# Algebraic Expressions Worksheets 3

Lesson Plans > Mathematics > Algebra > Expressions## Algebraic Expressions Worksheets 3

In the first set of worksheets we focused on converting simple phrases to algebraic expressions. For example:

- Four less than a number = n - 4

- Two more than half a number = + 2n2
- The product of a number with a third of that number = n()n3

In this set, we are dealing with expressions which contain two variables, but one of the variables can be written in terms of the other, which allows us to do simple substitutions. For example:

- The product of two numbers, if one of the numbers is five less than the other

In this problem, we encourage students to choose two variables, since there are two unknowns. Some students balk at this, because they can mentally replace "five less than the other" with "x - 5". I tell them it's wonderful that they can do this step mentally, and eventually I won't force them to write it out, but it helps to cement the process in their minds to write it out. So I expect to see the following:

x = the first number; y = the second number = x - 5; xy = x(x - 5)

Forcing them to write it out in this manner reinforces the idea of an algebraic substitution, which, of course, is a crucial step as students advance through algebra. If you don't want to force your students to write out the substitution, simply tweak the worksheet instructions.

Note that some of these problems will involve the possibility of introducing exponents into solutions. Consider this example:

- The area of a triangle in which the height is half the base

^{2}

Depending on where you are in the progression of your course when you use this worksheet, you may choose not to expect your students to rewrite the expression with an exponent.

Similarly, if the answer is something like w(w + 1), you may or may not expect your students to distribute the result. The answer key for the worksheet shows multiple forms, to make it easier to grade, depending on what you expect of your students.

These worksheets make use of some of the formulas highlighted in worksheet #1.1. Areas, perimeters, interest, and speed. Reviewing these formulas with students before beginning the worksheet may be appropriate.

The second worksheet contains some problems with three variables instead of two. It also introduces "consecutive integer" and "consecutive odd/even integer" expressions.

The solution key for this worksheet makes use of subscripted variables. For example, one problem involves two investments which had different *times*, so we used t_{1} and t_{2}. This may be a good time to introduce the concept of subscripted variables to your students, if you haven't already. However, since the subscripted variables show up only in the answer key, this topic can be skipped.

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## Handouts/Worksheets

## Algebraic Expressions Worksheet #3.1

For each phrase below, pick variables to represent the unknowns. State what the variables represent, and then write the phrase as an algebraic expression.

Example: The perimeter of a rectangle, if the width is twice the length

Solution: L = length; W = the width = 2L; 2L + 2W = 2L + 4L = 6L

- The area of a rectangle in which the length is two more than the width

- The sum of two numbers, if the second number is double the first

- The area of a triangle, if the height is two more than twice the base

- The perimeter of a rectangle in which the length is two-thirds the width

- The cost of 5 pens and 10 markers, if a marker costs $0.75 more than a pen

- The sum of Ted's and Ed's ages, if Ed is twice as old as Ted

- The product of seven and the sum of two numbers, if the second number is five less than the first

- The difference between a number and twice another number, if the second number is one more than the first

- The difference between the interests of two loans, if they have the same principal and rate, but one loan's term is 5 years less than the other

- The total distance traveled, if a car travels from city A to city B, and then to city C, if the first leg of the journey is half the length of the second

- The difference between the sum of two numbers and their product, if one number is 2 more than the other

- Half the sum of two numbers, if one of them is one more than twice the other

## Algebraic Expressions Worksheet #3.1: Answer Key

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## Algebraic Expressions Worksheet #3.2

For each phrase below, pick variables to represent the unknowns. State what the variables represent, and then write the phrase as an algebraic expression.

Example: The sum of three numbers if the second is twice the first and the third is half the first.Solution: A, B, C = the numbers; B = 2A; C =

- The sum of Mark, Anthony, and Bruno's ages, if Bruno is twice as old as Mark, and Anthony is two years older than Mark

- The perimeter of a triangle if one of the sides is two units longer than the shortest side, and the other side is three units longer than the shortest side

- The area of a rectangle in which the length is ten more than the width

- The product of two consecutive integers

- The difference between the larger and smaller of two consecutive odd integers

- The sum of three consecutive integers

- The area of a triangle in which the base length is five less than two-thirds the height length

- The price of two bananas, three oranges, and 5 apples, if the price of an orange is twice the price of a banana, and the price of an apple is $1.00 more than the price of a banana

- The difference between the price of a watermelon and the price of two canteloupes, if a canteloupe costs $1.50 less than a watermelon

- The sum of the father's age, his son's age, and his daughter's age, if the father is five times as old as the son, and the daughter is 3 years older than the son

- Two more than the product of two numbers, if one number is two more than twice the other

- The total cost of a loaf of bread, a muffin, and two donuts, if the cost of a bread loaf is twice the cost of a muffin, the cost of a donut is two-thirds the cost of a muffin

## Algebraic Expressions Worksheet #3.2: Answer Key

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