  • Envelopes or paper clips (optional)


15 Minute(s)

Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Go to slide 5. Assign each student a partner or have students find a partner and pass out a copy of the What’s My Function? handout to each pair. Have students work together to determine the pattern or rule for the function machine. Have students use the table to determine which two operations have been applied to each input to get each output. Then, have them write the rule or pattern that would work for any input into the function machine, using the words "input" and "output."

Once students have completed their handouts, display slide 6. Have students compare their results from question 1 with what is on the screen. Give students time to discuss and ask questions. Then, ask students how to translate their words into an algebraic equation, still using the words input and output. Ask for some volunteers to share their ideas. Then, display the algebraic equation on slide 7.

Repeat this with question 2 using slides 8–9 and question 3 using slides 10–11.


20 Minute(s)

Go to slide 12 and give each pair a set of Card Matching cards. Have students complete the Card Matching activity by matching cards from three categories: scenarios, equations, and function notation.

Show slide 13 and reveal the correct matches to the students. Give them time to make any necessary revisions.

Display slide 14 and ask the class: “What do you think f(x) means?” Have students write their answers on the back of their What’s My Function? handout.

Have students set aside their handouts. They will reflect on their answers to this question later in the lesson.


20 Minute(s)

Show slide 15 and have students think about the three functions machines from earlier in the lesson. Tell them that instead of using the words "output" and "input", they also could have used “y” and “x”, as shown on the slide. Ask for a volunteer to share why that is true. Use student responses to determine if students need a reminder that, traditionally, x is the input, and y is the output.

Now ask the class, "What does y equal when x equals 3?" As students start asking questions, try to stay casual and tell them that they just need to plug in x = 3.

Confess that you asked a ridiculous question and transition to slide 16. Explain that naming functions would have made your question clearer.

Show slide 17 and ask students to recall the Card Matching activity from earlier in the lesson. Now that students have a formal definition of function notation, have students reread the function notation card (Card J on the screen) and interpret the meaning. Have students share with an Elbow Partner their interpretation of the meaning. If time allows, ask for a couple of volunteers to share.

Display slide 18 to reveal the meaning of Card J. Give students time to reflect and ask questions.

Show slide 19. Use this slide to have students recall the function machine from earlier in the lesson and how they could write the equation algebraically but using function notation.

Now have students reflect on what they thought f(x) meant, referring to the back of their What’s My Function? handout (used at the end of the Explore portion of the lesson). Ask the class if their current understanding matches their prior understanding. Have students modify their definitions and express to them that now is the time to ask any questions about function notation. Use student responses to determine if students need more examples or if they are ready to evaluate functions using function notation.

Display slide 20 and ask students how they think they can find f(3). Encourage students to share their approaches. There are two common approaches that both have the same steps but not the same order: Simplifying then substituting, or substituting then simplifying. Show students both approaches for example 1. Ask students what they think the pros and cons of each approach are.

Continue by going through examples 2–3 on slides 21–22.


20 Minute(s)

Display slide 23 and prepare students for a digital breakout treasure map hunt. Have students find a new partner or assign them new partners. Capture their interest by having them imagine that they are with a group of sailors who are on a voyage to find the missing treasure map. It is their job as their ships' captains to find the keys that unlock the three locks, retrieving the map that leads their crews to their destiny.

Show slide 24 and preview the activity without giving away too many hints and share the vocabulary shown on the slide.

Display slide 25 and instruct pairs to open the Treasure Map Hunt digital breakout by going to this link. In addition, pass out a copy of the Treasure Map Hunt handout to each pair. Students should use the handout to track their progress on the hunt. Have them document where they found each clue and the key that they identified, and have them describe how they found each key in mathematical terms(i.e., show their work). For your reference, a copy of the Treasure Map Hunt Guide is attached.

Extend 2

10 Minute(s)

Show slide 26. Have students find a new partner or assign new partners for them. Give each pair of students a copy of the Using Function Notation handout. Have pairs work together to apply their knowledge of function notation to the variety of questions.


5 Minute(s)

Display slide 27 and use the Exit Ticket strategy to individually assess what students have learned from the lesson. Give students a sticky note, an index card, etc. for them to write their response or have students write on the back of their Using Function Notation handout.

Collect student responses and use them to determine if your students need additional practice or are ready for the next topic. Use the hidden slide 28 for a sample response.