522: Where to start with factoring

You are asked to find the roots and factors of the following polynomial function:
f(x) = x⁴ – 2x³ – 18x² + 6x + 45

By the rational root theorem, possible rational roots are
± 1, 3, 5, 9, 15, or 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 one is most likely?

517: What is the function?

Algebra 2:
Write a function for this graph ...

Estimation:
What are some possible values for a,b,c,d?
If I told you it also went through (0,54), what might the leading coefficient be?

I would not use the phrase "in factored form", if I were using this in a review or summative assessment since students should be selecting the form that's most appropriate.  I might use it if we are in the middle of learning about functions for the first time and we hadn't really made the case for the utility of various forms of the equation. 

source:

How would your Ss write a function in factored form for this graph? Nice #MP7 #CthenC problem from Bossé #ncctm16 pic.twitter.com/DoUZUzNR7L

— Jennifer Wilson (@jwilson828) October 28, 2016

502: Powerful Question

It's not included in the PEMDAS Order of Operations ...

Should $a^{b^c} = ({a^b})^c$ or should it be $a^{b^c} = a^{(b^c)}$ ??

Does $3^{2^0}$ equal 1 or 3?
 
Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?

For the record,  $a^{b^c} = a^{(b^c)}$ is the accepted order of operations here.

496: Determining the Discriminant

How many different solutions can we find?

492: Sneak in Some Algebra

Marilyn Burns pointed out:

Well, does it?
Is there a pattern that always works?


source.

489: Which Quadratic Formula?

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?

458: Is That You, Pythagoras?

If we stipulate that $(x+12)^2+(y+4)^2+(z+3)^2=0$, then

$\sqrt{x^2+y^2+z^2}=?$

Should we brute-force this or is there a more beautiful or subtle way of getting what we want?

Source:

(x+12)²+(y+4)²+(z+3)²=0 √(x²+y²+z²)=? #SATPrep #mathchat #algebra

— David Marain (@dmarain) May 13, 2015

455: Another Proof Without Words

n³ - 1 is always a multiple of n - 1 #ProofWithoutWords pic.twitter.com/CkOKg0wD6O

— Roy Wright (@Sted304A) April 14, 2015

444: Floorlamp

The floor lamp casts a shadow.

Straight line AB is drawn on a wide, spacious floor. A lamp is at a height of 8  m above point C, which is located on line AB.  Line CD is perpendicular to line AB.  A rectangular solid 5 m by 3 m by 4 m is on the floor as shown.  The box is 2  m from line AB and a meters from line CD. Assume the lamp is a point source and will cast a perfect shadow. Given that the rectangular solid casts a shadow with area 90  m², determine the distance a (in meters).

• Does the 2m distance from the wall matter to the shape of the shadow?
• How does the distance a change the area of the shadow?
• What shape is that shadow on the floor? 
• Are the edges of the shadow guaranteed to parallel with the edges of the box?
• How far away is that box from the light? 

It's another Five Triangles puzzle

442: Square Root

x² = 81 has two solutions, -9 and 9.

But does √81 have one solution or two? Is it correct to say that √81 = +9 and -9?

Or should we be saying that √81is an expression and that 9, 18/2, 27^(2/3), and 1+6+2, are equivalent expressions?

It is really alarming how often I see materials written by math teachers claiming that for example sqrt(81) is both 9 and -9.
— Kate Nowak (@k8nowak) April 22, 2015

434: Integer Quotient

Is it possible to find a positive integer value of  n so that this quotient is an integer?

429: Least Possible Value of y

Is there enough information to determine the least possible value of y?

424: Point-point-slope

Challenges

1. Use four different digits (from 2-9) to create two points which determine a line with the greatest possible slope.
2. Use four different digits (from 2-9) to create two points which determine a line with the least possible slope.
3. Use four different digits (from 2-9) to create two points which determine a line with a slope as close to zero as possible.

source.

What does "greatest" slope mean?
What does "least" slope mean ... most negative?

419: The Value of Why

Can you find values for y that will make each answer true?
Generalize the rules here.

417: What can we do with this?

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

402: Plugging in to Solve the SAT

This is from the end of a section of an SAT test, and is therefore a bitch to solve. At least, the results from the test seem to indicate so.  On a 5-choice multiple choice question, only 8% of respondents got this one right ... if they had covered their eyes, refused to read the question, and randomly guessed, they would have more than doubled their chances of getting it right. We can do better than that!

What numbers should I plug into the equations to test for correctness?

25. A watch loses x minutes every y hours. At this rate, how many hours will the watch lose in 1 week?

1. $7xy$
2. $\dfrac{7x}{y}$
3. $\dfrac{x}{7y}$
4. $\dfrac{14y}{5x}$
5. $\dfrac{14x}{5y}$

401: Which Question is Harder?

As we all know, the questions in an SAT test get harder as you get to the end of a section. I warn my students repeatedly that, at the end of a section,

"If you can't see what all those people thought was the obvious answer and clearly see why that obvious answer is wrong, then you are one of those who will jump to the wrong conclusion ... and you'll get it wrong, too."

 You see, the question at the end aren't really difficult usually. They can be badly worded but they're rarely HARD. They are usually four-step problems that everyone else is so sure about, and the answer is so obvious that they get suckered in. These two were rated the same level of difficulty, and were answered correctly by about 3%-5% of the students.

Which one is harder?

25. A woman drives to work each day at an average speed of 40 miles per hour and returns along the same route at 30 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the trip?

1. 30 
2. $30\dfrac{1}{7}$ 
3. $34\dfrac{2}{7}$ 
4. 35
5. 40

24. A 25 foot ladder is placed against a vertical wall of a building with the bottom of the ladder standing on concrete 7 feet from the base of the building. If the top of the ladder slips down 4 feet, then the bottom of the ladder will slide out.

1. 4ft 
2. 5ft 
3. 6ft 
4. 7ft 
5. 8ft

And can you say why the obvious answer is wrong?

400: Factoring

This problem, from Robert Kaplinsky, asks for you to fill in the spaces.

Being the ornery sort, I wondered if there were other coefficients that we could choose for the cubic that might give multiple sets of answers?

397: Substitutions

Based on a couple of posts I made a while back, Don Steward made these:

383: Selecting Thoughtfully

source: Don Steward
"this resource follows a fine series of posts on MathArguments180 'which values of x do we choose?' (number 330)