Practice Representation in Math

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

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Quick Recap

A mathematical representation is any format — diagram, equation, table, graph, or symbolic expression — used to encode and communicate a mathematical idea or relationship between quantities.

The same idea can be shown in multiple ways—each reveals different aspects.

Showing a random 20 of 50 problems.

Example 1

easy

A pie chart and a bar chart show the same survey data. What does this illustrate about representation?

Example 2

medium

A system of two linear equations can be a pair of lines (graph) or a matrix equation

A\mathbf{x}=\mathbf{b}

. Which representation generalizes better to

100

equations, and why?

Example 3

hard

The map

f

sends

1 \to a

2 \to b

3 \to c

. Write

f

as a set of ordered pairs.

Example 4

easy

To show that two angles are equal, which representation usually helps most: a labeled diagram or a paragraph of words?

Example 5

challenge

A graph appears to show

f(x)=\frac{x^2-1}{x-1}

as the line

y=x+1

. Explain how the graphical representation can mislead, and which representation corrects it.

Example 6

medium

Convert

\frac{5}{8}

to a percent.

Example 7

challenge

Convert the complex number

1+i

from rectangular to polar form.

Example 8

medium

Represent the line

y=3x-2

as a

y

-intercept and slope pair.

Example 9

medium

A motion problem is easier in a position-vs-time graph than in a paragraph. Name the representation switch and the feature it exposes.

Example 10

challenge

Counting lattice paths from

(0,0)

(m,n)

can be represented as binomial coefficients or as a grid-walk picture. Show both representations give

\binom{m+n}{m}

and explain why the symbolic one scales.

Example 11

easy

Write the solution set of

x^2 = 9

in (a) roster notation, (b) set-builder notation.

Example 12

medium

Express '

x

is at most

3

' as an inequality.

Example 13

medium

Convert

\tfrac{7}{20}

to a decimal.

Example 14

medium

Convert the recurring decimal

0.\overline{3}

to a fraction.

Example 15

easy

What kind of representation is a Venn diagram?

Example 16

medium

A function doubles its input and adds

1

. Write the rule symbolically.

Example 17

medium

Complex number

z

can be written

a+bi

r(\cos\theta + i\sin\theta)

. Which representation makes multiplying two complex numbers simplest?

Example 18

hard

Express the parabola

y=x^2-4x+5

in vertex form.

Example 19

medium

A probability problem is stated in words. Drawing a tree diagram changes the representation. What does the tree make explicit?

Example 20

medium

Which numerical representation makes it easiest to add

\tfrac{1}{4}

and

0.5

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Related Concepts

Multiple Viewpoints Abstraction