RATIO AND PROPORTION

Graphing Ratios
(Student Copy)

Activity 7:  Graphing Ratios

•  completed copy of Activity 6
•  calculator
•  metric ruler
•  graph paper (4 squares to the inch preferred)

Part A

Ratios can be written in a variety of forms.  The ratio of 2 to 3 can be written using a colon (2:3) or a fraction (2/3).  In order to answer the following questions, look back at the table in Part B of Activity 6 for the number of teeth on the front sprocket of a bicycle compared to the number of teeth on the rear sprocket.

Write the gear ratios in Activity 6 (Part B) in simplest form using a colon and using a fraction.

To determine the speed of the bicycle in each combination of gears, you divided the number of teeth in the rear sprocket into the number of teeth in the front sprocket.  Explain, in writing, why we did not give these answers in colon or fraction form.

Part B

In manufacturing, it is important to keep track of the number of defective parts or products being produced.  Rather than simply counting the number of defective parts, company officials usually consider the number of defects per hundred, thousand, or million parts.

Suppose that the Lightning Electric Company was making complex light switches and had a defect rate of 25 parts per hundred.   Why would the figure 25 parts per hundred be considered a ratio?

Fill in the table below to show the number of parts produced in the top row and the number of defective parts in the bottom row.  The top row should have the entries 100, 200, 300, 400, and 500.

 Parts produced Defective parts

What number must you multiply the bottom row by to get the top row?  (If you are not sure, take a guess and check it with your calculator.)

What number must you multiply the top row by to get the bottom row?

Part C

Draw an x-y graph on your graph paper for the data in the table; each line on the graph paper should represent 25 units.  Label the x-axis "Number of Parts" and the y-axis "Number of Defects".  Plot the points listed in your table and then connect them with a line.

Next, find a point on the line that is between two points.  Carefully, draw a horizontal line (to the left) from your new point to the y-axis and measure it in millimeters.  Record the distance below.  Also, draw a vertical line (down) from your new point to the x-axis.  Measure and record.

Horizontal distance:

Vertical distance:

What number must you multiply the horizontal distance by to get the vertical distance?

Explain how the numbers you just found relate to the top and bottom rows of the table.

Do you think that choosing a different point on the line and repeating the measurements would result in a different ratio of the horizontal distance to the vertical distance?  Why or why not?

Activity 8:  Using Ratio to Approximate

•  standard-units (English) ruler
•  graph paper (4 squares to the inch preferred)
•  calculator
•  Reference Copy

Part A

Ö2 = _______
Draw squares with sides of 3 inches, 4 inches, 6 inches and 8 inches on graph paper.  Draw and measure (to the nearest quarter of an inch) the length of one diagonal (a diagonal connects two opposite corners) for each square.  In the space below, make a table showing the length of the side and the length of the diagonal for each square.

 Length of the side (inches) 3 4 6 8 Length of the diagonal (inches) Divide the diagonal by side

Now, complete the table by dividing the length of each diagonal by the length of the side.  Do you think the ratio will be the same for any square?  Write your answer below.

Part B

Before people had sophisticated measure instruments, distances had to be approximated.  Distances across a river or a canyon could be particularly troublesome.  Even so, precise measurements were often desired.  How would you judge the amount of rope needed to construct a bridge made of rope and wood that needed to be built over a rushing river if you didn't have any sophisticated tools?  You will soon find out.

Refer to the Reference Sheet. Let's say that the picture represents the place where a bridge is to be built.  You have noticed two prominent landmarks—a tree and a rock—and have marked two spots directly across the river from them.  Next you measured the distance between the two marked spots and found that they were 300 feet apart.

To make the picture as realistic as possible, you first drew the river with the tree and the Marked Spot A.  You located the rock in your picture at 28 degrees, the same angle that the rock is located from Marked Spot A.  Finally, you completed with picture as shown.

The two marked spots and the rock form a triangle.  Draw the triangle on the picture.  The triangle on the page is similar to a triangle formed by the actual tree and the two marked spots.  Your two triangles are similar, which means that the ratio of corresponding sides will be the same number.  You decide to measure the distances on the picture and use that ratio to find the same ratio that works with a side of 300 feet.

Use your ruler to measure to the nearest 1/8 inch (rounding up for mid-distances):  the distance from A to B (scaled-distance #1),  and from B to the closest edge of the rock (scaled-distance #2).  Next, divide scaled-distance #1 by scaled-distance #2 to get the ratio of length 1 to length 2.  Decide how you can use this information to find the approximate real-life distance across the river.

1. distance from A to B is  inches (scaled-distance #1)

2. distance from B to the rock is  inches (scaled-distance #2)

3. ratio of scaled-distance #2 to scaled-distance #1

4. actual distance across the river is feet

Summary Questions

1. If your speedometer needle is pointing to 55, what ratio is indicated?  What kind of ratio is it (same units or different units)?  How would you state this ratio formally?  How would you write it?

2. Most car speedometers have both English (miles per hour) and metric (kilometers per hour) units.  How could you make a table to show the ratio of miles per hour to kilometers per hour?