# Lesson 5

## 5.1: True or False: Squared (5 minutes)

### Warm-up

The purpose of this warm-up is for students to analyze symbolic statements about square roots and decide if they are true or not based on the meaning of the square root symbol.

### Launch

Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

### Student Facing

Decide if each statement is true or false.

$$\left( \sqrt{5} \right)^2=5$$

$$\left(\sqrt{9}\right)^2 = 3$$

$$7 = \left(\sqrt{7}\right)^2$$

$$\left(\sqrt{10}\right)^2 = 100$$

$$\left(\sqrt{16}\right)= 2^2$$

### Activity Synthesis

Poll students on their responses for each problem. Record and display their responses for all to see. If all students agree, ask 1 or 2 students to share their reasoning. If there is disagreement, ask students to share their reasoning until an agreement is reached.

## 5.2: Square Root Values (10 minutes)

### Activity

The purpose of this activity is for students to think about square roots in relation to the two whole number values they are closest to. Students are encouraged to use numerical approaches, especially the fact that $$\sqrt{a}$$ is a solution to the equation $$x^2=a$$, rather than less efficient geometric methods (which may not even work). Students can draw a number line if that helps them reason about the magnitude of the given square roots, but this is not required. However the reason, students must construct a viable argument (MP3).

### Launch

Do not give students access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner then a whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to a number line that includes rational numbers to support information processing.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

What two whole numbers does each square root lie between? Be prepared to explain your reasoning.

1. $$\sqrt{7}$$

2. $$\sqrt{23}$$

3. $$\sqrt{50}$$

4. $$\sqrt{98}$$

### Student Facing

#### Are you ready for more?

Can we do any better than “between 3 and 4” for $$\sqrt{12}$$? Explain a way to figure out if the value is closer to 3.1 or closer to 3.9.

### Activity Synthesis

Discuss:

• “What strategy did you use to figure out the two whole numbers?” (I made a list of perfect squares and then found which two the number was between.)
• “Did anyone use inequality symbols when writing their answers?” (Yes, for the first problem, I wrote $$2<\sqrt5<3$$.)

Once the class is satisfied with which two whole numbers the square roots lie between, ask students to think more deeply about their relationship. Give 1–2 minutes for students to pick one of the last two square roots and figure out which whole number the square root is closest to and to be ready to explain how they know. One possible misconception that could be covered here is that if a number is exactly halfway between two perfect squares, then the square root of that number is also halfway between the square root of the perfect squares. For example, students may think that $$\sqrt{26}$$ is halfway between 4 and 6 since 26 is halfway between 16 and 36. It’s close, since $$\sqrt{26}\approx5.099$$, but it’s slightly larger than “halfway.”

This is a good opportunity to remind students of the graph they made earlier showing the relationship between the side length and area of a square. The graph showed a non-proportional relationship, so making proportional assumptions about relative sizes will not be accurate.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their strategies for figuring out the two whole numbers that each square root lies between, present an incorrect solution based on a misconception about the definition of exponents. For example, “$$\sqrt{7}$$ is in between 2 and 4, because $$2^2$$ is 4, and $$4^2$$ is 8”; or “$$\sqrt{23}$$ is in between 11 and 12, because $$11^2$$ is 22, and $$12^2$$ is 24.” Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who clarify the meaning of a number raised to the power of 2. This routine will engage students in meta-awareness as they critique and correct the language used to relate square roots to the two whole number values they are closest to.
Design Principles(s): Cultivate conversation; Maximize meta-awareness

## 5.3: Solutions on a Number Line (10 minutes)

### Activity

The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line and to reinforce the definition of a square root as a solution to the equation of the form $$x^2=a$$. This is also the first time that students have thought about negative square roots.

### Launch

Do not provide students with access to calculators. Students in groups of 2. 2 minutes of quiet work time followed by a partner, then a whole-class discussion.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, use a kinesthetic representation of the number line on a clothesline. Students can place and adjust numbers on folder paper or cardstock on the clothesline in a hands-on manner.
Supports accessibility for: Conceptual processing
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have had time to plot $$x$$, $$y$$, and $$z$$ on the number line, ask them to write a brief explanation of their reasoning for each number on their paper. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “How do you know that $$z = \sqrt{30}$$?” and “How do you know that $$\sqrt{30}$$ is between 5 and 6?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students revise and refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness

### Student Facing

The numbers $$x$$, $$y$$, and $$z$$ are positive, and $$x^2 = 3$$, $$y^2 = 16$$, and $$z^2 = 30$$.

1. Plot $$x$$, $$y$$, and $$z$$ on the number line. Be prepared to share your reasoning with the class.
2. Plot $$\text- \sqrt{2}$$ on the number line.

### Activity Synthesis

Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.

Conclude the discussion by asking students to share how they placed -$$\sqrt{2}$$ and why.

## Lesson Synthesis

### Lesson Synthesis

To approximate a square root, start by finding the whole numbers it lies between, and then try to get more accurate approximations.

• “How can we find the whole numbers that a square root lies between?” (Look at the squares of whole numbers whose squares are greater than and less than the number inside the square root symbol, like 121 and 144 for $$\sqrt{130}$$.)
• “How can we get a better approximation?” (Test values between those two whole numbers.)

## Student Lesson Summary

### Student Facing

In general, we can approximate the values of square roots by observing the whole numbers around it, and remembering the relationship between square roots and squares. Here are some examples:

• $$\sqrt{65}$$ is a little more than 8, because $$\sqrt{65}$$ is a little more than $$\sqrt{64}$$ and $$\sqrt{64}=8$$.
• $$\sqrt{80}$$ is a little less than 9, because $$\sqrt{80}$$ is a little less than $$\sqrt{81}$$ and $$\sqrt{81}=9$$.
• $$\sqrt{75}$$ is between 8 and 9 (it’s 8 point something), because 75 is between 64 and 81.
• $$\sqrt{75}$$ is approximately 8.67, because $$8.67^2=75.1689$$.
If we want to find a square root between two whole numbers, we can work in the other direction. For example, since $$22^2 = 484$$ and $$23^2 = 529$$, then we know that $$\sqrt{500}$$ (to pick one possibility) is between 22 and 23.