## List of Objectives Involving Change, Rates, and Rate of Change

List for Math 503

Objectives (1):

1. Traditional proportion problem. Three of the four numbers in the proportion are given and the student is to find the fourth. [two corresponding amounts and a third amount (which corresponds to an unknown amount)]
• •The bakery can produce 84 loaves of bread in 3 hours. How many can they produce in 5 hours?
• •A 10 ounce serving has 225 calories. How many calories are there in a 12 ounce serving?
1. Find the rate, given two change amounts (one change for the numerator and one change for the denominator, "d/t = rate"). Could be, but need not be, a rate of change. [delta-x and delta-y] (2)
• •Over a three hour period yesterday morning the temperature increased 15. What was the rate of temperature increase?
• •The car traveled 378 miles on 14 gallons of gasoline. What was the gas mileage?
1. Convert units. (The units may be a "rate unit," e.g., miles per hour.)
• •66 inches is how many feet?
• •The secretary can type 30 pages per hour. How many pages can she type per minute?
1. Find the rate, given two amounts and a time (or other amount for "the denominator"). Could be, but need not be, a rate of change. [x1, x2, and delta-y]
• •Over a three hour period yesterday morning the temperature went from 68 to 83. What was the rate of temperature increase?
• •At the last gasoline fill-up the odometer reading was 23,450 miles. At the current fill-up the odometer reading is 23,828 miles. It took 14 gallons to fill up the tank. What was the gas mileage?
1. State the rate in a different form, given the rate. (Find the reciprocal.) [rate]
• •The car is getting 25 miles/gallon. How much gasoline is needed to travel 1 mile?
• •If the temperature is increasing at a rate of 5 per hour, how long does it take for the temperature to increase 1?
1. Find rate of change, given two ordered pairs (for example, two amounts at two different times) [x1, y1, x2, and y2]
• •At 10:00 a.m. the temperature was 68. At 1:00 p.m. it was 83. What was the rate of temperature increase from 10 a.m. until 1 p.m.?
• •In 1980, the population of Ft. Collins was 60,000. In 2000, the population of Ft. Collins was 1,000,000. What was the average yearly increase?
1. Find the average-which usually can be interpreted as a rate-given several successive amounts. [successive amounts]
• •On Monday through Friday, the farmer harvested 355, 300, 325, 340, and 380 acres respectively. What was the average number of acres harvested per day?
• •Four servings were measured and had weights of 6.0, 6.2, 5.9, and 6.2 ounces. What was the average weight per serving?
1. Multiply a rate and an amount to get a result ("r×t = d"). [rate and delta-y]
• •If the car drove 55 mph for 4 hours, how far did it travel.
• •If the factory puts out 180 lbs of air pollutants per hour, how many lbs does it put out in an 8-hour work day?
• •The farmer is to use 6 gallons of insecticide per acre. He has 800 acres. How much insecticide does he need?
• •There is to be a 5% increase in the library budget for this year. Last year's budget was \$800. What is the amount of the increase?
1. Find an amount given a rate and a result ("d ÷ r = t"). [rate and delta-x]
• •The family is planning a 400 mile trip. They can average 50 mph. How long with the trip take?
• •The farmer is to use 6 gallons of insecticide per acre. He has 720 gallons of insecticide. How many acres can he treat?
• •We are making trail mix. Each bag is to contain 2 cups of the trail mix. We have 56 cups of mix. How many bags can we prepare?
2.
3. Multiply rates to find a rate. [two rates]
• •The car uses .05 gallon per mile. Gasoline costs \$1.45 per gallon. Find the cost per mile.
• •The printer uses .5 pounds of toner per day. Toner costs \$20 per pound. What is the cost per day?
• •The health food has 700 calories per serving. The container has 6 servings per container. What is the number of calories per container?
• A farmer is harvesting corn. He gets 46 bushels for each acre harvested. Each hour he can harvest 20 acres. How many bushels are harvested each hour?

4. Find a weighted average. [multiple rates and corresponding amounts]
• •Caitlyn had three babysitting jobs. The first was 3 hours at \$2.50 per hour. The second was 6 hours at \$2.00 per hour. The third was one hour for \$3.25. What was her average pay per hour?
• •Three rooms in the new house had flooring installed. Kitchen linoleum: 150 sq. ft at \$2.25 per sq ft.; Living room carpet: 350 sq ft at \$3.25 per sq ft.; Family room carpet: 275 sq ft at \$3.00 per sq ft. Find the average cost per square foot of flooring.
1. Combination of objectives.
• •The secretary can type 20 pages per hour. She gets paid \$7.50 per hour. What is the typing cost per page? (Find the reciprocal and multiply rates)
• •Three trucks in the fleet got the following gas mileages: 14 mpg for 300 miles; 15 mpg for 450 miles; 12 mpg for 250 miles. Gasoline costs \$1.58 per gallon. What was the gasoline cost for the fleet? (Weighted average and multiply rates)

### Categories of Uses of Rates with Examples

Flow rate or speed

• •cc/min (IV)
• •gal/min
• •lb/hr (pollutant)
• •words/min (typed or read)

Use Rate

• •gal/acre (insecticide)
• •cups/batch (cookies)
• •gal/mile
• •oz weed killer/oz of water

Production Rate

• •loaves/hr
• •bu/acre

Density or pressure

• •people/sq mile
• •lb/cu. in.
• •lb/sq. in.

Pay Rate

• •\$/hr.
• •\$ commission/\$ sales

Cost

• •\$/sq. ft
• •\$/oz.
• •\$/family
• •\$ insurance cost/\$ value insured

Size of a portion or subpart

• •beats/measure (music)
• •oz/box
• •calories/serving
• •people/pew

Rate of Change

• •F/hr
• •mile/hr *
• •gal/mile *
• •% increase

(*The categories are not mutually exclusive. Some of these examples can fall under other categories.)

Equivalencies (these equal 1)

• •3 ft/yd
• •5280 ft/mile

Footnotes:

1. School districts need to decide where (when) students will be introduced to these ideas, and when the ideas will be developed, mastered, and reviewed.

2. For those that like notation, the information that is given in the problem will be put in square brackets. When dealing with change, rates, and rates of change one works with the following quantities: x1; y1; x2; y2; delta-x (x2 - x1); delta-y (y2 - y1); rate = delta-y ÷ delta-x.

J.R.Olsen ~ W.I.U.