Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.7Represent Polygons on the Coordinate Plane

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can draw polygons from their vertex coordinates on the coordinate plane, and use coordinates to find side lengths and solve perimeter and area problems.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
vertexSpanish: vértice A corner point of a polygon, named by an ordered pair on the coordinate plane. named by an ordered pair
polygonSpanish: polígono A closed figure made of line segments; on the plane you draw it by plotting the vertices and connecting them in order. triangle, rectangle, square
perimeterSpanish: perímetro The total distance around a polygon — the sum of all its side lengths. add every side length
distance between verticesSpanish: distancia entre vértices When two vertices share an x- or y-coordinate, the distance between them is the absolute difference of the other coordinates. (2, 4) and (5, 4) share y, so the side is |5 − 2| = 3 units.
scaleSpanish: escala What one grid unit stands for in the real situation. 1 unit = 1 meter
Side lengthSpanish: Longitud del lado The distance between two vertices that are joined by a side. found with absolute value

How it worksWorked examplefinding the perimeter of a rectangle from its vertices

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A rectangle has vertices at (−8.5, 7.5), (5.5, 7.5), (5.5, −3.5), and (−8.5, −3.5). Find its perimeter.

  1. Plot the four vertices and connect them in order to draw the rectangle.
  2. Top side: the y-coordinates match, so use the x-values: −8.5 + 5.5 = 14
  3. Left side: the x-coordinates match, so use the y-values: 7.5 + −3.5 = 11
  4. Perimeter adds all four sides: 14 + 11 + 14 + 11 = 50

Answer: The rectangle measures 14 units by 11 units, so its perimeter is 50 units.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A rectangle has vertices at (−6.25, 9.75), (2.75, 9.75), (2.75, −3.25), and (−6.25, −3.25). Find its perimeter.

  1. Plot the four vertices and connect them in order to draw the rectangle.
  2. Top side: the -coordinates match, so use −6.25 + 2.75 =
  3. Left side: the -coordinates match, so use + =
  4. Perimeter adds all four sides: + + + =
Answer:perimeter of the rectangle, in units

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • Vertices and share the -coordinate, so that side runs .
  • The length of that side is units because .
  • The perimeter is units because I added .
  • This polygon is a because .

Word bank vertexpolygonperimeterside lengthcoordinate planeabsolute valuehorizontalverticalrectanglesquareaddunits

Watch outA common mistake

Subtracting coordinates without the absolute value — from y = 3 down to y = −3.75 the distance is 3 + 3.75 = 6.75 units, not 3.75 − 3 = 0.75. Distances that cross zero are the whole reason the absolute value is there.

Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.7Represent Polygons on the Coordinate Plane

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can draw polygons from their vertex coordinates on the coordinate plane, and use coordinates to find side lengths and solve perimeter and area problems.

Part 1Apply itreal-world problems
  1. 1Circle the letter of the best answer. Show how you know.

    Vertices (-4.5, 3) and (5, 3) share y = 3. What is the distance between them?

    1. A0.5 units
    2. B1.5 units
    3. C8 units
    4. D9.5 units

    Hint Which coordinate is the same? Which differs?

    My work — one step per lineWhat I did to both sides
    1. 1LookWhat is done to the variable?
    2. 2UndoDo the opposite — both sides.
    3. 3SimplifyOne side at a time.
    4. 4CheckPut it back in.
  2. 2Circle the letter of the best answer. Show how you know.

    Vertices (5, 3) and (5, -3.75) share x = 5. What is the distance between them?

    1. A0.75 units
    2. B5 units
    3. C6.75 units
    4. D8.75 units

    Hint Which coordinate differs?

    My work — one step per lineWhat I did to both sides
    1. 1LookWhat is done to the variable?
    2. 2UndoDo the opposite — both sides.
    3. 3SimplifyOne side at a time.
    4. 4CheckPut it back in.
  3. 3Circle the letter of the best answer. Show how you know.

    What is the perimeter of the rectangle with side lengths 9.5 and 6.75 units?

    1. A16.25 units
    2. B26 units
    3. C32.5 units
    4. D64.125 units

    Hint A rectangle has four sides.

    Formula
    Put the numbers in
    Work it out
    Answer with its unit
    1. 1Name itWhich measure is asked?
    2. 2Write the formulaBefore any numbers.
    3. 3SubstituteMatch each letter.
    4. 4Unitunits, sq units, or cubic?
  4. 4Circle the letter of the best answer. Show how you know.

    Why are two of the terms multiplied by 2 in the perimeter calculation?

    1. ABecause perimeter always doubles the area
    2. BBecause there are two coordinates in each ordered pair
    3. CIt is a rounding rule
    4. DA rectangle's opposite sides are equal, so each length appears on two sides

    Hint Walk the rectangle: how many 9.5 sides?

    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    To draw a polygon from a list of vertices, you:

    1. APlot only the first vertex
    2. BConnect the points in any random order
    3. CAdd all the coordinates together
    4. DPlot each ordered pair, then connect them in order with segments

    Hint What did the deck say to do first?

    Show your work
Part 2Explain and correctreasoning
  1. 6Answer in complete sentences.

    A square has vertices (1, 2), (6, 2), (6, -3), (1, -3). Find one side length and the perimeter.

    Sentence starter The vertices ___ and ___ share ___ .
    So one side is ___ , and the perimeter is ___ .
    Write your answer

Explain your thinkingTake a side you both put under 'Slanted.' Why does subtracting coordinates give the wrong length there, and what has to be true about two vertices for subtraction to work?

Sentence starter Subtracting works when the two vertices share ___, so for ___ I cannot ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.7Represent Polygons on the Coordinate Plane

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can draw polygons from their vertex coordinates on the coordinate plane, and use coordinates to find side lengths and solve perimeter and area problems.

Part 1Apply itreal-world problems
  1. 1Circle the letter of the best answer. Show how you know.

    The park's marked vertices are 3 units apart and each unit is 100 feet. What is the park's width?

    1. A3 feet
    2. B30 feet
    3. C103 feet
    4. D300 feet
    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    The park's area must be 240,000 sq ft and its width is 300 ft. What is its length?

    1. A720 feet
    2. B800 feet
    3. C80,000 feet
    4. D240,300 feet
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  3. 3Circle the letter of the best answer. Show how you know.

    The length is 800 feet and each unit represents 100 feet. How many units long is the park on the map?

    1. A0.8 units
    2. B8 units
    3. C80 units
    4. D800 units
    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    The known vertices of the park are at (2, 4) and (5, 4), and the length runs DOWNWARD 8 units. Where are the other two vertices?

    1. A(2, -4) and (5, -4)
    2. B(2, 12) and (5, 12)
    3. C(2, -8) and (5, -8)
    4. D(10, 4) and (13, 4)
    Show your work
  5. 5Circle the letter of the best answer. Show how you know.

    A rectangle has vertices (-3, 2), (4, 2), (4, -1), (-3, -1). What is its perimeter?

    1. A10 units
    2. B13 units
    3. C20 units
    4. D21 units
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
Part 2Explain and correctreasoning
  1. 6Answer in complete sentences.

    Sheng plans a zoo visit using a map where exhibits sit at grid points. Explain how identifying WHAT the problem asks (his shortest route? total distance?) changes what you compute.

    Write your answer

Explain your thinkingTake a side you both put under 'Slanted.' Why does subtracting coordinates give the wrong length there, and what has to be true about two vertices for subtraction to work?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.7Represent Polygons on the Coordinate Plane

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can draw polygons from their vertex coordinates on the coordinate plane, and use coordinates to find side lengths and solve perimeter and area problems.

Part 1Understand the ideamodels and meaning
  1. 1Plot and label each point on the grid.

    Plot the vertices of rectangle PQRS, then use the coordinates to find its perimeter.

    012345678012345678xy
    Show your thinking
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Maria is choosing between two store spaces on a grid where each unit is 1 foot and each grid line is 5 feet apart. Space A is a rectangle 25 ft by 40 ft; Space B is 30 ft by 30 ft. Which has more floor area?

    1. ASpace A — 1,000 sq ft vs 900 sq ft
    2. BSpace B — 900 sq ft vs 850 sq ft
    3. CThey are equal
    4. DSpace B — because a square always beats a rectangle
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    A square has vertices (a, b) and (a, b + s) on one side. Which operation gives the side length for ANY values?

    1. AAdd all four coordinates
    2. BSubtract the y-coordinates and take the absolute value
    3. CMultiply a × b
    4. DAverage the coordinates
    Show your work
Part 3Explain and correctreasoning
  1. 4Answer in complete sentences.

    The park design used width from coordinates, then length from the area, then the scale to place the missing vertices. Design your own version: pick an area and a scale, mark two vertices, and show where the other two go.

    Write your answer
  2. 5Answer in complete sentences.

    Check the deck's perimeter of 32.5 units a DIFFERENT way than adding the four sides in order.

    Write your answer

Explain your thinkingTake a side you both put under 'Slanted.' Why does subtracting coordinates give the wrong length there, and what has to be true about two vertices for subtraction to work?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help