# 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.

### Problem 1

#### Pre-unit

Practicing Standards:  3.G.A.1

1. Which shapes are rectangles? ________________________

2. Which shapes are rhombuses? ________________________

3. Which shapes are squares? ________________________

### Solution

For access, consult one of our IM Certified Partners.

### Problem 2

#### Pre-unit

Practicing Standards:  4.G.A.1

1. Name two lines in the drawing that are parallel.___________________

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

### Solution

For access, consult one of our IM Certified Partners.

### 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?

### Solution

For access, consult one of our IM Certified Partners.

### Problem 4

#### Pre-unit

Practicing Standards:  4.G.A.2

Which of the triangles are right triangles?

### Solution

For access, consult one of our IM Certified Partners.

### Problem 5

1. How would you describe the point labeled $$P$$?

2. How would you describe the rectangle R?

### Solution

For access, consult one of our IM Certified Partners.

### Problem 6

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

2. Locate and label point $$A$$ with coordinates $$(7, 1)$$, point $$B$$ with coordinates $$(2, 8)$$, and point $$C$$ with coordinates $$(6, 6)$$.

### Solution

For access, consult one of our IM Certified Partners.

### 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?

2. Locate the points $$(1,1)$$, $$(2,2)$$, $$(3,3)$$, and $$(4,4)$$ on the graph. What do you notice about the points?

### Solution

For access, consult one of our IM Certified Partners.

### 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?

2. Describe, using coordinates, how to trace a letter from your name.

### Solution

For access, consult one of our IM Certified Partners.

### 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.

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.

### Solution

For access, consult one of our IM Certified Partners.

### 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.

### Solution

For access, consult one of our IM Certified Partners.

### Problem 2

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

For access, consult one of our IM Certified Partners.

### 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

### Solution

For access, consult one of our IM Certified Partners.

### 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

For access, consult one of our IM Certified Partners.

### 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

### Solution

For access, consult one of our IM Certified Partners.

### 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

For access, consult one of our IM Certified Partners.

### 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.

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?

### Solution

For access, consult one of our IM Certified Partners.

### 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

For access, consult one of our IM Certified Partners.

### 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

For access, consult one of our IM Certified Partners.

### 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.

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

For access, consult one of our IM Certified Partners.

### Problem 4

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

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

For access, consult one of our IM Certified Partners.

### Problem 5

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

For access, consult one of our IM Certified Partners.

### Problem 6

#### Exploration

area of base (square inches) height (inches)

1. The volume of a box is 240 cubic inches. List some possible values for the area of 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 measurements for the box? Explain your reasoning.

### Solution

For access, consult one of our IM Certified Partners.

### 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

For access, consult one of our IM Certified Partners.