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Transcript
Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑1 Rational Numbers A fraction is in simplest form when the greatest common factor (GCF) of the numerator and denominator is 1. 24 Example 1: Write in simplest form. 36 Use prime factorization and circle the common factors. 24 = 2 · ·2 36 = 2 · ·2 So, · 32· · 33· 24 2 = 36 3 To write a fraction as a decimal: (1) Divide numerator by decimal: (2) Divide until the remainder is 0 or until the remainder repeats. (3) Use a bar to show digits repeating. Example 2: Write 5 as a decimal 6 0.833 6 5.000 −48 20 −18 è So 5 = 0.833..., or 0.833. 6 20 −18 2 ← Remainder repeats. To write a decimal as a fraction: Example 3: Write 0.375 as a fraction. 0.375 (1) Write as a fraction 0.375 = with the denominator 1. 1 375 (2) Since there are 3 digits = to the right of the decimal point, multiply the numerator 1000 and the denominator by 1,000. 375÷125 (3) Divide the numerator = and denominator by the GCF 1000 ÷125 3 3 (4) Simplify. = è So 0.375 = . 8 8 Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑2 Irrational Numbers and Square Roots • The square of 5 is 25. 5 · 5 = 52 = 25 • The square root of 25 is 5 because 52 = 25. 25 = 5 12 = 1 ! # 22 = 4 # # 32 = 9 " perfect squares # 42 = 16# 52 = 25#$ 62 = 36 ! # 72 = 49 # # 82 = 64 " perfect squares # 92 = 81 # 102 = 100#$ Example: You can use a calculator to find square roots. Find tenth. 36 =6 21 36 and 21 to the nearest ≈ 4.5825757 ≈ 4.6 You can estimate square roots like 49 = 7 52 ≈ 7 64 = 8 112 = 121 ! # 122 = 144# # 132 = 169 " perfect squares # 142 = 196# 152 = 225#$ 52 and 61 . 49 = 7 Estimate 61 ≈ 8 64 = 8 Remember to look and see what two perfect squares the square must fall between. Which number is it closer to? Multiply a number by itself and repeat until you find something that gets close to the number you want. Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑3 The cube of 1 is 1. 1×1×1 = 13 = 1 The cube of 3 is 27. 3× 3× 3 = 33 = 27 perfect cubes !##### #"###### $ 3 3 3 3 1 = 1 2 = 8 3 = 27 4 = 64 x3 = 3 = 3 3 27 216 x= Cube Roots The cube of 5 is 125. 5× 5× 5 = 53 = 125 perfect cubes !########"######## $ 3 3 3 3 5 = 125 6 = 216 7 = 343 8 = 512 Example: You can solve cube root equations x3 = 3 27 . 216 27 ← Find the cube root of each side. 216 ← Find the cube root of the numerator and denominator. 3 1 = ← Simplify. 6 2 Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑4 The Pythagorean Theorem The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. Example 1: Find the length of the hypotenuse. a 2 + b2 = c 2 32 + 42 = c 2 9 +16 = c 2 25 = c 2 25 = c 2 5= c The length, c, of the hypotenuse is 5 cm. The Pythagorean Theorem Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑5 Using the Pythagorean Theorem You can use the Pythagorean Theorem to find the length of a leg in a right triangle. Example: Find the length of the unknown side. a 2 + b2 = c 2 62 + b2 = 102 36 + b2 = 100 b2 = 100 − 36 b2 = 64 b=8 The length b of the unknown leg is 8 cm. Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑6 Converse of the Pythagorean Theorem You can use the Pythagorean Theorem to determine whether a triangle is a right triangle. ? a 2 + b2 = c 2 ← Use the Pythagorean Theorem. ? 32 + 42 = 52 ← Substitute 3 for a, 4 for b, and 5 for c. ? 9 +16= 25 ← Simplify. 25 = 25 ✓ The equation is true, so the triangle is a right triangle. You can use the Triangle Inequality to determine whether given lengths can make a triangle. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Name _________________________________ Period ___________ Date _________________ Ch. 1 Review Guide Reteaching 1-‐‑7 Distance in the Coordinate Plane You can graph a point on a coordinate plane. Use an ordered pair (x, y) to record the coordinates. The first number in the pair is the x-coordinate (left/right on the x – axis). The second number is the y-coordinate (up/down on the y – axis). To graph a point, start at the origin, O. Move horizontally according to the value of x. Move vertically according to the value of y. Example 1: (4, – 2) Example 2: (– 3,2) Start at O, move right 4, then down 2. Start at O, move left 3, then up 2. **To find the distance between two coordinates you have to find the horizontal distance and the vertical distance. To find the horizontal distance, subtract the two x – coordinates from the ordered pairs. Next, in order to find the vertical distance, subtract the two y – coordinates from the ordered pairs. Once you have the vertical & horizontal distances, you can make a right triangle between the coordinates; of which the hypotenuse is the shortest, straight-line distance between the two coordinates. You can then use the Pythagorean theorem to find the distance between the two points.