1. Real Numbers and the Coordinate Plane
Chapter 3
Real Numbers and the Coordinate Plane

2. 3-1Exploring Square roots and Irrational Numbers
Objectives:
To simplify the square root of a perfect square.
To estimate without a calculator the square root of a number. (non perfect square)

3. vocab Perfect Square: a number made by squaring a whole number.
Square Root: The square root of a number is a value that, when multiplied by itself, gives the number.
Irrational Number: a real number that cannot be written as a simple fraction
Real Numbers: Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.

4. A perfect Square
Multiply a number times itself to get a perfect square:
1 x 1 = is a perfect square
2 x 2 = is a perfect square
3 x 3 = is a perfect square

5. You try…
Write the perfect squares up to 225 (15 x 15)

6. Answers
4x4 = 16
5x5 = 25
6x6 = 36
7x7 = 49
8x8 = 64
9x9 = 81
10x10 = 100
11x11 = 121
12x12 = 144
13x13 = 169
14x14 = 196
15x15 = 225

7. Square Roots Squares roots are the opposites of perfect squares.
16 =4 because 4 x 4 = 16

8. You try… 36
100
169

9. Estimating Square roots (when its not perfect)27
The falls between 𝑎𝑛𝑑 25
Therefore the answer will fall between 4 (which is the 16 ) and 5 (which is the 25 )

10. Lets try another one! 98
First lets decide what perfect square 98 is close to.
Well 98 is very close to 100.
We know is 10.
Since 98 is smaller than 100, then the must be smaller than
Therefore must be 9.8 or 9.9
Both of these options are less than 10 but very close to it, just as 98 is very close to 100.

11. You try! 51 27
125 18

12. Practice
Complete pages 109 – 110 #’s 9-21, 32-34, 36, 38-47

13. 3-2 The Pythagorean Theorem
Objective:
To use the Pythagorean theorem to determine the length of the hypotenuse of a right triangle given the lengths of the legs.

14. Vocab: Legs: Sides of the right triangle opposite the acute angles.
Hypotenuse: Longest side of the triangle, opposite the right angle.
Pythagorean theorem: a2 + b2 = c2
(leg) 2 + (leg) 2 = (Hypotenuse) 2

15. Example 1 Leg 1 = 3 Leg 2 = 4 Hypotenuse = x 32 + 42 = x2 9 + 16 = x2? 3 4

16. Example 2 Find the missing side (the hypotenuse) 242 + 122 = x2

17. You try
Find the missing side (the hypotenuse) 12 x 5

18. Practice
Page 114 – 115 #’s 1 – 20, 23

19. 3-3 Using the Pythagorean theorem
Objective:
To use the Pythagorean theorem to determine the length of a leg of a right triangle given the lengths of a leg and the hypotenuse.

20. Note* You should use the same formula to start.
(leg) 2 + (leg) 2 = (Hypotenuse) 2
Make sure you are putting the hypotenuse in the correct spot!!!

21. Example 1
x = 102
x = 100
x2 = 64
x = 8 x 10 6

22. Example 2
x = 152
x = 225
x2 = 144
x = 12 x 15 9

23. you Try!
(leg) 2 + (leg) 2 = (Hypotenuse) 2 15 39 6

24. Practice
Page 120 – 121 #’s 1- 19

25. 3-4 graphing in the Coordinate Plane
Objective:
To graph points and use the Pythagorean theorem to determine the distance between two points.

26. Vocab:
Coordinate Plane: a grid formed by the intersection of two number lines.
Y – Axis: vertical number line.
X – Axis: horizontal number line.
Quadrants: the axes divide the plane into 4 sections called quadrants.
Origin: the point where the number lines intersect, (0, 0).
Ordered Pairs: (x, y) gives the coordinates of the location of a point.
X-Coordinate: the first number tells the horizontal units from the origin.
Y-Coordinate: the second number tells the vertical units from the origin.

27. Example 1
Graph the following:
A (3, 5)
B (-3, 5)
C (5, -3)
D (-5, -3)

28. Example 2 x 8 6 Find the distance between: (2, 4) and (-6, -2)
Count your horizontal distance (leg 1)
Count your vertical distance (leg 2)
Find the distance (the hypotenuse) x 6 8

29. You try!
Find the distance between (6, 1) and (-6, -4) x 5 12

30. Practice
Page 126 #’s 1 – 25

31. 3-5 Equations, Tables and Graphs
Objectives:
To create a table (chart) given an equation
To create a graph given a chart or an equation

32. vocab
Solution: a number or pair of numbers that creates a correct answer.
Linear Equation: a set of points that forms a line.

33. Notes Goal: to have y = mx + b Step 1: add/subtract “x” term
Step 2: divide by the number with the ‘y” term
Step 3: create a chart
Step 4: graph the points from the chart

34. Example 1 3x + 6y = 18 -3x -3x 6y = 18 – 3x /6 /6 /6 y = 3 – 3/6 x
3 – ½ (0) = 3 – 0 = 3 3 2
3 – ½ (2) = 3 – 1 = 2 4
3 – ½ (4) = 3 – 2 = 1 1
3x + 6y = 18
-3x x
6y = 18 – 3x
/ /6 /6
y = 3 – 3/6 x
y = 3 – ½ x

35. Example 2 10x - 2y = 8 -10x -10x -2y = 8 – 10x /-2 /-2 /-2 y = -4 +5 x
-4 + 5(0) = = -4 -4 1
-4 + 5(1) = = 1 2
-4 + 5(2) = = 6 6
Example 2
10x - 2y = 8 -10x -10x -2y = 8 – 10x /-2 /-2 /-2 y = x

36. You try
Graph -4x + 2y = 6

37. Practice
page 132 #’s 1-4, worksheet

38. 3-6 Translation Objective:
To translate a figure a given amount horizontally and vertically.
To determine the amount a figure was translated.

40. Notes
Link:

41. Practice
Page 137 – 139 #’s

42. 3-7 reflections Objectives:
To graph the reflection of an object in a coordinate plane.
To determine the line of reflection given the original and final object positions.

43. Vocab
Reflection: a flip Line of Reflection: the line that the figure is flipped over.

44. Example 1

45. Practice
Page #’s 1 – 16,

46. 3-7 Symmetry
Objective: to identify if a shape has reflectional symmetry or rotational symmetry.
Vocab:
Reflectional symmetry is when a shape can be reflected onto itself.
Lines of Symmetry: the line that divides the shape into two pieces that are reflections of each other.

47. eXAMPLE

48. Practice
Page 143 – 144 #’s