Summary
In this lesson, students will use their knowledge of factoring to simplify radical expressions. Students will practice simplifying square roots and cube roots through a game of Bingo. This is an introductory lesson to simplifying radicals.
Essential Question(s)
What does it mean to simplify a radical expression?
Learning Objectives
Simplify radical expressions.
Snapshot
Engage
Students identify numbers as perfect squares, perfect cubes, both, or neither through a Card Sort activity.
Explore
Students recall factoring to complete factor trees and practice writing expressions in multiple forms.
Explain
Students learn the vocabulary for radical expressions and relate the process for simplifying radicals to that of factoring.
Extend
Students practice simplifying radical expressions with a game of Bingo.
Evaluate
Students apply their learning to simplify a radical expression and then reflect on their learning.
Materials
Lesson Slides (attached)
Card Sort cards (attached; one set per group; print one-sided)
Factor Trees handout (attached; one per student; print two-sided)
Guided Notes handout (attached; one per student; print one-sided)
Guided Notes (Model Notes) document (attached)
Radical Bingo Cards handout (attached; one set per class; print one-sided)
Radical Bingo Slides (attached)
Radical Bingo Tracking Sheet (attached)
Radical Thinking handout (attached; one half page per student; print one-sided)
Preparation
Before you begin, print the attached Card Sort cards (one copy per group of 2–3 students) and cut them out. If you plan to reuse these cards, consider printing on cardstock paper.
During the Extend phase of the lesson, students are going to practice simplifying radical expressions through a game of Bingo. Determine how you would like students to use the attached Radical Bingo Cards. If you would like to reuse the cards, whether that be for multiple rounds within a class or for multiple class periods, then consider printing one set of these on cardstock and/or laminate the cards. Students can place Bingo chips, tokens, coins, etc. on their cards. If you chose to laminate the cards, then students could place items on the cards or use dry-erase markers to write on (and wipe off). If you do not plan to reuse the cards, then you will need to print one set per class. Each set contains 30 bingo cards. And each card contains the simplified expressions for all 24 given problems.
Engage
10 Minute(s)
Introduce the lesson using the attached Lesson Slides. Show the essential question on slide 3, then move to slide 4 to identify the lesson's learning objective. Review each of these with students to the extent you feel necessary.
Display slide 5 and have students get into small groups of 2–3 or assign groups. Give each group a set of the attached Card Sort cards. Introduce the Card Sort strategy and explain to students that they are to work in their group to sort the cards with numbers into one of the following categories: perfect squares, perfect cubes, both, or neither. Give students approximately 8 minutes to complete this task. If they need a hint, you can tell students that there are only three cards in the “Both” category, and five in each of the others.
Use the hidden slide 6 for quick reference or unhide and show it for students to check their work.
Explore
20 Minute(s)
Show slide 7. Give each student a copy of the attached Factor Trees handout and tell them to not yet write anything until you are finished with all of the directions, as you are going to complete the first problem together. Have students find a partner or assign partners. Explain to students that for each problem, they are going to need to create a factor tree to find all of the prime factors, then write the expression in its expanded form and its exponential form.
Move to slide 8 and introduce the Pass the Problem strategy. Explain to students how they will use this strategy to create the factor trees. Each student will create the first pair of branches, then trade papers with their partner, who will continue their work with the next step. They will repeat this process until the factor tree is complete. If students are familiar with this strategy, use the following slides to describe the process. If students are not familiar with this strategy, consider the following approach.
Have each pair determine who wants to be Student A and who wants to be student B, then show slide 9. Tell Student A that they wanted to start their factor tree for 12 using the branches of 2 and 6 (so Student A should write this like they see on the slide). And tell Student B that they wanted to start their factor tree for 12 using the branches of 3 and 4 (similarly, writing that down). Direct partners to trade papers.
Display slide 10 and point out to students that even though their friend started the factor tree differently than how they would have, they still did it correctly, and it is now their job to continue where their partner stopped.
Show slide 11 and point out that even though their work looks different, their final answers for both the expanded and exponential formats will be the same.
Give students a chance to ask questions about the process, then move to slide 12. Tell them to continue this process but stop after finishing the fifth problem.
As students work, circulate the room. Listen to discussions and questions being asked. Make note of misunderstandings. Since this activity is a review of prior knowledge, use this as a formative assessment to determine if your students are ready for the lesson or need to pause for intervention. As you notice pairs completing the fifth problem, have them stop using the Pass the Problem strategy and instead work collaboratively on the remaining problems. Once the majority of students are done with the first five problems, display slide 13 and announce that they should work with their partner on the remaining three problems, no longer taking turns and passing the problem.
Explain
20 Minute(s)
Show slide 14 and give each student a copy of the attached Guided Notes handout. Introduce the vocabulary: radical, radicand, and index.
Move to slide 15 and share how to read aloud radical notation. Share that if they do not see anything written for the index that it is an index of 2. Ask volunteers to share why they think that is the case. If needed, ask guiding questions to help students see that a square root, which they have seen before, relates to “square,” which is a power of 2.
Display slide 16 and use the slide to explain to students how to simplify radicals.
Transition through slides 17–22 to walk through the first example, which finds the square root of 24. Then walk through the second example using slides 23–26, which finds the cube root of 24. Emphasize to students the similarities and differences between these two problems.
Show slide 27 and ask students to try the first step on their own. Here students are asked to find the square root of x4y3. Depending on your class needs, either transition through slides 28–33 to help walk students through the problem or to allow them to check their work as they work ahead.
Display slide 34 and ask students to try the fourth example on their own. Once students are done, move to slide 35 so that students can check their work.
Repeat this using slides 36–37 for Example 5 and slides 38–39 for Example 6.
Use the attached Guided Notes (Model Notes) document as needed.
Extend
20 Minute(s)
Show slide 40 and introduce the Radical Bingo activity. Allow students to choose their own Radical Bingo Card to avoid any suspicions of bias. Each card has all 24 possible simplified expressions.
Clearly communicate your expectations for this activity, especially regarding how you would like students to show their work and how you would like them to mark their Bingo Card.
Use the attached Radical Bingo Slides to facilitate this activity. Display and read aloud slide 3, which is the first radical expression they are to simplify: the square root of 9a6. The simplified expression is on the hidden slide 4. As students are simplifying the expression and looking for the result on their Radical Bingo Card, mark that problem in the first row of the attached Radical Bingo Tracking Sheet. This paper is to help you keep track of which expressions have been called, and make it easier to check when a student calls Bingo.
Evaluate
5 Minute(s)
Display slide 41 and give each student a half-page of the attached Radical Thinking handout. Here students are asked to simplify the fourth root of 81 and then reflect on their learning using the Muddiest Point strategy.
After students have a couple minutes to simplify the given radical, move to slide 42 and have students write their responses to the prompts on the slide:
Crystal Clear: What do you think is the easiest (clearest) part of simplifying radicals?
Muddiest Point: What do you think is the most confusing (muddiest) part of simplifying radicals?
Resources
K20Center. (n.d.). Card sort. Strategies. https://learn.k20center.ou.edu/strategy/147
K20Center. (n.d.). Muddiest point. Strategies. https://learn.k20center.ou.edu/strategy/109
K20Center. (n.d.). Pass the problem. Strategies. https://learn.k20center.ou.edu/strategy/151