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INVESTIGATION #1 * Identify Locations (points) on a coordinate grid by using coordinate pairs. (x, y) For example: (5, 3) To draw parallel lines use the same slope (rise / run) To draw perpendicular lines use slopes that are “Opposite Reciprocals” B (-1/3) and 3 are A Both lines have a slope of ( – 1/3 ) Opposite Reciprocals Line A has a slope of ( – 1/3 ) Line B has a slope of 3 INVESTIGATION #1 cont. * Know the defining characteristics of the following geometric shapes... Quadrilateral: a four sided shape Parallelogram: - the opposite sides are parallel to each other Rectangle: - opposite sides are the same length - the opposite sides are parallel to each other - all four corners are 90 degree angles Square: - all four sides are of equal length - the opposite sides are parallel to each other - all four corners are 90 degree angles Right Triangle: a triangle with a 90 degree angle INVESTIGATION #1 cont. * Know the difference between the following types of measurement… Length: the measurement of the distance from one point to another Perimeter: the measurement of the distance around a shape Area: the measurement of the space inside a shape (in square units) INVESTIGATION #2 Length vs. Area 1 unit 2 units Strategies to find the area of shapes 1 square unit (1 unit2 ) 4 square units (4 units2 ) “Divide & Count” - divide into square units and count the number of squares inside the shape “Cut & Paste” - Fit partial units together to make complete units “Area Formulas” - Rectangle: A= L x W / Triangle: A = B x H 2 “Surround & Conquer” - Surround the shape with a rectangle. Subtract the area of the ‘empty space’ from the area of the rectangle. INVESTIGATION #2 cont. * You need to be able to draw a square on dot paper * Be able to explain the relationship between the length (s) of one side, and the area (A) of the square. S2 = A S = √A 32 = 9 3 = √9 * Be able to find the precise length of a tilted line on dot paper without a ruler Strategy 1 - Create a square so that the line is one side of the square. Find the area of the square. Take the square root of the area to get the side length. Strategy 2 - Create a right triangle so that the line is the hypotenuse. Solve using the Pythagorean Theorem. INVESTIGATION #3 Hypotenuse – The longest side of a right triangle. It will always be opposite of the 90 degree angle. Hypotenuse Leg Leg c2 Pythagorean Theorem – a2 + b2 = c2 a2 b2 INVESTIGATION #3 cont. If given two side lengths of a right triangle, you can solve for the third side by using the Pythagorean Theorem. Example 1 To solve for the hypotenuse (c) c 9 14 a2 + b 2 = c 2 92 + 142 = c2 81 + 196 = c2 277 = c2 c = √277 c ≈ 16.64 Example 2 To solve for a leg length (a or b) 7 √84 b a2 + b2 = c 2 72 + b2 = √842 49 + b2 = 84 -49 -49 b2 = 35 b = √35 b ≈ 5.92 INVESTIGATION #3 cont. Is it a Right Triangle? If given three side lengths, you can use the Pythagorean Theorem to check if the triangle is a right triangle. Example 1 5, 12, 13 a2 + b2 = c 2 52 + 122 = 132 25 + 144 = 169 169 = 169 Yes, this is a Right Triangle! Example 2 7, √110, 14 a2 + b2 = c 2 72 + √110 2 = 142 49 + 110 = 196 159 ≠ 196 No, this is a NOT a Right Triangle! INVESTIGATION #4 SPECIAL TRIANGLES 45 - 45 – 90 (isosceles triangle) • Legs are the same length • The hypotenuse is equal to the leg length times the √2 30 - 60 – 90 (bisected equilateral) • The short leg is half of the hypotenuse • The long leg is equal to the short leg length times the √3 6√2 6 12 6√3 6 6 Perimeter: Perimeter: 6 + 6 + 6√2 ≈ 20.49 12 + 6 + 6√3 ≈ 28.39 INVESTIGATION #5 RATIONAL vs. IRRATIONAL NUMBERS definite length precise value Infinite number of decimal places • Terminating and Repeating decimals are rational numbers and can be written as a fraction Repeating Decimal Patterns 1 digit repeat - Denominator is 9 ex. 2/9 = .2222… 8/9 = .8888… 2 digit repeat - Denominator is 99 ex. 61/99 = .616161… 7/99 = .070707… 3 digit repeat - Denominator is 999 ex. 538/999 = .538538… 84/999 = .084084…