How Far Away is That? Ratios, Proportions, Maps and Medicine

Transcription

1 38 How Far Away is That? Ratios, Proportions, Maps and Medicine Maps A ratio is simply a fraction; it gives us a way of comparing two quantities. A proportion is an equation that has exactly one ratio on each side. Map scales are an example of the use of proportions that everyone needs occasionally. I measured the distance from the west end of St. Croix to the southern tip of Domenica on a map. It was 1.25 inches. Then I looked in the corner of the map and learned that the scale used on my map was 3 inches for 800 miles. x miles 800 miles So, I wrote this proportion: inches 3 inches x represents the actual distance from St. Croix to the south end of Domenica in miles. In the first ratio, we are comparing the actual distance (in miles) to the distance on the map, in inches. The second ratio is the scale given to me in the map s legend, also in miles to inches. Here, x is divided by I can solve this equation by undoing that division: multiply both sides of the equation by 1.25: x x 800 (1.25) (1.25) About how far is it from St. Croix to Domenica? Suppose you plan to go to Jamaica. You are going to fly to Kingston, then rent a car and drive to Negril. How long will that take? Well, first you need to know how far it is from Kingston to Negril. Measure the distance on the map. Use the scale printed on the map to find the actual distance. You will need to use your ruler to see what the scale is numerically. Assuming that you average 70 km/hour, how long will it take you to drive from Kingston to Negril?

2 39 Medicine dosage Another important use for ratios and proportions is for determining the correct dose of a medicine according to the patient s body weight. This is not done with all medicines, but for some it is an important consideration in determining how much to prescribe. Care is especially important in the case of small children. Consider the following: 1.5 milliliters of a certain drug is prescribed for every 10 pounds of the patient s weight, to be administered once every 6 hours for 5 days. In the same way that the scale on a map might tell us that for every 1.5 cm on the map we will get another 20 km on the ground, this information about the medicine dosage says that for every 10 pounds of patient(!) we ll need another 1.5 ml of medicine. 1. Find the correct dosage for little Yamika, who weighs 38 pounds, by writing a proportion and solving it. First, give a name to the quantity you want to find: Let x number of milliliters of the medicine Yamika should have. a. Then write the ratio of the amount she should have (in ml) to her weight (in lbs). b. Write another ratio to represent the dosage information (1.5 ml of the drug for every 10 lbs of the patient s weight). Be sure to use the same order as before. c. To get the child s prescription right, these two ratios will have to be equal to each other. Write the equation and solve it. Then write the prescription for Yamika. 2. When you are finished writing the prescription for Yamika, please prepare a prescription for Mr. Jno-Lewis for Medicine X. He weighs 185 lbs. Medicine X requires a dosage of 2.1 mg for every 50 lbs of body weight. It is to be taken once a day for 10 days.

3 40 Teaching Guide for How Far Away is That? Ratios, Proportions, Maps and Medicine Introduction: The mathematics in this lesson is focused on proportional reasoning. It is important for students to develop a deep understanding of the concept of proportionality. They need to see it used in a variety of situations. Mapping is an important one. The mathematics in the lesson includes solving a proportion using the scale of the map and determining how long it would take to travel between two points, given the speed of travel (another proportion concept). Explanations of the words ratio and proportion are included in the lesson. Students should learn to use both words appropriately. The medicine dosage example is intended to reinforce the concept of the equality of the two ratios in a proportion. If you think your students lack feel for the notion of scale, get them to make a map of a small area or a scale drawing of a building or a room before doing this lesson. The lesson begins with a worked example that is not completed. To find the distance x from St. Croix to the southern tip of Domenica, we solve the proportion: x miles 800 miles inches 3 inches Students should get x 333 miles. In our solution, we avoid the cross multiplication that students are commonly encouraged to do and recommend solving this equation by undoing the division by However, students may proceed with a variety of approaches. Encourage them to understand the process they choose to use. Finally, students are asked: About how long will it take you to drive from Kingston to Negril if your average speed is 45 mph? Since the distance is 104 miles, write t and solve to get t 2.3 hours, or 2 hrs. and 18 minutes. Use a map of Jamaica to determine how far it is from Kingston to Negril. Measure the distance on the map. Use the scale printed on the map to find the actual distance. To use the scale printed on the map, students must first measure to see how many cm (or other unit) represents 30 km. Then they must measure the map distance from Kingston to Negril. Because the scale of the map is likely to vary proportionally as a result of printing and photocopying, you will need to make all these measurements yourself (or trust your students results!). Students should write a proportion equivalent to x kilometers 30 km. measured distance on map in cm measured length of scale indicator in cm

4 41 The calculated distance should be about 170 km, depending upon accuracy of measurements. 170 km The time it will take to drive would thus be about 2.4 hrs., or about 2 hours and km/hr minutes.

5 42 Medicine Dosage Be sure students understand the following statement from the lesson. In the same way that the scale on a map might tell us that for every 1.5 cm on the map we will get another 20 km on the ground, this information about the medicine dosage says that for every 10 pounds of patient we ll need another 1.5 ml of medicine. 1. Find the correct dosage for little Yamika, who weighs 38 pounds, by writing a proportion and solving it. First, give a name to the quantity you want to find: Let x number of milliliters of the medicine Yamika should have. x ml a. The ratio of the amount she should have to her weight is. 38 lbs b. The ratio that tells us the correct amount to prescribe, the dosage information, is 1.5 ml 10 lbs c. To get the child s prescription right, these two ratios will have to be equal to each other. x ml 1.5 ml Thus, we write: 38 lbs 10 lbs x 1.5 Or simply: Thus, x 5. 7 There is value in getting students to write a prescription for the child, both to clarify for them what they have done and to reinforce the practical nature of the mathematics of proportions. A student might write: Yamika s prescription. Take 5.7 ml of this medicine every 6 hours for 5 days. 2. The student should solve an equation equivalent to: x The prescription might read: Take 7.77 ml once a day for 10 days.