Practice Multiple Viewpoints 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 practice of analyzing the same mathematical object or problem from several different representations, frameworks, or perspectives.

Looking at the same thing from different angles reveals different truths.

Showing a random 20 of 50 problems.

Example 1

medium

Evaluate

\sum_{i=1}^{5} i

by direct addition or by the formula

\frac{n(n+1)}{2}

. Give the value.

Example 2

easy

Compute

\tfrac{3}{4}+\tfrac{1}{4}

as a fraction or as decimals. Give the value.

Example 3

hard

Compute

\sum_{k=1}^{100}k

via the closed-form

n(n+1)/2

or via pairing

1+100,2+99,\ldots

Give the value.

Example 4

easy

The number

\frac{1}{2}

can be viewed as a fraction, a decimal, a probability, and a ratio. Describe each viewpoint and what it emphasises.

Example 5

easy

View

|{-5}|

as a distance from

0

on the number line and also as the algebraic definition

\max(x,-x)

. Give the value.

Example 6

medium

The number

0.5

as a fraction or as

50\%

of a whole: a pizza is cut so one person gets

0.5

. Out of

8

slices, how many do they get?

Example 7

hard

View the equation

x^3=8

as 'find a cube root of

8

' (real-number viewpoint) or as a degree-

3

polynomial

x^3-8=0

(complex viewpoint). Give all real and complex roots.

Example 8

medium

Compute the distance from

(1,2)

(4,6)

via the distance formula or via the Pythagorean theorem with legs

3

and

4

. Give the distance.

Example 9

medium

View the Pythagorean theorem

a^2+b^2=c^2

from three different perspectives: algebraic, geometric, and physical. Give one application for each.

Example 10

easy

Find the area of a right triangle with legs

6

and

8

via

\frac12 bh

(give the value).

Example 11

challenge

Compute

1+2+4+8+16

as a direct sum or as

2^5-1

(geometric viewpoint). Give the value.

Example 12

easy

Count the dots in a

3\times4

grid two ways (rows times columns, or columns times rows). Give the count.

Example 13

medium

View

0.\overline{9}

as the limit of

0.9,0.99,0.999,\ldots

or as the fraction obtained by

x=0.\overline{9}, 10x=9.\overline{9}

. Give the exact value.

Example 14

easy

View the vector

(3,4)

as Cartesian coordinates or as magnitude-direction. Give its magnitude.

Example 15

easy

Find

\frac{1}{2}+\frac{1}{4}

as a fraction or as a decimal. Give the fraction.

Example 16

hard

Prove the inequality

a^2+b^2\ge2ab

for real

a,b

by viewing it as

(a-b)^2\ge0

(algebra) or as a fact about non-negative squares (geometry). Give the key identity.

Example 17

medium

Compute

\binom{6}{3}

using the formula and by listing 3-subsets of

\{1,2,3,4,5,6\}

. Give the value.

Example 18

medium

Solve

x^2-4=0

as a difference of squares or by the quadratic formula. Give the solutions.

Example 19

medium

Compute the slope of the line through

(2,3)

and

(5,9)

as rise/run or as

\tan

of the angle of inclination. Give the slope.

Example 20

challenge

View the identity

\sum_{k=0}^{n}\binom{n}{k}=2^n

combinatorially (subsets of an

n

-set) or via the binomial theorem at

x=y=1

. Give the value at

n=10

← Back to Multiple Viewpoints Worked Examples

Related Concepts

Representation