12 of 21 Items .... Course: ALG

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These questions are from the NY Regents test, and discussed by Patrick Honner in his long-running series, "Are These Tests Any Good?".

Can your students find what is wrong with them?



and

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Everyone older than ten knows that there is a way to play tic-tac-toe so that you can never lose. Any game that can be played to a draw unless someone makes a newbie's mistake, is boring once you know the secret.

Let's look at this one and see if you can find a strategy for it:

Players alternate writing a number from 1 - 9 (once used, it’s dead). First one with a set of three numbers that sum to 15 wins.

Is there a guaranteed winning strategy for Player 1?
Is there a guaranteed winning strategy for Player 2?

How does his later comment help? "It’s equivalent to tic-tac-toe since any row/col/diagonal in a 3x3 magic square sums to 15, but more mathematically interesting."

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How is this problem




related to this one from a few days back?



Is any square number times four another square number?

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Always True, Sometimes True, or Never True?



Cut out the cards and have students sort them into one of the three categories.

In each case, students need to say WHY its always true or NEVER true, or give an example and a counterexample for when it is SOMETIMES true.

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The process seems to be the interesting thing here. How would you begin to work on this?

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If a, b are positive integers greater than 1 and \(b^a=2^{12}\) then what is the largest possible value of b?

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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?

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Why are coordinate axes perpendicular?

Do they have to be?

What if there are more than two dimensions?

My question of the day: what's so great about axes being perpendicular?

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If anything to the power of 0 is 1, and 0 to any power is 0, then what is \( 0^0 \)?

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Can you show if this is true?

\(2^6 * 2^6 = 2^{11} + 2^{11}\)

Can we generate more like this one ... simple and elegant?
How about one with 3 as a base?

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How would you split these lines into 2 groups?




What other lines would fit into your groups? What wouldn't fit?
Which ones do belong?