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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative.

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  1. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org

  2. 8th Grade Math Numerical Roots & Radicals 2012-12-03 www.njctl.org

  3. Setting the PowerPoint View • Use Normal View for the Interactive Elements • To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. • On the Slide Show menu, confirm that Resolution is set to 1024x768. • Use Slide Show View to Administer Assessment Items • To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 8for an example.)

  4. Click on topic to go to that section. Numerical Roots and Radicals Squares, Square Roots & Perfect Squares Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating Square Roots Rational & Irrational Numbers Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Properties of Exponents Solving Equations with Perfect Square & Cube Roots Common Core Standards: 8.NS.1-2; 8.EE.1-2

  5. Squares, Square Roots and Perfect Squares Return to Table of Contents

  6. Area of a Square The area of a figure is the number of square units needed to cover the figure. The area of the square below is 16 square units because 16 square units are needed to COVER the figure...

  7. Area of a Square The area (A) of a square can be found by squaring its side length, as shown below: A = s2 The area (A) of a square is labeled as square units, or units2, because you cover the figure with squares... A = 42 = 4 4 = 16 sq.units Click to see if the answer found with the Area formula is correct! 4 units

  8. 1 What is the area of a square with sides of 5 inches? A 16 in2 B 20 in2 25 in2 C 30 in2 D

  9. 2 What is the area of a square with sides of 6 inches? A 16 in2 B 20 in2 24 in2 C 36 in2 D

  10. 3 If a square has an area of 9 ft2, what is the length of a side? A 2 ft B 2.25 ft C 3 ft 4.5 ft D

  11. 4 What is the area of a square with a side length of 16 in?

  12. What is the side length of a square with an area of 196 square feet? 5

  13. When you square a number you multiply it by itself. 52 = 5 5 = 25 so the square of 5 is 25. You can indicate squaring a number with an exponent of 2, by asking for the square of a number, or by asking for a number squared. What is the square of seven? What is nine squared? 49 81

  14. Make a list of the numbers 1-15 and then square each of them. Your paper should be set up as follows: NumberSquare 1 1 2 4 3 (and so on)

  15. NumberSquare 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 The numbers in the right column are squares of the numbers in the left column. If you want to "undo" squaring a number, you must take the square root of the number. So, the numbers in the left column are the square roots of the numbers in the right column.

  16. Square RootSquare 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 The square root of a number is found by undoing the squaring. The symbol for square root is called a radical sign and it looks like this: Using our list, to find the square root of a number, you find the number in the right hand column and look to the left. So, the 81 = 9 What is 169?

  17. Square Perfect RootSquare 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 When the square root of a number is a whole number, the number is called a perfect square. Since all of the numbers in the right hand column have whole numbers for their square roots, this is a list of the first 15 perfect squares.

  18. Find the following. You may refer to your chart if you need to.

  19. 1 6 What is ?

  20. 81 What is ? 7

  21. What is the square of 15 ? 8

  22. 256 What is ? 9

  23. What is 132? 10

  24. 196 What is ? 11

  25. 12 What is the square of 18?

  26. 13 What is 11 squared?

  27. 14 What is 20 squared?

  28. Squares of Numbers Greater than 20 Return to Table of Contents

  29. Think about this... What about larger numbers? How do you find ?

  30. It helps to know the squares of larger numbers such as the multiples of tens. 102 = 100 202 = 400 302 = 900 402 = 1600 502 = 2500 602 = 3600 702 = 4900 802 = 6400 902 = 8100 1002 = 10000 What pattern do you notice?

  31. For larger numbers, determine between which two multiples of ten the number lies. 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 Next, look at the ones digit to determine the ones digit of your square root.

  32. 2809 Examples: Lies between 2500 & 3600 (50 and 60) Ends in nine so square root ends in 3 or 7 Try 53 then 57 532 = 2809 Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then 88 822 = 6724 NO! 882 = 7744 7744

  33. 15 Find.

  34. 16 Find. 42

  35. Find. 17

  36. Find. 18

  37. Find. 19

  38. Find. 20

  39. Find. 21

  40. Find. 22

  41. Find. 23

  42. Simplifying Perfect Square Radical Expressions Return to Table of Contents

  43. Can you recall the perfect squares from 1 to 400? 12 = 82 = 152 = 22 = 92 = 162 = 32 = 102 = 172 = 42 = 112 = 182 = 52 = 122 = 192 = 62 = 132 = 202 = 72 = 142 =

  44. Square Root Of A Number Recall: If b2 = a, then b is a square root of a. Example: If 42 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100? 8 5 10

  45. Square Root Of A Number Square roots are written with a radical symbol Positive square root: = 4 Negative square root:- = - 4 Positive & negative square roots: = 4 Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal -16.

  46. Is there a difference between & ? Which expression has no real roots? Evaluate the expressions:

  47. Evaluate the expression is not real

  48. 24

  49. ? 25

  50. = ? 26

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