12 of 24 Items .... Type: Which Would You Choose?

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You are asked to find the roots and factors of the following polynomial function:

\( f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45 \)

By the rational root theorem, possible rational roots are
\( \pm ( 1, 3, 5, 9, 15, 45 ) \).

In order to minimize your effort, you know that you should begin with the possibility that is most likely to be a root. Which roots are most likely?

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What car will have the shorter path?

How could you tell for certain?

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So ... I've seen a couple of YouTube videos that feature songs about the Quadratic Formula. I often see it written like this:

\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
\)

and it occurs to me that I've always written it this way:

\(x = \dfrac{-b}{2a} \pm \dfrac{\sqrt{b^2-4ac}}{2a}\)

Which one is better?

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We all know that playing the middle of a tic-tac-toe game ensures that you can never lose if you play it right, but this one seems to be a bit trickier.

Where is the best place to start?

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What do you notice about these three equations?

\(10 * 9 * 8 * 7 \div 6 \div 5 * 4 * 3 - 2 +1 = 2015\)

\((4 * 7 * 69 ) + 83 = 2015\)

\((69 - 4) * (38 - 7) = 2015\)

Which one do you like best?

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Would you guess? $250,000 if correct and $100,000 if she refuses to try for it.

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So, students. You've had a chance to weigh in on addition and multiplication. What is the best way to do division?

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A small company wants to give raises to their 5 employees. They have $10,000 available to distribute. Imagine that you are in charge of deciding how the raises should be determined.

1. What are some variables you should consider?


2. Describe mathematically as many different methods you can think of to distribute the raises. (We came up with nine; can you beat that?)


3. What information will you need to compute those raises according to your various methods?


4. Which of your methods do you feel is the most fair?

1. zm vjfzo znlfmg, gdl gslfhzmw vzxs.


2. vjfzo kvixvmgztv yzhvw lm hzozib.


3. mvklgrhn: 1p, 1p, 1p, 1p, Ldmvi'h hlm: 6p.


4. R'n lmv lu gsv vnkolbvvh: 0p, 0p, 0p, 0p, nv: 10p


5. kilwfxgrergb tlzoh: 1p, 1p, 2p, 2p, 4p


6. olggvib, zoo li mlgsrmt.


7. olggvib, 0p, 1p, 2p, 3p, 4p.


8. zokszyvgrxzo liwvi, 0p, 1p, 2p, 3p, 4p.


9. vnkolbvv'h xsrow'h gvhg hxlivh rm gsv olxzo hxsllo ... ru gsviv'h ml rnkilevnvmg, ml ylmfh.

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Which would you press ... and why?

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This system of equations has a peculiar characteristic ... I think it is easier to solve it by analytical means than by using Desmos or a TI-84. 

Do you agree?

What about it makes a graphical solution difficult?

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What method works best to find the area of the triangle created by these three lines?

Organized list (a la excel)
Substitution?
Linear Combination?
Plug and Pray?

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"The sum of two positive integers is 216. The greatest common factor of the two numbers is 24. What are all the possible pairs of numbers?"

What approach seems the easiest here?

I can see using solution methods such as:

- algebra

- guess and check

- organized list

- visual representation

Which (or which other) method rings true for you? Which of the two sentences eliminates the most numbers?