RATIO AND PROPORTION

Graphing Ratios
(Teacher Copy)

### Activity 7:  Graphing Ratios

Students identify ratios of quantities with like and unlike units, write ratios in colon and fraction formats, and make x-y line graphs of ratios.  .

To understand the concept of ratio using tables, graphs, and a formal definition.

Introduction:

In the previous activities, we constructed a number of tables.  Once we knew the first numbers in the table, we were often able to predict what the next numbers would be.  Whenever we can predict numbers in one row of a table by multiplying numbers in another row of a table by a given number, we call the relationship between the numbers a ratio.  There are ratios in which both items have the same units (they are often called proper ratios).  For example, when we compared the diameter of a circle to its circumference, both measured in centimeters, we were using a same-units ratio.  Miles per gallon is a good example of a different-units ratio.  If we did not specifically state that we were comparing miles to gallons, there would be no way to know what was being compared!

When both quantities in a ratio have the same units, it is not necessary to state the unit.  For instance, let's compare the quantity of chocolate chips used when Mary and Quinn bake cookies.  If Mary used 6 ounces and Quinn used 9 ounces, the ratio of Mary's usage to Quinn's would be 2 to 3 (note that the order of the numbers must correspond to the verbal order of the items they represent).  How do we get this?   One way would be to build a table where the second row was always one and a half times as much as the first row.  This is the method we used in the first two lessons.  Another way is to express the items being compared as a fraction complete with units:

6 ounces
9 ounces
Notice that both numerator and denominator have the same units and thus we can "cancel out" the units.  Notice also that both numerator and denominator have values that are divisible by three.  When expressing ratios, we generally treat them like fractions and "reduce" or simplify them to the smallest numbers possible (fraction and colon forms use two numbers, as a 3:1 ratio, whereas the decimal fraction form uses a single number—for example, 3.0—that is implicitly compared to the whole number 1).

Part A

 Gear Ratio Gear Ratio Gear Ratio 1 1:1,  1/1 5 7:4,  7/4 9 38:21,  38/21 2 7:6,  7/6 6 2:1,  2/1 10 19:9,  19/9 3 4:3,  4/3 7 19:14,  19/14 11 19:8,  19/8 4 14:9,  14/9 8 19:12,  19/12 12 19:7,  19/7

The gear ratios in Part B of Activity 6 were written as decimal fractions.  Ratios expressed as decimal fractions implicitly use the whole number 1 as the number of comparison.  These ratios make it easier to compare the relative size of different ratios.

Part B

Twenty-five parts per hundred (25:100 or 1:4) represents the ratio of  the number of defective parts to the number of parts produced.

 Parts produced 100 200 300 400 500 Defective parts 25 50 75 100 125

Numbers in the bottom row are multiplied by 4 to get the corresponding number in the top row.  Numbers in the top row are multiplied by 0.25 (or divided by 4) to get the corresponding numbers in the bottom row.

Part C

Sample Graph:  Lightning Electric Company's Defect Rate

Horizontal distance:  38 mm (on 4-squares-per-inch graph paper)

Vertical distance:  9.5 or 10 mm

The vertical distance must be multiplied by about 4 to get the horizontal distance.

The coordinates for the new point (or any point) on the drawn line will be in a ratio of 1:4, the same as the ratio in the table for number of defective parts to number of parts produced.

No.  Because the graphed line is straight, the horizontal and vertical distances from any point on the line to the x and y axes will be proportionate.

Closing Discussion

• Can all ratios that are represented by tables be represented by graphs?  Is the graph of a ratio always a straight line?  Yes, all ratios represented by tables can be represented by x-y graphs.  No, only constant ratios are straight lines; geometric and inverse ratios are not straight lines.  All ratios shown so far in this unit are constant ratios and thus do form straight lines.
• How could you make a graph from the colon form of a ratio without making a table first?  Write several instances of the ratio (e.g., 1:2; 2:4; 3:6, etc.).  Then, plot the ratios as points on the graph by using the first number of each ratio for the distance along one axis and the second number of each ratio for the second axis.
• How is the slope of a line related to the ratio between the vertical and horizontal distance between two points on the line?  The slope of the line is the ratio of the y-axis difference between two values to the  difference between the corresponding x-axis values.

### Activity 8:  Using Ratio to Approximate

Students approximate Ö2 using ratios and squares, and approximate distance using similar triangles.

To understand the concept of ratio using tables, graphs, and a formal definition.

Introduction:

Who knows how to find the length of the hypotenuse of a right triangle?  Does anyone know the Pythagorean Theorem?  With the students, find the length of the hypotenuse of a right triangle with one-unit sides.

Part A

Using their calculators, students should find that Ö2 » 1.4.

 Length of the side (inches) 3 4 6 8 Length of the diagonal (inches) » 4.2 in. » 5.6 in. » 8.5 in. » 11.3 in. Divide the diagonal by side » 1.4 » 1.4 » 1.4 » 1.4

The ratio of the length of the diagonal to one of the sides of a square will always be Ã2.  Students may say that the length of the diagonal gets shorter/longer proportionally as the length of a side gets shorter/longer.

Part B

1. » 4 1/4 (4.25) inches
2. » 2 inches
3. » 2.125
4. » 141 feet

Summary Questions

1. When the speedometer registers 55, the ratio is 55 miles to one hour.  The units are different:  miles and hours.  Ratios might be stated as "55 to 1" and written 55:1 or 55/1.
2. Answers may vary.  Students should recognize that one row of the table would have different speeds in miles per hour and another would show corresponding speeds in kilometers.

Closing Discussion

In the first lesson in this unit, we referred to the number p.  Does p represent a ratio?  Why or why not?  Although the term ratio was never used in the first lesson, p represents the ratio of circumference to diameter of a circle.