Using "SAT-Type" Problems to Develop Understanding of Quadratic Functions in Algebra

f(x) = t-2(x+4)² where t is a constant.
If f(-8.3) = f(a) and a > 0, what is the value of a?


This type of question is of the Grid-in type (or short constructed response) that now appears on standardized testing like the SAT-I and ADP Algebra 2.

I administered it to a group of strong SAT students recently and the students who completed Alg II struggled with it. As our president might say, this was a "teachable moment!"

A few thoughts...
Should textbooks include more questions of this type both as examples and regular homework exercises? As you might guess, I'm very much opposed to having questions labeled as Standardized Test Practice in texts or appear in a separate section of the text or in ancillaries.

By the way, by including the label "SAT-type problems" in the title of this post I'm trying to engender both positive and negative response. Those of you who have followed this blog for 2- 1/2 years know that what I'm really referring to are "conceptually-based questions." Some of you react adversely to the idea that standardized test questions should influence our curriculum or how we teach. N'est-ce pas?

Your comments...

Updates, ODDS AND EVENS and some Geometry Packing Problems

Enjoying your summer hiatus or as busy as ever? I know that feeling!

1. MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus for now will be on Geometry, Algebra II and Precalculus. Several other ideas are running through my head but I need the time to bring them to fruition. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' geeeemail dot com."

2. CNNMoney.com Article - Something to tell your students in September!
Here is the link. The 2nd paragraph says it all:

The top 15 highest-earning college degrees all have one thing in common -- math skills.

3. Silly Instruments for Math Teachers to Play
I always told my students that I'm predominantly left-brained -- analytical, organized, detailed, process-oriented, algebraic -- as opposed to most of my children and my wife who are creative, spatial, mechanical, who see the forest more than the trees. One of my sons is a musician and another is a dancer so we are not always on the same wavelength! So I mentioned to my SAT students that I wish I had a more creative side and perhaps be able to play an instrument, but, in fact, the only thing I can "play" is my iPod! One of my students in the front row immediately responded, "I know an instrument you can play, Mr. M -- the triangle! I congratulated her for the cleverness and told her that maybe I will learn how to play the "cymbals." (the class actually applauded that lame attempt at word play!). In fact, I've read that many famous mathematicians were also musicians, so let us know: Do you play an instrument or are passionate about music or do you have a silly instrument for a mathematician to play?

4. Circle Packing Problems
Even though I am dominantly left-brained, I still enjoy challenging spatial geometry problems. I find these questions have improved my creativity and my spatial sense and they often involve multi-faceted thinking. Here are a couple of famous 'packing' problems which are accessible to geometry students. More important than solving these is to give our students a sense of the importance of packing problems and the ongoing research in this area. There are still unsolved problems here!

Although you can easily research packing problems on MathWorld and Wikipedia, the diagrams below come from an exceptional website I discovered. The author, Peter Szabo (missing accents), provides diagrams for packing 2-100 circles with accompanying data (radii, density, etc).


PROBLEM I


The two congruent circles at the left are actually enclosed in a unit square which is not shown.The circles are tangent to each other and to the sides of the square. If these circles have the maximum radius possible, determine the radius.
Note: The indicated square (assume it is a square) is helpful in solving the problem. Trig is not necessary here.

Answer (Yes, I'm providing this since the objective is to discuss the method):
[The following is the diameter, not the radius, of each circle. Thanks to watchmath for correcting this error].









PROBLEM II



Again, imagine that the three congruent circles at the left are enclosed in a unit square and are tangent to each other and to the sides of the square. If the circles have the maximum radius possible, determine this radius.
Notes: The indicated square again may be helpful to solve this problem. Trig can be used but clever use of special right triangles is preferred.

Answer:
[The diameter is given below, not the radius. Thanks to watchmatch for correcting this]

A Morning Warmup for Middle and High Schoolers - No Calculators Please!

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...


Comments
1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!

...Read more

Taking Middle Schoolers Beyond Procedures To The Next Level...

Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...

Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.

BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...

What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...


1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).

3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?


These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...

To see a more challenging version of the examples above, click Read more...



You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!

The lcm of 100 and N is 500. What is the least positive integer value of N?

...Read more

Dorothy Revisited -- Another View...

Mathmom contributed some insightful thoughts about how most middle school students might feel about the probability investigation from the other day. I agree with her that some would be able to compute the results or even devise a general formula but "proving" it in the general case might be too ambitious. In my reply, I suggested there might be another way of deriving the formula 1/N for the probability of losing the game. Here's what I came up with. It still requires some careful development to show that the outcomes are equally likely but I will indicate how it could be done in the particular case where N = 10.

Brief Explanation of Method:
There are N equally likely (to be shown) ways for the game to end (i.e., when the red card is selected). Of these, only one will result in a loss -- when the red is the last card chosen. Therefore, the probability of losing is 1/N, hence the probability of winning is 1 - 1/N or (N-1)/N.

Demonstrating "Equally Likely" for N = 10:
P(game ending after one card) = 1/10
P(game ending after 2 cards) = P(black selected followed by red) = (9/10)(1/9) = 1/10
P(ending after 3 cards) = P(black,black,red) = (9/10)(8/9)(1/8) = 1/10
etc...

The general case is similar using N in place of 10. I do think that students with some understanding of algebra could follow it but deriving it on their own is another story!

I also indicated that I might provide a program for the TI-83 or -84 which could be used to simulate the game. The programming skills needed are not that advanced and some high schoolers or even middle schoolers can pick up on the code and begin writing their own programs - I've seen it happen! Here it is...











T represents the number of times the game is played with 3 cards. I entered 100 for the number of trials. K stores the number of times Dorothy won when playing 100 times. Can you make sense of the rest of the code?

The experimental probability of 0.68 is reasonably close to the theoretical probability of 2/3. I often feel more confident of my reasoning in difficult probability problems when my simulation approximates my answer. This doesn't prove anything but it does have value IMO. There is also the opportunity to demonstrate some important stat concepts by running the program several times and having students plot the experimental probabilities and observing their distribution.

Please Help Dorothy Go Home - A Probability Fantasy for Middle School and Beyond

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!



The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.


The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!

"On The Road Again" With 'TC' -- A Real World Application of Geometry

As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:

• How to introduce this
• Asking questions to provoke deeper thought
• Drawing conclusions and further generalizations
• Connecting this problem to other circle or geometry problems
  
• Maximizing student involvement
  

I told TC I would need some time to rework the original problem for the younger students so here goes...





Diagram for Parts I and II







Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.

If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).

Show that Option 2 is shorter than Option 1.

Part II (middle and secondary students)
Same diagram but now the radii are R and r with R > r.
Show algebraically that Option 2 is shorter.


Part III (secondary students)












To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R > r.

As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).

Show that Option 2 will be shorter provided π ≥ θ > 2.

Click Read More for further discussion...


Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?

Thanks TC for providing us with another stimulating challenge!

...Read more

An Equation Which May Be More 'Complex' Than It Appears!

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a^-3/2 - a^-1/2 - a^1/2 = 0

Preliminary Comments/Questions/Issues

• Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  
• Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  
• It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  
• Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
• What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  
• Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...