MathNotations

Posted by Dave Marain at 11:11 AM 0 comments

Labels: algebra, Common Core, explorations, investigations, iteration, precalculus, recursion

Here are the last two math challenges I just tweeted for middle schoolers and beyond. You may want to use this as a fifteen minute activity to improve reading, review basic terms and concepts, develop reasoning and writing in math. There was an error on the 2nd question as it originally appeared on Twitter. I then corrected it.

List the nine 2-digit primes which produce prime numbers when their digits are reversed.




List the SIX 3-digit primes which produce primes when their digits are written in ALL possible orders. 137 fails b/c 371 is not prime.

For both questions students should work in teams of 2-4.

For the first question, students should not be allowed to use a calculator!

For the second one, have them experiment with a calculator for a few minutes. If a student thinks they found one, their teammates must verify it! After 3 minutes ask: "Have you noticed that the numbers you're looking for cannot contain certain digits like 2. What digits and why? Discuss it and one member of the team must record the team's findings and provide a written explanation!

After 3-4 more minutes, have them refer to a table of primes online (or print it and hand out a copy to each team). If they don't find it within the 15 min time limit, have them finish it for extra credit for the next day.

Here is one of the numbers: 113. Good luck!


"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught." --from South Pacific

Posted by Dave Marain at 10:22 AM 2 comments

Labels: explorations, investigations, math challenge, middle school, primes

Consider the following problem:

If -5 ≤ x ≤ 4, and f(x) = 2x² - 3, how many integer values are possible for f(x)?


One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).

Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...


WITH YOUR LEARNING PARTNER(S):

1. Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.

WRITE your conjecture for the answer to the problem: ____

2. Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table. Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1? YES NO

3. The following statement is plausible but FALSE.

The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.

Explain why this is wrong. There is more than one error!

4. The correct answer is 51. Depending on the class, a few, if not several, students should be able to come up with the correct answer and provide a thorough explanation.

5. Group Discussion:

• Ask students how they might have approached this question if it appeared on a standardized test? Plug in x-values? Use the graphing calculator? Guess? Skip it?
• Ask the group what made this questionable formidable for some students? How important was understanding what was asked for?
• Review one successful approach to solving the problem by calling on individual students to give the "next" step.

Posted by Dave Marain at 9:35 AM 3 comments

Labels: advanced algebra, explorations, investigations, SAT strategies, SAT-type problems

Since pi day fell on a Sunday this year, we should still be celebrating it today. Besides, March should be declared pi-Month!


It is always fascinating to see how readership (or should I say one-time viewership) always picks up around March 14th every year! I feel obligated to add another pi Day activity or exploration in addition to those I've posted the past three years. By the way, the pi Day Scavenger Hunt is the most popular post by far and I'm not even the one who thought of that idea!


Despite the title of this post, I did not upload a video for this activity. However, there is another video on the MathNotationsVids Channel on YouTube.

Here is an investigation/exploration/activity for middle and secondary:

Part (A)
(i) List all ordered pairs of positive integers (m,n) such that
(1) 1 ≤ m ≤ 10 and 1 ≤ n ≤ 10
(2) m and n are divisible by the same prime p

For example, (m,n) could be (6,9) since 6 and 9 are each divisible by the prime 3.

(ii) Should (9,6) also be counted?

(iii) Another way of expressing Condition (2) is:
The _______________ of m and n is ________ one.
Answer: gcf; not equal to or greater than

(iv) If you listed and counted correctly, you should have found there are 37 ordered pairs which satisfy both conditions. If not, have a partner check your list. Each of you should be checking each other's lists routinely.

Part (B)
(i) Explain, using the multiplication principle, why there are 100 ordered pairs which satisfy Condition (1) above.

(ii) ) What % of all the possible ordered pairs from Condition (1) are relatively prime. If you have immediate access to the internet, research this term before asking your teacher what it means!

(iii) In probability terms, you could say:

If one of the 100 ordered pairs (m,n) from Part (A) is selected at random, the probability that
m and n are relatively prime is ____%.

Part (C) (more advanced)

If you have access to a graphing calculator, such as the TI-84 or TI-Inspire, enter the following program into memory (call it RELPRIME):

:ClrHome
:Prompt N
:0 → K
:For (X,1,N)
:For (Y,1,N)
:If gcd(X,Y) ≠ 1
:K+1 → K
:End
:End
:Disp K
:Stop

Using this program, complete the following table:

N..........Total # ord. prs..........# of not rel prime prs........% rel prime prs

10.........100.............................37....................................63%

20.........400............................ 145.................................

30

40

50

100

Notes:
K represents the count of ordered pairs which are not relatively prime
N represents the greatest value for the integers
gcd is found by going to MATH, then NUM, then 9:gcd(
The program slows down considerably as N increases. For N = 10, it checks 100 ordered pairs which may take only 2-3 seconds. For N = 100, it checks 100^2 pairs, which could take up to 4-5 minutes. Be patient!!

Conclusion: So what does all of this have to do with π ?
Well, as N increases without bound in the program, the probability that a randomly chosen ordered pair of positive integers (with values up to an including N) will be relatively prime approaches 60.7% rounded.

From out of the blue, compute 6/π²...
Want to know why? Well, that requires some advanced machinery involving infinite products, infinite series, and the Riemann Zeta Function! Perhaps, I'll do an informal development in a video. I love this stuff...


----------------------------------------------------------------------------------
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Posted by Dave Marain at 9:07 AM 6 comments

Labels: activities, explorations, investigations, middle school, pi, pi day, probability, relatively prime