6.NS.C.6: Rational Numbers on the Number Line and the Coordinate Plane
In plain English: 6.NS.C.6 is the Common Core grade 6 math standard that asks students to see every rational number, including negative numbers, as a point on a number line. Students place numbers on horizontal and vertical number lines, find opposites (so -(-3) = 3), use the signs of an ordered pair to name its quadrant, and see that pairs differing only in sign are reflections across the axes.
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
a.Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
b.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.
c.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.
Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Apply and extend previous understandings of numbers to the system of rational numbers. Also written as 6.NS.6 · Official standard
Students extend the number lines and first-quadrant graphs they know from grade 5 to include negative numbers. A rational number is a number that can be written as a fraction of two integers, such as 3, -2, 1/2, -3/4 or -2.5. (Integers are the whole numbers and their opposites: ..., -2, -1, 0, 1, 2, ....) Every rational number is a point on the number line. Students place integers, fractions and decimals on horizontal and vertical number lines, and learn that the opposite of a number is the number the same distance from 0 on the other side. The opposite of 3 is -3, the opposite of -3 is 3, so -(-3) = 3, and 0 is its own opposite.
Next, students extend both axes of the coordinate plane (the grid formed by a horizontal x-axis and a vertical y-axis that cross at 0) past 0. The axes split the plane into four quadrants, numbered I to IV counterclockwise, starting at the top right. The signs of an ordered pair (x, y), the two coordinates that locate a point, tell which quadrant a point is in. When two ordered pairs differ only by signs, such as (4, 2) and (-4, 2), the points are reflections (mirror images) of each other across one or both axes. Comparing and ordering numbers (6.NS.C.7) and distances between points (6.NS.C.8) come in the next lessons.
Learning Objectives
By the end of this lesson, students will be able to:
Place integers, fractions and decimals, positive and negative, on horizontal and vertical number lines
Explain that a number and its opposite are on opposite sides of 0, that -(-a) = a, and that 0 is its own opposite
Plot and name ordered pairs of integers and other rational numbers in all four quadrants
Use the signs of an ordered pair to name its quadrant
Recognize that ordered pairs that differ only by signs are reflections across one axis or both axes
Prior Knowledge Required
Students should already be comfortable with:
Placing fractions on a number line 3.NF.A.2
Reading and writing decimals to hundredths 4.NF.C.6
Plotting points in the first quadrant 5.G.A.15.G.A.2
Using positive and negative numbers for temperature, elevation and money 6.NS.C.5
Draw a number line from 0 to 6 on the board, the kind students used in grade 5, and ask:
Warm-Up Prompt
"This number line starts at 0. A thermometer in the morning read 2 degrees below zero. How could we change the number line to show that temperature? Where would 2 degrees below zero go, and where would half a degree below zero go?"
Let students suggest extending the line to the left of 0. Mark -1, -2, -3 at the same spacing as 1, 2, 3, so that -2 is as far left of 0 as 2 is right of it. Then place -1/2 halfway between 0 and -1. Point out that the new marks follow the same rule as the old ones: equal spaces for equal amounts. Tell students that every fraction and decimal, positive or negative, has its own point on this line.
Direct Instruction20 minutes
Part 1: Number lines (standards a and c). Use Diagram 1. On a horizontal number line, positive numbers are to the right of 0 and negative numbers to the left. On a vertical number line, like a thermometer, positive numbers are above 0 and negative numbers below. To place a fraction, split each unit into equal parts and count from 0. Then introduce opposites: 3 and -3 are both 3 units from 0, on opposite sides. The minus sign can be read as "the opposite of", so -(-3) means "the opposite of the opposite of 3", which is 3 again. Ask: what is the opposite of 0? Nothing moves, because 0 is the only point that is 0 units from 0.
Fractions and decimals on a horizontal number line
Place -2 1/4 and 1.5 on a horizontal number line from -3 to 3 marked in fourths.
Equation: 1.5 is 6 fourths to the right of 0, halfway between 1 and 2. -2 1/4 is 9 fourths to the left of 0: go to -2, then 1 more fourth to the left
A vertical number line
A vertical number line is marked in thirds. Point W is 4 tick marks above 0, and point Z is 2 tick marks below 0. What numbers are at W and Z?
Equation: W = 4/3 = 1 1/3 and Z = -2/3
Opposites and the opposite of the opposite
Find the opposite of 8, the opposite of -8, the value of -(-8) and the opposite of 0.
Equation: The opposite of 8 is -8, the opposite of -8 is 8, -(-8) = 8, and the opposite of 0 is 0
Signs and quadrants
Name the quadrant or axis of each point: (-3, 5), (2, -4), (-1.5, -2.5) and (0, 6).
Equation: (-3, 5): Quadrant II; (2, -4): Quadrant IV; (-1.5, -2.5): Quadrant III; (0, 6): on the y-axis, in no quadrant
Reflections across one or both axes
Start with P(3, -1). Change the sign of the x-coordinate, then of the y-coordinate, then of both. How is each new point related to P?
Equation: (-3, -1) is P reflected across the y-axis; (3, 1) is P reflected across the x-axis; (-3, 1) is P reflected across both axes
Part 2: The coordinate plane (standards b and c). Extend both axes past 0, as in Diagram 2. The x-coordinate tells how far to move right (positive) or left (negative) from the origin, the point (0, 0). The y-coordinate tells how far to move up (positive) or down (negative). The signs decide the quadrant: Quadrant I is (+, +), II is (-, +), III is (-, -) and IV is (+, -). Points with a 0 coordinate lie on an axis and are in no quadrant. Then show the reflections in Diagram 2: changing only the sign of y flips a point over the x-axis, changing only the sign of x flips it over the y-axis, and changing both flips it over both axes. Fold a sheet of grid paper along an axis to show that the two points land on each other.
Guided Practice15 minutes
Pairs work on four-quadrant grid paper. For each row of the table, they plot the point, write its quadrant, and say which point it reflects. Then they answer a vertical number line question.
Plot K(-2, 3), then fill in each missing point
Point
How to get it from K
Coordinates
Quadrant
K
start
(-2, 3)
II
L
reflect K across the x-axis
?
?
M
reflect K across the y-axis
?
?
N
reflect K across both axes
?
?
Answers: L(-2, -3) in Quadrant III, M(2, 3) in Quadrant I, N(2, -3) in Quadrant IV. Vertical number line question: on a vertical number line marked every 0.25, what numbers are 3 tick marks above 0 and 5 tick marks below 0? (0.75 and -1.25.) Listen for pairs who move up for the x-coordinate: remind them that x always comes first and always means left or right.
Independent Practice10-15 minutes
Students work alone and check with a partner at the end. (1) Name the opposite of -6.5 and the opposite of 0. (6.5 and 0.) (2) Find -(-11). (11.) (3) Name the quadrants of (-9, -2) and (4, -8). (III and IV.) (4) Reflect (-5, 1) across the x-axis. ((-5, -1).) (5) Draw a number line from -1 to 1 marked in tenths and place -0.4 and 0.7. (-0.4 is 4 tenths left of 0 and 0.7 is 7 tenths right of 0.) (6) Plot (0.5, -3.5) on grid paper and name its quadrant. (IV.)
Closure5 minutes
Exit ticket: (1) In which quadrant is (-1.5, 6)? (II.) (2) What is the reflection of (7, -4) across both axes? ((-7, 4).) (3) What is the opposite of the opposite of -13? (-13.) Collect the tickets and sort them into "ready" and "needs another look" for the next lesson on ordering (6.NS.C.7).
Differentiation Strategies
For Struggling Students
Give number lines that are already marked in halves or fourths, so students count parts instead of drawing them
Put a sign card, "x: left or right" and "y: up or down", next to the grid, and have students trace the path from the origin with a finger before plotting
Use a small mirror or a folded sheet of grid paper to check every reflection
For Advanced Students
Ask students to plot a triangle with one vertex in each of three quadrants, then write the vertices of its reflection across both axes
Ask what happens to a point on the x-axis, such as (5, 0), when it is reflected across the x-axis, and why
Challenge students to find the value of -(-(-(-2.5))) and to explain a rule for any number of minus signs
Assessment Guidance
What to Look For
Check that students keep equal spacing on both sides of 0 and count fractional parts from 0, not from the nearest whole number in the wrong direction. On the coordinate plane, watch for x and y reversed: ask students to say "right or left first, then up or down". Students should name quadrants from the signs alone, without plotting, and should explain why a point with a 0 coordinate is on an axis. For reflections, ask which sign changed and which axis the point flipped across.
02
Classroom Activities
3 Activities
1
Masking Tape Number Line and Opposite Partners
15 minWhole class
Tape a number line on the floor from -5 to 5 with marks every half unit. Twelve students each get a number card, stand on their number, and then find their "opposite partner" (standards a and c).
Number Cards (12 cards)
-4 and -(-4)
1/2 and -1/2
3 1/4 and -3 1/4
3.5 and -3.5
2 and -2
0 and "the opposite of 0"
Procedure
Each student with a card finds the matching point and stands on it; the class checks each spot
Students with fractions between marks, such as 3 1/4, explain how they found the spot
Each student finds the partner the same distance from 0 on the other side and holds hands across 0
Turn the line to stand upright on the wall (tape it vertically) and repeat with half the cards
Discussion Questions
Where did the student with -(-4) stand? Why is that the same spot as 4? (-(-4) = 4)
Which two cards stand on the same point? (0 and "the opposite of 0")
How did you place 3 1/4 when the tape only has marks every half unit?
Modification for Distance Learning
Share a slide with a number line from -5 to 5 and 12 movable number boxes. Students drag each box to its point and draw a line connecting opposites.
2
Quadrant Card Sort
15 minGroups of 3-4
Each group sorts 12 ordered pair cards into five piles: Quadrant I, II, III, IV and "on an axis". Then they plot every card on one sheet of four-quadrant grid paper to check (standards b and c).
Ordered Pair Cards (12 cards)
(3, 6) and (1.5, 2.5): Quadrant I
(-2, 7) and (-0.5, 4): Quadrant II
(-5, -5) and (-3.5, -1.5): Quadrant III
(6, -4), (4.5, -6) and (2, -1/2): Quadrant IV
(0, -3), (-7, 0) and (0, 0): on an axis
Procedure
Sort the cards using only the signs, without plotting
Write the sign pattern, such as (-, +), on a sticky note for each quadrant pile
Plot all 12 points on one grid and check the sort; fix any card that landed in the wrong pile
Discussion Questions
How many cards are on an axis, and why are they in no quadrant? (3 cards: each has a 0 coordinate)
Which quadrant pile has the most cards? (Quadrant IV, with 3 cards)
Which point is on both axes at once? (the origin, (0, 0))
Challenge Variation
Each group writes 4 new cards, one for each quadrant, using fractions or decimals for at least two coordinates, and trades them with another group to sort.
3
Fold-and-Check Mirror Flags
20 minPairs
Pairs draw a flag in Quadrant I on four-quadrant grid paper, then reflect it across the y-axis, the x-axis and both axes by changing signs. They fold the paper along the axes to check (standards b and c).
Procedure
Plot the flag's four vertices (corner points) and connect them in order: (1, 1), (1, 5), (3.5, 4) and (1, 3). The pole runs from (1, 1) to (1, 5)
Change the sign of each x-coordinate and draw the new flag: (-1, 1), (-1, 5), (-3.5, 4), (-1, 3). This is the reflection across the y-axis
Change the sign of each y-coordinate of the first flag: (1, -1), (1, -5), (3.5, -4), (1, -3). This is the reflection across the x-axis
Change both signs: (-1, -1), (-1, -5), (-3.5, -4), (-1, -3). This is the reflection across both axes
Fold the paper along the y-axis, then along the x-axis, and hold it to the light: each flag should land exactly on its partner
Discussion Questions
Which quadrant is each of the four flags in? (I, II, IV and III)
The flag's point (3.5, 4) has a decimal coordinate. Did that change how you reflected it?
Which fold matches the first flag with the flag in Quadrant III? (Neither fold alone: you need both folds)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Opposites on a Horizontal Number Line, and a Vertical Number Line
Top: a horizontal number line from -4 to 4, drawn to scale with marks every half unit. 3 and -3 are both 3 units from 0, on opposite sides. Taking the opposite twice returns to the start, so -(-3) = 3, and 0 is its own opposite. Bottom: a vertical number line marked in fourths. 1 1/2 is six fourths above 0, and -1 1/4 is five fourths below 0.
Diagram 2: The Four Quadrants and Reflections
A coordinate plane drawn to scale, with each quadrant labeled by its sign pattern. A(4, 2) is in Quadrant I. B(-4, 2) differs only in the sign of x, so it is A reflected across the y-axis. D(4, -2) differs only in the sign of y, so it is A reflected across the x-axis. C(-4, -2) differs in both signs, so it is A reflected across both axes. E(-2.5, -5) shows that coordinates can be decimals.
04
Homework Assignment
~30 min
6.NS.C.6 Homework: Number Lines, Quadrants and Reflections
Directions: Use a ruler and grid paper. Draw every number line with equal spacing and label 0. For every point on a coordinate plane, write its coordinates and its quadrant.
Part 1: Number Lines (Problems 1-2)
Draw a horizontal number line from -3 to 3 marked in halves. Place -2.5, 1 1/2, -1 1/2, 0 and the opposite of 2. Which two of your points are opposites of each other? Explain how you know from the number line.
Draw a vertical number line from -2 to 2 marked in fourths. Place 1 1/4, -3/4, -(-1/2) and the opposite of 0. Explain where -(-1/2) goes and why.
Part 2: Opposites and Quadrants (Problems 3-4)
Find each value: (a) the opposite of -7; (b) -(-12); (c) the opposite of the opposite of 4.5; (d) the opposite of 0. For part (d), explain your answer using a number line.
Without plotting, name the quadrant or axis of each point: (-6, 1), (2.5, -3), (-4, -1/2), (0, -5) and (7, 7). Then plot all five on one coordinate plane to check. Write a rule that tells the quadrant from the signs.
Part 3: The Coordinate Plane and Reflections (Problems 5-6)
Plot A(2, 5). Then find and plot B, the reflection of A across the x-axis, C, the reflection of A across the y-axis, and D, the reflection of A across both axes. Write the coordinates and the quadrant of each point.
Point P is at (-1.5, 3.5). (a) Plot P and name its quadrant. (b) Point Q is at (1.5, 3.5). Across which axis is Q a reflection of P? (c) Point R is at (1.5, -3.5). How is R related to P? (d) Write the point that is P reflected across the x-axis.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Number Lines
Equal spacing, 0 labeled, every number placed correctly
One number misplaced or uneven spacing
Several numbers misplaced
Opposites
All opposites and -(-a) values correct, with a clear explanation
Values correct but no explanation
Two or more values incorrect
Quadrants
Every quadrant or axis named correctly, with a sign rule
One point misnamed, or no rule
Quadrants named from the wrong signs
Reflections
All reflected points and axes correct
One reflected point or axis incorrect
Reflections missing or x and y swapped
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
On a horizontal number line, each unit from -2 to 0 is split into 4 equal parts. Point K is 3 parts to the left of -1. What number is at K?
Answer: A
Each part is 1/4 of a unit. Starting at -1 and moving 3 fourths to the left gives -1 3/4, or -1.75, which is between -1 and -2. Choice B moves 3 fourths to the right of -1 instead of the left, which gives -1/4. Choice C moves 3 whole units instead of 3 fourths. Choice D counts the 3 parts from 0 instead of from -1, which gives -3/4.
Question 2 of 20 · Multiple Choice
Which statement about 5 and -5 on a number line is true?
Answer: B
5 is 5 units to the right of 0 and -5 is 5 units to the left, so they are opposites: opposite sides, same distance from 0. Choice C uses 10, which is the distance between the two points, not the distance from each point to 0. Choices A and D put them on the same side, but the opposite signs mean opposite sides.
Question 3 of 20 · Multiple Choice
What is the value of -(-9)?
Answer: C
-(-9) means the opposite of -9. The opposite of -9 is the number the same distance from 0 on the other side, which is 9. Choice A ignores the outer minus sign. Choices B and D confuse the opposite of a number with its reciprocal (1 divided by the number).
Question 4 of 20 · Multiple Choice
Which number is its own opposite?
Answer: D
0 is 0 units from 0, so the point on the other side at the same distance is 0 itself. Choices A and B are opposites of each other: the opposite of 1 is -1, a different number. Choice C has the opposite -1/2, which is also a different point.
Question 5 of 20 · Multiple Choice
A point has a negative x-coordinate and a positive y-coordinate. In which quadrant is the point?
Answer: B
A negative x means left of the y-axis, and a positive y means above the x-axis. Left and up is Quadrant II, with signs (-, +). Choice D swaps the roles of x and y: (+, -) is Quadrant IV. Choice C would need both coordinates negative. Choice A would need both positive.
Question 6 of 20 · Multiple Choice
In which quadrant is the point (2.5, -4.5)?
Answer: D
The x-coordinate 2.5 is positive (right) and the y-coordinate -4.5 is negative (down). Right and down is Quadrant IV, (+, -). Choice B reads the pair as (y, x). Choice A ignores the minus sign. Choice C treats both coordinates as negative.
Question 7 of 20 · Multiple Choice
The point (6, -2) is reflected across the x-axis. What are the coordinates of the new point?
Answer: A
A reflection across the x-axis flips the point up or down, so only the sign of the y-coordinate changes: (6, -2) becomes (6, 2). Choice B changes the x sign, which is a reflection across the y-axis. Choice C changes both signs, a reflection across both axes. Choice D swaps the two coordinates, which is not a reflection across either axis.
Question 8 of 20 · Multiple Choice
How are the points (-3, 7) and (3, -7) related?
Answer: C
Both coordinates change sign, so the points are reflections across both axes: one in Quadrant II and the other in Quadrant IV. Choice A would change only the y sign, giving (-3, -7). Choice B would change only the x sign, giving (3, 7). Choice D is false: different signs mean different points.
Question 9 of 20 · Multiple Choice
Which point is the reflection of (-4.5, 1) across the y-axis?
Answer: D
A reflection across the y-axis flips the point left or right, so only the x sign changes: (-4.5, 1) becomes (4.5, 1). Choice A changes both signs. Choice B changes only the y sign, which is the reflection across the x-axis. Choice C only swaps the two coordinates, writing (y, x).
Question 10 of 20 · Multiple Choice
A vertical number line has tick marks every 0.5 unit. Point M is 3 tick marks below 0. What number is at M?
Answer: C
Three tick marks of 0.5 each is 1.5 units, and below 0 is negative, so M is at -1.5. Choice A drops the sign and puts M above 0. Choice B counts tick marks as whole units. Choice D writes the 3 tick marks as 3 tenths.
Question 11 of 20 · Multiple Choice
Which ordered pair names the point 4 units to the left of the origin and 3 units up?
Answer: B
Left is the negative x direction and up is the positive y direction, so the point is (-4, 3), in Quadrant II. Choice A treats left as positive. Choice D swaps the numbers: it is 3 left and 4 up. Choice C writes the y-coordinate first, as (y, x).
Question 12 of 20 · Multiple Choice
Start at the origin. Move 1/2 unit to the right and then 2 1/2 units down. Which ordered pair names the point?
Answer: A
Right 1/2 gives x = 1/2, and down 2 1/2 gives y = -2 1/2, so the point is (1/2, -2 1/2) in Quadrant IV. Choice B reverses both directions. Choice D keeps the signs in place but swaps the numbers 1/2 and 2 1/2. Choice C writes the y-coordinate first, as (y, x).
Question 13 of 20 · Multiple Choice
Point P is on a number line halfway between -5 and -4. Which number is at P?
Answer: D
The point halfway between -5 and -4 is half a unit to the left of -4, which is -4.5. Choice A drops the sign and gives the point halfway between 4 and 5. Choice B moves half a unit to the left of -5, outside the two numbers. Choice C takes half of the 1-unit distance between -5 and -4, which is a length, not the point halfway between them.
Question 14 of 20 · Multiple Choice
On a horizontal number line, point A is at 2 3/4. Where is the opposite of A?
Answer: C
The opposite of 2 3/4 is -2 3/4, which is the same distance from 0, 2 3/4 units, on the other side: to the left. Choice A is point A itself. Choice B keeps only the fraction part and drops the 2. Choice D measures from A instead of from 0, which lands on 0.
Question 15 of 20 · Short Answer
Find each value and explain: (a) -(-15); (b) -(-(-2)); (c) the opposite of 0.
(a) 15: the opposite of -15 is 15. (b) -2: -(-2) = 2, and the opposite of 2 is -2. Each minus sign flips the number to the other side of 0, so three flips end on the negative side. (c) 0: 0 is 0 units from 0, so it is its own opposite.
Question 16 of 20 · Short Answer
Name the quadrant or axis for each point: (-8, -3), (0, 4), (5, -1) and (-2, 9).
(-8, -3): Quadrant III (-, -). (0, 4): on the y-axis, in no quadrant, because x = 0. (5, -1): Quadrant IV (+, -). (-2, 9): Quadrant II (-, +).
Question 17 of 20 · Short Answer
Point G is at (2.5, -1.5). Give the coordinates of its reflection across the y-axis, its reflection across the x-axis, and its reflection across both axes. Name the quadrant of each.
Across the y-axis: (-2.5, -1.5), Quadrant III. Across the x-axis: (2.5, 1.5), Quadrant I. Across both axes: (-2.5, 1.5), Quadrant II. G itself is in Quadrant IV. Each reflection changes the sign of the coordinate that crosses the axis.
Question 18 of 20 · Short Answer
Draw a vertical number line from -3 to 3 marked in halves. Mark -2.5, 0.5 and the opposite of 0.5. Which two of your points are opposites?
-2.5 is 5 halves below 0, 0.5 is 1 half above 0, and the opposite of 0.5 is -0.5, 1 half below 0. The opposites are 0.5 and -0.5: they are the same distance from 0 on opposite sides. -2.5 has its opposite, 2.5, which was not asked for.
Question 19 of 20 · Short Answer
Explain how to find the point for -7/3 on a horizontal number line. Then explain how to find 3/4.
-7/3 = -2 1/3. Split each unit into thirds, start at 0 and count 7 thirds to the left: the point is 1 third to the left of -2, between -2 and -3. For 3/4, split each unit into fourths and count 3 fourths to the right of 0. Both are rational numbers, so each has exactly one point on the line.
Question 20 of 20 · Short Answer
Mia says that (5, -3) and (-5, 3) are reflections of each other across the y-axis. Is she right? Explain.
No. Both signs change, so the points are reflections across both axes: (5, -3) is in Quadrant IV and (-5, 3) is in Quadrant II. The reflection of (5, -3) across the y-axis changes only the x sign: (-5, -3), in Quadrant III.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.NS.C.6 mean?
6.NS.C.6 means students treat every rational number, including negatives, as a point on the number line and in the coordinate plane. It has three parts: (a) opposites, including -(-3) = 3 and 0 as its own opposite, (b) signs of ordered pairs and quadrants, with reflections across the axes, and (c) placing numbers and points on number lines and the coordinate plane.
Is 6.NS.C.6 taught before or after 6.NS.C.5?
After. 6.NS.C.5 introduces positive and negative numbers in real situations, such as temperature and elevation. 6.NS.C.6 puts those numbers on number lines and extends the coordinate plane to four quadrants. Next, 6.NS.C.7 orders numbers and introduces absolute value, and 6.NS.C.8 uses the four quadrants to solve problems.
What is a rational number, in simple words?
A rational number is any number that can be written as a fraction of two integers, with a denominator that is not 0. Whole numbers, fractions, terminating decimals (decimals that end) such as 0.25, and their negatives are all rational: -7 = -7/1 and -2.5 = -5/2. In grade 6, students place these on the number line.
Why is -(-3) equal to 3?
Because the minus sign can mean "the opposite of". The opposite of 3 is -3, 3 units to the left of 0. The opposite of -3 flips back to the right side, to 3. So -(-3), the opposite of the opposite of 3, is 3. The same flip works for any number, including fractions and decimals.
How do students remember the signs in each quadrant?
Think of the path from the origin: x first, right (+) or left (-), then y, up (+) or down (-). Quadrant I is right and up, and the quadrants are numbered counterclockwise from there. Students who sketch a small plus-and-minus chart at the top of their paper rarely mix them up.
What does it mean that two points are reflections across an axis?
It means one point is the mirror image of the other, with the axis as the mirror. If you fold the grid along the axis, the two points land on each other. In coordinates, a reflection across one axis changes the sign of one coordinate, and a reflection across both axes changes both signs.
What are common mistakes with 6.NS.C.6?
A common mistake is reversing x and y when plotting. Another is placing a negative mixed number such as -1 1/2 on the wrong side of -1, by counting the fraction toward 0 instead of away from it. Students also sometimes put points with a 0 coordinate, such as (0, 7), in a quadrant, when they are on an axis.
Is 0 a positive or a negative number?
Neither. Positive numbers are to the right of 0 (or above it) and negative numbers are to the left (or below). 0 is the point that separates them. That is also why a point with a 0 coordinate lies on an axis, which is the boundary between two quadrants.
How do you plot a point with fractions or decimals, such as (-1.5, 2.25)?
Use the same moves as with integers, and count parts of units. For x = -1.5, move 1 1/2 units to the left. For y = 2.25, move 2 1/4 units up. Grid paper with 1 unit = 4 squares makes fourths and halves easy to count.
How does 6.NS.C.6 connect to later math?
Four-quadrant graphs are used in almost every later course. In grade 7, students add and subtract on the number line (7.NS.A.1). In grade 8, reflections return as transformations described with coordinates (8.G.A.3), and graphs of lines use all four quadrants.
07
Related Standards
6 standards
These standards connect to 6.NS.C.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
3.NF.A.2Prerequisite
Understand a fraction as a number on the number line
Lesson coming soon
5.G.A.1Prerequisite
Use perpendicular number lines (axes) with an origin at 0 to locate points
Lesson coming soon
Alongside
6.NS.C.5Parallel
Use positive and negative numbers for opposite quantities and explain 0
