Sunday, September 28, 2014

Implement The Core: A Variation on a Classic Rate Problem

Twitter Problem 9-28-14
(twitter.com/dmarain)

Trip to work took 90 min including stopping b/c of accident for x min. Avg'd 30 mph overall; apart from delay, avg'd 50 mph. x=?

Answer: 36 min
Solution??

REFLECTIONS
Do rate-time-distance problems still appear on SAT and other standardized tests? Yes!

Do our students get enough practice with these? In Prealgebra? Algebra 1?

Do you view these as applied problems?

3 days left for half-price subscription for detailed solutions to all Twitter problems. See right sidebar...

Friday, September 26, 2014

Implement The Core: Mean of 3 scores=90%,Range=30%,Median??

A little more detail from the Twitter Math Problem 9-26-14
(twitter.com/dmarain)

Mean of 3 tests:90
Range:30
Explain why median must be 100.

Note: Assume all tests are based on 100 pts. The % info could be misleading, aka wrong!

REFLECTIONS...
1) Emphasis here is on explanation/reasoning rather than giving a numerical answer. That's why the problem is different from the title. This is at the "core" of the Mathematical Practices of the Common Core.
2) As any dedicated professional knows:
Finding challenging problems to promote collaboration and maximize participation is a daunting task. But isn't that what the Common Core is all about?
3) As educators would you promote an algebraic explanation or feel equally comfortable with one that uses a number-sense approach like "the lowest score has to be 70% or less because...", etc???
4) I've given away over a thousand original higher-order problems over 7 years on this blog and, more recently, on Twitter. And we know everyone is looking for freebies on the web. But writing detailed solutions/strategies/Common Core Implementation is labor-intensive. Creating new nonroutine problems every day is my passion but all good things must come to an end. Hope you RE2PECT that! Pls note the special offer in the sidebar which ends on 9-30-14.

Thursday, September 25, 2014

Least positive integer with 2014 factors - Detailed Solution

Actual Twitter Problem from 9-23-14 had additional restrictions which didn't fit in the title:

Explain why (3^52)(7^18)(11) is the smallest positive integer with 2014 factors and which doesn't end in 5 or an even digit.

Before the solution, a few
REFLECTIONS...
1) This is not an SAT or typical Common Core Problem. It's more challenging than that. But it does apply a fundamental principle of arithmetic which is often overlooked.
2) The solution below is more detailed than most but these are the kinds of solutions I will be emailing to you when you subscribe. See details at top of sidebar to the right.


Very Very Very Detailed Solution:
There's a fundamental rule about the number of factors of any positive integer > 1. I'll demo it with 12...
Step 1. Prime factorization of 12 is (2^2)•(3^1)
Step 2. Each factor of 12 is then of the form (2^a)(3^b) where a=0,1,2 and b=0,1
Step 3. Using the multiplication principle of counting there are (3)(2)=6 possible combinations of the exponents, each one producing a unique factor of 12:
1=(2^0)(3^0) (0,0) pair
2=(2^1)(3^0) (1,0) pair
3=(2^0)(3^1) (0,1) pair etc...
4=(2^2)(3^0)
6=(2^1)(3^1)
12=(2^2)(3^1)
So think of this as the
"Add 1 to the exponents and multiply" Rule!
Back to 2014 factors...
From Wolfram Alpha (enter "factor 2014")
2014=53•19•2
To construct an integer with this many factors we reverse the previous procedure, I.e., we SUBTRACT 1:
If p1,p2,p3 are different primes then
((p1)^52)•((p2)^18)•((p3)^1) will have
53•19•2 =2014 factors!
From the conditions we want to use the 3 smallest primes excluding 2 and 5, namely 3,7,11:
(3^52)(7^18)(11^1).
QED
(Mathematician's way of saying "I'm done!")

Wednesday, September 24, 2014

The sum of 2 pos int is 216, gcf=24 -- PUFM and Common Core

The sum of 2 positive integers is 216 and their gcf is 24. Find all possibilities.
Parents/Teachers/Students...
Want higher-order thinking questions (with detailed solutions/strategies) like this sent to your inbox almost every day for the 2014-15 school year? Just subscribe to MathNotations Twitter problems. Details and ordering instructions at top right of sidebar. Special pricing ends 9-30-14.
REFLECTIONS
To solve the problem above without Guess-Test-Revise requires a more (P)rofound (U)nderstanding of (F)undamental (M)athematics - thus the acronym in the title. (Research Liping Ma for more info).
Students may find solutions by playing around with multiples of 24 on their calculator and that is a good thing. That's how we learn. But...
How many  will discover without our guidance a systematic approach to finding the 3 pairs of numbers. A method which makes sense to them and can be applied to more complex problems...
I believe there are BIG IDEAS, aka Fundamental Arithmetic Principles, embedded in this innocent question. I've said enough...

Which one of these might be on your child's math HW tonight? IMPLEMENTING THE CORE...

From my Twitter feed today...
(twitter.com/dmarain)

And I'm not talking about that so-called Challenge Problem at the bottom of the worksheet. The one where your child says, "Oh, we don't have to do that one!"

1) Remainder when 999 is ÷ by 30?

2) Largest multiple of 30  less than 1000?

3) Largest 3-digit integer div by 2,3 &5?

Which of these require more reasoning and conceptual understanding?

Mathematical Practices and Core Reflections...

1) How often do we just throw a challenge problem at a class knowing that only a couple will actually try it. You know, the "smart" ones. Not really for everyone else...

2) If we  don't seriously value the importance of such a question, WHY ASK IT? Because it's on the worksheet? Really? Are you going to review it carefully or is there no time for that?

3) What are the BIG IDEAS OF DIVISIBILITY underlying these questions? Are they identified in the Common Core? Where?

Oh yes...
The answer to #2 & #3 above is 990. See, that was easy. Guess that's all way can say about this problem, right?
Case closed...

Actually NOT...
The Common Core will not raise the bar by itself. Only we can do that. Teachers, parents and everyone in our society...

Do you sense an "edge" to these remarks? Then my message is getting through...

Tuesday, September 23, 2014

ImplementTheCore...It's 11:15 am. In 4hr 55min it will be?

Common Core Considerations...

The question in the title is appropriate for which grade levels?

To teachers/parents...

Which of your students/children

Think fast--get it right? wrong?
Need to write it out?
Need a clearly taught method?
Need less repetition? Extensive reps?

How do you make a variety of thinking/learning styles work in a collaborative setting when one of the children  in a group thinks
12-1-2-3
15+55=70=1 hr + 10 min
4:10

Do YOU show them that adding 55 min can be done by adding 60 then "backing off" 5? OR
Do you ask THEM who found another way?

Yup, teaching is the easiest job....

Monday, September 22, 2014

Implementing The Core: This is not a parenthetical remark...


-3a^2+4b+c; a=-3,b=-2,c=-1
Step One:  -3(  )^2+4(  )+(  )

Note that I'm recommending this BEFORE the numbers go in!

Do you share my belief in the critical role of (  ) in evaluating algebraic expressions?

OR

Are you thinking this is too much detail and most students don't need to do this?

And I haven't even gotten to replacing -7-3 by (-7)+(-3)!

Sunday, September 21, 2014

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