Download Changes in Quantities

Functions Homework due at Saturday Workshop #3
Grades 5, 6
Changes in Quantities
1. Let x and y represent the values of two different quantities.
a. If the value of x goes from x = 3 to x = 12, what is the change in x?
b. If the value of y goes from y = 23 to y = 11, what is the change in y?
c. If the value of y goes from y = 9 to y = b, what is the change in y?
Changes in Quantities
If x is the value of Quantity A and y is the value of Quantity B, we use Δx to represent a
change in the value of x and Δy to represent a change in the value of y.
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Ex 1: If we let the value of x change from x = 2 to x = 14, then !x = 12 which we
calculate with the expression 14 ! 2 .
Ex 2: If we let the value of y change from y = 11 to y = 3 , then !y = "8 which we
calculate with the expression 3! 11 .
2. Fill in the following tables showing the appropriate changes in the variable values.
a.
b.
y
x
Δy
Δx
2
6
17
32
15
11.9
6.1
−1.3
3. Each of the following graphs shows two ordered pairs ( x, y) . Find Δx and Δy from the point on the
left to the point on the right.
a.
b.
Functions Homework due at Saturday Workshop #3
Grades 5, 6
4. Suppose t represents the temperature, in oF, in Winter, Wisconsin. We start with a value of
t = 7 (which we call our reference point).
a. What expression will calculate the change in t from t = 7 to any possible value of t?
b. Repeat part (a) if our initial reference point is t = 42 instead of t = 7.
c. Repeat part (a) if our initial reference point is t = –12 instead of t = 7.
5. Suppose a represents the amount of money in your checking account. What do each of the following
expressions calculate when we substitute a value for the variable? (In terms of change in quantities)
a. a ! 340
b. a + 25
6. Conrad is at a park a mile from his house and he begins jogging in the direction away from his home.
Let d = Conrad’s distance (in miles) from his house. What’s the difference in meaning between d = 3
and !d = 3 ?
7. Students often have trouble understanding the difference between a quantity (say “time elapsed”) and
a change in a quantity (“change in time elapsed). What are questions you could ask or examples you
could use in class to help students understand the difference between the two?
Functions Homework due at Saturday Workshop #3
Grades 5, 6
Constant Rate of Change
8. Suppose that you are driving from Flagstaff to Albuquerque. During part of the trip you traveled at a
constant speed, covering 42 miles in 50 minutes.
The following line segments represent “a change in distance traveled of 42 miles” and “a change in
time elapsed of 50 minutes.” [You can use these segments to assist you in answering parts (a)
through (e).]
a. During this time, how far did you travel in 25 minutes?
b. Does your answer to part (a) depend on which 25-minute interval we’re talking about? Explain.
c. How far did you travel
i) every 5 minutes?
iii) every 30 minutes
ii) every 10 minutes?
iv) every 12 minutes?
9. Assume that a baseball and a tennis ball are both traveling at constant speeds (but not necessarily the
same constant speed as each other). Recall that speed = !d
.
!t
What does it mean to say that the baseball is traveling slower than the tennis ball? Be descriptive.
Functions Homework due at Saturday Workshop #3
Grades 5, 6
10. Suppose we want to landscape our backyard by adding some river rocks.
a. Explain why it makes sense to say there is a constant rate of change between the total weight of
the river rocks in the yard and the amount of river rocks.
b. When we add 8 cubic feet of river rocks to the yard, the weight increases by 1200 pounds. What
are some conclusions we can draw from this?
c. How much will the weight of river rocks in the yard change if we add 14 cubic feet of rocks?
What if we remove 3.2 cubic feet of rocks?
11. In the Saturday workshop, recall the problem where we were asked to determine the order of objects
from fastest to slowest. In that problem, we saw that knowing the unit rate is very useful.
For example, if we know that the Matt is jogging at a constant speed of 0.1 miles per minute, then it’s
easy to determine the change in total distance Matt has jogged (in miles) if we change the number of
minutes he’s been jogging. Let d represent the total distance (in miles) Matt has traveled during his
jog and let t represent the total number of minutes he’s been jogging.
a. Suppose Matt has jogged for some amount of time and then he jogs for 3 more minutes.
Represent the change in the number of minutes Matt has jogged t using ∆ notation.
b. How much does Matt’s total distance jogged change over the 3-minute interval? Represent your
answer using ∆ notation.
c. Repeat parts (a) and (b) if instead of a 3-minute time period we look at a
i) 7.8-minute time period
ii) 0.3-minute time period
iii) any x-minute time period
d. For any change in t, what happens to the change in d? Explain, then represent your thinking using
∆ notation.