Making Connections, 2006, Montana Team
Jenny McNulty, Johnny Lott, Terri Dahl
Let x be a positive number. Let a be a positive number which is decreasing. What
happens to the following expressions?
xa
x-a
-x + a
a  x  2  a
ax  1
Increases
Decreases
Remains Unchanged
For each expression, explain why you made the choice you did.

xa

xa

-x + a

a  x  2  a

ax  1
Nate tried several approaches in simplifying 7  23  8x . Listed below are several
expressions he could have used. Which are mathematically correct?
7  2 5x
7   23  8x
7  6  16x
53  8x
7  6  16 x 
7  6  16x
7  3  8x  3  8x
Correct
For each expression, explain why you made the choice you did.
 7  2 5x

7   23  8x

7  6  16x

53  8x

7  6  16 x 

7  6  16x

7  3  8x  3  8x
Incorrect
Determine whether each equation has a positive solution, a negative solution, a zero
solution or no solution.
7x  5
5  3x  7
5x  3  7
8x  11  2x  3
8x  3  8x  11
8x  3x  2x  11x
A
B
C
D
E
F
Solution is
positive
Solution is
negative
Solution is
zero
No solution
Explain how you could have predicted the answers for each of the equations above
without solving them.

A

B

C

D

E

F
You are saving money to buy an iPod. You currently have d dollars, not enough to
purchase the iPod. Your mom, a math teacher, will add to your savings if you can
determine which of the following offers, if any, is better and why? To answer your
mother's question, write an algebraic expression to represent the total amount of money
of have after each offer.
 Offer 1
You have d dollars. Your mother will give you half of what you have.
 Offer 2
You have d dollars. Your mother invests your money, it triples in value. You may spend
½ of it on the iPod.
 Offer 3
You have d dollars. Your mother will give you the same amount of money that you
currently have, if you give ¼ of your new total to charity.