Practice Structure Recognition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The skill of identifying that a given mathematical expression or problem belongs to a known family or matches a recognizable pattern.

Seeing 'Oh, this is really a quadratic' or 'This has the same structure as...'

Showing a random 20 of 50 problems.

Example 1

easy
What type of equation is 3x+2=113x + 2 = 11?

Example 2

medium
Recognize the hidden quadratic in 9x−4⋅3x+3=09^x - 4\cdot 3^x + 3 = 0 and solve.

Example 3

medium
Recognize the structure to solve 1x+1x+1=1x⋅1x+1⋅k\frac{1}{x}+\frac{1}{x+1}=\frac{1}{x}\cdot\frac{1}{x+1}\cdot k ... instead, factor x2+5x+6x^2+5x+6.

Example 4

easy
Recognize the structure and factor x2−25x^2 - 25.

Example 5

easy
What known sequence type is 2,5,8,11,…2, 5, 8, 11, \dots?

Example 6

medium
Recognize the structure: what is eln⁡7e^{\ln 7}?

Example 7

challenge
Recognize the structure to solve x4−5x2+4=0x^4 - 5x^2 + 4 = 0.

Example 8

challenge
Recognize the structure of 11⋅2+12⋅3+⋯+19⋅10\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\dots+\frac{1}{9\cdot10} and evaluate.

Example 9

medium
Recognize the structure to sum 1+2+3+⋯+1001 + 2 + 3 + \dots + 100.

Example 10

medium
Recognize the structure: factor x3−8x^3 - 8.

Example 11

hard
Recognize the structure: compute (100)+(101)+⋯+(1010)\binom{10}{0} + \binom{10}{1} + \dots + \binom{10}{10}.

Example 12

easy
What known identity does a2−2ab+b2a^2 - 2ab + b^2 match?

Example 13

medium
Recognize the structure and simplify x2−9x2+6x+9\frac{x^2 - 9}{x^2 + 6x + 9}.

Example 14

easy
Recognise and name the algebraic structure in x2−6x+9x^2 - 6x + 9, then factorise.

Example 15

medium
Recognize the structure: solve x+1x=2x + \frac{1}{x} = 2 for x>0x > 0.

Example 16

hard
Recognize the structure: factor 4x2−12x+94x^2 - 12x + 9.

Example 17

easy
Factor x2−9x^2 - 9 by recognizing its structure.

Example 18

medium
Recognize the structure of 1+3+9+27+811 + 3 + 9 + 27 + 81 and compute via formula.

Example 19

easy
Is y=mx+by = mx + b structurally linear or quadratic?

Example 20

easy
Identify the structure of the sequence 5,10,20,40,…5, 10, 20, 40, \dots.
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