Practice Transfer of Ideas in Math

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

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Quick Recap

The ability to recognize that a technique or concept from one area of mathematics applies, possibly in adapted form, to a different area.

Seeing that the same mathematical structure appears in two apparently different contexts โ€” then using what you know about one to solve the other.

Showing a random 20 of 50 problems.

Example 1

medium
The distributive law transfers from numbers to algebra. Expand 3(x+4)3(x+4) and give the constant term.

Example 2

hard
The 'change of basis' idea transfers from linear algebra to coordinate systems. Convert polar (r,ฮธ)=(2,ฯ€/3)(r,\theta)=(2,\pi/3) to Cartesian.

Example 3

challenge
Recursion transfers from the Fibonacci sequence to many problems. With F1=F2=1F_1=F_2=1 and Fn=Fnโˆ’1+Fnโˆ’2F_n=F_{n-1}+F_{n-2}, give F6F_6.

Example 4

easy
The factorisation a2โˆ’b2=(aโˆ’b)(a+b)a^2-b^2=(a-b)(a+b) transfers to factoring x4โˆ’16x^4-16. Apply it.

Example 5

easy
The notion of multiplying as repeated addition transfers to multiplying polynomials. Compute 3(x+5)3(x+5).

Example 6

easy
The factoring identity a2โˆ’b2=(aโˆ’b)(a+b)a^2-b^2=(a-b)(a+b) transfers. Factor x2โˆ’25x^2-25.

Example 7

easy
Adding fractions ac+bc=a+bc\tfrac{a}{c}+\tfrac{b}{c}=\tfrac{a+b}{c} transfers to algebraic fractions. Simplify x5+2x5\tfrac{x}{5}+\tfrac{2x}{5}.

Example 8

medium
The 'area under a curve' idea transfers from geometry to physics. A particle moves at constant velocity 77 m/s for 44 s. Find the distance.

Example 9

easy
The idea of completing the square to solve x2+6x+5=0x^2+6x+5=0 transfers to converting x2+6x+5x^2+6x+5 to vertex form. Show both applications.

Example 10

medium
The proof technique 'assume the hypothesis and derive the conclusion' (direct proof) from logic transfers to proving: 'If ff and gg are continuous at aa, then f+gf+g is continuous at aa.' Sketch the transferred argument structure.

Example 11

medium
Logarithms transfer multiplication into addition. Using logโก(ab)=logโกa+logโกb\log(ab)=\log a+\log b, compute logโก28+logโก24\log_2 8 + \log_2 4.

Example 12

medium
The idea of balancing an equation transfers to balancing a chemical reaction's counts. To keep xโˆ’3=5x-3=5 balanced, add 33 to both sides; give xx.

Example 13

hard
The 'difference of cubes' identity a3โˆ’b3=(aโˆ’b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2) transfers across variables. Factor x3โˆ’27x^3-27.

Example 14

easy
The Pythagorean theorem from geometry transfers to the distance between two numbers on a line. The distance between โˆ’2-2 and 55 is?

Example 15

challenge
The 'continuity argument' transfers from intermediate-value theorem to root finding. If f(x)=x3+xโˆ’1f(x)=x^3+x-1, justify that a real root lies in (0,1)(0,1).

Example 16

easy
The slope idea from lines transfers to rates of change. A line through (0,0)(0,0) and (2,6)(2,6) has what slope?

Example 17

medium
The 'inverse function' idea transfers from arithmetic to logs. If f(x)=10xf(x)=10^x, find fโˆ’1(1000)f^{-1}(1000).

Example 18

medium
AM-GM transfers across number counts. Apply it to two positive numbers 99 and 1616: 9+162โ‰ฅ9โ‹…16\tfrac{9+16}{2}\ge \sqrt{9\cdot 16}. Compute both sides.

Example 19

easy
The idea of solving an equation by inverse operations transfers across operations. To solve x+5=12x+5=12, what operation undoes +5+5? Give the result xx.

Example 20

easy
The notion of inverse transfers across operations. The inverse of multiplying by 55 is what?
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