G06.U07 FAL
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G06.U07
Integers on the Coordinate Plane
Unit 07 – Rational Explorations, Numbers and Their Opposites
Graphing in the Coordinate Plane
GEORGIA MDC-CREATED FORMATIVE ASSESSMENT LESSON.
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G06.U07 FAL
Georgia Department of Education
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INTRODUCTION TO THIS FORMATIVE ASSESSMENT LESSON
MATHEMATICAL GOALS What mathematical knowledge will your students have addressed at the conclusion of this FAL?
This lesson unit is intended to help you assess how well students are able to plot points in the coordinate plane and
apply transformations across the x-axis and the y-axis. In particular, the lesson will help you identify and help
students with the following:
 Plot points in the four quadrants of the coordinate plane
 Work with reflections in the coordinate plane
GEORGIA STANDARDS OF EXCELLENCE
This lesson involves mathematical content in the standards from across the grades, with emphasis on:
 MGSE6.NS.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the
coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are
related by reflections across one or both axes.
 MGSE6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line
diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
This lesson emphasizes the following standards for mathematical practice:
 Reason abstractly and quantitatively
 Look for and make use of structure
 Look for/express regularity in repeated reasoning.
INTRODUCTION This is a brief layout of this FAL’s flow.
This lesson is structured in the following way:
Before the Lesson:
Students work individually on a pre-assessment task that is designed to reveal their current understandings and
difficulties. The pre-assessment should be given in class a day or more before the formative assessment lesson. The
teacher then reviews and assesses student work noting the specific difficulties students have with the content. The
teacher creates guiding questions for students to answer in order to improve their solutions.
At the Start of the Lesson:
Students will work independently on a “Collaborative Discussion” activity at the beginning of the lesson.
During the Lesson:
The teacher leads a whole-class interactive introduction of the content. The lesson should include aspects from the
pre-assessment task that build on and/or re-teach concepts needed for student understanding.
After the Whole-Group Class Discussion:
After a whole-class interactive introduction, students work in pairs on a collaborative activity. Students are paired
based on their misunderstandings or understandings for the pre-assessment task. The teacher moves around the
classroom asking guiding questions to help students understand the concepts and help to clarify misunderstandings
that were revealed from the pre-assessment task.
MATERIALS REQUIRED What kind of materials will you need to gather?
Your “shopping list” is below.
Each individual student will need:
 Two copies of the pre-assessment task, Graphing in the Coordinate Plane
 a ruler
 pencil and eraser
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G06.U07 FAL

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coordinate graph paper (optional)
TEACHER PREP REQUIRED What kind of prep items are there for this FAL?
Teacher, be advised that prior to the lesson, the following preparations/copies will need to be made:
 2 copies of the pre-assessment
 Teacher will need to assess and group students according to misunderstandings.
 Copy of the Collaborative Activity for each student
TIME NEEDED: How much time should you allot to this FAL?
For PreAssessment:
15 minutes
For
Lesson:
How many days?
40-50 minutes
For
Post:
15 minutes
FRAMING FOR THE TEACHER:
Why is this FAL important? When (sequentially) would you give it, and what can you
expect? A brief discussion about common misconceptions here, what to look for, and what this FAL aims to do.
The purpose of this FAL is to help the teacher identify and correct any misconceptions that students have when
reflecting points across the x-axis, the y-axis or both axes. With the pre-assessment, students are asked about these
reflections three different ways, which should provide insight into any misunderstandings.
FRAMING FOR THE STUDENTS:
This is what to say to the STUDENTS at the beginning of this lesson. It helps them to know
why they are doing this, and what they can expect of this FAL experience. Word-for-word is not crucial; but the overall message, “This
is formative”; “I’m checking your understanding”; and “You will have a chance to prove yourself” are important ideas for students to
understand.
Say to the students:
This activity will take about 1 ½ days for us to complete. The reason we are doing this is to be sure you understand
that when ordered pairs differ only by signs, the locations of the ordered pairs are reflections of one another across
one or both axes.
You will have a chance to work with a partner to help correct any misconceptions that you may have with reflecting
points in the coordinate plane.
After your partner work, you will have a chance to show me how much you have learned!
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PRE-ASSESSMENT BEFORE THE LESSON
Name of Assessment Task: Graphing in the Coordinate Plane
Time This Should Take: 15-20 minutes
Have the students do this task in class or for homework, a day or more before the formative assessment lesson.
This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it.
You will them be able to target your help more effectively in the follow-up lesson.
Give each student a copy of the Pre-Assessment task:
Graphing in the Coordinate Plane
Briefly introduce the task and help the class to understand the
problem and its context.
Spend 15 minutes working individually on this task. Read
through the task and try to answer it as carefully as you
can. Show all your work so that I can understand your
reasoning. Don’t worry if you can’t complete everything.
There will be a lesson that should help you understand
these concepts better. Your goal is to be able to
confidently answer questions similar to these by the end
of the next lesson.
Students should do their best to answer these questions, without teacher assistance. It is important that students are
allowed to answer the questions on their own so that the results show what students truly do not understand.
Students should not worry too much if they do not understand or do everything on the pre-assessment, because in
the next lesson they will engage in a task which is designed to help them. Explain to students that by the end of the
next lesson, they should expect to be able to answer questions such as these confidently.
This is their goal.
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COLLABORATION TIME/READING STUDENTS RESPONSES
You Will Not “Grade” These!
Collect students’ :responses to the task. It is helpful to read students’ responses with colleagues who are also
analyzing student work. Make notes (on your own paper, not on their pre-assessment) about what their work
reveals about their current levels of understanding, and their approaches to the task. You will find that the
misconceptions reveal themselves and often take similar paths from one student to another, and even from one
teacher to another. Some misconceptions seem to arise very organically in students’ thinking. Pair students in the
same classes with other students who have similar misconceptions. This will help you to address the issues in fewer
steps, since they’ll be together. (Note: pairs are better than larger groups for FAL’s because both must participate
in order to discuss!)
You will begin to construct Socrates-style questions to try and elicit understanding from students. We suggest you
write a list of your own questions, however some guiding questions and prompts are also listed below as a jumpingoff point.
GUIDING QUESTIONS Here are some questions that you may find useful as you address misconceptions in class:
COMMON ISSUES
SUGGESTED QUESTIONS AND PROMPTS
Student has difficulty getting started


Student incorrectly graphs the data.
For example, student reverses x and y coordinates.


Student has difficulty with x and y axes.
For example, student reverses the axes.


Student adds little or no explanations as to why
answers are formed.


Choose one point on the graph to try reflecting
across the y-axis.
Can you identify the y-axis?
Can you explain what the ordered pair (2, 3)
means?
Think of this point (3, 5). Give me “driving”
instructions from the origin.
Can you point out the x-axis? (or y-axis)
Teacher could use transparency or wax paper. The
points can be plotted on the transparency or wax
paper. The student would ‘fold’ the
paper/transparency along the x- or y- axis to
determine which axis the point reflects across.
How would you explain this to someone unfamiliar
with these questions?
What does opposite mean?
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LESSON DAY
SUGGESTED LESSON OUTLINE:
Part 1: Whole-Class Introduction:
Time to Allot: ( 10-12 minutes)
Independent Work: (5-8 minutes)

Give each student a sheet of coordinate graph paper.
Have them plot the following points:
(6, 2); (2, 5); (0, 2); (-2, 5); (-6, 2); (-3, -3) and (3, -3)
Have them label the following points with the designated
letters:
A = (6, 2); B = (2, 5); C = (0, 2); D = (-2, 5); E = (-6, 2);
F = (-3, -3) and G = (3, -3)









Lesson Warm-up
P lot the following points on your coordinate plane.
Label each ordered pair with the given letter below.
 A = (6, 2)
 E = (-6, 2)
 B = (2, 5)
 F = (-3, -3)
 C = (0, 2)
 G = (3, -3)
 D = (-2, 5)
D o you notice anything special about the ordered
pairs? If yes, what do you notice? Be ready to discuss.
Students should observe plotted ordered pairs and be able
to discuss what they notice about the graphed ordered
pairs. Allow students to share with a partner.
Whole-Class Discussion:
Say, Using the coordinate graph, let’s try the following…
Darken every ordered pair you graphed.
Darken the x- and y- axis on the coordinate graph.
Fold your paper along the x-axis. (Fold the paper with the images on the inside of the paper.)
Hold your paper up to the light.
What points appear to ‘match-up’ when you fold along the x-axis? Make a list of these points.
Fold your paper along the y-axis. (Fold the paper with the images on the inside of the paper.)
Hold your paper up to the light.
What points appear to ‘match-up’ when you fold along the x-axis? Make a list of these points.
Part 2: Collaborative Activity: Secret Message
Put students into their pairs according to your analysis of student errors.
Do/Say the Following:
 Distribute the Secret Message activity to each pair of
students.
 Students should work with a partner to write coordinate
pairs that represent each situation. After graphing each
coordinate pair, students should use the Secret Decoder at
the top of the page to locate the corresponding letter.
 Students will find the word that matches each graph.
 As students work through the decoding puzzles, have them
explain strategies used to each other. (Teacher should
make note of any unusual or different strategies used by
the students to help with the plenary discussion.)
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Time to Allot: (40-45
minutes)
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During the Collaborative Activity, the teacher has 3 tasks:
1. Circulate to students’ whose errors you noted from the pre-assessment and support their reasoning with
your guiding questions.
2. Circulate to other students also to support their reason in the same way.
3. Make a note of student approaches for the summary (plenary discussion). Some students have
interesting and novel solutions!
Part 3: Plenary (Summary) Discussion: Secret Messages
Time to Allot: (15-20
minutes)
Gather students together, share solutions. Discussion prompts should be made up of your original guiding
questions and notes about student approaches. Some other discussion prompts are listed below:
NOTE: “Scribing” helps to increase student buy-in and participation. When a student answers your question,
write the student’s name on the board and scribe his/her response quickly. You will find that students volunteer
more often when they know you will scribe their responses – this practice will keep the discussions lively and
active!
Do/Say the Following:


Have students share their secret for being able to decode the puzzles. Be sure to call on students
who had unique approaches to the collaborative activity or unusual approaches to solving the
decoding problems. Encourage sharing a variety of strategies used and explanations of the
reflections.
Allow students to listen to each other to determine “better” or different methods used for
accomplishing the same task. Allow them to evaluated each other’s approaches and determine
their validity.
Part 4: Improving Solutions to the Assessment Task:
Graphing in the Coordinate Plane
Time to Allot: (10-15 minutes)
Do/Say the Following:



Display questions and prompts for each error type.
Return students’ work on the assessment task and a new copy of the task sheet. Students need
to read, review, and correct all assessment questions. (Some teachers prefer not re-distributing
the original assessment because students tend to interpret “no x’s to show incorrect answers” as
meaning that they did everything correctly the first time.)
Note: If you are running short on time you may do the post-assessment the following day.
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PRE/POST-ASSESSMENT (Answer Key)
ASSESSMENT TASK:
Graphing in the Coordinate Plane
1. Betsy wants to know what will happen to the pentagon if she changes each x-coordinate to its opposite.
Draw the new pentagon below on the graph next to it. Record the ordered pairs on the table.
x
8
-1
-7
0
8
y
1
6
3
-5
-1
x
-8
1
7
0
-8
y
1
6
3
-5
-1
Explain how you knew where the new vertices should be. The y-coordinate remains the same, but the sign of the
x-coordinate changes, unless the x-coordinate is 0. (Then it remains 0.)
2. For the following points, write the coordinates of the “new” point after it has been:
a. (2, 8) The new point has the same x- coordinate and the opposite of the y-coordinate: (2,-8)
b. (-3, 5) The new point has the same y- coordinate and the opposite of the x-coordinate: (3,5)
c. (-5, 0) The new point is the opposite of each coordinate: (5,0)
3. You are recording your savings account balance as an ordered pair using the x-coordinate to represent
years starting when you were born and the y-coordinate to represent your savings in dollars on each birthday.
You see that you wrote (11, 26) on your last birthday.
What does that represent? On your 11th birthday you had $26 in savings.
Can either coordinate ever be negative? Yes, the y-coordinate. Explain. If you overdraw at the bank you
will have a negative balance. The x-coordinate cannot be negative because you cannot go back in time
before you were born.
4. Evil Dr. Pain trapped you in an alternate universe where locations are represented by ordered pairs
related to your school. If John lives at (2, 4), that means that he lives 2 miles east and 4 miles north of the
school. What would be the location for Charlotte if she lives at (-3,-1)? Charlotte would live 3 miles west and
1 mile south of the school.
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Collaborative Activity (Answer Key)
Secret Message
Using the Secret Decoder, discover the secret message below.
1. Read each description below of Sam’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each
point will match a letter in the secret decoder.
Description
DAY
(X)
MOOD
(y)
4 days ago Sam was
feeling OK.
-4
1
The next day Sam
was feeling NOT
great.
Today Sam is the
opposite of NOT
great.
On Day 2 he will be
NOT OK.
-3
-3
0
3
2
-1
1a. Write the letters you found: __________
1b. Unscramble the letters to make a word:
____________________________________LOVE________________________________
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2. Read each description below of Terri’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY
(X)
MOOD
(y)
4 days ago Terri came
in to class feeling OK.
-4
1
On Day 3 Terri will feel
great.
3
3
On Day 4 Terri will
come to class feeling
OK.
When Terri leaves class
on Day 4 she will be
NOT great.
4
1
4
-3
2a. Write the letters you found: __________
2b. Unscramble the letters to make a word: ___MORE_
3. Read each description below of Mark’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY
(X)
MOOD
(y)
Four days ago Mark came
to class feeling great.
-4
3
When Mark left class he
was the opposite of how he
entered.
Yesterday Mark was not
feeling OK.
-4
-3
-1
-1
Today Mark came in the
same as he left 4 days ago.
0
-3
When Mark left class today
he was BLAH.
0
0
3a. Write the letters you found: __________
3b. Unscramble the letters to make a word: _SNACK
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4. Read each description below of Madison’s moods. Write an ordered pair that represents each day as the x-coordi
each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point will match a letter in the
decoder.
Description
DAY
(X)
MOOD
(y)
4 days ago Madison was
feeling OK.
-4
1
On Day 2 Madison will be
the same as 4 days ago.
2
1
On Day 3 Madison will be
the opposite of great.
3
-3
On Day 4 Madison will be
the same as the day before.
4
-3
4a. Write the letters you found: __________
4b. Unscramble the letters to make a word: __TIME__
5. What is your final message? __________LOVE MORE SNACK TIME_____________
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COLLABORATIVE ACTIVITY
Secret Message
Using the Secret Decoder, discover the secret message below.
1. Read each description below of Sam’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY (X)
MOOD (y)
4 days ago Sam was feeling OK.
The next day Sam was feeling NOT great.
Today Sam is the opposite of NOT great.
On Day 2 he will be NOT OK.
Write the letters you found: __________ Unscramble the letters to make a word: __________
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2. Read each description below of Terri’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY (X)
MOOD (y)
4 days ago Terri came in to class feeling OK.
On Day 3 Terri will feel great.
On Day 4 Terri will come to class feeling OK.
When Terri leaves class on Day 4 she will be NOT great.
Write the letters you found: __________ Unscramble the letters to make a word: __________
3. Read each description below of Mark’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY
(X)
MOOD
(y)
Four days ago Mark came to class feeling great.
When Mark left class he was the opposite of how he
entered.
Yesterday Mark was not feeling OK.
Today Mark came in the same as he left 4 days ago.
When Mark left class today he was BLAH.
Write the letters you found: __________ Unscramble the letters to make a word: __________
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4. Read each description below of Madison’s moods. Write an ordered pair that represents each day as the xcoordinate and each mood as the y-coordinate. Graph the ordered pairs on the graph provided. Each point
will match a letter in the secret decoder.
Description
DAY (X)
MOOD (y)
4 days ago Madison was feeling OK.
On Day 2 Madison will be the same as 4 days ago.
On Day 3 Madison will be the opposite of great.
On Day 4 Madison will be the same as the day before.
Write the letters you found: __________ Unscramble the letters to make a word: __________
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Pre/Post Assessment Directions
Graphing on the Coordinate Plane Task
•
•
•
•
Spend 15 minutes working individually on this task.
Read through the task and try to answer it as carefully as you can.
Show all your work so that I can understand your reasoning.
Don’t worry if you can’t complete everything. There will be a lesson
that should help you understand these concepts better.
• Your goal is to be able to confidently answer questions similar to
these by the end of the next lesson.
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Lesson Day Collaborative Discussion
Graphing Review & Lesson
Plot the following points on your coordinate plane.
Label each ordered pair with the given letters below.




A = (6, 2)
B = (2, 5)
C = (0, 2)
D = (-2, 5)




E = (-6, 2)
F = (-3, -3)
G = (3, -3)
H = (6, -2)
Do you notice anything special about the ordered pairs?
If yes, what do you notice? Be ready to discuss.
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Whole-Class Lesson
Using the coordinate graphs, let’s try the following:
•
•
•
Darken every ordered pair you graphed.
Darken the x- and y- axis on the coordinate graph.
Fold your paper along the x-axis. (Fold the paper with the images on the inside of
the paper.)
•
•
•
Hold your paper up to the light.
What points appear to ‘match-up’ when you fold along the x-axis?
Make a list of these points.
Fold your paper along the y-axis. (Fold the paper with the images on the inside of
the paper.)
•
•
Hold your paper up to the light.
What points appear to ‘match-up’ when you fold along the x-axis?
Make a list of these points.
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Collaborative Activity Instructions:
Secret Message Activity
• Work with a partner to identify the
coordinate pair that represents each
situation. Graph your coordinate pair.
Using the Secret Decoder at the top of the
page locate the letter that coordinates with
your point.
• Find the word that matches each student by
unscrambling the letters.
• Work through the decoding activity and make sure that you and your partner
understand how each ordered pair changes when reflected.
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Collaborative Plenary Discussion Questions:
1. How were you able to decode the message?
2. What did you notice about ordered pairs when the mood was
the opposite?
3. How are yesterday and tomorrow related on your coordinate
plane?
4. Describe how you can tell an opposite when graphing.
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