The Problem Solving Power of Units

1. 2A The Problem Solving Power of Units

2. Basics Units of quantity describe what is being measured or counted. We can only add values that have the same unit of measure. We can multiply and divide values that have different units. 

3. Examples It wouldn’t make sense to combine 3 apples with 2 bananas…5 banapples? If you walk 20 miles in 3 hours, what is your average speed?

4. Examples If you drive 60 miles per hour, how far did you travel in 2 hours?

5. Examples • To find the area of a room we multiply length times width • A 10 foot by 8 foot room is 80 square feet or 80 ft2 • To find the volume of a box we take length x width x height • 2cm x 3cm x 4cm = 24 cm cubed • To find the amount of energy used by a light bulb we multiply its power rating by the length of time it is turned on. • 60 kilowatt x 2 hours = 120 kilowatt-hours

6. Examples • Area and Volume Conversions- Page 97 #48 • A warehouse is 40 yards long and 25 yards wide and piled with cartons to a height of 3 yards. What is the area of the warehouse floor? What is the total volume of the cartons? (Assume there is no space between the cartons) • The bed of a pickup truck is 3.5 feet deep, 12 feet long, and 5 feet wide. What is the area of the bed’s floor? What is the volume of the bed? • A can has a circular base with an area of 6 square inches and is 4 inches tall. What is the total volume?

7. Reading units Operation Keyword or Symbol Example Miles / hours Division per “miles per hour” Raising to a square ft x ft or ft^2 Second power “square feet” or “feet squared” Raising to a third power cube or cubic ft x ft x ft or ft^3 “cubic feet” or “feet cubed” Multiplication hyphen kilowatt x hours “kilowatt-hours”

8. Example • Identifying Units • Identify the units you would expect in each of the following. State the units in both words and mathematically. • The price of fabric, found by dividing its cost in dollars by its area in square feet. • The gas mileage of a car, found by dividing the distance in miles it travels by the amount of gas in gallons that it uses. • The cost for grass seed when you buy enough to cover 80 square yards at a total price of $160. • The density of a rock, found by dividing its weight in grams by its volume in cubic centimeters. • A car engine torque calculated by multiplying a force in pounds by a distance in feet.

9. Working with fractions • A fraction represents division • Numerator- top of the fraction • Denominator- bottom of the fraction • Integers can be written as fractions • 5 = 5/1 

10. Adding and Subtracting Fractions

11. Multiplying Fractions

12. Dividing Fractions

13. Fraction to Decimal

14. Decimal to Fraction

15. Example • Evaluate each of the following

16. Unit Conversions • The trick is to find a “well chosen 1” • Multiplying the numerator and the denominator by the same number doesn’t change the value of the original fraction. • When the numerator and the denominator are the same value the fraction is equal to 1…hence the name “well chosen 1”

17. Examples-Well Chosen 1’s 60 seconds1 minute 1 minute 60 seconds 7 days1 week 1 week 7 days 1 foot 12 inches 12 inches 1 foot

18. Convert the following 3 feet to inches 108 inches to feet

19. Chain of Conversions How many seconds are there in 1 week?

20. Conversions with Powers How many square feet in a square yard? 3 ft 3 ft

21. Example Convert 150 sq ft to sq yds

22. Examples • Unit Conversions- • Convert a distance of 7 miles into yards; there are 1760 yards in a mile. • Using the fact that there are 1760 yards in a mile and 3 feet in a yard, convert a distance of 3 miles into feet. • Use a chain of conversions with familiar measures of time to convert 4 weeks into minutes • Convert a park size of 3.5 square miles to acres. (1 acre = 1/640 mi2) • A car is driving 100 kilometers per hour. What is its speed in kilometers per second?

23. Examples • Cubic Units- • Find a conversion factor between cubic inches and cubic feet. Write it in three forms. • How many cubic inches are in 3 cubic yards? • A cargo container is 50 feet long, 10 feet wide, and 8 feet tall. Find its volume in cubic feet and cubic yards.

24. Example • Currency Conversions- • Use Table 2.1 on page 90 • Which is worth more, 1 Mexican peso or 1 Japanese yen? Explain. • How many Canadian dollars can you buy for $100? • You return from a trip with 75 British pounds. How much are your pounds worth in dollars? • Apples in Japan sell for about 250 yen each. If you buy 4 apples, how much have you spent in dollars?

25. Using Units to Help You Solve Problems By looking at what kind of answer we are trying to get this can help us determine what operation we need to perform.

26. Examples • Page 98- #70 • You are buying 2.8 kilograms of cherries priced at $3.50 per kilogram. How much will you pay? • Page 98- #66 • An airplane travels 95 miles in 10 minutes. How fast is it going in miles per hour?

27. Examples • A 40 acre orchard produces 12,000 apples. What is the yield in apples per acre? • You are buying floor tile to cover a room that measures 20 feet by 25 feet. The tile is priced at $7.50 per square foot. How much will the tile cost?

28. Examples • You are buying artificial turf to cover a game field that is 150 feet long and 100 feet wide. The turf costs $7.50 per square yard. How much will the turf cost? • You work 40 hours per week and are paid $13.50 per hour. If you work all 52 weeks in a year, how much will you earn?

29. Examples • Page 98- #78 • An average human heart beats 60 times per minute. If an average human being lives to the age of 75, how many times does the average heart beat in a lifetime?