Summary
In this activity, students will explore function notation and operations. Students will use function composition and other operations to simplify and evaluate functions. This is the sixth activity in a 10-week "Power Up" series for ACT prep.
Essential Question
How can I increase my ACT score?
Learning Objectives
Use function notation to simplify and evaluate functions.
Perform operations using function notation.
Materials List
Activity Slides (attached)
Notation Exploration handout (attached; one per pair; printed front only)
Function Notation handout (attached; one per student; printed front/back)
Exit Ticket handout (attached; one-half per student; printed front only)
Pencil
Paper
Calculators
Introduction
5 Minute(s)
Introduce the activity using the attached Activity Slides. Share the essential question on slide 3 and the learning objectives from slide 4 to the extent you see fit. Have students get their calculator out; follow regular classroom procedures for this.
Display slide 5 and give each pair of students a copy of the attached Notation Exploration handout. Direct students to work with a partner to answer each question in the order it was given. As students work, circulate the room and monitor students’ discussions, but remember that this is the time for students to try these problems on their own and later receive more guidance from you.
As students finish the last few problems, show slide 6 and encourage the class to check their work for questions 1–5.
Because some students need to hear a pattern more than see a pattern, use slide 6 to read the question and first step aloud, ("f of 3 equals 3 squared minus 5; f of –1 is –1 squared minus 5; …; f of a star is a star squared minus 5."). Then show slide 7 and continue reading aloud through question 9. Ask the class what they think f of g of x equals. Click to display the answer for question 10, again reading it aloud. Guide the class to check their work, discuss with their partner, and ask any questions.
Activity
20 Minute(s)
Display slide 9 and give each student a copy of the attached Function Notation handout. Explain to students how the composition operation symbol and how to read f(g(x)). Emphasize that the composition symbol between the letters is not a multiplication symbol, which is a common misconception.
Show slide 10 and preview the activity with the class. Their handout has the following three sections: Making Observations, Verbalizing Observations, and Applying Observations. Tell students that they are to continue working in pairs to complete the handout. For the Making Observations section, students are to try to notice patterns among the given four worked-out problems. Then they are to write their observations and try to make generalizations for the Verbalizing Observations portion, and lastly they are to apply what they have learned to answer the ACT-style question in the Applying Observations section.
After previewing the activity, give students approximately five minutes to make observations and verbalize their observations. Then bring the class together and use the guiding questions on the slide to facilitate a brief discussion. Make sure students understand the input and output relationship before moving on. Consider having students rewrite f(–1) = 4 as an ordered pair, (–1, 4), to help with understanding the meaning of function notation, if needed.
Transition to slide 11 and ask for volunteers to share what they wrote.
Show slide 12 and have pairs answer the given ACT-style question.
After approximately one minute, move to slide 13 and share the sample response with the class. Give students time to correct their work and answer questions.
Direct students’ attention to back side of their handout and show slide 14. Explain to students how to read f(g(x)) and how to algebraically represent the sum, difference, and product of f(x) and g(x).
Show slide 15 and have students repeat the procedure just as they did on the front side of their paper.
After approximately five minutes, show slide 16 and facilitate a brief discussion with the class.
Show slide 17 and have pairs answer the given ACT-style question.
After approximately one minute, move to slide 18 and share the sample response with the class. Give students time to correct their work and answer questions.
Wrap-Up
5 Minute(s)
Display slide 19 and use the Exit Ticket strategy to individually assess what students have learned. Explain to students that they will have two minutes to answer two questions. Give each student a copy of the Exit Ticket handout and have students keep the paper face down until you start the timer. Once everyone has a copy of the handout, tell them to turn their paper over. Start the 2-minute timer on the slide.
If time allows, unhide slide 20 and review the answers with the class. Use the hidden slides 21-22 for sample student responses.
Remind the class that if they answered one of the two questions correctly that they did great. Remind them that they do not need to answer every question correctly on the ACT to do well.
Collect the Exit Ticket handout to use as a formative assessment.
Before you dismiss, show slide 23: You Powered Up! and remind students to practice the action they selected on their Goal Setting handout from week 1.
Next Step
Complete the following week’s activity, “Power Up: Math ACT Prep, Week 7,” where students will take an online practice ACT.
Research Rationale
Standardized testing in high schools has long stood as a metric for assessing college readiness and school accountability (McMann, 1994). While there has been debate surrounding the accuracy of such metrics, as well as concerns regarding equity, many institutions of higher education continue to make these scores part of the admissions process (Allensworth & Clark, 2020; Black et al., 2016; Buckley et al., 2020). Aside from admissions, it is also important to keep in mind that standardized test scores can also provide students with scholarship opportunities they wouldn’t otherwise have (Klasik, 2013). Though the topic of standardized testing continues to be debated, effective test prep can ensure that our students are set up for success.
With several benefits to doing well on college admissions tests, it is important to consider how best to prepare students for this type of high stakes test. Those students from groups that may historically struggle to find success, such as those in poverty or first generation college students, especially stand to benefit from effective test preparation (Moore & San Pedro, 2021). The American College Test (ACT) is one option students have for college admissions testing that is provided both at national centers and school sites. Taking time to understand this test including the timing, question types, rigor, and strategies for approaching specific questions can help to prepare students to do their best work on test day and ensure their score is a more accurate representation of what they know (Bishop & Davis-Becker, 2016).
Resources
Allensworth, E. M., & Clark, K. (2020). High school GPAs and ACT scores as predictors of college completion: Examining assumptions about consistency across high schools. Educational Researcher, 49(3), 198-211.
Bishop, N.S. & Davis-Becker, S. (2016). Preparing examinees for test taking: Guidelines for test developers and test users. 2nd edition. Crocker, L. (Ed). In Handbook of test development (pp. 129-142). Routledge.
Black, S. E., Cortes, K. E., & Lincove, J. A. (2016). Efficacy Versus Equity: What Happens When States Tinker With College Admissions in a Race-Blind Era? Educational Evaluation and Policy Analysis, 38(2), 336–363. http://www.jstor.org/stable/44984542
Buckley, J., Baker, D., & Rosinger, K. (2020). Should State Universities Downplay the SAT?. Education Next, 20(3).
K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (2021, September 21). K20 Center 2 minute timer. [Video]. YouTube. https://youtu.be/HcEEAnwOt2c
Klasik, D. (2013). The ACT of Enrollment: The College Enrollment Effects of State-Required College Entrance Exam Testing. Educational Researcher, 42(3), 151–160. http://www.jstor.org/stable/23462378
McMann, P. K. (1994). The effects of teaching practice review items and test-taking strategies on the ACT mathematics scores of second-year algebra students. Wayne State University. https://www.monroeccc.edu/sites/default/files/upward-bound/McMannP.-the-effects-of-teaching-practice-review-items-ACT-mathematics-second-year-algebra.pdf
Moore, R., & San Pedro, S. Z. (2021). Understanding the Test Preparation Practices of Underserved Learners. ACT Research & Policy. Issue Brief. ACT, Inc. https://files.eric.ed.gov/fulltext/ED616526.pdf
