6.G.A.3: Drawing Polygons on the Coordinate Plane and Finding Side Lengths
In plain English: 6.G.A.3 is the Common Core grade 6 math standard that asks students to draw polygons on the coordinate plane from the coordinates of their vertices and to find the length of any side that is horizontal or vertical. When two points share a coordinate, students use the other coordinates to find the distance, then solve problems about maps, floor plans and gardens.
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-world and mathematical problems involving area, surface area, and volume. Also written as 6.G.3 · Official standard
Students draw polygons on the coordinate plane, the flat grid made by a horizontal number line (the x-axis) and a vertical number line (the y-axis) that cross at the origin, (0, 0). Each point is named by an ordered pair (x, y): the first coordinate tells how far to move left or right, and the second tells how far to move up or down. A polygon is a closed shape made of straight sides (line segments, straight paths from one point to another), and each corner is a vertex (plural: vertices). Students plot the vertices they are given and connect them in order to close the shape.
Next, students find side lengths without counting squares. When two vertices have the same first coordinate, the side between them is vertical (straight up and down), and its length comes from the second coordinates. When they have the same second coordinate, the side is horizontal (straight across), and its length comes from the first coordinates. Points can sit in any of the four quadrants (the four regions the axes make), so students use absolute value, a number's distance from 0: subtract the absolute values when the two numbers have the same sign, and add them when the signs are opposite. Students then apply these lengths to perimeter (the distance around a shape), area (the number of unit squares inside), maps and garden plans. Slanted sides are left for grade 8, where the Pythagorean Theorem gives their length.
Learning Objectives
By the end of this lesson, students will be able to:
Plot the vertices of a polygon from their coordinates in all four quadrants and connect them in order
Find the length of a vertical side joining two points with the same first coordinate
Find the length of a horizontal side joining two points with the same second coordinate
Use side lengths found from coordinates to solve perimeter, area and map problems
Prior Knowledge Required
Students should already be comfortable with:
Graphing points in the first quadrant of the coordinate plane 5.G.A.15.G.A.2
Finding the perimeter and area of rectangles 4.MD.A.3
Locating points with negative coordinates in all four quadrants 6.NS.C.6
Understanding absolute value as distance from 0 6.NS.C.7
Hand out coordinate grid paper. Ask students to plot two points and answer without counting squares first:
Warm-Up Prompt
"Plot M(2, 7) and N(2, 1). Connect them. Is the segment horizontal, vertical or slanted? How long is it? Can you find the length from the coordinates alone?"
Collect answers. Both points have the same first coordinate, 2, so the segment is vertical. Its length is the change in the second coordinate: 7 - 1 = 6 units. Some students may count the grid dots from 1 to 7 and get 7. Point out that length counts the spaces between the dots, not the dots. Introduce the words horizontal, vertical and vertex, and say that today students will use coordinates to find the length of any horizontal or vertical side, even when the points are on different sides of an axis.
Direct Instruction20 minutes
Part 1: Drawing a polygon from its vertices. Plot each vertex, label it with its letter, then connect the vertices in the order given and connect the last one back to the first. The order matters: connecting A to C instead of A to B in a rectangle makes the sides cross. Draw rectangle ABCD from Diagram 1 with the class.
Part 2: Lengths from coordinates. Look at which coordinate stays the same. If the first coordinates match, the side is vertical, and you compare the second coordinates. If the second coordinates match, the side is horizontal, and you compare the first coordinates. Then use signs:
Same sign (same side of the axis): subtract the absolute values. From (6, -2) to (1, -2) is 6 - 1 = 5 units, because both points are to the right of the y-axis.
Opposite signs (different sides of the axis): add the absolute values. From (-4, 3) to (5, 3), the point (-4, 3) is 4 units left of the y-axis and (5, 3) is 5 units right of it, so the length is 4 + 5 = 9 units.
Check the picture: length is never negative, and it counts spaces between grid lines, not dots.
Drawing a polygon across four quadrants
Draw rectangle ABCD with A(-4, 3), B(5, 3), C(5, -3) and D(-4, -3). Find the length of each side, the perimeter and the area.
Equation: AB = 4 + 5 = 9 units, BC = 3 + 3 = 6 units; perimeter 9 + 6 + 9 + 6 = 30 units; area 9 × 6 = 54 square units
Vertical side, same first coordinate
P(-3, -1) and Q(-3, -7) share the first coordinate -3. How long is side PQ?
Equation: Both second coordinates are negative, so subtract: 7 - 1 = 6 units
Real-world map
On a town map, each unit is 1 block. The bakery is at (-5, 2), the bank is at (4, 2) and the post office is at (4, -6). How far is the walk from the bakery to the bank and then to the post office along the streets?
Equation: Bakery to bank: 5 + 4 = 9 blocks; bank to post office: 2 + 6 = 8 blocks; total 17 blocks
Finding a missing vertex
Three vertices of a rectangle are (1, 1), (1, 6) and (7, 6). Find the fourth vertex and the perimeter.
A garden plan uses 1 unit = 1 meter. The garden is the polygon (0, 0), (7, 0), (7, 2), (3, 2), (3, 5), (0, 5). How much fence goes around it, and how much ground does it cover?
Equation: Sides 7, 2, 4, 3, 3 and 5 m; fence 24 m; area 7 × 2 + 3 × 3 = 23 square meters (Diagram 2)
Part 3: Using the lengths. The perimeter is the distance around a polygon, so add all the side lengths. The area is the number of unit squares inside, so for a rectangle multiply length by width. For the garden in Diagram 2, split the shape into two rectangles, as in 6.G.A.1. When a map or plan has a scale, such as 1 unit = 5 meters, find the length in units first, then multiply by the scale. Finally, show a triangle with one slanted side and ask why the rule does not work for that side (neither coordinate stays the same).
Guided Practice15 minutes
Pairs plot each polygon on grid paper, label the vertices, and write the length of every horizontal or vertical side next to it. After each problem, one pair shows its drawing and explains whether it added or subtracted.
Guided practice polygons
Problem
Vertices (connect in order)
Answers
1
J(-2, 4), K(3, 4), L(3, -1)
Right triangle (a triangle with a square corner); JK = 2 + 3 = 5 units, KL = 4 + 1 = 5 units; JL is slanted
2
(-1, -1), (3, -1), (3, 3), (-1, 3)
Square; each side 4 units; perimeter 16 units; area 16 square units
3
(-4, -3), (4, -3), (2, 2), (-2, 2)
Trapezoid (a four-sided shape with one pair of parallel sides, sides that run in the same direction and never meet); bottom side 8 units, top side 4 units; the other two sides are slanted
For Problem 1, ask: is JL longer or shorter than 5 units? (Longer: it cuts across the corner of a 5-by-5 square, so it is the longest side of the triangle.) For Problem 3, ask how tall the trapezoid is. (The top is at y = 2 and the bottom at y = -3, so 2 + 3 = 5 units.) Listen for students who subtract 4 - 4 = 0 for the bottom side because they ignore the signs.
Independent Practice10-15 minutes
Students work alone and draw each problem on grid paper. (1) How long is the segment from (-6, 5) to (-6, -4)? (5 + 4 = 9 units.) (2) Plot the rectangle (-7, 0), (-7, 3), (2, 3), (2, 0). Find its perimeter and area. (Sides 9 and 3 units; perimeter 24 units; area 27 square units.) (3) A soccer field is drawn on a grid where 1 unit = 5 meters. Its corners are (-10, -6), (10, -6), (10, 6) and (-10, 6). How long and wide is the real field? (20 units, or 100 meters, long and 12 units, or 60 meters, wide.) (4) Three corners of a rectangle are (-2, -5), (-2, 1) and (4, 1). Where is the fourth corner? ((4, -5).)
Closure5 minutes
Exit ticket: (1) Which pair makes a vertical segment: (2, 5) and (5, 2), or (-3, 4) and (-3, -1)? (The second pair: the first coordinates match.) (2) How long is the segment from (3, -2) to (-5, -2)? (3 + 5 = 8 units.) (3) In one sentence, explain when you add the absolute values instead of subtracting them. (When the two points are on opposite sides of an axis, so the numbers that change have opposite signs.)
Differentiation Strategies
For Struggling Students
Start in the first quadrant only, then add one negative coordinate at a time
Give a sentence frame: "The ___ coordinates are the same, so the side is ___. I compare the ___ coordinates."
Let students count grid spaces to check each length after they compute it, and circle the spaces, not the dots
For Advanced Students
Give the perimeter and three vertices of a rectangle, such as perimeter 20 units with (-2, 1) and (3, 1), and ask for every possible pair of other vertices
Ask students to design a six-sided room on grid paper with an area of 32 square units and list its vertices
Challenge: reflect a rectangle across the x-axis and explain why the side lengths do not change (reflections of points are part of 6.NS.C.6)
Assessment Guidance
What to Look For
Check that students plot each point with the first coordinate moving left or right and the second moving up or down, and that they connect the vertices in the order given. When they find a length, they should first name which coordinate stays the same and say whether the side is horizontal or vertical. Watch for subtracting when the points are on opposite sides of an axis, for counting dots instead of spaces, and for trying to use the rule on a slanted side. In word problems, look for the scale applied at the end and for units in the answer.
02
Classroom Activities
3 Activities
1
Mystery Polygon Cards
20 minPairs
Each pair gets 6 cards. Each card lists the vertices of a mystery polygon. Pairs plot the points on grid paper, connect them in order, name the shape, and find the length of every horizontal or vertical side.
Polygon Cards (6 cards)
Card 1: (-5, 1), (-1, 1), (-1, 5), (-5, 5). (Square; each side 4 units)
Card 2: (1, -6), (6, -6), (6, -3), (1, -3). (Rectangle; 5 units by 3 units)
Card 3: (-4, -5), (-4, -1), (3, -5). (Right triangle; sides of 4 and 7 units and one slanted side)
Card 4: (0, 2), (6, 2), (4, 5), (2, 5). (Trapezoid; bottom 6 units, top 2 units, two slanted sides)
Card 5: (-3, 3), (2, 3), (4, 6), (-1, 6). (Parallelogram, a four-sided shape whose opposite sides are parallel; top and bottom 5 units each, two slanted sides)
Card 6: (2, -1), (7, -1), (7, 1), (4, 1), (4, 4), (2, 4). (Hexagon, a six-sided polygon, shaped like an L; sides 5, 2, 3, 3, 2 and 5 units; perimeter 20 units)
Procedure
Partner A plots the points and connects them in order; Partner B checks each point before the lines are drawn
Together, mark each side H (horizontal), V (vertical) or S (slanted)
Find the length of every H and V side from the coordinates, then check by counting grid spaces
Switch roles for the next card
Discussion Questions
Which cards have exactly two slanted sides? (Cards 4 and 5)
On Card 3, which side can you not find with coordinates, and why? (The side from (-4, -1) to (3, -5): neither coordinate stays the same)
On Card 2, did you add or subtract to find the 5-unit side? Why? (Subtract: 1 and 6 are both positive)
Modification for Distance Learning
Share the cards as a slide with a coordinate grid. Students drag dots to the vertices and draw the sides with the line tool, then type each length next to its side.
2
Neighborhood Map Walk
20 minGroups of 3
Groups get a printed neighborhood map on a coordinate grid, where each unit is 1 block and all streets run along grid lines. Groups find walking distances between places and plan a delivery loop.
Places on the Map
Library (-6, 4), school (0, 4), store (5, 4)
Home (-6, -3), park (0, -3), fire station (5, -3)
Tasks
Find the walking distance from the school to the store (5 blocks) and from the library to home (4 + 3 = 7 blocks)
Find the distance from the park to the school (3 + 4 = 7 blocks) and from home to the fire station (6 + 5 = 11 blocks)
A delivery bike rides the loop library, store, fire station, home, and back to the library. Draw the loop as a polygon and find its total length (11 + 7 + 11 + 7 = 36 blocks)
Discussion Questions
Which trip is longer: library to store, or store to fire station? (Library to store: 11 blocks, compared with 7 blocks)
Why did you add 6 and 5 to go from home to the fire station?
The school and the park are both on the y-axis. How does that make their distance easy to find?
Challenge Variation
Each group adds one new place to the map, gives its coordinates, and writes a trip question for another group. The trip must use only horizontal and vertical streets.
3
Design a Dog Pen
15 minPairs
Pairs have 24 units of fencing to build a rectangular dog pen on grid paper, with sides along grid lines and one corner at (-3, -2). They record each pen's vertices, side lengths and area, and look for the pen with the most space.
Procedure
Draw a rectangle with one corner at (-3, -2) that uses exactly 24 units of fence, and list its four vertices
Find each side length from the coordinates and check that the sides add up to 24
Find the area, then draw a different rectangle with the same corner and the same 24 units of fence
Record every pen in a table: vertices, length, width, perimeter, area
Discussion Questions
Which pen gives the dog the most space? (The 6-by-6 square, with vertices (-3, -2), (3, -2), (3, 4) and (-3, 4) and an area of 36 square units)
Why is the 1-by-11 pen a poor choice for a dog, even though it uses the same fence?
If a length and a width add up to 12, why does the perimeter always equal 24?
Extension Variation
Allow six-sided pens shaped like an L that still use 24 units of fence. Pairs compare the area of their best L-shaped pen with the area of the square pen.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Side Lengths from Coordinates in Four Quadrants
Rectangle ABCD drawn to scale. Side AB is horizontal because A and B share the second coordinate 3. The y-axis splits it into 4 units and 5 units, so AB = 9 units. Side BC is vertical because B and C share the first coordinate 5. The x-axis splits it into 3 units and 3 units, so BC = 6 units.
Diagram 2: A Six-Sided Garden Plan
A garden plan in the first quadrant, drawn to scale with 1 unit = 1 meter. Every side is horizontal or vertical, so each side length comes from the coordinates. The fence is the perimeter, 24 meters. The area is found by splitting the garden into a 7-by-2 rectangle and a 3-by-3 square.
04
Homework Assignment
~30 min
6.G.A.3 Homework: Polygons and Side Lengths on the Coordinate Plane
Directions: Use coordinate grid paper. Plot and label every vertex, and connect the vertices in the order given. For each length, name the coordinate that stays the same and show whether you added or subtracted. Write units with every answer.
Part 1: Drawing Polygons (Problems 1-2)
Plot A(-3, 2), B(4, 2), C(4, -5) and D(-3, -5), and connect them in order. Find the length of each side. What kind of polygon is ABCD? Find its perimeter.
Plot the five points (-4, -1), (2, -1), (2, 3), (-1, 6) and (-4, 3), and connect them in order to make a shape like a house. Which sides are horizontal or vertical? Find their lengths. Which sides can you not find with this method, and why?
Part 2: Side Lengths (Problems 3-4)
Find each length and say whether the segment is horizontal or vertical: (a) from (5, -8) to (5, -1); (b) from (-9, 4) to (6, 4); (c) from (-2, -3) to (-2, 7).
Three vertices of a rectangle are (-6, -2), (3, -2) and (3, 5). Find the fourth vertex. Then find the perimeter and the area of the rectangle.
Part 3: Real-World Problems (Problems 5-6)
On a town map, each unit is 1 block and the streets run along the grid lines. Mia's home is at (-4, -3), the soccer field is at (-4, 5) and the ice cream shop is at (6, 5). How many blocks does Mia walk from home to the soccer field and then to the ice cream shop?
A vegetable garden plan uses 1 unit = 1 foot. The garden has vertices (0, 0), (10, 0), (10, 4), (6, 4), (6, 8) and (0, 8). Find every side length, the length of fence needed to go around the garden, and the area of the garden.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Plotting Polygons
All vertices plotted and labeled correctly, connected in order
One point misplaced or one side connected out of order
Several points misplaced or shape not closed
Side Lengths
Every horizontal and vertical length correct, with add or subtract explained
Most lengths correct, or one sign error
Lengths counted incorrectly or missing
Real-World Use
Map distance, fence and area correct with units
Correct method with one computation or unit error
Method does not fit the problem
Explanation
Clearly says why slanted sides cannot be found this way
Partial explanation
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, and use grid paper to sketch 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
Kai plots (1, 1), (5, 1), (5, 4) and (1, 4) and connects them in order. What does he draw?
Answer: A
The bottom side goes from (1, 1) to (5, 1), so it is 5 - 1 = 4 units wide. The right side goes from (5, 1) to (5, 4), so it is 4 - 1 = 3 units tall. Choice C uses the largest coordinates, 5 and 4, as the lengths. Choice D adds the coordinates (5 + 1 and 4 + 1) instead of subtracting. Choice B assumes every side matches the 4-unit width.
Question 2 of 20 · Multiple Choice
The points are A(-2, 1), B(3, 1), C(3, -2) and D(-2, -2). In which order should you connect them to draw a rectangle whose sides do not cross?
Answer: C
A to B is the top side, B to C the right side, C to D the bottom side and D to A the left side, so no sides cross. Choice B goes from B to D and from C to A: both are diagonals of the rectangle (segments joining opposite corners), and they cross in the middle, making a bow-tie shape. Choices A and D also use two crossing diagonals.
Question 3 of 20 · Multiple Choice
How long is the segment from (4, 9) to (4, 2)?
Answer: B
The first coordinates are both 4, so the segment is vertical. Compare the second coordinates: 9 - 2 = 7 units. Choice A adds 9 + 2 even though both numbers are positive. Choice C subtracts the first coordinates, 4 - 4, which are the ones that stay the same. Choice D counts the grid dots from 2 to 9 (there are 8) instead of the spaces between them.
Question 4 of 20 · Multiple Choice
How long is the segment from (-5, 3) to (7, 3)?
Answer: D
The second coordinates are both 3, so the segment is horizontal. -5 and 7 have opposite signs, so the points are on opposite sides of the y-axis: add the absolute values, 5 + 7 = 12 units. Choice A subtracts 7 - 5 and ignores that -5 is on the other side of 0. Choice B subtracts the second coordinates, 3 - 3. Choice C counts 13 grid dots from -5 to 7 instead of 12 spaces.
Question 5 of 20 · Multiple Choice
How long is the segment from (-2, -6) to (-2, 4)?
Answer: B
The first coordinates are both -2, so the segment is vertical. The second coordinates, -6 and 4, have opposite signs, so add the absolute values: 6 + 4 = 10 units. Choice A subtracts 6 - 4 as if both points were on the same side of the x-axis. Choice C subtracts the matching first coordinates. Choice D counts 11 grid dots instead of 10 spaces.
Question 6 of 20 · Multiple Choice
How long is the segment from (-9, -4) to (-3, -4)?
Answer: A
The second coordinates match, so the segment is horizontal. -9 and -3 are both negative, so both points are left of the y-axis: subtract the absolute values, 9 - 3 = 6 units. Choice B adds 9 + 3, which only works when the signs are opposite. Choice C counts 7 grid dots. Choice D adds the absolute values of the matching second coordinates, 4 + 4, which do not measure this side.
Question 7 of 20 · Multiple Choice
Which two points are joined by a vertical segment?
Answer: D
A vertical segment joins two points with the same first coordinate. In choice D, both first coordinates are -1. Choice B has the same second coordinate, so its segment is horizontal, not vertical. In choices A and C, neither coordinate matches, so those segments are slanted. Choice A swaps the same two numbers, which does not make any coordinate match.
Question 8 of 20 · Multiple Choice
A rectangle has vertices (-3, -2), (5, -2), (5, 4) and (-3, 4). What is its perimeter?
Answer: C
The bottom side goes from -3 to 5, opposite signs, so it is 3 + 5 = 8 units. The right side goes from -2 to 4, so it is 2 + 4 = 6 units. Perimeter: 8 + 6 + 8 + 6 = 28 units. Choice A is the area, 8 × 6 = 48, not the distance around. Choice B adds only one length and one width. Choice D subtracts without the signs, 5 - 3 = 2 and 4 - 2 = 2, and then adds four sides of 2.
Question 9 of 20 · Multiple Choice
A rectangle has vertices (-4, 1), (2, 1), (2, 6) and (-4, 6). What is its area?
Answer: A
The width goes from -4 to 2, opposite signs, so it is 4 + 2 = 6 units. The height goes from 1 to 6, so it is 6 - 1 = 5 units. Area: 6 × 5 = 30 square units. Choice B is the perimeter, 6 + 5 + 6 + 5 = 22. Choice C finds the width as 4 - 2 = 2 by ignoring the signs, then multiplies 2 × 5. Choice D adds 6 + 5 instead of multiplying.
Question 10 of 20 · Multiple Choice
Three vertices of a rectangle are (-5, 2), (1, 2) and (1, -3). What is the fourth vertex?
Answer: D
The missing corner lines up under (-5, 2), so its first coordinate is -5, and it lines up with (1, -3), so its second coordinate is -3: the point is (-5, -3). Choice A has the right numbers in the wrong order. Choice B drops the negative sign from the first coordinate. Choice C drops the negative sign from the second coordinate.
Question 11 of 20 · Multiple Choice
On a map, each unit is 1 block. The pool is at (-6, 1) and the gym is at (4, 1). How many blocks apart are they along the street?
Answer: B
The second coordinates match, so the street between them is horizontal. -6 and 4 have opposite signs, so add: 6 + 4 = 10 blocks. Choice A subtracts 6 - 4, as if both places were on the same side of the y-axis. Choice C counts 11 street corners instead of 10 blocks. Choice D uses the matching second coordinate, 1, as the distance.
Question 12 of 20 · Multiple Choice
A park plan uses 1 unit = 5 meters. A straight path goes from (-3, -4) to (-3, 6). How long is the real path?
Answer: D
The first coordinates match, so the path is vertical. On the plan it is 4 + 6 = 10 units, and each unit is 5 meters, so the path is 10 × 5 = 50 meters. Choice B stops at 10 units and forgets the scale. Choice A subtracts 6 - 4 = 2 and forgets the scale. Choice C multiplies the first coordinate 3 by the scale.
Question 13 of 20 · Multiple Choice
Which list of vertices, connected in order, makes a square?
Answer: C
In choice C, the left side goes from -1 to 4, so it is 1 + 4 = 5 units, and the top side goes from -2 to 3, so it is 2 + 3 = 5 units. All four sides are 5 units, so it is a square. Choice A is 5 units wide but only 1 + 3 = 4 units tall: a student who counts the 5 grid dots on its left side may think it is a square. Choice B is 3 units wide and 5 units tall. Choice D is 7 units wide and 4 units tall.
Question 14 of 20 · Multiple Choice
A six-sided polygon has vertices (-3, -2), (4, -2), (4, 1), (1, 1), (1, 4) and (-3, 4). What is its perimeter?
Answer: A
The sides are 3 + 4 = 7, 2 + 1 = 3, 4 - 1 = 3, 4 - 1 = 3, 1 + 3 = 4 and 4 + 2 = 6 units. Their sum is 7 + 3 + 3 + 3 + 4 + 6 = 26 units. Choice B leaves out one of the 3-unit sides. Choice C is the area of the shape (7 × 3 + 4 × 3 = 33 square units), not the distance around. Choice D is the area of the whole 7-by-6 rectangle around the shape.
Question 15 of 20 · Short Answer
Plot the triangle with vertices (-3, -2), (4, -2) and (4, 5). Find the lengths of the two sides you can find with coordinates. Which side can you not find this way, and why?
The side from (-3, -2) to (4, -2) is horizontal: 3 + 4 = 7 units. The side from (4, -2) to (4, 5) is vertical: 2 + 5 = 7 units. The side from (-3, -2) to (4, 5) is slanted: neither coordinate stays the same, so subtracting coordinates does not give its length. It is longer than 7 units; its exact length comes in grade 8 with the Pythagorean Theorem.
Question 16 of 20 · Short Answer
Find the length of the segment from (-8, -3) to (-8, 5). Explain whether you added or subtracted, and why.
The first coordinates are both -8, so the segment is vertical. The second coordinates, -3 and 5, have opposite signs, so the points are on opposite sides of the x-axis. Add the absolute values: 3 + 5 = 8 units.
Question 17 of 20 · Short Answer
Three vertices of a rectangle are (-7, -5), (-7, 1) and (4, 1). Find the fourth vertex, the perimeter and the area.
The fourth vertex is (4, -5). The top side goes from -7 to 4: 7 + 4 = 11 units. The left side goes from -5 to 1: 5 + 1 = 6 units. Perimeter: 11 + 6 + 11 + 6 = 34 units. Area: 11 × 6 = 66 square units.
Question 18 of 20 · Short Answer
A school garden plan uses 1 unit = 2 feet. A rectangular flower bed has corners (1, 1), (8, 1), (8, 4) and (1, 4). How many feet of edging go around the bed?
On the plan, the bed is 8 - 1 = 7 units long and 4 - 1 = 3 units wide. With the scale, that is 7 × 2 = 14 feet and 3 × 2 = 6 feet. Edging: 14 + 6 + 14 + 6 = 40 feet. (Or find the perimeter in units, 20 units, then multiply by 2.)
Question 19 of 20 · Short Answer
On a map where each unit is 1 block, Ana walks from (-5, -4) east to (3, -4), then north to (3, 2). How many blocks does she walk in all?
East: the second coordinates match, and -5 and 3 have opposite signs, so 5 + 3 = 8 blocks. North: the first coordinates match, and -4 and 2 have opposite signs, so 4 + 2 = 6 blocks. Total: 8 + 6 = 14 blocks.
Question 20 of 20 · Short Answer
Leo says the distance from (-7, 1) to (3, 1) is 4 units, because 7 - 3 = 4. Explain his mistake and find the correct distance.
Leo subtracted, but -7 and 3 have opposite signs, so the points are on opposite sides of the y-axis. The point (-7, 1) is 7 units left of the axis and (3, 1) is 3 units right of it, so the distance is 7 + 3 = 10 units. Subtracting only works when both numbers have the same sign.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.G.A.3 mean?
6.G.A.3 means students can draw a polygon on the coordinate plane when they are given the coordinates of its vertices, and can find the length of any horizontal or vertical side from the coordinates. They then use those lengths in real problems, such as distances on a map or the fence around a garden.
Is 6.G.A.3 a grade 6 or grade 7 standard?
It is a grade 6 standard in the Geometry domain. It builds on graphing points in all four quadrants (6.NS.C.8), which is taught in the same year. In grade 7, students use coordinates and scale in scale drawings (7.G.A.1), and in grade 8 they find slanted distances with the Pythagorean Theorem (8.G.B.8).
How do you find the length of a side from its coordinates?
Find the coordinate that stays the same, then compare the other one. If the first coordinates match, the side is vertical and you compare the second coordinates. If the second coordinates match, the side is horizontal and you compare the first coordinates. For (2, -1) and (2, 5), the length is 1 + 5 = 6 units.
Why do you add the numbers when the points are on opposite sides of an axis?
Because the side crosses the axis, so it is made of two pieces. Each piece is the distance from a point to the axis, which is the absolute value of its coordinate. For (-3, 2) and (6, 2), one piece is 3 units and the other is 6 units, so the side is 3 + 6 = 9 units. When both points are on the same side, the shorter distance is inside the longer one, so you subtract.
Can students find the length of a slanted side in grade 6?
Not with 6.G.A.3. The standard only covers sides joining points with the same first coordinate or the same second coordinate. Students can say that a slanted side is longer than the horizontal or vertical distance it spans, but its exact length needs the Pythagorean Theorem, which comes in grade 8.
What mistakes do students often make with 6.G.A.3?
A common mistake is subtracting the absolute values when the points are on opposite sides of an axis, for example 5 - 2 = 3 for (-5, 1) and (2, 1) instead of 5 + 2 = 7. Other mistakes are counting grid dots instead of spaces, plotting (x, y) as (y, x), and connecting the vertices out of order so that the sides cross.
Why does the order of the vertices matter?
The polygon is drawn by connecting the vertices in the order they are listed, and then back to the first. The same four points can make a rectangle or a crossed bow-tie shape depending on the order. When you are given the vertices of a named shape, such as rectangle ABCD, connect A to B, B to C, C to D and D back to A.
Do students need a formula to find area on the coordinate plane?
Only the area formulas they already know. Once they find the length and width from coordinates, they multiply for a rectangle, or split a polygon into rectangles and triangles as in 6.G.A.1. The coordinates give the lengths; the area methods are the same as on paper.
Where is 6.G.A.3 used in real life?
Anywhere a grid describes places or plans: city maps with streets on a grid, floor plans, garden plans, sports fields drawn to scale and video game levels. For example, a map with 1 unit = 1 block turns coordinates into walking distances, and a plan with 1 unit = 2 feet turns them into real lengths of fence or edging.
How can parents help with coordinate plane geometry at home?
Play a grid game: one person names the vertices of a secret shape, and the other plots them and names the shape. Ask your child to find each side length from the numbers before counting squares, and to say whether they added or subtracted and why. Grid paper and a ruler are all you need.
07
Related Standards
6 standards
These standards connect to 6.G.A.3: 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.6Prerequisite
Understand rational numbers as points on the number line and the coordinate plane
