12 of 30 Items .... Type: Epiphany

Ask your class if there are any Pythagorean anti-Triples: \(\dfrac{1}{a^2} + \dfrac{1}{b^2} = \dfrac{1}{c^2}\) ?

If a, b, c are integers, the question seems harder, since 1/3, 1/4, 1/5 would be (to me) obvious answers - leading to the rules for Pythagorean triples and just using their reciprocals.

.: [ALG1], [James Tanton], [Epiphany].

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I might have posted this puzzle before.

What do your students think?
Can they generalize it?
What is the overlapped area?

.: [GEOM], [David Wees], [Epiphany].

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Seems like a tough problem.

"Wait a minute. That would make it easier ..."

.: [MS Math], [Five Triangles], [Epiphany].

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What can we do with this? There doesn't seem to be enough information.

3

.: [ALG2], [Curmudgeon], [Epiphany].

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.: [GEOM], [James Pearce], [Epiphany].

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If you can ...

.: [ALL], [internet], [Epiphany].

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Make similar questions ...

If \(x^2+y^2 = 196\) and \((x-y)^2 = 36\), what is the value of \(xy\)?

-116

-8

.: [SAT], [SAT], [Epiphany].

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Three squares of equal but unknown size.

Is this a fair question to ask a fifth-grader?

.: [ALG1], [Numberphile], [Epiphany].

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Take a moment to consider the rules and the methods that we use with powers and exponents.

\(2^x + 2^x + 2^x +2^x = 2^{2014}\)

x = ?

What is the "Aha!" thought, the epiphany, in this problem?

.: [ALG1], [David Marain], [Epiphany].

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If you're a junior or senior, you've seen versions of this problem before, perhaps on the SAT (the source of this problem). As I've said before, the SAT is designed in a way that calculators are not necessary and each question must be solvable in less than a minute. Often, the student is expected to change the form of the question: text to algebra, or algebra to visual (graphical); or rearrange the terms, or work backwards from the known.

The sum of the positive odd integers less than 200 is subtracted from the sum of the positive even integers less than or equal to 200. What is the resulting difference?

As it stands, that question would take you far too long to find an answer for, so it must have a simplification somewhere. What can we do to make this quicker to answer? Or simpler to understand? How can we change it?

.: [SAT], [SAT], [Epiphany].

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Town A is 300 km from Town B and Town B is 200 km away from Town C.

- What is the closest A and C could be?

- How far apart could they be, if they were as far apart as possible?

- How many integer distances between A&C are possible?

.: [SAT], [David Marain], [Epiphany].

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Why don't we need a calculator for this?

.: [ALG2], [UVM], [Epiphany].