Symbolic Abstraction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Symbolic Abstraction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using letter symbols to represent mathematical concepts in a form that holds independent of any specific numerical values.

Instead of $2+3=3+2$ and $5+7=7+5$ , write $a+b=b+a$ for ALL numbers.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Symbolic abstraction states a fact once with letters instead of re-checking it for specific numbers.

Common stuck point: The procedure for symbolic abstraction is the easy part; the trap is trying to solve a universal identity for a value. Asking "Am I making a claim meant to hold for every value, not just the numbers in front of me?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I making a claim meant to hold for every value, not just the numbers in front of me?

Worked Examples

Example 1

medium

The area of a circle is

A = \pi r^2

. Without knowing

r

, what happens to

A

r

is doubled?

Answer

The area is multiplied by 4.

First step

Replace

r

with

2r

A_{\text{new}} = \pi(2r)^2 = \pi \cdot 4r^2 = 4\pi r^2

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Example 2

hard

f(x) = ax^2 + bx + c

and

f(0) = 5

, what is

c

Example 3

medium

The perimeter of a rectangle with length

l

and width

w

P=2l+2w

. Find

P

when

l

is increased by

3

and

w

stays the same. Express your answer in terms of

l

and

w

Example 4

medium

The area of a square is

A=s^2

. If the side length is tripled, by what factor does

A

change?

Example 5

medium

Write the sum of three consecutive even integers starting with

2k

Example 6

hard

The sum of

n

consecutive integers starting with

a

equals

na+\tfrac{n(n-1)}{2}

. Use this to find the sum of integers from

10

20

inclusive.

Example 7

hard

Prove symbolically that the sum of any two even integers is even.

Example 8

hard

Show symbolically that

(a+b)^2-(a-b)^2 = 4ab

Example 9

challenge

Show that for any three consecutive integers, the difference between the square of the middle one and the product of the other two equals

1

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

y = kx

and

y = 15

when

x = 3

, find

k

Example 2

medium

V = lwh

, what happens to

V

when all three dimensions are halved?

Example 3

easy

Write 'a number plus five' as an algebraic expression.

Example 4

easy

Write the commutative law of addition for all numbers

a

and

b

Example 5

easy

n

is any integer, write an expression for the next integer.

Example 6

easy

Write 'twice a number

x

' symbolically.

Example 7

easy

Write an even number symbolically using integer

k

Example 8

easy

If a pencil costs

p

dollars, write the cost of 4 pencils.

Example 9

easy

Write 'the sum of a number and its square' for variable

x

Example 10

easy

Express 'three less than a number

y

' symbolically.

Example 11

medium

Use symbols to prove the sum of two consecutive integers is odd.

Example 12

medium

Write the area of a square of side

s

, then the area if the side doubles.

Example 13

medium

For any number

a

, simplify

\sqrt{a^2}

stating the condition.

Example 14

medium

Write 'a two-digit number with tens digit

t

and units digit

u

' symbolically.

Example 15

medium

f

adds 3 then doubles, write the rule for input

x

Example 16

medium

Write the perimeter of a rectangle with length

\ell

and width

w

Example 17

medium

Express 'the average of

a

b

, and

c

' symbolically.

Example 18

challenge

Disprove the claim '

\sqrt{a^2+b^2}=a+b

for all

a,b

' using symbols.

Example 19

challenge

Show that

n^2-n

is always even for integer

n

, symbolically.

Example 20

challenge

Write a symbolic expression for the sum

1+2+\dots+n

and verify it for

n=4

Example 21

medium

Write a symbolic expression for an odd number using integer

k

Example 22

medium

Write the distributive law symbolically for all

a,b,c

Example 23

easy

Express 'half of a number

n

' symbolically.

Example 24

easy

Write an odd integer using integer

k

Example 25

easy

If a number is

n

, write the next two consecutive integers.

Example 26

medium

f(x)=3x+1

, find an expression for

f(2x)

Example 27

medium

Sarah has

\$x

and Tom has

\$10

more. Write Tom's amount.

Example 28

medium

g(x)=x^2-1

, find

g(a+1)

in simplified form.

Example 29

medium

Write 'the square of the sum of

a

and

b

Example 30

hard

The volume of a cylinder is

V=\pi r^2 h

. What happens to

V

r

is doubled and

h

is halved?

Example 31

hard

f(x)=ax+b

and

f(1)=5

f(3)=11

, find

a

and

b

Example 32

hard

Each side of an equilateral triangle is

s

. Write its perimeter and area in terms of

s

Example 33

hard

f(x)=x^2

and

g(x)=x+1

, write

f(g(x))

in simplified form.

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Related Concepts

Variables Proofs Generalization

Background Knowledge

These ideas may be useful before you work through the harder examples.

variables