Lesson 5
Reasoning About Square Roots
Let’s approximate square roots.
Problem 1
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Explain how you know that \(\sqrt{37}\) is a little more than 6.
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Explain how you know that \(\sqrt{95}\) is a little less than 10.
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Explain how you know that \(\sqrt{30}\) is between 5 and 6.
Problem 2
Plot each number on the number line: \(\displaystyle 6, \sqrt{83}, \sqrt{40}, \sqrt{64}, 7.5\)
![A number line with 6 evenly spaced tick marks and the integers 5 through 10 are indicated.](https://staging-cms-im.s3.amazonaws.com/rqm9geeGdG7Dy1VGX3i6PLYT?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.07.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.07.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T155910Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=18fe2b2bb7072cfacf3f69f2381e85f2ca7ff3bbc212c69d12f6e165c2d974fc)
Problem 3
The equation \(x^2=25\) has two solutions. This is because both \(5 \boldcdot 5 = 25\), and also \(\text-5 \boldcdot \text-5 = 25\). So, 5 is a solution, and also -5 is a solution.
Select all the equations that have a solution of -4:
\(10+x=6\)
\(10-x=6\)
\(\text-3x=\text-12\)
\(\text-3x=12\)
\(8=x^2\)
\(x^2=16\)
Problem 4
Find all the solutions to each equation.
- \(x^2=81\)
- \(x^2=100\)
- \(\sqrt{x}=12\)
Problem 5
Select all the irrational numbers in the list. \(\displaystyle \frac23, \frac {\text{-}123}{45}, \sqrt{14}, \sqrt{64}, \sqrt\frac91, \text-\sqrt{99}, \text-\sqrt{100}\)
Problem 6
Each grid square represents 1 square unit. What is the exact side length of the shaded square?
![A square, not aligned to the horizontal or vertical gridlines, is on a square grid.](https://staging-cms-im.s3.amazonaws.com/kDWLUAsSM7vRWFC9TACRXfPN?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T155910Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6e3b23b851bff3d612377b69153919d393564b8569ae9034e60708cb0a2a83ad)
Problem 7
For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.
- \(0.37 \boldcdot 10^6\) and \(700 \boldcdot 10^4\)
- \(4.87 \boldcdot 10^4\) and \(15 \boldcdot 10^5\)
- \(500,000\) and \(2.3 \boldcdot 10^8\)
Problem 8
The scatter plot shows the heights (in inches) and three-point percentages for different basketball players last season.
![Scatter plot with line of best fit drawn. Horizontal axis, height in inches, scale 70 to 90, by 5’s. Vertical axis, 3 point percentage, scale 0 to 60, by 15’s.](https://staging-cms-im.s3.amazonaws.com/tynEJzZgwP1yT33wHqFD927i?response-content-disposition=inline%3B%20filename%3D%228-8.6.PP.B.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%278-8.6.PP.B.Image.12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T155910Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d6e0735dc1e9c0e15f29df1b7c6cd99671985051ebf4f55e932e09827ef62f41)
- Circle any data points that appear to be outliers.
- Compare any outliers to the values predicted by the model.