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

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 #2Length vs. Area

1 unit

2 units

1 square unit (1 unit2 )

4 square units (4 units2 )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 H2

“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 32 = 9S = √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.

INVESTIGATION #3Hypotenuse – The longest side of a right triangle. It will always be opposite of the 90 degree angle.

Leg

Leg

Hypotenuse

Pythagorean Theorem –

a2 + b2 = c2

a2

b2

c2

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 19

14

c

To solve for the

hypotenuse (c)a2 + b2 = c2

92 + 142 = c2

81 + 196 = c2

277 = c2

c = √277c ≈ 16.64

Example 27 √84

bTo solve for a leg length (a or b)

a2 + b2 = c2

72 + b2 = √842

49 + b2 = 84-49 -49 b2 = 35

b = √35b ≈ 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 Example 2

5, 12, 13a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169169 = 169

a2 + b2 = c2

72 + √110 2 = 142

49 + 110 = 196159 ≠ 196

7, √110, 14

Yes, this is a Right Triangle! No, this is a NOT a Right Triangle!

INVESTIGATION #4SPECIAL TRIANGLES45 - 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

INVESTIGATION #5RATIONAL vs. IRRATIONAL NUMBERSdefinite length

precise valueInfinite 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…