Lesson 5

Reasoning About Square Roots

Let’s approximate square roots.

Problem 1

  1. Explain how you know that \(\sqrt{37}\) is a little more than 6.

  2. Explain how you know that \(\sqrt{95}\) is a little less than 10.

  3. 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.

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:













Problem 4

Find all the solutions to each equation.

  1. \(x^2=81\)
  2. \(x^2=100\)
  3. \(\sqrt{x}=12\)

Problem 5

Select all the irrational numbers in the list. \(\displaystyle \frac23, \frac {\text{-}123}{45}, \sqrt{14}, \sqrt{64}, \sqrt\frac91, \text-\sqrt{99}, \text-\sqrt{100}\)

(From Unit 8, Lesson 3.)

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. 
(From Unit 8, Lesson 2.)

Problem 7

For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.

  1. \(0.37 \boldcdot 10^6\) and \(700 \boldcdot 10^4\)
  2. \(4.87 \boldcdot 10^4\) and \(15 \boldcdot 10^5\)
  3. \(500,000\) and \(2.3 \boldcdot 10^8\)
(From Unit 7, Lesson 10.)

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.
  1. Circle any data points that appear to be outliers.
  2. Compare any outliers to the values predicted by the model.
(From Unit 6, Lesson 4.)