Summary
In this activity, students will focus on probability and Venn diagrams. Students will be given a brief introduction of probability, use Venn diagrams to help with counting, and then practice probability-related ACT-style math questions. This is the fourth activity in a 10-week "Power Up" series for ACT prep.
Essential Question
How can I increase my ACT score?
Learning Objectives
Determine the probability of an event and the probability of its complement.
Use a Venn diagram to calculate probability.
Apply the concept of probability to unique situations.
Materials List
Activity Slides (attached)
Probability handout (attached; one per student; printed front/back)
Exit Ticket handout (attached; one per student; printed front only)
Pencil
Paper
Calculators
Introduction
5 Minute(s)
Introduce the activity using the attached Activity Slides. Use the Bell Ringer strategy to begin class. Have students get their calculator; follow regular classroom procedures for this.
Once students have their calculator, move to slide 3 and have students independently answer the question on a piece of notebook paper or elsewhere if you have a classroom norm for bell work. After a minute, ask for a volunteer to share their thinking. Facilitate a brief discussion covering that there are five options for a math question on the ACT, and since one of those options is the correct answer, one-fifth or 0.2 or 20% is the likelihood of guessing correctly on a question from the math portion of the ACT.
Move to slide 4 and have students answer the second bell ringer question. This question should take less time since they already found the chance of guessing correctly. Since four of the five options are incorrect, then four-fifths or 0.8 or 80% is the chance of guessing incorrectly on a question from the math portion of the ACT.
Show slide 5 and ask the class if they think that the likelihood of guessing correctly would increase or decrease if they could eliminate some of the options before guessing. Facilitate a brief discussion about not leaving questions blank, and the benefits of guessing, especially educated guessing (eliminating some of the wrong options).
Share the essential question on slide 6 and the learning objectives on slide 7.
Activity
20 Minute(s)
Give each student a copy of the attached Probability handout, then show slide 8. Review the definition of outcomes on the slide and share the example using the illustration of a bag of gems. Have students write the example on their handout.
Then display slide 9 and review the definition of event and share the example on the slide. As time allows, ask for students to share another example of an event that may or may not be related to the bag of gems.
Show slide 10 and review the definition of probability and share the example on the slide. As time allows, ask for students to share another example of probability that may or may not be related to the bag of gems. Again, encourage students to write an example on their handout.
Display slide 11 and explain the idea of events being mutually exclusive. As time allows, ask for students to share another example of a mutually exclusive event that may or may not be related to the bag of gems. Again, encourage students to write an example on their handout.
Transition to slide 12 and define G as the event of drawing a green gem from the given bag. Direct students to talk with a partner about what they notice from the bag of gems. Give students 2-3 minutes to take some notes.
Then move to slide 13 and define probability as the ratio of the number of ways event A can occur over the total number of possible outcomes. Have students work in pairs to find the probability and complete the corresponding Venn diagram.
After a couple of minutes, move to slide 14 and explain to students how to label the Venn diagram and how to find the probability of drawing a green gem.
Next, introduce the complement of event A with slide 15. Have pairs work together to find the probability of the complement of G and complete the corresponding Venn diagram.
After a couple of minutes, move to slide 16 and ask for a volunteer to explain how to label the Venn diagram and how to find the probability of drawing a gem that is not green. Let students know that their Venn diagram labeling will feel repetitive, as they are referring to the same scenario, but have them pay attention to what is and what is not shaded on the Venn diagram and how it relates to the corresponding question.
Move to slide 17 and explain the idea of finding the probability of event A or event B is the sum of the probabilities of each. Let students know that this is true for mutually exclusive events, but that it is not always true. They will see an example on the back of their handout where the events are not mutually exclusive. Have pairs work together to find the probability of the drawing a green or red gem and complete the corresponding Venn diagram.
After a couple of minutes, move to slide 18 and ask for a volunteer to explain how to label the Venn diagram and how to find the probability of drawing a green or red gem.
Display slide 19 and direct students’ attention to the back of the handout: NOT Mutually Exclusive Events. Read the problem aloud:
A survey asked 100 students on what device they played video games. The results showed that 50 students play video games on their console, 45 students play video games on their PC, and 15 students play video games on their console and PC.
Then ask pairs to label their given Venn diagram. After a couple of minutes, ask the class what is different about this Venn diagram than the ones on the front of their handout. Facilitate a brief discussion that the overlapping portion of the circles represents those who play video games on both their console and PC, so this event is not mutually exclusive.
As students work, circulate the room and transition through slides 20-21 slowly so that students can check their work and ask questions. For example, after a minute or two of students labeling their Venn diagram, display slide 20. Then as they are finishing labeling their Venn diagram, show slide 21. As pairs complete their Venn diagrams, bring the class together for a brief discussion of how to label the Venn diagram.
After the class agrees on the labeling of the Venn diagram, direct pairs to answer questions 1-4 on their handout. Transition through slides 22-25 so students can check their work.
Wrap-Up
10 Minute(s)
Show slide 26 and ask students to think back to the bell ringer questions. Were they expected to put their final answers as fractions or decimals or percentages? The questions were not multiple-choice and the directions did not specify. Share the test-taking tip of glancing at the multiple-choice options to see how to format the final answer. Advise students to start the math problem using pencil and paper, then glance at the answers to see if their final answer should be a decimal, fraction, etc. or if they even need to simplify their work. Then they will have more direction as they finish the problem.
Display slide 27 and use the Exit Ticket strategy to individually assess what students have learned. Explain to students that they will have five minutes to answer five 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 5-minute timer on the slide.
After the time expires, show slide 28 and review the answers with the class. Remind students that the ACT is not designed for everyone to earn a perfect score and that it is okay if they only answered approximately half of the questions correctly on this assessment.
Use slides 29-33 as needed to review the work for the given questions.
Before you dismiss, show slide 34: 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 5,” which will focus on pacing.
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 5 minute timer. [Video]. YouTube. https://youtu.be/EVS_yYQoLJg?si=fJvuvFWH3vJ3B0z9
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
