Grade 7, Unit 3 - Practice Problems (2024)

Select all quantities that are proportional to the diagonal length of a square.

1. Area of a square
2. Perimeter of a square
3. Side length of a square

Diego made a graph of two quantities that he measured and said, “The points all lie on a line except one, which is a little bit above the line. This means that the quantities can’t be proportional.” Do you agree with Diego? Explain.

The graph shows that whileit was being filled, theamount of water in gallons in a swimming pool was approximately proportional to the time that has passed in minutes.

1. About how much water was in the pool after 25 minutes?
2. Approximately when were there 500 gallons of water in the pool?
3. Estimate the constant of proportionality for the number of gallons of water per minute going into the pool.

Here is a circle with center $H$ andsome line segments and curves joining points on the circle.

Identify examples of the following. Explain your reasoning.

1. Diameter

2. Radius

Diego measured the diameter and circumference of several circular objects and recorded his measurements in the table.

One of his measurements is inaccurate. Which measurement is it? Explain how you know.

Complete the table. Use one of the approximate values for $\pi$ discussed in class (for example 3.14, $\frac{22}{7}$, 3.1416). Explain or show your reasoning.

1. Name a segment that is a radius. How long is it?
2. Name a segment that is a diameter. How long is it?

1. Consider the equation $y=1.5x +2$. Find four pairs of $x$ and $y$ values that make the equation true. Plot the points $(x, y)$ on the coordinate plane.
2. Based on the graph, can this be a proportional relationship? Why or why not?

Here is a picture of a Ferris wheel. It has a diameter of 80 meters.

1. On the picture, draw and label a diameter.

2. How far does a rider travel in one complete rotation around the Ferris wheel?

Identify each measurement as the diameter, radius, or circumference of the circular object. Then, estimate the other two measurements for the circle.

1. The length of the minute hand on a clock is 5 in.

2. The distance across a sink drain is 3.8 cm.

3. The tires on a mining truck are 14 ft tall.

4. The fence around a circular pool is 75 ft long.

5. The distance from the tip of a slice of pizza to the crust is 7 in.

6. Breaking a cookie in half creates a straight side 10 cm long.

7. The length of the metal rim around a glass lens is 190 mm.

8. From the center to the edge of a DVD measures 60 mm.

A half circle is joined to an equilateral triangle with side lengths of 12 units. What is the perimeter of the resulting shape?

The length of segment $AE$ is 5 centimeters.

Find the area of the polygon.

1. Draw polygons on the map that could be used to approximate the area of Virginia.
   
2. Which measurements would you need to know in order to calculate an approximation of the area of Virginia? Label the sides of the polygons whose measurements you would need. (Note: You aren’t being asked to calculate anything.)

Here are several recipes for sparkling lemonade. For each recipe describe how many tablespoons of lemonade mix it takes per cup of sparkling water.

Recipe 1: 4 tablespoons lemonade mix and 12 cups of sparkling water

Recipe 2: 4 tablespoons of lemonade mix and 6 cups of sparkling water

Recipe 3: 3 tablespoons of lemonade mix and 5 cups of sparkling water

Recipe 4: $\frac12$ tablespoon os lemonade mix and $\frac34$ cups of sparkling water

The $x$-axis of each graph has the diameter of a circle in meters. Label the $y$-axis on each graph with the appropriate measurement of a circle: radius (m), circumference (m), or area (m²).

1. Here is a picture of two squares and a circle. Use the picture to explain why the area of this circle is more than 2 square units but less than 4 square units.

   

2. Here is another picture of two squares and a circle. Use the picture to explain why the area of this circle is more than 18 square units and less than 36 square units.

   

Find the circumference of this circle.

The picture shows a circle divided into 8 equal wedges which are rearranged.

The radius of the circle is $r$ and its circumference is $2\pi r$. How doesthe picture help toexplain why the area of thecircle is $\pi r^2$?

The Carousel on the National Mall has 4 rings of horses. Kiran is riding on the inner ring, which has a radius of 9 feet. Mai is riding on the outer ring, which is 8 feet farther out from the center than the inner ring is.

1. In one rotation of the carousel, how much farther does Mai travel than Kiran?

2. One rotation of the carousel takes 12 seconds. How much faster does Mai travel than Kiran?

Here are the diameters of four coins:

1. A coin rolls a distance of 33cm in 5rotations. Which coin is it?
2. A quarter makes 8rotations. How far did it roll?
3. A dimerolls 41.8cm. How many rotations did it make?

Find the area of the shaded region. Express your answer in terms of $\pi$.

Which of these pairs of quantities are proportional to each other? For the quantities that are proportional, what is the constant of proportionality?

1. Radius and diameter of a circle
2. Radius and circumference of a circle
3. Radius and area of a circle
4. Diameter and circumference of a circle
5. Diameter and area of a circle

Find the area of this shape in two different ways.

Elena and Jada both read at a constant rate, but Elena reads more slowly. For every 4 pages that Elena can read, Jada can read 5.

1. Complete the table.

   row 1	pages read by Elena	pages read by Jada
   row 2	4	5
   row 3	1
   row 4	9
   row 5	$s$
   row 6		15
   row 7		$j$

2. Here is an equation for the table: $j = 1.25e$. What does the 1.25 mean?
3. Write an equation for this relationship that starts $e = \text{...}$

For each problem, decide whether the circumference of the circle or the area of the circle is most useful for finding a solution. Explain your reasoning.

1. A car’s wheels spin at 1000 revolutions per minute. The diameter of the wheels is 23 inches. You want to know how fast the car is travelling.

2. A circular kitchen table has a diameter of 60 inches. You want to know how much fabric is needed to cover the table top.

3. A circular puzzle is 20 inches in diameter. All of the pieces are about the same size. You want to know about how many pieces there are in the puzzle.

4. You want to know about how long it takes to walk around a circular pond.

The city of Paris, France is completely contained within an almost circular road that goes around the edge. Use themap with its scale to:​

Here is a diagram of a softball field:

While in math class, Priya and Kiran come up with two ways of thinking about the proportional relationship shown in the table.

1. What value of $k$ would each student get using their own method?
2. How are Priya and Kiran's approaches related?
3. Explain how each student might approach solving the equation $\frac23 k=50$.