Summary
In this lesson, students will recall properties of exponents and how to simplify square roots and cube roots. Then, students will learn how to write radical expressions as expressions with rational exponents and vice versa. Students will use this knowledge to simplify expressions using either approach. This is the first lesson of three in the "Radical Yet Rational" lesson series.
Essential Question(s)
Why do we use rational exponents to represent radicals?
Snapshot
Engage
Students recall properties of exponents through a Card Matching activity.
Explore
Students simplify square roots and cube roots to solve a square puzzle.
Explain
Students complete guided notes with the class and formalize their understanding of why and how radical expressions can be written as expressions with rational exponents and vice versa.
Extend
Students apply what they have learned to simplify expressions using rational exponents and radicals.
Evaluate
Students demonstrate and reflect on how to select an approach when simplifying expressions that contain radicals or rational exponents.
Materials
Lesson Slides (attached)
Square Puzzle handout (attached; one per pair; printed front only)
Guided Notes handout (attached; one per student; printed front/back)
Different Methods, Same Result handout (attached; one per student; printed front only)
Pass the Problem handout (attached; one per pair; printed front only)
Card Matching handout (optional; attached; one per pair; printed front only)
Desmos account
Pencils
Scissors
Student devices with internet access
Engage
15 Minute(s)
Introduce the lesson using the attached Lesson Slides. Display slide 3 to share the lesson’s essential question with students. Go to slide 4 to share the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.
Assign student pairs or ask students to find their own partners. Inform students they are going to complete a Card Matching activity in Desmos.
Display slide 5 and provide students with your session code. Then, have students go to student.desmos.com and enter the session code.
On screen 1 of the Desmos activity, students work in pairs to match cards that contain equivalent expressions and examples with the exponent property they demonstrate. On screen 2, students receive feedback on how many cards they matched correctly.
Explore
15 Minute(s)
Display slide 6. Pass out scissors and the attached Square Puzzle handout to each pair of students. Have students cut out the nine square tiles and rearrange them into one large square by matching each side with an equivalent value. Inform students that when they finish, every pair of touching sides should be equivalent.
As students work together to solve the puzzle, use the image below as a quick way to check students’ work. The symbol in the top-left corner matches one of the tiles; once that symbol is oriented the same way on students’ puzzles, the letters on their tiles will be oriented as shown below if placed correctly.
After students have completed their puzzles, display slide 7. Have students use the I Notice, I Wonder strategy to examine their finished puzzles. Ask for volunteers to share what they notice and what they wonder.
Explain
25 Minute(s)
After students have had time to discuss what they notice and wonder from the Square Puzzle activity, display slide 8. Walk students through the question posed on the slide: "If we know that the square root of x squared is x, then x to what power squared would also equal x?"
Give students a few minutes to discuss what they think that exponent is and why. Ask students to share their thoughts. Then, go to slide 9 to demonstrate why those expressions with rational exponents are equivalent to the radical expressions.
Display slide 10 and pass out the attached Guided Notes handout to each student. You may have students add this handout to their math notebooks if that is a classroom norm.
Complete the handout as a class.
Extend
15 Minute(s)
Display slide 11 and inform students it’s time for them to apply what they have learned.
Pass out the attached Different Methods, Same Result handout to each student. Have students work in pairs to simplify each expression using different methods.
After finishing each problem, students should check their final answers with their partners. If their answers match, have students move on to the next question. If their answers do not match, have partners trade papers to check each other’s work and find any mistakes made.
Evaluate
10 Minute(s)
To assess students’ flexibility in working with both methods, use the Pass the Problem strategy. Display slide 16 and pass out the attached Pass the Problem handout to each pair of students.
Explain the procedure to students as follows: For question 1, student A writes the first step in the simplifying process, then passes the paper to student B, who writes the next step; students should continue taking turns until the expression is completely simplified. For question 2, student B starts instead.
The goal is for students to demonstrate their flexibility between methods, as they do not have control over how their partners start the simplifying process.
Once students finish simplifying, go to slide 17 and ask students to reflect on how they started each question and why.
After students have submitted their work, unhide and show slides 18–19. Give students time to reflect on their thought processes and solutions. Use student responses to see which misconceptions persist.
Resources
ElisaRiva. (2017, February 13). Brain, mind, psychology [Illustration]. Pixabay. https://pixabay.com/illustrations/brain-mind-psychology-idea-drawing-2062057/
K20 Center. (n.d.). Card Matching. Strategies. https://learn.k20center.ou.edu/strategy/1837
K20 Center. (n.d.). I Notice, I Wonder. Strategies. https://learn.k20center.ou.edu/strategy/180
K20 Center. (n.d.). Pass the Problem. Strategies. https://learn.k20center.ou.edu/strategy/151
MyWhyU. (2011, December 16). Pre-Algebra 31 - Simplifying Radical Expressions [Video]. YouTube. https://www.youtube.com/watch?v=Ef2gOQbDv7M
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081
