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How radical is this?

How radical is this?. How does the area of a square relate to its side lengths?. Review: What are the properties of a square? …remember a square has 4 equal side lengths What is the difference between perimeter and area? And now….onto the Dot Paper Activity. Dot Paper Activity. Step 1

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How radical is this?

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  1. How radical is this?

  2. How does the area of a square relate to its side lengths? Review: • What are the properties of a square? …remember a square has 4 equal side lengths • What is the difference between perimeter and area? And now….onto the Dot Paper Activity

  3. Dot Paper Activity Step 1 In the first 5 x 5 grid, connect the dots so that the length of the side of the square has a length of 1 and on the other grid, create another square that has a side length of 2. This square has a side length of 1! This square has a side length of 2!

  4. Continue the process, until you have drawn 4 squares • Square with a side length of 1 • Square with a side length of 2 • Square with a side length of 3 • Square with a side length of 4

  5. Dot Paper Activity Step 2 What is the area? This has a side length of 1 1 1 2 3 4 What is its area? This square has an area of 4

  6. Let’s make a TABLE The lengths of the side of the squares The area of each square 1 1 4 2 9 3 4 16 5 25 36 6 7 49 64 8 9 81

  7. Make a conjecture... • With your elbow partner, study the chart. Create a list of any patterns you notice. • Share with the class. • How does the area of a square relate to its side lengths? REVIEW

  8. Why are these “Perfect Squares”? 64 225 1 81 256 4 100 289 9 121 16 324 144 25 400 169 36 196 49 625

  9. Simplify The SIMPLIFIED answer is like the length of the side of the square – So if a square has an area of 4, what is the length of the sides of the square? = 2 This is called theRADICAND.Think of theRADICANDas theArea of a square

  10. Simplify the following……. = 5 = 4 =10 = 12

  11. How many solutions does a perfect square have? and -5 = 5 What is -5 (-5)? = 25, so for each perfect square there are 2 solutions. We write this as 5.

  12. Square Root of a Number If b 2 = a, then b is a square root of a. square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares. Principal square root: the positive square root of a number; the principal square root of 9 is 3. negative square root: the negative square root of 9 is –3 and is shown like radical: the symbol which is read “the square root of a” is called a radical. radicand: the number or expression inside a radical symbol 3 is the radicand. perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares. irrational number: a number whose decimal form is nonteminating and nonrepeating. is an irrational number. Rational number: a number that can be written in the form a/b, where a and b are integers (b cannot equal 0) radical expression: an expression that contains a radical. an sbl original ’07, modified by GLF

  13. Understanding Square Roots

  14. Estimate the radicals between two consecutive integers. Estimate the √10 • Can we make a square with an area of 10 with the same side lengths? • No, so we must estimate the radical. 3. Think of a perfect square that is less than the square root of ten and greater than the root of 10.

  15. How do I estimate a non perfect square? 1 2 3 4 5 Steps: • Find 2 perfect squares that are closest to • For example, the closest perfect square that is less than the square root of 10 is the • The closest perfect square that is greater than the square root of 10 is • Place both square roots above their solutions on the number line. • The falls between 3 and 4. 3 4

  16. EQ: How do I simplify a radical?

  17. Simplifying Radicals Simplest form is when the radical expression has no perfect square factors other than 1 in the radicand Simplifying Radicals

  18. Product Property of Radicals *a number inside a radical can be separated into parts by finding its factors 9.3 – Simplifying Radicals

  19. 1. Look for perfect squarefactors 2. Separate into 2 parts 3. Simplify the perfect square Ex. 1. Simplify the expression: 9.3 – Simplifying Radicals

  20. Ex. 2. Simplify the expression: 9.3 – Simplifying Radicals

  21. Ex. 3. Simplify the expression: 9.3 – Simplifying Radicals

  22. Ex. 4. Simplify the expression: 9.3 – Simplifying Radicals

  23. Ex. 5. Simplify the expression: 9.3 – Simplifying Radicals

  24. Ex. 6. Simplify the expression: *If the number is too big, break it down in steps 9.3 – Simplifying Radicals

  25. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  26. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  27. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  28. How do I add and subtract radical expressions? + To combine radicals: combine the coefficients of like radicals

  29. What is a radical term? It is a term which contains a radical.

  30. But what is a radical? A radical is another name for a square root.

  31. Okay—so a radical term . . . . . . is an term that contains a radical, or square root. Look! There goes one now!

  32. Consider these two expressions:

  33. What do they have in common? What makes them different?

  34. You may have noticed that the two expressions are really the same, if . . .

  35. If what? Under what condition would the two expressions be identical?

  36. The two expressions are identical when

  37. That means since you already know how to simplify the first expression . . .

  38. . . . then you also know how to simplify the radical expression .

  39. The rules that apply to combining like terms

  40. also apply to combining radical terms.

  41. also apply to combining radical terms.

  42. You can only combine radical terms when the radicands are identical. When what are identical? What is a radicand?

  43. The radicand is the number underneath the square root sign.

  44. When two (or more) terms have exactly the same radicand,

  45. we call them like radical terms, and we can combine them .

  46. But when the radicands are not identical . . .

  47. . . . the terms cannot be combined.

  48. Practice combining radical terms:

  49. Practice combining radical terms:

  50. What is the rule for adding and subtracting radicals? ? Just like As long as the radicands are exactly the same, you can treat them as “liketerms.” distributive property! Your turn! If the radicands are different, simplify first (if possible), then collect your “___________ _________.” Your turn! an sbl original ‘07

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