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Exploring Square Roots and Perfect Squares in Math: Calculate Areas and Side Lengths

Dive into the world of perfect squares and square roots in mathematics. Learn how to calculate the area of shaded squares, express it as a product, and relate side length to area using square roots. Explore squaring and square rooting operations, including rational numbers. Discover strategies for estimating square roots of non-perfect squares and practice using benchmarks and calculators. Apply the Pythagorean theorem to solve for missing sides in right triangles, estimate lengths of objects, and calculate areas. Embrace the challenge of working with non-perfect squares and sharpen your math skills!

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Exploring Square Roots and Perfect Squares in Math: Calculate Areas and Side Lengths

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  1. 1.1 Square Roots of Perfect Squares Math 90

  2. For each shaded square: What is its area? Write this area as a product. How can you use a square root to relate the side length and area?

  3. Calculate the Area:

  4. Calculate the side length

  5. For the area of each square in the table… • Write the area as a product. • Write the side length as a square root.

  6. Squaring vs. Square Rooting • Squaring and square rooting are opposite, or inverse operations. • Eg. • When you take the square root of some fractions you will get a terminating decimal. • Eg. • These are all called RATIONAL numbers.

  7. When you take the square root of other fractions you will get a repeating decimal. • Eg. • These are all called RATIONAL numbers

  8. 1.2 Square Roots of Non-Perfect Squares d

  9. Introduction... • Many fractions and decimals are not perfect squares. • A fraction or decimal that is not a perfect square is called a non-perfect square. • The square roots of these numbers do not work out evenly! • How can we estimate a square root of a decimal that is a non-perfect square?

  10. Here are 2 strategies... Ask yourself: “Which 2 perfect squares are closest to 7.5?” 7.5 2.5 2 3 7.5 is closer to 9 than to 4, so is closer to 3 than to 2. What would be a good approximation?

  11. Strategy #2... • Use a calculator!  • But, of course, you must be able to do both!

  12. Example #1 • Determine an approximate value of each square root. We call these 2 numbers ‘benchmarks’. close to 9 close to 4 What does this mean?

  13. Example #2 • Determine an approximate value of each square root. Your benchmarks! 0.30 0.36 0.20 0.25 0.40 Of course, you can always use a calculator to CHECK your answer!

  14. What’s the number? • Identify a decimal that has a square root between 10 and 11. If these are the square roots, where do we start? 121 110 100 120 or 10 11

  15. Mr. Pythagoras Recall: a2 + b2 = c2 • Junior High Math Applet Remember, we can only use Pythagorean Theorem on RIGHT angle triangles!

  16. Practicing the Pythagorean Theorem First, ESTIMATE each missing side and then CHECK using your calculator. 7 cm x 13 cm 5 cm 8 cm x

  17. Applying the Pythagorean Theorem 1.5 cm 2.2 cm 6.5 cm The sloping face of this ramp needs to be covered with Astroturf. • Estimate the length of the ramp to the nearest 10th of a metre • Use a calculator to check your answer. • Calculate the area of Astroturf needed.

  18. Let’s quickly review what we’ve learned today... • Explain the term non-perfect square. • Name 3 perfect squares and 3 non-perfect squares between the numbers 0 and 10. • Why might the square root shown on a calculator be an approximation?

  19. Assignment Time! • Complete the following questions in your notebook. • Be prepared to discuss your answers in class. • Show all of your work!

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