Using Variables To Represent Co-Varying Quantities & Define Formulas

1. Using Variables To Represent Co-Varying Quantities & Define Formulas Module 1 Investigation #2 Day 1

2. Today’s goal • Identify quantities in a given situation • Define variable to quantities in a given situation • Write a formula using variables for a give situation

3. In the first investigation we learned that it’s useful to define variables to represent all the possible values that a specific quantity (like the length of the side of a square or the amount of wire left on a spool of wire) can take on. Once variables have been defined we are able to create general formulas to describe (or define) how the values of the quantities (or variables) change together. As we continue to practice defining formulas we will focus on the meaning of addition, subtraction, multiplication, and division so we know when to use each of these operations when writing expressions and defining formulas. What is an expression? What is a formula?

4. When reading a word problem the first thing you need to do is identify what quantities are described and what quantities you are being asked to relate. This is because formulas represent relationships between quantities and we cannot write formulas to relate objects such as Bob and car, or foot and shoe. A quantity is some attribute of an object that has a measure or can take on numerical values. Some examples of quantities include the length of Bob’s foot (a fixed quantity), the distance Bob has traveled since leaving home (a varying quantity). Fixed Quantity - is a measurement that will not change during the word problem. Varying Quantity – is a measurement that will change during the word problem.

5. Name the varying and fixed quantities in each of the given situations. • Bob runs 3 miles on a straight road at a constant speed of 4.5 miles per hour. Fixed Quantities: • 3 miles • 4.5 miles per hour Varying Quantities • The amount of time Bob has been running on the straight road. • Distance Bob has from completing his 3 mile run. • Distance Bob has already ran on the straight road at a given time. • A 40 gallon fish tank is draining. Fixed Quantities: • 40 gallons Varying Quantities: • Gallons of water drained from the tank at a given time. • Gallons of water in the tank at a given time. • Amount of time the water has been draining. Is the fish tank a quantity? Is Bob a Quantity? Is the Straight Road a Quantity?

6. Mike was born on the same day of the year as Jose, but was born 2 years later so is always 2 years younger than Jose. Let j = the number of years since Jose was born Let m = the number of years since Mike was born • Express Mike’s age in terms of Jose’s age. m = j - 2 • Express Jose’s age in terms of Mike’s age. j = m + 2 • Use your formulas in (a) and (b) to determine Jose’s age when Mike is 14. Do you get the same answer? Why does this make sense? Explain. 14 = j – 2 Jose = 16 j = 14 + 2 Jose = 16 Yes it makes sense. If Jose is 2 years older than Mike, Mike will be 2 years younger than Jose. (14 and 16)

7. If Mary receives $1 for every $2 of pay that John receives, • Write a formula to represent the amount of pay that Mary receives in terms of the amount of pay that John receives. Define your variables before writing your formula. y = pay that Mary receives in dollars x = pay that John receives in dollars y = ½ x • Check your answers by substituting $330 for the variable that represents John’s pay. Does you’re answer make sense? Explain. y = ½ (330) ; 165 yes it makes sense. Mary make ½ of what John does and 165 is half of 330.

8. A cabinet-maker is building cabinets for a kitchen. The cabinet-maker has determined that the depth of every cabinet he builds for this kitchen must be 1/3 times as large as the cabinet’s height. • Define variables to represent the cabinets depth and height. (When defining variables be sure to include units.) d= the depth of the cabinet in inches h = the height of the cabinet in inches

9. A cabinet-maker is building cabinets for a kitchen. The cabinet-maker has determined that the depth of every cabinet he builds for this kitchen must be 1/3 times as large as the cabinet’s height. • Fill in the values in the given table, then explain how to determine the depth of the cabinet when the height of the cabinet is known. 6 10 11.67 15 Explain: The depth is 1/3 times as large as the cabinet’s height. Therefore whatever the cabinet’s height is, I must multiply it by 1/3 in order to get the depth.

10. A cabinet-maker is building cabinets for a kitchen. The cabinet-maker has determined that the depth of every cabinet he builds for this kitchen must be 1/3 times as large as the cabinet’s height. • Use your variables to write an expression that defines the depth of various size cabinets in terms of the height of those cabinets. • Write a formula to determine the cabinet’s depth in terms of the height of various size cabinets. Which is the input variable, which is the output variable? d = (1/3)h The input variable is the height of the cabinet. The output variable in the depth of the cabinet. (1/3)h

11. A cabinet-maker is building cabinets for a kitchen. The cabinet-maker has determined that the depth of every cabinet he builds for this kitchen must be 1/3 times as large as the cabinet’s height. • What is the difference between an expression and a formula? An expression is a mathematical statement with only one variable whereas a formula includes an equal sign and relates two varying quantities. • Determine the cabinet’s depth when the height is: • 48 inches 16 inches • 42.5 inches 14.17 inches • 38 ½ inches 12.83 inches

12. The local candy store sells bulk candy by the pound. The cost of all candy in the store is $7.50 per pound. • Write a formula to express the total amount of a purchase (before tax) in terms of the number of pounds of candy that is purchased. Define your variables first, then write the formula. x = the number of pound of candy purchased C = the total cost of candy purchased in dollars C = 7.50x • What amount is to be paid for 2.4 pounds of candy? C = 7.50(2.4) ; $18

13. The local candy store sells bulk candy by the pound. The cost of all candy in the store is $7.50 per pound. • Write a formula to determine the number of pounds of candy that can be purchased if a customer has C dollars to spend on candy. • Determine the number of pounds of candy that can be purchased if a child has: • $5.00 2/3 lb candy or .67 lb candy • $2.50 1/3 lb candy or .33 lb candy • $12.00 1.6 lb candy x = C 7.50

14. Homework In Homework Packet: • #8, #9, #10