12 of 16 Items .... Source: Grant Wiggins

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

Navigation:

. . . View This Fullsize

Pythagorean theorem: \(a^2 +b^2 = c^2\)



“In a right triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two legs.”

Here is the problem: does the figure whose areas we compare, drawn on the triangle's legs, have to be square?

Can there be other shapes – triangles, rhombuses, regular pentagons, etc. – that make the Pythagorean Theorem more generally true?


.: [ALL], [Grant Wiggins], [New Understanding].

. . . View This Fullsize




Question 15: Negative times a negative is ... ?

Why is a negative times a negative equal to a positive?


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

True or false? .999999… = 1
To the best of your understanding, explain why you think so.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

“In geometry we assume many axioms.”

To the best of your understanding, explain the difference between valid and goofy axioms. What gives us the right to assume the axioms we do in Euclidean geometry?


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

“In geometry, we begin with undefined terms.”


Here’s what’s odd, though: every Geometry textbook always draws points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually.

To the best of your understanding, define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

There's an “accurate” answer and “an appropriately precise” answer.

To the best of your understanding, explain the difference between an “accurate” answer and “an appropriately precise” answer.

(HINT: when is the answer on your calculator inappropriate?)


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize




As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.).
The order of operations is a convention.

X(A + B) = XA + XB is the distributive property. It is a law.

To the best of your understanding, explain the difference between a convention and a law. Give another example of each.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize




Why were imaginary numbers invented?

12th graders: Why was the calculus invented?


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

Create the equations that a website or a calculator might use to convert between feet and yards. What are the equations that describe the mathematical relationship?

HINT: all you need as parts of the equation are F, Y, =, and 3.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize




Most teachers assign final grades by using the mathematical mean (the “average”) to determine them, but this measure hides information. What information is hidden and what, if anything, should we do about it?

To the best of your understanding, explain what the mean hides and give at least two reasons why the mean may not be the best measure of achievement.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize




A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

Explain your logic.


.: [ALL], [Grant Wiggins], [Explainer].

. . . View This Fullsize

“Multiplication is just repeated addition.”

To the best of your understanding, explain why this statement is false, giving examples.


.: [ALL], [Grant Wiggins], [Explainer].