5.7 Shapes on the Coordinate Plane

Unit Goals

  • Students plot coordinate pairs on a coordinate grid and classify triangles and quadrilaterals in a hierarchy based on properties of side length and angle measure. They generate, identify, and graph relationships between corresponding terms in two numeric patterns, given two rules, and represent and interpret real world and mathematical problems on a coordinate grid.

Section A Goals

  • Locate points on a coordinate grid.

Section B Goals

  • Classify triangles and quadrilaterals in a hierarchy based on angle measurements and side lengths.

Section C Goals

  • Generate, identify, and graph relationships between corresponding terms in two patterns, given a rule.
  • Represent and interpret real world and mathematical problems on a coordinate grid.
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Section A: The Coordinate Plane

Problem 1

Pre-unit

Practicing Standards:  3.G.A.1

4 shapes on grid. All have 2 sets of parallel sides. B and C each have 4 right angles. B, 3 by 5. C, 3 by 3. Diagonals of A, 2 and 4. D, base, 5, height, 3.

  1. Which shapes are rectangles? ________________________

  2. Which shapes are rhombuses? ________________________

  3. Which shapes are squares? ________________________

Solution

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Problem 2

Pre-unit

Practicing Standards:  4.G.A.1

lines M and N. Intersected by lines L and K.
  1. Name two lines in the drawing that are parallel.___________________

  2. Name two lines in the drawing that are perpendicular.___________________

Solution

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Problem 3

Pre-unit

Practicing Standards:  3.MD.D.8

  1. Draw a rectangle on the grid.
  2. What is the perimeter of the rectangle?
grid, 8 by 12 units.

Solution

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Problem 4

Pre-unit

Practicing Standards:  4.G.A.2

Which of the triangles are right triangles?

Solution

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Problem 5

  1. How would you describe the point labeled \(P\)?

  2. How would you describe the rectangle R?

Coordinate plane. Rectangle R, 0 comma 2, 0 comma 5, 7 comma 2, 7 comma 5. Point, P, 8 comma 9.

Solution

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Problem 6

  1. What are the coordinates of the point on the grid?

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's. Point graphed at 9 comma 3.
  2. Locate and label point \(A\) with coordinates \((7, 1)\), point \(B\) with coordinates \((2, 8)\), and point \(C\) with coordinates \((6, 6)\).

Solution

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Problem 7

  1. Fill in the blanks with 4 different numbers between 0 and 10

    and plot the points on the graph.

    \((\underline{\hspace{1 cm}},0)\)  \((\underline{\hspace{1 cm}},0)\)

    \((\underline{\hspace{1 cm}},0)\)  \((\underline{\hspace{1 cm}},0)\)

    What do you notice about the points?

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.
  2. Locate the points \((1,1)\), \((2,2)\), \((3,3)\), and \((4,4)\) on the graph. What do you notice about the points?

Solution

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Problem 8

Exploration

  1. One of the Illustrative Math characters describes their first initial like this: “Start at \((2,4)\) and go to \((2,8)\), then go to \((3,6)\), then to \((4,8)\), and to \((4,4)\).” Which character is it?

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.
  2. Describe, using coordinates, how to trace a letter from your name.

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.

Solution

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Problem 9

Exploration

Work with a partner.

Student 1:

Draw a rectangle on the coordinate plane. Make sure the area of your rectangle is at least 20 square units. Don't show your partner your rectangle.

Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.

Student 2:

Your goal is to figure out which rectangle your partner drew. You name points in the coordinate plane and your partner will tell you whether the point is on their rectangle.

Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.

Solution

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Section B: The Hierarchy of Shapes

Problem 1

Determine whether the statement is true or false. Explain or show your reasoning.
  1. The shape is a rectangle.
  2. The shape is a square.
  3. The shape is a rhombus.
Quadrilateral on grid. all sides equal length. no right angles. 

Solution

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Problem 2

blank grid
blank grid
  1. Draw a trapezoid that is also a parallelogram. Explain how you know it is a trapezoid and a parallelogram.
  2. Draw a trapezoid that is not a parallelogram. Explain how you know it is a trapezoid but is not a parallelogram.

Solution

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Problem 3

Determine if you can make each given shape so that it contains these two sides. Explain your reasoning.
  1. a square
  2. a rectangle
  3. a rhombus
2 line segments of equal length on grid. do not meet at a right angle.

Solution

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Problem 4

Decide if each statement is true or false. Explain or show your reasoning.
  1. A parallelogram is sometimes a rhombus.
  2. A rhombus is always a parallelogram.
  3. A trapezoid is never a rectangle.
  4. A rectangle is never a square.
  5. A parallelogram is always a trapezoid.

Solution

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Problem 5

For each description, draw a right triangle with the described side lengths on the grid or explain why there is no such right triangle.
  1. 2 equal side lengths
  2. 3 equal side lengths
  3. 3 different side lengths
blank grid

Solution

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Problem 6

Exploration

  1. Jada cut a quadrilateral in half, from one vertex to the opposite vertex, and she got two isosceles triangles. What kind of quadrilateral could Jada have cut in half? Explain or show your reasoning.

  2. Elena put together two right triangles to make a quadrilateral. What kind of quadrilateral could Elena have made? Explain or show your reasoning.

Solution

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Problem 7

Exploration

  1. Can you find a square on the grid that does not have a vertical or horizontal side? Explain or show your reasoning.

    blank grid

  2. Draw the line segment from \((4, 4)\) to \((6, 5)\). Can you find a square that contains this segment as one of its sides?

    Coordinate grid. horizontal and vertical axis, 0 to 10, by 1's.

Solution

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Section C: Numerical Patterns

Problem 1

  1. List the first ten numbers starting at 0 and counting by 5s.
  2. List the first ten numbers starting at 0 and counting by 10s.
  3. What patterns do you observe between your two lists of numbers?

Solution

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Problem 2

  1. List the first ten numbers starting at 0 and counting by 6.
  2. List the first ten numbers starting at 4 and counting by 6.
  3. When the first list has the number 222, what number will be on the second list? Explain or show your reasoning.

Solution

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Problem 3

The points on the graph, starting in the bottom left and moving up and to the right, represent how Han and Mai counted.

Coordinate plane. Horizontal axis, Han's count, 0 to 25, by 1's. Vertical axis, Mai's count, 0 to seventy 5, by 5's. 

  1. How much is Han adding each time in his count? Explain how you know.

  2. How much is Mai adding each time in her count? Explain you know.

  3. Name and locate 3 more points on the graph.

Solution

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Problem 4

The points on the graph show the results Lin and Tyler got when they tossed a coin.

Coordinate plane. Horizontal axis, number of heads, 0 to 10, by 1's. Vertical axis, number of tails, 0 to 10, by 1's. Lin, 3 comma 5. Tyler, 6 comma 3. 

  1. Who tossed the coin more times, Lin or Tyler? Explain or show your reasoning.
  2. Who got more tails, Lin or Tyler? Explain or show your reasoning.
  3. Toss a coin 7 times and plot the point on the graph. Explain or show your reasoning.

Solution

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Problem 5

Coordinate plane. Horizontal axis, length in centimeters, 0 to 10, by 1's. Vertical axis, width in centimeters, 0 to 10, by 1's. point at 3 comma 7. 

  1. The point on the graph shows the length and width of a rectangle. What is the perimeter of the rectangle?
  2. Plot 4 more points for different rectangles with the same perimeter as the given rectangle.
  3. Which point would represent a square with the same perimeter as the given rectangle?

Solution

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Problem 6

Exploration

area of base (square inches) height (inches)

Coordinate plane. Horizontal axis, area of base in square inches, 0 to 3 hundred, by 20's. Vertical axis, height in inches, 0 to 10, by 1's. 
  1. The volume of a box is 240 cubic inches. List some possible areas for the base of the box and for its height in the table.
  2. Plot several different possible area and height pairs on the graph.
  3. What do you notice about the points on the graph?
  4. Which point do you think represents the most reasonable side lengths for the box? Explain your reasoning.

Solution

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Problem 7

Exploration

  • Andre starts from 2 and counts by 6s.
  • Clare starts at 1,000 and counts back by 7s.
  1. List the first 6 numbers Andre and Clare say.
  2. Do Andre and Clare ever say the same number in the same spot on their lists? Explain or show your reasoning.

Solution

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