Unit 6, Lesson 6
Squares and Square Roots
Integrated Math 2
6.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on finding solutions to equations of the form . It encourages students to think about square roots and to rely on what they know about numbers to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students examine roots more closely. To find all solutions, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneFind the solutions of each equation mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for reasoning about solutions to equations of the form .
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
6.2
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
The purpose of this activity is for students to consider some reasons for the convention that means the positive square root of . Students think about what it would mean if the square root function were not a function. In the synthesis, the convention that makes a function is introduced. It should be emphasized that there are still two square roots of every positive number, but the square root function gives us only one of those. Students will explore the consequences of this convention in future activities.
6.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students use a table of values to draw the graph of and consider whether it is a function. Then students use the graph to reinforce the idea that an equation like has one solution while has two solutions.
Lesson Synthesis
The purpose of the discussion is for students to recognize that is a single, positive value (when is positive), but that there are two square roots of as solutions to the equation . Here are some questions for discussion:
- “What do you know about the number ?” (It is a number that squares to make 17. It is positive. It is a little greater than 4.)
- “What are all the numbers that square to make 17? In other words, what are the square roots of 17?” ( and )
- “Let’s say that is a positive number. What do you know about the number ?” (It is a number that squares to make . It is positive.)
- “What are all the numbers that square to make ? That is, what are the square roots of ?” ( and )
- “Is it possible for to be equal to -10? Explain your reasoning.” (No, it’s not possible because is a positive number and -10 is not.)
Student Lesson Summary
To avoid confusion, we use the convention that represents a single positive number (when is positive). This allows us to easily describe both solutions to the equation . The solutions are and .
The equation has two solutions, because , and also.
The equation only has one solution, namely 121.
The equation only has one solution, namely 11.
The equation doesn’t have any solutions, because the left side is positive and the right side is negative, which is impossible, because a positive number cannot equal a negative number.