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Use of Variables

These problems involve using a variable to represent an unknown quantity and making manipulations of the variables at an introductory algebra level. The problems are suitable for the beginning
of an introductory algebra class.
 Related Resources

Interactive resources from our Math Tools project:
Algebra: Use of Variables
The closest match in our Ask Dr. Math archives:
High School: Algebra
NCTM Standards:
Algebra Standard for Grades 912
Access to these problems requires a Membership.

A 2for1 Lunch
 Marsha Hurwitz

Algebra, difficulty level 2. Two boys had lunch on a twoforone coupon, plus taxes and tips. What was their percent of savings on the transaction?
... more>>

Alvin's Theorem
 Terry Trotter

Algebra, difficulty level 2. Alvin notices that the difference between any two consecutive square
numbers can also be expressed as the sum of consecutive integers.
Can you use algebra to show that this pattern is always true?
... more>>

Apple Orchard
 Steve Risberg

Algebra, difficulty level 2. How many of each kind of apple did the Nomial family pick?
... more>>

Are You Ready for Some Football?
 Steve Risberg

Algebra, difficulty level 2. The Algenauts football team racked up 229 yards rushing in their first
game. Given descriptions of how many yards each of the players ran
for in relation to each other, determine the exact yardage each one had.
... more>>

As the Crow Flies
 Terry Trotter

Algebra, difficulty level 1. A percentage of crows sitting on a telephone line leaves, then returns. Find how many crows are present at the end.
... more>>

Baffling Brother
 Steve Risberg

Algebra, difficulty level 2. Brandon wows his sister by having her choose a secret number and make
a series of calculations, then telling her what her final answer is.
Can you use variables and algebra to explain how Brandon did it?
... more>>

Battle of the Bands
 Terry Trotter

Algebra, difficulty level 3. Students at Rahkenrole High School vote to decide which band should
play at the endofyear dance. After some students change their
minds about which band to vote for, the result is a tie. Can you
figure out how many students voted?
... more>>

Bowling Buddies
 Steve Risberg

Algebra, difficulty level 2. Four friends go bowling and have a small contest. From the clues
given about their scores, can you determine who won the contest?
... more>>

Boxes of Cereal
 Steve Weimar

Algebra, difficulty level 2. Lily and Mikey consume their boxes of cereal at different rates. Use
the hints to determine how much cereal is in each child's box.
... more>>

Calendar Calculations
 Steve Risberg

Algebra, difficulty level 1. Write variable expressions for dates in a calendar pattern and use your expressions to explore a numeric relationship among those dates.
... more>>

Count Your Change
 Steve Risberg & Terry Trotter

Algebra, difficulty level 3. I have three times as many nickels as dimes in my pocket. If the
number of coins is reversed, by what percent will the value of the
money in my pocket increase?
... more>>

Eduardo's Equations
 Steve Risberg Annie Fetter

Algebra, difficulty level 1. Find all possible integer solutions to a system of four equations in
two variables.
... more>>

Explaining a Number Trick
 Annie Fetter

Algebra, difficulty level 1. Use a variable to explain how this number trick works.
... more>>

Fraction Debate
 Steve Risberg

Algebra, difficulty level 3. Three students disagree about what happens when you perform a certain operation on a positive proper fraction. Which student do you agree with? Why?
... more>>

Generosity
 Ethel Breuche

Algebra, difficulty level 2. Three generous friends, each with some cash, redistribute their money.
... more>>

Make the Save!
 Steve Risberg

Algebra, difficulty level 1. A hockey goalie has saved 112 out of 131 shots so far this season.
How many consecutive shots does she need to save to raise her save
percentage to .900?
... more>>

Mind Reading
 Annie Fetter

Algebra, difficulty level 1. Use variables to show how a numberbased "magic trick" works.
... more>>

The Operations Grid
 Annie Fetter

Algebra, difficulty level 2. Use variables to explore whether this grid works for the four basic operations.
... more>>

Reverse It!
 Annie Fetter Steve Risberg

Algebra, difficulty level 1. Find a twodigit number such that when you reverse the digits and add
the new number to the original one, the sum is 154.
... more>>

September Squares
 Cynthia Lanius and Steve Risberg

Algebra, difficulty level 2. Use a variable to explain how you can be given the sum of four dates
that form a block on a calendar and determine what the four dates must be.
... more>>

A Surprising Sequence
 Steve Risberg

Algebra, difficulty level 2. In a Fibonacci sequence, the first number is 1 and the tenth number is
111. Find the second number.
... more>>

TeeterToddlers
 Steve Risberg

Algebra, difficulty level 1. Two kids of different weights balance on a seesaw. How far from the
center is each one sitting?
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Vexing Variables
 Steve Risberg

Algebra, difficulty level 3. Use two algebraic statements involving three variables to find the
numeric value of the ratio of two of those variables.
... more>>

What Time Is It?
 Steve Risberg

Algebra, difficulty level 3. Determine the angle between the hands of a clock at certain times,
and
find the times that generate a particular angle.
... more>>
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