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UNIT 6

2017 – 2018

Coordinate Plane CCM6+

Name: ________________

Math Teacher:___________ Main Concept Page(s)

Descartes and the Fly / Unit 6 Vocabulary 2

Daily Warm-Ups 3 – 5

Graphing on the Coordinate Plane 6 – 11

Reflections (x & y axis) 12 – 14

Distance Between Points on the Coordinate Plane 15 – 19

Unit 6 Study Guide 20 – 21

Projected Test Date: ______________ Some work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M2-TE-1.3.0-08.2015

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Descartes and the Fly

Who invented the coordinate plane? Sam knows a math legend that answers this question:

René Descartes was a French man who lived in the 1600s. When he was a child, he was often sick, so the

teachers at his boarding school let him stay in bed until noon. He went on staying in bed until noon for

almost all his life. While in bed, Descartes thought about math and philosophy.

One day, Descartes noticed a fly crawling around on the ceiling. He watched the fly for a long time. He

wanted to know how to tell someone else where the fly was. Finally he realized that he could describe the

position of the fly by its distance from the walls of the room. When he got out of bed, Descartes wrote

down what he had discovered. Then he tried describing the positions of points, the same way he described

the position of the fly. Descartes had invented the coordinate plane! In fact, the coordinate plane is

sometimes called the Cartesian plane, in his honor.

Sam likes this story, because it is about flies. Sam spends lots of time trying to find flies, just like

Descartes does in the story. But is the story true? Or is it just a legend?

The story of the coordinate plane turns out to be a long story, with many parts. It starts long before

Descartes, in Ancient Greece.

Unit 6: Coordinate Plane Vocabulary

coordinate plane A plane formed by the intersection of the x-axis and the y-axis.

x-axis The horizontal number line

y-axis The vertical number line

quadrants The x- and y-axes divide the coordinate plane into four regions. Each region is called a quadrant.

origin The point where the x-axis and y-axis intersect on the coordinate plane.

ordered pairs A pair of numbers that can be used to locate a point on a coordinate plane.

x-coordinate The first number in an ordered pair; it tells the distance to move right or left from the origin.

y-coordinate The second number in an ordered pair; it tells the distance to move up or down from the origin.

reflection a transformation of a figure that flips the figure across a line

integers The set of whole numbers and their opposites.

opposites Two numbers that are equal distance from zero on the number line.

absolute value The distance of a number from zero on a number line; shown by the symbol: │ │

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Unit 6 - Coordinate Plane

Daily Warm-Ups

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1. Label the axes.

2. A coordinate is written in the form: ( _____ , _____ )

3. Graph the following coordinates on the plane above:

A. (-6, 2) B. (0, 8) C. (-10, 5) D. (3, -7)

E. (9, 3) F. (4, 0) G. (9, -9) H. (-2, -8)

4. Label the quadrants.

5. Label the origin - the ordered pair is ( _____ , _____ )

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

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What are the new coordinates when these points are reflected over the x-axis?

(-4, 3) and (-5, 3)

(2, 9) and (2, 5)

What are the new coordinates when these points are reflected over the y-axis?

(-4, 3) and (-5, 3)

(2, 9) and (2, 5)

Describe why we use absolute value to find the distance between two points. (Focus on the concept of ‘distance’ not being negative)

What is the distance between (3, −51

2) and (3, 2

1

4)?

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Components of the Coordinate Plane

To describe locations of points in the coordinate plane, we use __________________________

of numbers.

The first number of an ordered pair is called the _____________________________________.

The second number of an ordered pair is called the ___________________________________.

Order is important, so on the coordinate plane, we use the form ( ).

The first coordinate represents the point’s location from zero on the -axis.

The second coordinate represents the point’s location from zero on the -axis.

All points on the coordinate plane are described with reference to the origin. What is the origin,

and what are its coordinates?

Exercises 1–3

1. Use the coordinate plane to answer

parts (a)–(c).

a. Graph three points on the 𝑥-axis,

and label their coordinates.

b. What do the coordinates of your

points have in common?

c. What must be true about any

point that lies on the 𝑥-axis? Explain.

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2. Use the coordinate plane to answer

parts (a)–(c).

a. Graph three points on the 𝑦-axis,

and label their coordinates.

b. What do the coordinates of your

points have in common?

c. What must be true about any point

that lies on the 𝑦-axis? Explain.

3. If the origin is the only point with 0 for both coordinates, what must be true about the origin?

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Quadrants of the Coordinate Plane

Exercises 4–6

4. Locate and label each point described by the ordered pairs below. Indicate which of the

quadrants the points lie in.

A(7, 2)

B(3, −4)

C(1, −5)

D(−3, 8)

E(−2, −1)

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5. Write the coordinates of another point in each of the four quadrants.

a. Quadrant I

b. Quadrant II

c. Quadrant III

d. Quadrant IV

6. Do you see any similarities in the points within each quadrant? Explain your reasoning.

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1. Name the quadrant in which each of the points lies. If the point does not lie in a quadrant,

specify which axis the point lies on.

a. (−2, 5)

b. (8, −4)

c. (−1, −8)

d. (9.2, 7)

e. (0, −4)

2. Jackie claims that points with the same 𝑥- and 𝑦-coordinates must lie in Quadrant I or

Quadrant III. Do you agree or disagree? Explain your answer.

3. Locate and label each set of points on the coordinate plane. Describe similarities of the

ordered pairs in each set, and describe the

points on the plane.

𝑎. {(−2, 5), (−2, 2), (−2, 7), (−2, −3), (−2, −0.8)}

𝑏. {(−9, 9), (−4, 4), (−2, 2), (1, −1), (3, −3), (0, 0)}

𝑐. {(−7, −8), (5, −8), (0, −8), (10, −8), (−3, −8)}

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1. Label the second quadrant on the

coordinate plane, and then answer the

following questions:

a. Write the coordinates of one

point that lies in the second

quadrant of the coordinate plane.

b. What must be true about the

coordinates of any point that lies

in the second quadrant?

2. Label the third quadrant on the

coordinate plane, and then answer the

following questions:

a. Write the coordinates of one point that lies in the third quadrant of the coordinate

plane.

b. What must be true about the coordinates of any point that lies in the third quadrant?

3. An ordered pair has coordinates that have the same sign. In which quadrant(s) could the

point lie? Explain.

4. Another ordered pair has coordinates that are opposites. In which quadrant(s) could the point

lie? Explain.

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Reflecting a Point over x- or y-axis

Give an example of two opposite numbers, and describe where the numbers lie on the number

line. How are opposite numbers similar, and how are they different?

Extending Opposite Numbers to the Coordinates of

Points on the Coordinate Plane

Locate and label your points on the coordinate plane to the right. For each

given pair of points in the table below, record your observations and

conjectures in the appropriate cell. Pay attention to the absolute values of the

coordinates and where the points lie in reference to each axis.

(𝟑, 𝟒) and (−𝟑, 𝟒) (𝟑, 𝟒) and (𝟑, −𝟒) (𝟑, 𝟒) and (−𝟑, −𝟒)

Similarities of

Coordinates

Differences of

Coordinates

Similarities in

Location

Differences in

Location

Relationship Between

Coordinates and

Location on the Plane

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Exercises

In each column, write the coordinates of the points that are related to the given point by the

criteria listed in the first column of the table. Point 𝑆(5, 3) has been reflected over the 𝑥- and 𝑦-

axes for you as a guide, and its images are shown on the coordinate plane. Use the coordinate

grid to help you locate each point and its corresponding coordinates.

Given Point: 𝑺(𝟓, 𝟑) (−𝟐, 𝟒) (𝟑, −𝟐) (−𝟏, −𝟓)

The given point is

reflected across

the 𝑥-axis.

The given point is

reflected across

the 𝑦-axis.

The given point is

reflected first

across the 𝑥-axis

and then across

the 𝑦-axis.

The given point is

reflected first

across the 𝑦-axis

and then across

the 𝑥-axis.

1. When the coordinates of two points are (𝑥, 𝑦) and (−𝑥, 𝑦), which axis did you reflect over?

Explain.

2. When the coordinates of two points are (𝑥, 𝑦) and (𝑥, −𝑦), which axis did you reflect over?

Explain.

S

M

A

𝒙

A

𝒚

𝒙

𝒚

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1. Locate a point in Quadrant IV of the coordinate plane. Label the point 𝐴, and write its ordered pair next

to it.

a. Reflect point 𝐴 over an axis so that its

image is in Quadrant III. Label the

image 𝐵, and write its ordered pair

next to it. Which axis did you reflect

over? What is the only difference in

the ordered pairs of points 𝐴 and 𝐵?

b. Reflect point 𝐵 over an axis so that its

image is in Quadrant II. Label the

image 𝐶, and write its ordered pair

next to it. Which axis did you reflect

over? What is the only difference in

the ordered pairs of points 𝐵 and 𝐶?

How does the ordered pair of point 𝐶

relate to the ordered pair of point 𝐴?

c. Reflect point 𝐶 over an axis so that its

image is in Quadrant I. Label the

image 𝐷, and write its ordered pair next to it. Which axis did you reflect over? How does the

ordered pair for point 𝐷 compare to the ordered pair for point 𝐶? How does the ordered pair for

point 𝐷 compare to points 𝐴 and 𝐵?

2. If (8, -5) was reflected over both axes, what is the new ordered pair? ____________________________

3. What is the rule for crossing over both axes? _______________________________________________

4. Graph the following coordinates and

connect each point:

A (-3, 2)

B (-6, 2)

C (-6,-2)

D (-3,-2)

5. Describe the figure shown.

6. Reflect the object across the y-axis.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

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Distance between Points

Consider the points (−4, 0) and (5, 0).

What do the ordered pairs have in common, and what does that mean about their location in the

coordinate plane?

How did we find the distance between two numbers on the number line?

Use the same method to find the distance between (−4, 0) and (5, 0).

Consider the line segment with end points (0, −6) and (0, −11).

What do the ordered pairs of the end points have in common, and what does that mean about

their location in the coordinate plane?

Find the distance between between (0, −6) and (0, −11).

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POINTS IN SAME QUADRANT:

____________________________________________________________

____________________________________________________________

(5, 3) and (5, 8) Find the Distance

Between the

Points

(-2, 3) and (-8,3)

38)

(-4, -6) and (-1, -6) (5, -10) and (5, -2)

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POINTS IN DIFFERENT QUADRANTS:

____________________________________________________________

____________________________________________________________

Find the Distance

Between the

Points

(8, -6) and (8, 6) (-2, 10) and (5, 10)

(-3, 8) and (-3, -3) (4, -5) and (-3, -5)

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Think…same or different quadrants…so what should I do?

Find the Distance

Between the

Points

(-3, 4) and (-3, 8) (5, -2) and (-6, -2)

(9, -4) and (3, -4) (2, -7) and (-2, -7)

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Distance Between Points

Use the graph below to help solve the following problems.

Find the distance between the following

points:

1. (4, 5) and (4, -8)

2. (10, -7) and (10, 3)

3. (-9, 6) and (4, 6)

4. (-2, 5) and (-3, 5)

Find the distance without using the

graph.

1. (9, 5) and (9, -2)

2. (-6, 3) and (-7, 3)

3. (8, −41

4) and (8, 3

1

2)

4. (82

3, 4) and (-6

1

4, 4)

5. Tammy started at home at (4, 5) and then went to the store at (4, 2). She decided to then stop for gas

at (4, -3) and then to pick up her printed photos at (4, -5). She then went home. What was Tammy’s

total distance?

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Unit 6 Study Guide

Use the coordinate plane on the left for part I and II.

For part I, name each graphed point

and tell what quadrant the point is

in.

1. A ( , ) is in Quadrant _____

2. B ( , ) is in Quadrant _____

3. C ( , ) is in Quadrant _____

4. D ( , ) is in Quadrant _____

5. E ( , ) is in Quadrant _____

For part II, answer each question

about coordinate graphing.

6. What is the origin?

7. On the graph above right, label the x-axis and the y-axis.

8. For the star, reflect it across the x-axis. Where is it now? ( , )

9. For the lightning bolt, reflect it across the y-axis. Where is it now? ( , )

10. Find the distance between points E and F. _________

11. Find the distance between points A and J. _________

12. How can you remember the order of the quadrants? _________________

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For part III, answer each question about coordinate graphing without a plane.

13. Starting at (0,0) if you were to go 5 units left and 10 units down what coordinates would

you end up at? What quadrant would you be in?

14. Starting at (0,0) if you were to go 3 units right and 1 unit up what coordinates would you

end up at? What quadrant would you be in?

15. If two points are in a line and are in the same quadrant, just ___________ the absolute

values of the coordinates that are not alike.

16. If two points are in a line and are in different quadrants, just __________ the absolute

values of the coordinates that are not alike.

17. Find the distance between points (7 , -5) and (7 , 4). _________

18. Find the distance between points (1 , 10) and (8 , 10). _________