6.NS.C.8: Graphing in Four Quadrants and Finding Distances
In plain English: 6.NS.C.8 is the Common Core grade 6 math standard that asks students to solve real-world and math problems by graphing points in all four quadrants of the coordinate plane. Students use coordinates and absolute value to find the distance between two points that share a first coordinate or a second coordinate. It prepares for adding and subtracting negative numbers in grade 7.
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
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.8 · Official standard
Students extend the coordinate plane they used in grade 5 to all four quadrants. The coordinate plane is a grid made by two number lines that cross at 0: the horizontal (left-right) x-axis and the vertical (up-down) y-axis. They cross at the origin, (0, 0). An ordered pair (x, y) names a point: the first coordinate tells how far to move right (positive) or left (negative), and the second tells how far to move up (positive) or down (negative). The axes split the plane into four quadrants (regions), numbered I, II, III and IV.
Students plot and read points in every quadrant, including points with fraction or decimal coordinates, and use them to solve map and design problems. Then they find distances between two points that share a first coordinate (a vertical segment) or a second coordinate (a horizontal segment, a straight path between the two points). They use absolute value, a number's distance from 0 (for example, |-4| = 4): when the two points are on opposite sides of an axis, add the absolute values; when they are on the same side, subtract the smaller from the larger. Slanted distances wait until grade 8, and subtracting negative numbers waits until grade 7.
Learning Objectives
By the end of this lesson, students will be able to:
Plot and read points with positive, negative and zero coordinates in all four quadrants
Name the quadrant or axis of a point from the signs of its coordinates
Find the distance between two points with the same first coordinate or the same second coordinate by using absolute value
Solve map, path and rectangle problems by graphing points and finding distances along grid lines
Prior Knowledge Required
Students should already be comfortable with:
Graphing and reading points in the first quadrant 5.G.A.15.G.A.2
Placing positive and negative numbers on horizontal and vertical number lines 6.NS.C.6
Absolute value as a number's distance from 0 6.NS.C.7
Adding and subtracting whole numbers, fractions and decimals
Draw a plus sign on the board and label its center "flagpole". Tell students that east is right and north is up.
Warm-Up Prompt
"The school garden is 3 steps east and 2 steps north of the flagpole, so we write it as (3, 2). The bike rack is 4 steps west and 5 steps south of the flagpole. How could you write the bike rack's location with two numbers?"
Collect ideas. Guide students to (-4, -5): west is the negative direction on the horizontal number line, and south is the negative direction on the vertical one. Some students may write (-5, -4); ask them which number tells east or west. The first number always tells left or right. Tell students that negative numbers let us name every spot around the flagpole, not only the spots to the north and east.
Direct Instruction20 minutes
Part 1: The four quadrants. Draw the x-axis and the y-axis from -8 to 8 and mark the origin. To plot an ordered pair, start at the origin, move along the x-axis by the first coordinate, then move up or down by the second coordinate. Number the quadrants I to IV, starting at the upper right and turning the opposite way from a clock's hands (counterclockwise). Diagram 1 shows the sign pattern in each quadrant. A point with a 0 coordinate lies on an axis, so it is in no quadrant.
Plotting points and naming quadrants
Plot A (3, -4), B (-5, 2), C (-2, -6) and D (0, 4). Name the quadrant or axis for each point.
Equation: A is in Quadrant IV (+, -), B is in Quadrant II (-, +), C is in Quadrant III (-, -), and D is on the y-axis, so it is in no quadrant
Same second coordinate, opposite sides
On a town map, 1 unit is 1 block. The library is at (-3, 3) and the park is at (6, 3). How many blocks apart are they?
Equation: Both points have y = 3, so the segment is horizontal. They are on opposite sides of the y-axis: |-3| + |6| = 3 + 6 = 9 blocks
Same first coordinate, same side
Find the distance between P (2, -3) and Q (2, -7).
Equation: Both points have x = 2, so the segment is vertical. Both are below the x-axis: |-7| - |-3| = 7 - 3 = 4 units
Finding a point at a given distance
A coach places a cone at (-3, 2) on a field map, where 1 unit is 1 meter. The second cone must be 6 meters away on the same horizontal line. Where can it go?
Equation: 6 meters right of -3 is 3, and 6 meters left of -3 is -9, so the cone can go at (3, 2) or (-9, 2)
Rectangle on a map
A playground is a rectangle with corners at (-6, 5), (2, 5), (2, -2) and (-6, -2). Each unit is 1 meter. How much fence goes around it?
Equation: Top side: |-6| + |2| = 8 m. Right side: |5| + |-2| = 7 m. Perimeter (the distance around) = 8 + 7 + 8 + 7 = 30 m of fence
Part 2: Distances along a grid line. Two points with the same first coordinate lie on a vertical line, and two points with the same second coordinate lie on a horizontal line. Diagram 2 shows both cases. Think of each point's distance from the axis it is measured from: that distance is the absolute value of the coordinate that is different. Then use one of two steps:
Opposite sides of an axis: add the absolute values. The library and the park are 3 blocks and 6 blocks from the y-axis, on opposite sides, so they are 3 + 6 = 9 blocks apart.
Same side of an axis: subtract the smaller absolute value from the larger one. P and Q are 3 units and 7 units below the x-axis, so they are 7 - 3 = 4 units apart.
Check by counting: count the spaces between the points on the grid, not the grid points (the spots where grid lines cross) you touch. From x = -3 to x = 6 there are 10 grid points but only 9 spaces.
Part 3: Solving problems. Work the cone and playground examples above. In the cone example, the answer has two points, one on each side of the first cone. In the playground example, the perimeter (the distance around a shape) is the sum of the four side lengths, and each side length is a distance between two corners that share a coordinate.
Guided Practice15 minutes
Pairs plot four places on a school campus map on grid paper. Each unit is 10 meters, and the flagpole is at the origin. After each question, one pair shows its work.
Campus map (invented), 1 unit = 10 meters
Place
Coordinates
Quadrant
Gym
(-6, 4)
?
Cafeteria
(2, 4)
?
Office
(2, -5)
?
Bus stop
(7, -5)
?
Questions: (1) Name each quadrant. (Gym II, Cafeteria I, Office IV, Bus stop IV.) (2) How far is it from the gym to the cafeteria? (|-6| + |2| = 8 units, or 80 meters.) (3) From the cafeteria to the office? (|4| + |-5| = 9 units, or 90 meters.) (4) From the office to the bus stop? (Same side of the y-axis: 7 - 2 = 5 units, or 50 meters.) (5) A student walks gym, cafeteria, office, bus stop. How far is the whole walk? (8 + 9 + 5 = 22 units, or 220 meters.) Listen for pairs who add when both points are on the same side.
Independent Practice10 minutes
Students solve five problems on grid paper. (1) Name the quadrant or axis of (-8, -1), (6, 0), (-1, 5) and (4, -7). (III, x-axis, II, IV.) (2) Find the distance between (-3, -8) and (-3, -2). (Same side: 8 - 2 = 6 units.) (3) Find the distance between (-6, -5) and (4, -5). (Opposite sides: 6 + 4 = 10 units.) (4) A camp is at (1, -4) on a trail map. A lake is 7 units north of the camp. What are the lake's coordinates? (Count up 7 from -4: 4 units reach the x-axis and 3 more go above it, so (1, 3).) (5) On a pier diagram, the y-axis shows height in feet above or below the water. A pelican sits on a post at (5, 12), and a crab rests on the sea floor at (5, -9). How far apart are they? (12 + 9 = 21 feet.)
Closure5 minutes
Exit ticket: (1) In which quadrant is (-4, -9)? (Quadrant III.) (2) Find the distance between (8, -2) and (-4, -2). (8 + 4 = 12 units.) (3) In one sentence, explain when you add the absolute values and when you subtract them. (Add when the points are on opposite sides of an axis; subtract when they are on the same side.)
Differentiation Strategies
For Struggling Students
Give a printed plane with the axes already numbered and the quadrant numbers written in, and have students trace the moves with a finger: first left or right, then up or down
Let students count the spaces between two points first, then match the count to the absolute value steps
Use a sticky note arrow on the axis each point is measured from, so students see whether the points are on the same side or on opposite sides
For Advanced Students
Give the corners (-4.5, 3), (2.5, 3) and (2.5, -1.5) of a rectangle and ask for the fourth corner and the perimeter
Ask for every point on the grid that is exactly 5 units from (0, 0) along a horizontal or vertical line, and explain why there are four
Challenge: draw a path from (-6, -2) to (5, 6) that uses only horizontal and vertical moves, and show that every such path without backtracking has the same length
Assessment Guidance
What to Look For
Check that students move along the x-axis first and use the sign of each coordinate to choose the direction. When they name quadrants, look for the sign pattern, not guessing from the picture, and for points on an axis being named as on an axis. In distance problems, ask students to say which coordinate is the same, which axis they measure from, and whether the points are on the same side or on opposite sides. Watch for students who count grid points instead of spaces, and for answers written without units in map problems.
02
Classroom Activities
3 Activities
1
Four-Quadrant Battleship
15 minPairs
Partners play a guessing game on grid paper with both axes numbered from -6 to 6. Each player hides boats behind a file folder, and the other player calls ordered pairs to find them. After the game, players use coordinates and absolute value to find the length of each boat.
Setup
Each player secretly draws 3 boats on grid points: one horizontal boat covering 4 grid points, one vertical boat covering 5 grid points, and one boat of 3 grid points in either direction
At least one boat must cross an axis, and the boats may not touch
Stand a file folder between the players so they cannot see each other's grid
Procedure
Take turns calling an ordered pair and its quadrant, such as "(-3, 5), Quadrant II"; a wrong quadrant loses the turn
The partner answers "hit" or "miss", and the caller marks the result on a second grid
When a boat is found, both players write its end points and find its length with absolute values
Discussion Questions
A horizontal boat covers 4 grid points from (-2, 5) to (1, 5). How long is it? (|-2| + |1| = 3 units: 4 grid points make only 3 spaces)
A vertical boat covers 5 grid points from (4, -6) to (4, -2). How long is it? (Same side: 6 - 2 = 4 units)
How did you know which boats crossed an axis just from their end points?
Modification for Distance Learning
Partners play over a video call, each with a private grid on paper, or in a free online graphing tool with the points hidden. They type each call in the chat, so the game leaves a record of ordered pairs to check.
2
Town Map Card Sort
20 minGroups of 3-4
Groups receive 8 cards, each with two places on an invented town map (1 unit = 1 block). They sort the cards into three piles: same first coordinate, same second coordinate, and neither. Then they plot the points and find the distance for every card in the first two piles.
Map Cards (8 cards)
Card 1: pool (-7, 6) and zoo (-7, -1). (Same first coordinate: 6 + 1 = 7 blocks)
Card 2: bank (3, -8) and mall (3, -2). (Same first coordinate: 8 - 2 = 6 blocks)
Card 3: farm (-5, -4) and mill (7, -4). (Same second coordinate: 5 + 7 = 12 blocks)
Card 4: pier (-8, 2) and dock (-1, 2). (Same second coordinate: 8 - 1 = 7 blocks)
Card 5: clinic (4, 5) and diner (4, 1). (Same first coordinate: 5 - 1 = 4 blocks)
Card 6: barn (-6, -7) and camp (2, -7). (Same second coordinate: 6 + 2 = 8 blocks)
Card 7: fort (1, 3) and hut (-3, -2). (Neither)
Card 8: mine (-2, 8) and well (5, -6). (Neither)
Procedure
One student reads a card, a second sorts it and says which coordinate is the same, and a third plots both points; roles rotate
For each card in the first two piles, write whether the points are on the same side or on opposite sides of an axis, then find the distance
Check each distance by counting spaces on the grid
Discussion Questions
Which card has the longest distance? (Card 3, 12 blocks)
Cards 1 and 4 both give 7 blocks. Why do you add on one card and subtract on the other?
Why can't you find the distance on cards 7 and 8 with absolute values alone? (The segment is slanted; grade 8 finds such distances with the Pythagorean theorem)
Challenge Variation
Groups write a new card for each pile using a real place near school, trade cards with another group, and check each other's distances.
3
Floor Grid Walk
15 minGroups of 4
Tape a coordinate plane on the floor with masking tape, with axes from -5 to 5 and one floor tile (about 1 foot) per unit. Groups draw walk cards, predict the end point and the distance, and then one student walks it.
Walk Cards (6 cards)
Card 1: start at (-3, 2) and walk 5 units down. (End at (-3, -3))
Card 2: start at (4, -1) and walk 6 units left. (End at (-2, -1))
Card 3: start at (-4, -3) and walk 7 units right. (End at (3, -3))
Card 4: start at (1, 4) and walk 8 units down. (End at (1, -4))
Card 5: start at (-2, -4) and walk 6 units up. (End at (-2, 2))
Card 6: start at (5, 3) and walk 4 units left. (End at (1, 3))
Procedure
The group writes its predicted end point on grid paper before anyone walks
One student stands on the start point, walks the card, and calls out the end point and its quadrant
The others check the walk with absolute values, for example card 3: |-4| + |3| = 7 units
Discussion Questions
Which card is the only one where the walker stays on one side of an axis? (Card 6: from x = 5 to x = 1, and 5 - 1 = 4)
On card 2 the walker moves left 6 units. Why does the first coordinate change but not the second?
How can you tell from the start point and the move that the walker will cross an axis?
Modification for Small Rooms
Use a large grid on chart paper and a game piece instead of walking, or tape the plane in a hallway or outdoors with chalk.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Four Quadrants
The x-axis and the y-axis split the coordinate plane into four quadrants. The signs of the coordinates tell the quadrant: A (3, -4) is in Quadrant IV, B (-5, 2) is in Quadrant II and C (-2, -6) is in Quadrant III. D (0, 4) is on the y-axis, so it is in no quadrant. Drawn to scale, 1 grid square per unit.
Diagram 2: Distances Along a Horizontal and a Vertical Line
The library (-3, 3) and the park (6, 3) share a second coordinate and lie on opposite sides of the y-axis, so their distance is 3 + 6 = 9 blocks. P (2, -3) and Q (2, -7) share a first coordinate and lie on the same side of the x-axis, so their distance is 7 - 3 = 4 units. Drawn to scale.
04
Homework Assignment
~30 min
6.NS.C.8 Homework: Points, Quadrants and Distances
Directions: Use grid paper. Plot every point you use, label it with its coordinates, and show the absolute values you add or subtract for each distance. Write units with every map answer.
Part 1: Graphing in Four Quadrants (Problems 1-2)
Plot P (-7, 2), Q (4, -6), R (-3, -3) and S (0, -5). Name the quadrant or axis for each point. Which two points have a negative first coordinate, and what does that tell you about where they are?
An invented zoo map has the entrance at the origin, and 1 unit is 50 meters. The lions are at (-6, 6), the penguins are at (-6, -4) and the gift shop is at (3, -4). Plot the three places. How many meters is the walk from the lions to the penguins, and then from the penguins to the gift shop? What is the total walk?
Part 2: Distances with Absolute Value (Problems 3-4)
Find the distance between each pair of points. Say whether the points are on the same side or on opposite sides of an axis. (a) (-6, 1) and (-6, 8) (b) (-9, -2) and (4, -2) (c) (5, -1) and (5, -10) (d) (-2.5, 6) and (3.5, 6)
On a city map, 1 unit is 1 block. Mia's house is at (-8, -2) and her school is at (-8, 6). How many blocks does she walk if she goes straight along the street? Her friend Theo lives 6 blocks east of the school. What are the coordinates of Theo's house?
Part 3: Solving Problems (Problems 5-6)
A community garden is a rectangle with corners at (-5, 4), (7, 4), (7, -3) and (-5, -3), where 1 unit is 1 meter. Plot the rectangle. Find the length of each side and the total length of fence needed to go around the garden.
Jordan says the distance between (-4, -6) and (-4, 3) is 3 units, because |-6| - |3| = 6 - 3 = 3. Plot the points. Is Jordan right? Explain his mistake and find the correct distance. Then write a rule for when to add and when to subtract absolute values.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Plotting and Quadrants
All points plotted and labeled correctly, quadrants and axes named
One point or one quadrant wrong
Coordinates reversed or several points wrong
Distances
All distances correct, with the absolute values shown and same side or opposite sides stated
Correct distances but the reasoning is missing, or one error
Distances found by guessing or counting grid points
Map and Rectangle Problems
Correct answers with units, including Theo's house and the fence length
One answer wrong or units missing
Problems not attempted or mostly wrong
Explaining Errors
Jordan's mistake explained and a correct rule stated
Mistake found but the rule is incomplete
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, and keep grid paper nearby to plot the points. 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
In which quadrant is the point (-3, 7)?
Answer: B
The first coordinate is negative (left of the y-axis) and the second is positive (above the x-axis), so (-3, 7) is in Quadrant II, the upper left. Choice A ignores the negative sign. Choice D swaps the signs and reads the point as right and down, which is the pattern (+, -).
Question 2 of 20 · Multiple Choice
Starting at the origin, which moves plot the point (5, -2)?
Answer: C
The first coordinate, 5, is positive, so move 5 units right. The second coordinate, -2, is negative, so move 2 units down. Choice B swaps the order of the coordinates. Choice D ignores the negative sign of -2, and choice A uses the wrong direction for both numbers.
Question 3 of 20 · Multiple Choice
Which point is in Quadrant III?
Answer: D
In Quadrant III both coordinates are negative, and only (-2, -8) has two negative coordinates. (4, -1) is in Quadrant IV, (-6, 3) is in Quadrant II and (7, 5) is in Quadrant I. A student who chooses B may be thinking that any negative number puts a point in Quadrant III.
Question 4 of 20 · Multiple Choice
Which point lies on the y-axis?
Answer: A
Every point on the y-axis has a first coordinate of 0, because it is 0 units left or right of the origin. So (0, -6) is on the y-axis, 6 units below the origin. Choice B has a second coordinate of 0, so it lies on the x-axis: this mixes up the two axes. Choices C and D are in Quadrants III and IV.
Question 5 of 20 · Multiple Choice
What is the distance between (-5, 4) and (6, 4)?
Answer: C
Both points have a second coordinate of 4, so they lie on a horizontal line. They are on opposite sides of the y-axis, so add the absolute values: |-5| + |6| = 5 + 6 = 11 units. Choice A subtracts the absolute values (6 - 5), which only works when the points are on the same side. Choice B adds the second coordinates (4 + 4). Choice D counts the 12 grid points from -5 to 6 instead of the 11 spaces.
Question 6 of 20 · Multiple Choice
What is the distance between (3, -2) and (3, -9)?
Answer: A
Both points have a first coordinate of 3, so they lie on a vertical line. Both are below the x-axis, so subtract: |-9| - |-2| = 9 - 2 = 7 units. Choice B adds the absolute values, as if the points were on opposite sides. Choice C adds the first coordinates (3 + 3). Choice D counts the 8 grid points from -9 to -2 instead of the spaces.
Question 7 of 20 · Multiple Choice
On a town map, 1 unit is 1 block. The pharmacy is at (-7, -4) and the bakery is at (-7, 2). How many blocks apart are they?
Answer: D
Both places have a first coordinate of -7, so the street between them is vertical. The pharmacy is below the x-axis and the bakery is above it, so add: |-4| + |2| = 4 + 2 = 6 blocks. Choice A subtracts the absolute values (4 - 2). Choice B adds the absolute values of the first coordinates (7 + 7). Choice C counts the 7 grid points from -4 to 2.
Question 8 of 20 · Multiple Choice
Which point is 4 units directly above (-2, -1)?
Answer: B
Moving up changes only the second coordinate. From -1, 1 unit up reaches the x-axis and 3 more units go above it, so the point is (-2, 3). Choice A moves 4 units right instead of up. Choice C moves 4 units down. Choice D moves 4 units left.
Question 9 of 20 · Multiple Choice
Which pair of points is exactly 10 units apart?
Answer: C
The points (-6, 2) and (4, 2) share a second coordinate and lie on opposite sides of the y-axis: |-6| + |4| = 6 + 4 = 10 units. Choice A adds |-8| + |-2| = 10, but both points are on the same side, so the distance is 8 - 2 = 6 units. Choice B shares no coordinate: the segment is slanted and longer than 10 units. Choice D counts the 10 grid points from -3 to 6, but the distance is 6 + 3 = 9 units.
Question 10 of 20 · Multiple Choice
A rectangle has corners at (-5, 1), (4, 1), (4, -6) and (-5, -6). What is its perimeter (the distance around it)?
Answer: A
The top side goes from x = -5 to x = 4: 5 + 4 = 9 units. The right side goes from y = 1 to y = -6: 1 + 6 = 7 units. The perimeter is 9 + 7 + 9 + 7 = 32 units. Choice B adds only two sides (9 + 7). Choice C multiplies 9 × 7, which is the area, not the perimeter. Choice D subtracts the absolute values on both sides (5 - 4 = 1 and 6 - 1 = 5), so it gets 1 + 5 + 1 + 5 = 12.
Question 11 of 20 · Multiple Choice
On a map of a skate park, 1 unit is 1 meter. The ramp is at (-6, -7) and the bench is at (-6, 5). How far apart are they?
Answer: B
The points share the first coordinate -6, so the path is vertical. The ramp is below the x-axis and the bench is above it: |-7| + |5| = 7 + 5 = 12 meters. Choice A subtracts the absolute values (7 - 5). Choice C counts the 13 grid points from -7 to 5. Choice D uses the shared coordinate, |-6| = 6, which tells how far the path is from the y-axis, not how long it is.
Question 12 of 20 · Multiple Choice
Which statement about the points (-6, -1) and (-6, 7) is true?
Answer: D
The points share the first coordinate, so they lie on a vertical line (x = -6). One is below the x-axis and one is above, so the distance is 1 + 7 = 8 units. Choices A and C call the line horizontal, which mixes up the two cases: points with the same first coordinate are stacked up and down. Choice B subtracts the absolute values (7 - 1), which only works when the points are on the same side.
Question 13 of 20 · Multiple Choice
A robot starts at (-3, 1). It moves 4 units down, then 8 units right. Where does it stop?
Answer: D
Down 4 changes the second coordinate: from 1, 1 unit down reaches the x-axis and 3 more go below it, giving (-3, -3). Right 8 changes the first coordinate: from -3, 3 units reach the y-axis and 5 more go past it, giving (5, -3). Choice A moves up instead of down. Choice B moves left instead of right. Choice C swaps the moves: 4 units right and 8 units down.
Question 14 of 20 · Multiple Choice
Zara says the point (0, 5) is in Quadrant I. Is she right?
Answer: A
A first coordinate of 0 means the point is 0 units left or right of the origin, so it sits on the y-axis, 5 units up. Points on an axis are not in any quadrant. Choice B mixes up the axes. Choice C forgets that 0 is neither positive nor negative, and every point in Quadrant I has two positive coordinates. Choice D treats 0 as a negative number.
Question 15 of 20 · Short Answer
Name the quadrant or axis for each point: (-9, 4), (6, -8), (-1, -1) and (-7, 0).
(-9, 4): Quadrant II (-, +). (6, -8): Quadrant IV (+, -). (-1, -1): Quadrant III (-, -). (-7, 0): on the x-axis, because the second coordinate is 0, so it is in no quadrant.
Question 16 of 20 · Short Answer
Find the distance between (-9, -5) and (-9, 6). Explain how you used absolute value.
The points share the first coordinate -9, so they lie on a vertical line. (-9, -5) is 5 units below the x-axis and (-9, 6) is 6 units above it. They are on opposite sides, so add: |-5| + |6| = 5 + 6 = 11 units.
Question 17 of 20 · Short Answer
On a city map, 1 unit is 1 block. The fire station is at (6, -4) and a fire is reported at (6, 5). (a) How many blocks does the fire truck drive if it goes straight up the street? (b) A fire hydrant is 10 blocks west of the fire station. What are its coordinates?
(a) The points share the first coordinate 6. The station is 4 blocks below the x-axis and the fire is 5 blocks above it: 4 + 5 = 9 blocks. (b) West is the negative direction. From 6, 6 blocks west reach the y-axis and 4 more go past it, so the first coordinate is -4. The hydrant is at (-4, -4), and the second coordinate stays -4.
Question 18 of 20 · Short Answer
Find two points that are 7 units from (-1, -5) on the same horizontal line. Explain how you found them.
On a horizontal line the second coordinate stays -5. Move 7 units right: 1 unit reaches the y-axis and 6 more go past it, giving (6, -5). Move 7 units left: -1 is already 1 unit left of 0, so 7 more units left end 8 units left of 0, giving (-8, -5). Check: (6, -5) is on the other side of the y-axis, so |-1| + |6| = 7, and (-8, -5) is on the same side, so 8 - 1 = 7.
Question 19 of 20 · Short Answer
Three corners of a rectangle are (-8, 3), (1, 3) and (1, -2). Find the fourth corner, the side lengths and the perimeter.
The fourth corner shares a first coordinate with (-8, 3) and a second coordinate with (1, -2), so it is (-8, -2). The top side is |-8| + |1| = 9 units and the right side is |3| + |-2| = 5 units. The perimeter is 9 + 5 + 9 + 5 = 28 units.
Question 20 of 20 · Short Answer
Lee says the distance between (-2, 4) and (-8, 4) is 10 units, because |-2| + |-8| = 10. Is Lee right? Explain.
No. Both points have negative first coordinates, so they are on the same side of the y-axis. Lee should subtract: |-8| - |-2| = 8 - 2 = 6 units. Counting the spaces from x = -8 to x = -2 also gives 6. Adding the absolute values works only when the points are on opposite sides of the axis.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.NS.C.8 mean?
6.NS.C.8 means students graph points in all four quadrants of the coordinate plane and use them to solve problems. It also asks them to find the distance between two points that share a first coordinate or a second coordinate, using the coordinates and absolute value. Maps, game boards and floor plans are typical contexts.
Is 6.NS.C.8 taught in grade 6 or grade 7?
It is a grade 6 standard in The Number System domain. In grade 7, students learn to add and subtract negative numbers (7.NS.A.1), so they can write a distance as the absolute value of a difference. In grade 8, they find slanted distances with the Pythagorean theorem (8.G.B.8).
What are the four quadrants of the coordinate plane?
They are the four regions the x-axis and the y-axis cut the plane into. Quadrant I is the upper right (+, +), Quadrant II is the upper left (-, +), Quadrant III is the lower left (-, -) and Quadrant IV is the lower right (+, -). A point on an axis, such as (0, -3) or (8, 0), is not in any quadrant.
How do you find the distance between two points with the same x-coordinate?
Look at the second coordinates and use absolute value. The points lie on a vertical line, so the distance depends only on how far each point is above or below the x-axis. For (-1, 6) and (-1, -2), the points are on opposite sides, so the distance is 6 + 2 = 8 units. For (-1, 6) and (-1, 2), they are on the same side, so it is 6 - 2 = 4 units.
When do you add and when do you subtract the absolute values?
Add when the two points are on opposite sides of the axis, and subtract when they are on the same side. Opposite signs in the coordinate that changes mean opposite sides. Same signs mean the same side, so the shorter distance from the axis is inside the longer one and must be taken away.
Why do we use absolute value for distance?
Because a distance is never negative. The absolute value of a coordinate is that point's distance from the axis, whether the point is left, right, above or below. Walking from -5 to 0 on a number line is 5 steps, just like walking from 0 to 5.
What are common mistakes when graphing points in four quadrants?
A common mistake is swapping the coordinates, for example plotting (2, -6) at 2 down and 6 right. Another is ignoring a negative sign and plotting the point in the wrong quadrant. When finding distances, many students count the grid points they touch instead of the spaces between them, which gives an answer 1 too large.
Can students find the distance between any two points in grade 6?
No. In grade 6, students find distances only between points that share a first coordinate or a second coordinate, because those segments lie along the grid. A slanted segment, such as from (0, 0) to (3, 4), needs the Pythagorean theorem, which comes in grade 8.
How is 6.NS.C.8 used in real life?
Any map with a center point and streets on a grid works like a coordinate plane. City blocks, seat maps, game boards and computer screens all use coordinates. Finding how many blocks apart two places are on the same street is exactly the distance skill in this standard.
How can parents help with 6.NS.C.8 at home?
They can draw a simple map of the neighborhood or the house on grid paper, put home at (0, 0), and ask for the coordinates of other places. A good question is "How many blocks apart are these two places, and did you add or subtract?" Asking the child to explain the rule out loud builds the reasoning the standard asks for.
07
Related Standards
6 standards
These standards connect to 6.NS.C.8: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.G.A.2Prerequisite
Graph points in the first quadrant to represent real-world and math problems
Lesson coming soon
6.NS.C.7Prerequisite
Understand ordering and absolute value of rational numbers
