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 — 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) and give the constant term.

Example 2

hard
The 'change of basis' idea transfers from linear algebra to coordinate systems. Convert polar (r,θ)=(2,π/3) to Cartesian.

Example 3

challenge
Recursion transfers from the Fibonacci sequence to many problems. With F1=F2=1 and Fn=Fn−1+Fn−2, give F6.

Example 4

easy
The factorisation a2−b2=(a−b)(a+b) transfers to factoring x4−16. Apply it.

Example 5

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

Example 6

easy
The factoring identity a2−b2=(a−b)(a+b) transfers. Factor x2−25.

Example 7

easy
Adding fractions ac+bc=a+bc transfers to algebraic fractions. Simplify x5+2x5.

Example 8

medium
The 'area under a curve' idea transfers from geometry to physics. A particle moves at constant velocity 7 m/s for 4 s. Find the distance.

Example 9

easy
The idea of completing the square to solve x2+6x+5=0 transfers to converting x2+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 f and g are continuous at a, then f+g is continuous at a.' Sketch the transferred argument structure.

Example 11

medium
Logarithms transfer multiplication into addition. Using log⁡(ab)=log⁡a+log⁡b, compute log⁡28+log⁡24.

Example 12

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

Example 13

hard
The 'difference of cubes' identity a3−b3=(a−b)(a2+ab+b2) transfers across variables. Factor x3−27.

Example 14

easy
The Pythagorean theorem from geometry transfers to the distance between two numbers on a line. The distance between −2 and 5 is?

Example 15

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

Example 16

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

Example 17

medium
The 'inverse function' idea transfers from arithmetic to logs. If f(x)=10x, find f−1(1000).

Example 18

medium
AM-GM transfers across number counts. Apply it to two positive numbers 9 and 16: 9+162≥9⋅16. Compute both sides.

Example 19

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

Example 20

easy
The notion of inverse transfers across operations. The inverse of multiplying by 5 is what?
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