Guided Practice

- A point whose coordinates are both positive numbers lies in Quadrant I.
- A point with a negative x-coordinate and positive y-coordinate lies in Quadrant II.
- A point whose coordinates are both negative numbers lies in Quadrant III.
- A point with a positive x-coordinate and negative y-coordinate lies in Quadrant IV.

If you imagine folding this coordinate plane on the x-axis, perhaps you can see that points D and E would line up on top of each other.

We say that point E is a **reflection** of point D across the x-axis.

Try it out!

You can plot points on the coordinate plane by using information about the points. You can use the coordinates of the point, the quadrant it is in (if it is in a quadrant), and information about whether it is a reflection of another point.

For example, I'm thinking of a point in the third quadrant. It is a reflection, across the y-axis, of the point at (6, –2). Plot the point.

I'm thinking of a point in the second quadrant. It is a reflection, across the x-axis, of the point at

(–2, –2). Plot the point.

I'm thinking of a point not in a quadrant. It is a reflection, across the x-axis, of the point at (0, 4). Plot the point.

Write the ordered pairs for the new points in each quadrant.

Quadrant I: (, )

Quadrant II: (, )

Quadrant III: (, )

Quadrant IV: (, )

What are the coordinates of point A?

Point A: (, )

Point A has a x-coordinate and a y-coordinate.

Point A lies in Quadrant .

What are the coordinates of point B?

Point B: (, )

Point B has a x-coordinate and a y-coordinate.

Point B lies in Quadrant .

What are the coordinates of point C?

Point C: (, )

Point C has a x-coordinate and a y-coordinate.

Point C lies in Quadrant .

What are the coordinates of point D?

Point D: (, )

Point D has a x-coordinate and a y-coordinate.

Point D lies in Quadrant .

Point D is a reflection of point across the x-axis.

Point D: (, )

Point E: (, )

Point A is a reflection of point across the y-axis.

Point A: (, )

Point B: (, )

Point D is a reflection of point across the y-axis.

Point D: (, )

Point C: (, )

You can use properties to make operations simpler. Consider, for example, the multiplication:

21 × 14

Use the Distributive Property to make this simpler.21 × 14 = 21 × (10 + 4) = 210 + 84

The Associative and Commutative Properties can also be used to simplify calculations:13 + (5 + 10)

You can change the order of the 10 and 5 and then change the grouping of the numbers. What is the sum?You can use the Property to show that 5 + 6 = + 5.

You can use the Property to show that 8.5 × (10 + 2) = 8.5 × 10 + × 2.

The Property says that you can group numbers however you like when you add or multiply.

Complete the statement using one of the properties of operations.

(5 × 3) × = 5 × (3 × 6)

Complete the statement using one of the properties of operations.

+ 1 = 1 + 32

Complete the statement using one of the properties of operations.

+ (11.92 + 6.1) = (8.8 + 11.92) + 6.1

Use the Commutative and Associative Properties to make the calculation simpler. Determine the sum.

13 + (5 + 10) =

Use the Distributive Property to make the calculation simpler. Determine the product.

35 × 6 =

Use the Distributive Property to make the calculation simpler. Determine the product.

50 × 12 =

A relationship between two or more values can be described using an equation. For example, suppose you are traveling at 55 miles per hour. How far do you travel in 1.5 hours?

Set up an equation.

You can use a variable to represent an unknown value. Here, we use the variable d to represent the distance traveled.

To solve an equation for an unknown amount, you can use an opposite operation. The inverse operation for multiplication is division.

Imagine you travel 500 miles in 4 hours. What is your speed?

Divide both sides by 4: s × 4/4 = 500/4. So, s = 125.

You can solve an equation that uses any operation by using the opposite operation.

Solve for d in the example. What is the distance traveled?

d = miles

Suppose you are traveling at 60 miles per hour. How far do you travel in 2.5 hours?

Write the equation. Use the variable d for the unknown amount.

× =

d = miles

Suppose you are traveling at 10 miles per hour. How far do you travel in 0.5 hour?

Write the equation. Use the variable d for the unknown amount.

× =

d = miles

Now imagine that you travel 261 miles in 3 hours. What is your speed?

Complete the multiplication equation. Use the variable s for the unknown amount.

× = 261

s = miles per hour

Suppose you travel 341 miles in 5.5 hours. What is your speed?

Write a multiplication equation. Use the variable s for the unknown amount.

× = 341

s = miles per hour

Solve the equation.

2.5 + q = 11

q =

Solve the equation.

26 – m = 8.45

m =

Solve the equation.

1/2× n = 16

n =

Solve the equation.

82.46 ÷ w = 13.3

w =

Type coordinates into the ordered pair to plot a point and then change its location.

( , )

A point at (–35, 75) is in Quadrant .

A point at (14, 82) is in Quadrant –.

Now type coordinates into this ordered pair to show a reflection of the first point across the x‑axis.

( , )

Identify the location of the reflection of each point across the x-axis.

(84, 96): (, )

(–15, –31): (, )

When you type these coordinates, you will see a point and its reflection across the y-axis.

( , )

Identify the location of the reflection of each point across the y-axis.

(22, –6): (, )

(7, –40): (, )

Explore what happens when you reflect a point across the x- and the y-axis.

( , )

Identify the location of the reflection of each point across both the x- and y-axis.

(94, –67): (, )

(32, 12): (, )

The coordinate plane is formed by two lines dividing the plane into 4 regions called quadrants: I, II, III, and IV. The horizontal line is called the x-axis, and the vertical line is called the y-axis.

To locate a point on the coordinate plane, start at 0 on the x-axis and 0 on the y-axis: (0, 0). Count 2 to the right and 4 up to locate the point (2, 4). Count 2 to the left and 4 down to locate the point (–2, –4).

Enter 3 different points that are located in Quadrant II.

(, )

(, )

(, )

Enter 3 different points that are located in Quadrant I.

(, )

(, )

(, )

Enter 3 different points that are located in Quadrant IV.

(, )

(, )

(, )

Enter 3 different points that are located in Quadrant III.

(, )

(, )

(, )

Enter an ordered pair for a point located in Quadrant IV.

(, )

The point at (–1, –5) is in Quadrant .

The point at (1/4, 1/3) is in Quadrant .

The point at (62.1, –0.4) is in Quadrant .

The point at (–5, 10) is in Quadrant .

The point at (2, 5) is in Quadrant .