2. 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” Both lines have a slope of ( – 1/3 ) A B Line A has a slope of ( – 1/3 ) Line B has a slope of 3 (-1/3) and 3 are Opposite Reciprocals

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

4. 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)

5. INVESTIGATION #2 Length vs. Area 1 unit 2 units 1 square unit (1 unit 2 ) 4 square units (4 units 2 ) Strategies to find the area of shapes “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.

6. 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. S 2 = A 3 2 = 9 S = √A 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.

7. INVESTIGATION #3 Hypotenuse – The longest side of a right triangle. It will always be opposite of the 90 degree angle. Leg Hypotenuse Pythagorean Theorem – a 2 + b 2 = c 2 a2a2 b2b2 c2c2

8. 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 9 14 c To solve for the hypotenuse (c) a2 a2 + b2 b2 = c2c2 92 92 + 14 2 = c2c2 81 + 196 = c2c2 277 = c2c2 c = √277 c ≈ 16.64 Example 2 7 √84 b To solve for a leg length (a or b) a2 a2 + b2 b2 = c2c2 72 72 + b2 b2 = √84 2 49 + b2 b2 = 84 -49 b2 b2 = 35 b = √35 b ≈ 5.92

9. 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 1Example 2 5, 12, 13 a2 a2 + b2 b2 = c2c2 52 52 + 12 2 = 13 2 25 + 144 = 169 = a2 a2 + b2 b2 = c2c2 72 72 + √110 2 = 14 2 49 + 110 = 196 159 ≠ 196 7, √110, 14 Yes, this is a Right Triangle!No, this is a NOT a Right Triangle!

10. INVESTIGATION #4 SPECIAL TRIANGLES 45 - 45 – 90 (isosceles triangle) 30 - 60 – 90 (bisected equilateral) Legs are the same length The hypotenuse is equal to the leg length times the √2 The short leg is half of the hypotenuse The long leg is equal to the short leg length times the √3 6 6 6√2 6 6√3 12 Perimeter: 6 + 6 + 6√2 ≈ 20.49 Perimeter: 12 + 6 + 6√3 ≈ 28.39

11. 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…