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6th Grade · 6.G.A.3

Polygons on the coordinate plane in 6th grade

Polygons on the coordinate plane are flat shapes with straight sides, drawn on a grid by plotting their corners. In 6th grade your child plots these shapes from a list of points and finds the length of sides that run straight across or straight up and down. Those lengths then give the perimeter and the area.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.G.A.3 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.

What it means, in plain words

The coordinate plane is a grid with a horizontal number line, the x-axis, and a vertical one, the y-axis. They cross at (0,0)(0, 0). Every point has an address called an ordered pair, (x,y)(x, y). The first number says how far to go right or left, and the second says how far to go up or down. In 6th grade the grid includes negative numbers, so points can sit left of 0 or below it.

A polygon is a flat shape with straight sides, such as a rectangle or a triangle. Its corners are called vertices. To draw polygons in the coordinate plane, your child plots each vertex from its ordered pair and joins the points in the order given.

Then comes distance on the coordinate plane. When two vertices have the same x-coordinate, the side between them is vertical, and its length is the difference of the y-coordinates. When they have the same y-coordinate, the side is horizontal, and its length is the difference of the x-coordinates. When a side crosses an axis, your child adds the distance on each side of 0. With the side lengths, children find the perimeter (the distance around) and the area of polygons on the coordinate plane, such as a park on a map or a room on a floor plan.

Your child should be able to:

  • Plot a polygon from a list of vertices and join the points in order
  • Find the length of a side when its two end points share an x-coordinate or a y-coordinate
  • Find a length that crosses an axis, such as from −4-4 to 5
  • Use side lengths to find the perimeter and the area of a rectangle on the grid

Example

A rectangle has vertices at (2,3)(2, 3), (9,3)(9, 3), (9,8)(9, 8) and (2,8)(2, 8). What are its perimeter and its area?

  1. (2,3)(2, 3) and (9,3)(9, 3) share the y-coordinate 3, so that side is horizontal. Its length is 9−2=79 - 2 = 7 units.
  2. (9,3)(9, 3) and (9,8)(9, 8) share the x-coordinate 9, so that side is vertical. Its length is 8−3=58 - 3 = 5 units.
  3. Perimeter: 7+5+7+5=247 + 5 + 7 + 5 = 24 units. Area: 7×5=357 \times 5 = 35 square units.
  4. The rectangle has a perimeter of 24 units and an area of 35 square units.

Example

How long is the side from (−4,2)(-4, 2) to (5,2)(5, 2)?

  1. Both points have the y-coordinate 2, so the side is horizontal.
  2. From −4-4 to 0 is 4 units. From 0 to 5 is 5 units.
  3. 4+5=94 + 5 = 9, so the side is 9 units long.

Where children usually get stuck

  • To plot (3,5)(3, 5), your child goes up 3 and then right 5.

    The order of the two numbers is an agreement, and nothing in the numbers shows which comes first. The child guesses, and the point lands in the wrong place.

    What helps. Use the phrase "walk, then climb": walk along the floor to the first number, then climb up to the second. Say it out loud for the first few points.

  • For the side from (−4,2)(-4, 2) to (5,2)(5, 2), your child writes 5−4=15 - 4 = 1 unit.

    The minus sign on −4-4 gets dropped, so the child treats both points as if they were on the same side of the y-axis.

    What helps. Have your child count the grid squares between the two points. There are 9. Then show the two parts: 4 squares to reach 0 and 5 more after it.

  • For a side that runs from x=2x = 2 to x=9x = 9, your child counts the grid points and gets 8 instead of 7.

    Counting points includes both ends of the side. Length counts the spaces between the points, not the points themselves.

    What helps. Have your child put a finger on the first point and count each jump to the next point. The first count comes after the first jump, not on the starting point.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    A garden is shown on the grid below. Its corners, in order, are (1,1), (4,1), (4,4), and (1,4), where each unit on the grid represents 1 yard. How much fencing is needed to go around the garden?
    Give your answer in yards. Enter the total amount and choose the matching unit.
    AmountUnit
    Show the answer and how to solve it

    The answer is 12 yards because the four side lengths are 3, 3, 3, and 3, and fencing is a linear (1-dimensional) measurement in yards.

    1. 1Look at the four outside sides of the garden on the grid. Every side is horizontal or vertical, so each side length is the difference between the coordinates of its endpoints.
    2. 2Use the vertices in order to find each side length. Be sure to include the closing side from (1,4) back to (1,1).
    3. 3Add all four side lengths to get the total distance around the outside. Each grid unit represents 1 yard, and fencing measures distance around the outside—a length (1-dimensional)—so the unit is yards, not yards2(area)yards^{2} (area) or yards3(volume)yards^{3} (volume). The garden needs 12 yards of fencing.

    Common mistake. A common mistake is choosing yards2yards^{2} for the unit, which measures the area inside the garden rather than the distance around it. Another mistake is adding only two sides (3+3=6)(3 + 3 = 6) instead of all four sides.

    The idea underneath. For an orthogonal polygon, each side length is just the difference in one coordinate, so the total fencing is the sum of those horizontal and vertical differences. Fencing is a length around the outside, so the unit must be a 1-dimensional linear unit (yards), not a 2D unit (yards2)(yards^{2}) or 3D unit (yards3)(yards^{3}).

  2. 2On level
    The table below shows three vertices of a rectangle with horizontal and vertical sides and Maya's proposed fourth vertex.
    PointOrdered pair
    A(-5, 1)
    B(-5, 6)
    C(3, 1)
    Maya's point(3, -6)
    Select the best description of Maya's proposal.
    A
    Repeated a listed vertex
    B
    Swapped x and y
    C
    Correct
    D
    Changed a sign
    Show the answer and how to solve it

    The correct choice is Changed a sign because the missing vertex is (3, 6), but Maya wrote (3, −6).

    1. 1Look at the table. A rectangle with horizontal and vertical sides uses only two x-values and two y-values.
    2. 2The missing corner is directly above C, so its x-coordinate stays 3. It is level with B, so its y-coordinate is 6. The missing vertex is (3, 6).
    3. 3Maya proposed (3, −6). The x-coordinate matches, but the y-coordinate should be 6, not −6. Only the sign on the y-coordinate was changed.
    4. 4So the best description is Changed a sign.

    Common mistake. A common mistake is choosing Swapped x and y because the numbers 3 and 6 are both present. A true swap of (3, 6) would give (6, 3), not (3, −6).

    The idea underneath. The missing vertex is the unused combination of the rectangle's two x-values and two y-values.

  3. 3On level
    Three vertices of a rectangle with horizontal and vertical sides are shown on the coordinate plane below. Which ordered pair is the missing vertex?
    -5-4-3-2-1123451234560xyABC
    Select the missing ordered pair.
    A
    (2, 1)
    B
    (2, 4)
    C
    (4, 2)
    D
    (-2, 4)
    Show the answer and how to solve it

    The missing vertex is (2, 4) because it pairs the unmatched x-value (2) with the unmatched y-value (4) and completes the rectangle.

    1. 1Look at the graph. The three plotted points are A(-3, 1), B(-3, 4), and C(2, 1).
    2. 2A and B share x = -3, so they form one vertical side. A and C share y = 1, so they form one horizontal side.
    3. 3The missing corner must line up vertically with C, so it needs x = 2. It must line up horizontally with B, so it needs y = 4. The missing vertex is (2, 4).

    Common mistake. A common mistake is choosing (2, 1), which is already point C on the graph. The missing corner keeps x = 2 but needs y = 4, not y = 1.

    The idea underneath. An axis-aligned rectangle uses exactly two x-values and two y-values, so the missing vertex is made by pairing the x that appears once with the y that appears once.

  4. 4Stretch
    The table below lists a polygon's vertices in order.
    VertexOrdered pair
    A(4, 2)
    B(1, 2)
    C(1, 5)
    D(-3, 5)
    E(-3, -2)
    F(4, -2)
    Enter the perimeter of the polygon.
    Perimeter:
    Show the answer and how to solve it

    Answer. 28

    The perimeter is 28 because the 6 side lengths around the polygon add to 28.

    1. 1Look at the table. The vertices are already in boundary order, so use A to B, B to C, C to D, D to E, E to F, and F back to A.
    2. 2Each side is horizontal or vertical, so only one coordinate changes. For sides that cross zero, subtracting a negative adds: for example, 1−(−3)=41 - (-3) = 4 and 5−(−2)=75 - (-2) = 7. Always use the positive distance.
    3. 3Add all 6 side lengths.
    4. 4So the perimeter is 28.

    Common mistake. A common mistake is treating DE as 5+(−2)=35 + (-2) = 3 rather than the positive distance 5−(−2)=75 - (-2) = 7. Distance is always positive; never add the negative sign of a coordinate into your result.

    The idea underneath. On a horizontal or vertical side, the distance comes from one coordinate difference because the other coordinate stays the same.