11 of 11 Items .... Type: Smart Substitution

Problems, Questions, and Puzzles to spark discussion and argument in the maths classroom.

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This is from the end of a section of an SAT test, and is therefore a bitch to solve. At least, the results from the test seem to indicate so.  On a 5-choice multiple choice question, only 8% of respondents got this one right ... if they had covered their eyes, refused to read the question, and randomly guessed, they would have more than doubled their chances of getting it right. We can do better than that!

What numbers should I plug into the equations to test for correctness?

25. A watch loses x minutes every y hours. At this rate, how many hours will the watch lose in 1 week?


  1. \(7xy\)

  2. \(\dfrac{7x}{y}\)

  3. \(\dfrac{x}{7y}\)

  4. \(\dfrac{14y}{5x}\)

  5. \(\dfrac{14x}{5y}\)



.: [SAT], [SAT], [Smart Substitution].

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Based on a couple of posts I made a while back, Don Steward made these:




.: [ALG1], [Don Steward], [Smart Substitution].

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"This resource follows a fine series of posts on MathArguments 'which values of x do we choose?' (number 330)


.: [ALG1], [Don Steward], [Smart Substitution].

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For the following functions, think about "How to graph like a math teacher."
Math teachers want to sketch graphs quickly and efficiently and choose
values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?

\(y = \lvert x \rvert\)

\(y = \lvert x \rvert - x\)

\(y = 2\lvert x+2 \rvert\)

\(y = \dfrac{1}{2}\lvert x-3 \rvert\)


What are your favorite examples of this?


.: [ALG1], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?

\(y = \sqrt{x+2}\)

\(y = \sqrt{1-x}\)

\(y = \sqrt[3" />
{2x+1}\)

What are your favorite examples of this?


.: [ALG2], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher."
Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?


\(y = \dfrac{4}{(x+2)(x+11)}\)

\(y = \dfrac{1}{x^2+1}\)

\(y = \dfrac{1}{x^-1}\)

\( y = \dfrac{x+2}{x^2-4}\)

\( y=\dfrac{x^2-1}{x^2+1}\)

What are your favorite examples of this?


.: [ALG2], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher."
Math teachers want to sketch graphs quickly and efficiently and choose
values of x that work "nicely" in the equation and  generate integer
values of y, thus making it easier to graph.

Which points in the graph:
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?

\(y = sin(3\theta)\)
\(y = {sin(\theta)}^2\)
\(y = sin^{-1}(\theta)\)
\(r = 3sin(3\theta)\)

What are your favorite examples of this?


.: [PRE-CALC], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher."
Math teachers want to sketch graphs quickly and efficiently and choose
values of x that work "nicely" in the equation and  generate integer
values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?


\(y = \dfrac{1}{3}x + 7\)

\(y = \dfrac{2}{7}(x + 4)\)

\(y = \dfrac{x + 2}{5}\)


What are your favorite examples of this?


.: [ALG1], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher."
Math teachers want to sketch graphs quickly and efficiently and choose
values of x that work "nicely" in the equation and  generate integer
values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?


\(y = (x+2)(x-4)\)

\(y = -(x+1)(x+5)\)

\(y = x^2 + 6x + 9\)

\(y = \dfrac{1}{4}(x+5)^2\)


What are your favorite examples of this?


.: [ALG2], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

Which points in the graph
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?


\(x^2+y^2=25\)

\(\dfrac{(x-1)^2}{9}+\dfrac{(y+2)^2}{16} = 1\)

\(\dfrac{(x-4)^2}{9}-\dfrac{(y-4)^2}{16} = 1\)


What are your favorite examples of this?


.: [PRE-CALC], [T.R.Milne], [Smart Substitution].

. . . View This Fullsize

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

Which points in the graph:
- are best found by inspection?
- are best found by substitution?
- are best found by symmetry?

\(y = (x-2)^2(x+2)\)

\(y = (x-1)^3\)

\(y = x^3-8\)

\(y=x^3+3x^2\)

\(y=(x-3)^2(x+3)^2\)


What are your favorite examples of this?


.: [ALG2], [T.R.Milne], [Smart Substitution].
that's it.