10 of 10 Items .... Source: David Wees

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

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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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The problem speaks for itself.



.: [MS Math], [David Wees], [Explainer].

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You've been asked to shade one-third of this triangle.




And how do you know?

Here's a response:



  • Is this a valid way to get one-third of the triangle?

  • Does this technique require that we stipulate a right triangle?

  • Does this technique require that we stipulate an isosceles one?

  • Do the lines have to be parallel to each other?

  • Do the lines have to be parallel to a side for this to work? 

  • Perpendicular to a side for this to work?


  • How might we generalize this method (if it can be generalized)?


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

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Q1: How many different ways are there to shade this to show \(\frac{1}{4}\)?
Q2: How might kids shade this to show \(\frac{1}{4}\)?


.: [PREALG], [David Wees], [How Many Ways?].

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Mathematical Grammar ... I think a comma might be necessary in #2, maybe two of them. What do you think, students?







.: [ALL], [David Wees], [Find the Error].

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What do you see?
What do you wonder?


.: [PREALG], [David Wees], [Notice, Wonder].

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.: [GEOM], [David Wees], [Notice, Wonder].

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Here is an expression:  \(x² + 6x = x² - 8x + 4\)
Would you describe it as "quadratic" or "linear"?

If the \(x^2\) terms cancel out, does that change your thought?


Is this wolframalpha plot helpful?



Is this desmos response helpful?


.: [ALG], [David Wees], [Notice, Wonder].

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In a fascinating bit of testimony before a Michigan Senate Hearing, this slide was presented. The presenter (Dr. Ball) asked the legislators to identify where and how the 4th-grade solvers made their mistakes. 



David Wees asks:

"What is the mistake?"
"What is the thinking that led to this mistake?"

I saw both for (b) and (c) but the train of thought in (a) escaped me completely. What do you think happened in (a)?

What would you tell the students in each case?


.: [Teacher Education], [David Wees], [Find the Error].

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that's it.