Illustrative Mathematics Grade 8, Unit 8, Lesson 5: Reasoning About Square Roots

Learning Targets:

  • When I have a square root, I can reason about which two whole numbers it is between.

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Illustrative Math
Grade 8

Lesson 5: Reasoning About Square Roots

Let’s approximate square roots.

Illustrative Math Unit 8.8, Lesson 5 (printable worksheets)

Lesson 5 Summary

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:

  • √65 is a little more than 8, because √65 is a little more than √64 and √64 = 8.
  • √80 is a little more than 9, because √80 is a little more than √81 and √81 = 9.
  • √75 is between 8 and 9(it’s 8 point something), because 75 is between 64 and 81.
  • √75 is approximately 8.67, because 8.672 = 75.1689.

If we want to find a square root between two whole numbers, we can work in the other direction. For example, since 222 = 484 and 232 = 529, then we know that √500 (to pick one possibility) is between 22 and 23.

Many calculators have a square root command, which makes it simple to find an approximate value of a square root.

Lesson 5.1 True or False: Squared

Decide if each statement is true or false.

Lesson 5.2 Square Root Values

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

Are you ready for more?

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

  • Show Answer

    3.52 = 12.25
    So, √12 is closer to 3.1 than 3.9

Lesson 5.3 Solutions on a Number Line

The numbers x, y, and z are positive, and x2 = 3, y2 = 16, and z2 = 30.

  1. Plot x, y, and z on the number line. Be prepared to share your reasoning with the class.
  2. Plot √2 on the number line.

Lesson 5 Practice Problems

  1. a. Explain how you know that √37 is a little more than 6.
    b. Explain how you know that √95 is a little less than 10.
    c. Explain how you know that √30 is between 5 and 6.
  2. Plot each number on the number line:
  3. Mark and label the positions of two square root values between 7 and 8 on the number line.
  4. Select all the irrational numbers in the list.
  5. Each grid square represents 1 square unit. What is the exact side length of the shaded square?
  6. For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.
  7. The scatter plot shows the heights (in inches) and three-point percentages for different basketball players last season.
    a. Circle any data points that appear to be outliers.
    b. Compare any outliers to the values predicted by the model.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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