Lesson 6

Squares and Square Roots

6.1: Math Talk: Four Squares (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for solving equations in one variable and thinking about the graph of an associated function. The connection between the graph of \(y = x^2\) and solutions to quadratic equations should be familiar from earlier courses. These understandings help students develop fluency and will be helpful later in this lesson when students solve equations that involve squares and square roots.

Launch

Display one problem at a time alongside the image showing the graph of \(y =x^2\) and the 4 lines. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find the solutions of each equation mentally.

\(x^2 = 4\)

\(x^2 = 2\)

\(x^2 = 0\)

\(x^2 = \text{-}1\)

Graph of parabola y = x squares. Four horizontal dotted lines graphed with y intercepts of -1, 0, 2, and 4.

 

Student Response

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Activity Synthesis

Discuss the connections between the solutions of the equations and the graph of \(y=x^2\).

Although \(x^2=2\) has two solutions (\(\sqrt{2}\) and \(\text{-}\sqrt{2}\)), we don’t “take the square root” of each side of the equation, because by convention, \(\sqrt{2}\) means the positive square root of 2. If we want the negative square root, we need to write \(\text{-}\sqrt{2}\). Students will study this convention more closely in the following activity, so it does not need to be discussed in depth at this time.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

6.2: Finding Square Roots (15 minutes)

Activity

The purpose of this activity is for students to consider some reasons for the convention that \(\sqrt{x}\) means the positive square root of \(x\). Students think about what it would mean if the square root function were not a function. In the synthesis, the convention that makes \(\sqrt{x}\) a function is introduced. It should be emphasized that there are still two square roots of every positive number, but the square root function only gives us one of those. Students will explore the consequences of this convention in future activities.

Launch

Arrange students in groups of 2. Display the following for all to see:

\(\sqrt{4} + \sqrt{9} =\) ?

Tell students that in this activity, they will think about what the answer to this question is. Give students 2 minutes to read the statement, think, and write down their answers individually, and another 2 minutes for pairs to share their thoughts. Follow with a whole-class discussion.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for Clare’s question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “Why do you think only one of these could be the same as \(\sqrt{4}+\sqrt{9}\)?” or “Why do you think all of them could be the same as \(\sqrt{4}+\sqrt{9}\)?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for their responses to Clare’s question.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

Clare was adding \(\sqrt{4}\) and \(\sqrt{9}\), and at first she wrote \(\sqrt{4} + \sqrt{9} = 2+3\). But then she remembered that 2 and -2 both square to make 4, and that 3 and -3 both square to make 9. She wrote down all the possible combinations:

    2 + 3 = 5
    2 + (-3) = -1
    (-2) + 3 = 1
    (-2) + (-3) = -5

Then she wondered, “Which of these are the same as \(\sqrt{4} + \sqrt{9}\)? All of them? Or only some? Or just one?”

How would you answer Clare’s question? Give reasons that support your answer.

Student Response

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Student Facing

Are you ready for more?

  1. How many solutions are there to each equation?
    1. \(x^3=8\)
    2. \(y^3=\text -1\)
    3. \(z^4 = 16\)
    4. \(w^4 = \text -81\)
  2. Write a rule to determine how many solutions there are to the equation \(x^n=m\) where \(n\) and \(m\) are non-zero integers.

Student Response

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Activity Synthesis

Invite students to share their answers and reasons. Here are some questions for discussion if needed:

  • If only one of these numbers is the sum of \(\sqrt{4}\) and \(\sqrt{9}\), does that mean that 4 and 9 each have only one square root?
  • If all four of these are the sum of \(\sqrt{4}\) and \(\sqrt{9}\), what would happen if we added \(\sqrt{16}\) to them? How many answers would we get?
  • If more than one of these numbers is the same as \(\sqrt{4} + \sqrt{9}\), then are they the same as each other?

The goal of the discussion is for students to consider some reasons why we might want the operation of taking the square root to give us only one number. These questions do not have to be explored in depth at this time.

To conclude the discussion and preview the work ahead, display the following graphs and the functions they represent for all to see:

\(b = \sqrt{a}\)

Graph of the function \(b = \sqrt{a}\).

\(d^2 = c\)

Graph of \(d^2=c .\)

Tell students that these graphs represent the two possibilities that they have been thinking about. \(b=\sqrt{a}\) is a function, because each value of \(a\) has only one corresponding value of \(b\). But \(d^2=c\) is not a function, because many values of \(c\) have two corresponding \(d\) values. If we used a graph like \(d^2=c\) to find square roots, we would not get one unique value for each square root. This is why there is a convention in mathematics which says that the symbol \(\sqrt{x}\) means the positive square root of \(x\). All positive numbers do have two square roots, but when we write \(\sqrt{x}\), or \(x^{\frac12}\), that means only the positive square root. We indicate the negative square root by writing \(\text- \sqrt{x}\) or \(\text- x^{\frac12}\).

6.3: One Solution or Two? (15 minutes)

Activity

In this activity, students analyze the graphs of \(b=\sqrt{a}\) and \(t=s^2\) in order to think about solutions to equations like \(\sqrt{a}=4\) and \(s^2=5\). They build on the discussion from the previous activity to find that equations like \(\sqrt{a}=4\) have only one solution, while equations like \(s^2=5\) have exactly two solutions that are opposites of each other.

Student Facing

  1. The graph of \(b=\sqrt{a}\) is shown.

    B= the square root of a graphed on coordinate plane 
    1. Complete the table with the exact values and label the corresponding points on the graph with the exact values.
      \(a\) 1 4 9 12 16 20
      \(\sqrt{a}\)            
    2. Label the point on the graph that shows the solution to \(\sqrt{a} = 4\).
    3. Label the point on the graph that shows the solution to \(\sqrt{a} = 5\).
    4. Label the point on the graph that shows the solution to \(\sqrt{a} = \sqrt{5}\).
  2. The graph of \(t = s^2\) is shown.
    1. Label the point(s) on the graph that show(s) the solution(s) to \(s^2 = 25\).
    2. Label the point(s) on the graph that show(s) the solution(s) to \(\sqrt{t} = 5\).
    3. Label the point(s) on the graph that show(s) the solution(s) to \(s^2 = 5\).

    T = squared graphed on coordinate plane 

Student Response

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Anticipated Misconceptions

Since students usually see \(x\)-values on the horizontal axis and \(y\)-values on the vertical, they may look for \(a\) or \(s\) values on the wrong axis. Encourage students to annotate the graph by drawing horizontal or vertical lines that will intersect the curves at the point that represents the solution, or using some other method that is helpful for them. For example, when solving \(s^2=25\), students can draw a line representing \(t=25\) and see where it hits the graph of \(t=s^2\), since these points represent the solutions.

Activity Synthesis

The purpose of this discussion is to emphasize that equations like \(\sqrt{a}=5\) have only one solution while equations like \(s^2=11\) have two solutions. Students may find this confusing since we can talk about “the square roots” (plural) of a positive number, so it is important to remind students of the discussion in the previous activity and the convention that we use the symbol \(\sqrt{a}\) for the positive square root of \(a\) while \(\text-\sqrt{a}\) is used for the negative square root of \(a\). Display the two graphs and tables from the activity for all to see throughout the discussion. Here are some questions for discussion:

  • “How can you use the graph of \(b=\sqrt{a}\) to see that the equation \(\sqrt{a}=5\) has only one solution?” (The horizontal line \(b=5\) intersects the graph of \(b=\sqrt{a}\) at one point, \((25,5)\).)  
  • “How can you use the graph of \(t=s^2\) to see that the equation \(s^2=12\) has two solutions? What are the exact values of the two solutions?” (The horizontal line \(t=12\) intersects the graph of \(t=s^2\) at two points, \((\text- \sqrt{12},12)\) and \((\sqrt{12},12)\). The two solutions are \(\text- \sqrt{12}\) and \(\sqrt{12}\).)
Speaking: MLR8 Discussion Supports. As students share their responses to the first question, press for details by asking how they know that the intersection of the horizontal line \(b=5\) and the graph of \(b=\sqrt{a}\) represents the solution of \(\sqrt{a}=5\). Show concepts multi-modally by displaying the graph of \(b=\sqrt{a}\) and drawing the horizontal line \(b=5\). This will help students justify why equations such as \(\sqrt{a}=5\) have only one solution.
Design Principle(s): Support sense-making; Optimize output (for justification)
Representation: Internalize Comprehension. Use color and annotations to illustrate connections between representations in a problem. For example, circle \(s^2=25\) in a certain color and draw a horizontal line at 25 in the same color to show where it intersects the graph of \(t=s^2\).
Supports accessibility for: Visual-spatial processing; Conceptual processing

Lesson Synthesis

Lesson Synthesis

In this lesson, students encountered equations of the form \(x^2 = a\) and \(\sqrt{x} = a\) where \(a\) is positive. Here are some questions for discussion:

  • “What do you know about the number \(\sqrt{17}\)?” (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?” (\(\sqrt{17}\) and \(\text- \sqrt{17}\))
  • “Let’s say \(a\) is a positive number. What do you know about the number \(\sqrt{a}\)?” (It is a number that squares to make \(a\). It is positive.)
  • “What are all the numbers that square to make \(a\)? That is, what are the square roots of \(a\)?” (\(\sqrt{a}\) and \(\text- \sqrt{a}\))
  • “Is it possible for \(\sqrt{a}\) to be equal to -10? Explain your reasoning.” (No, it’s not possible because \(\sqrt{a}\) is a positive number and -10 is not.)

6.4: Cool-down - Squares and Roots (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

The symbol \(\sqrt{11}\) represents the positive square root of 11. If we want to represent the negative square root, we write \(\text{-} \sqrt{11}\).

The equation \(x^2 = 11\) has two solutions, because \(\sqrt{11}^2=11\), and also\(\left(\text{-}\sqrt{11}\right)^2=11\).

The equation \(\sqrt{x} = 11\) only has one solution, namely 121.

The equation \(\sqrt{x} = \sqrt{11}\) only has one solution, namely 11.

The equation \(\sqrt{x} = \text{-}11\) 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.