Sunday, September 28, 2014

Implement The Core: A Variation on a Classic Rate Problem

Twitter Problem 9-28-14

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

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

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

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

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
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...
So think of this as the
"Add 1 to the exponents and multiply" Rule!
Back to 2014 factors...
From Wolfram Alpha (enter "factor 2014")
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:
(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.
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.
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...

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
15+55=70=1 hr + 10 min

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?


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


For Teachers/Parents/Students

Subscribe now to start immediately receiving CCSS/SAT/ACT Twitter Challenge Problems with DETAILED Solutions, Extensions, Strategies and Common Core Implementation Ideas!

Subscribe in Sidebar to the right...

Half Price Until 9-30-14!

After 9-30-14, price will be $39.99
Subscription covers 2014-15 School Year through Summer 2015
Your subscription will also include
--Download of MathNotation's Challenge Math Workbook
--Additional Summer 2015 Problems
When payment is processed you will receive a confirmation email from MathNotations and Problems/Solutions will begin shortly. Also included will be my downloaded Workbook and passcode to open.
Duplication of problems for commercial use, forwarding/copying of emails is not permitted. Intended for classroom use by one teacher only.
Contact me for special site license pricing.

Tuesday, September 16, 2014

CCSS: One Less Than a Million - How Many Nines? Grade 2? 4? 6? 8?

Implementing The Core - Raising The Bar
One less than a million. How many 9's?
Developmentally inappropriate for 7 year olds?
***What questions should we be asking to develop this kind of arithmetic reasoning?***
What should children be writing on their paper to make conjectures about numerical patterns?
[1] less than 10: 9 [1 nine, 1 zero]
[1] less than 100: 99 [2 nines, 2 zeros]
How can this be EXTENDED to challenge the child who's ready for higher-order thinking?
Note that I didn't say *OLDER* children!
One less than a trillion? How many nines?
One less than a googol?
One less than 10^n?
One MORE than a billion? How many 1's?
One MORE than 10^10? What is the SUM OF THE DIGITS?
Discover a general rule and have them memorize it?
It's OK. I'm not expecting comments. I'm just planting seeds. Up to you to consider, modify, plant, add nutrients and illumination, watch growth
OR ignore all of this!

Monday, September 15, 2014

Typical 2nd Gr Assessment Questions and Your Thoughts...

Imagine that. I'm not promoting my problems/solutions!

As posted on my twitter account ( today...

Here are a couple of typical questions your 2nd grade child/student may be working on...

(Clock shows 1:00)
In 1/2 hour, it will be ___.

Mathematical Practices Reflections...
Why do you believe some children would struggle with this?
Possible teacher/parent interventions?

Write arrow rule. Fill in missing frames..

Mathematical Practices Reflections...
After child demonstrates proficiency, what can teacher/parent do to raise the bar? OR
Are you thinking this is ambitious enough for a 7 yr old?

Plugging in to avoid the algebra? Today's CCSS/SAT Twitter Problem

1/|8x-4| > 1
Possible value for x?
(SAT-type grid-in question)

"Plug in" - Easy?
Graphing calculator?

How do you think I devised this problem?

Want solution? Uh, you know what to do...

Sunday, September 14, 2014

More Solutions to Twitter CCSS/SAT Questions

The following is part of what everyone on my Twitter Problems mailing list has been receiving every day or two for the past 3 weeks. For free... You have 2 weeks left to sign up. Free...

As you can see I go beyond the answers. Way way beyond...
Just as the Math Practices of CCSS suggest we do...

1. I walked my daily path 25% slower than usual and took 5 min longer. How  many min does it usually take?

Let R=usual rate (mi/min); T=usual time (min)
One can infer that distances are equal from the phrase "daily path". Using D=RT and equating:
R's " cancel" leaving
15/4=1/4•T or T=15.

If slower rate is k•R (0<k<1) and extra time is m min then
kR(T+m)=RT --->
km=T(1-k) --->
Test it: k=3/4,m=5 ---> T=5((3/4)/(1/4))=5•3=15
Special case: If k=1/2 then T=m or if one walks half as fast trip will take double the usual time!

2. Test this "rule":
3 more than the square of an odd integer is a mult of 4.
Now prove it!
Devise a rule if "more" is repl'd by "less!"

An odd integer can be expressed as 1 more than even or 2n+1.
"3 more than the square of an odd integer" translates to
3+(2n+1)^2 = 3+4n^2+4n+1=4(n^2+n+1), a mult of 4.

Three less than the square of an odd becomes
(4n^2+4n+1)-3 = 4n^2+4n-2 which represents 2 less than a multiple of 4.
Thus "three less than the square of an odd" cannot be a multiple of 4 and in fact will always leave a remainder of 2. Why?

Saturday, September 13, 2014

Sample Solutions to Recent Twitter CCSS/SAT Problems

The following is copied from solutions I sent today to my mail list of those who have opted for free solutions for the rest of September...

Yes, I've been giving these away for weeks now. Hard to believe anyone would do this? There must be a catch, right?

I will continue this until the 30th then am considering a low fee subscription for the rest of the school year.

Subscribers will get detailed solutions which include strategies, big ideas, extensions, etc. Further I may include additional problems which will not appear on Twitter or this blog.

To sign up, provide all pertinent info in the Blogger Contact Form in the sidebar.

1. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

(a) 6
(b) Greatest perim:194; Least:40

Solution: 96 has 6 pairs of factors ---
Each pair has a different sum so there are 6 possible perimeters.
The greatest and least possible occur in the extreme cases, i.e., when the factors are farthest and closest apart. This is generally true.

Note: If integer condition is removed there would be no greatest perimeter and the least would be 16√6, a square!

2. Data:4,6;mean:5
(a) Avg diff from mean= ((4-5)+(6-5))/2=?
(b) v=((4-5)^2+(6-5)^2)/2=?
(c) √v=?
(d)Repeat for 3,7
Common name for √v?

(a) 0
Note: This is always true -- the avg difference or deviation from the mean is zero! This is why we square the differences to measure deviation!
(b) v=1
Note: The avg of the squared "deviations" from the mean is called the variance.
(c) √v=1
Note: The square root of the variance is called the standard deviation!
(d) For 3,7 ---
Mean is still 5
√v=2, the stand dev.
√v gives a measure of how dispersed the data is from the mean...

Friday, September 12, 2014

Six more Free Twitter CCSS/SAT questions...

Free Twitter CCSS/SAT Problems with complete solutions sent to your inbox ending in 18 days on 9-30-14. You know what you have to do...

1. List in order the ten 5-digit pos int containing 3 nines & 2 eights?
Explain connection to 5C2.
Is this open-ended?

2. The median of 100 different integers is 100. If the numbers are in increasing order and the 50th # is 83 what is the 51st #? Explain/show...

3. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

4. Data:4,6;mean:5
Avg diff from mean= ((4-5)+(6-5))/2=?
Repeat for 3,7
Common name for √v?

5. From the brilliant #xkcd...

Sneeze droplet: 200 million germs. Hand sanitizer kills 99.99%. How many live? No calculator-15 seconds!

6. Least positive odd integer with 8 factors?
Strategy: Make it ____
2 factors:3
4: 15 or 3×5

Wednesday, September 10, 2014

Seven More CCSS/SAT Twitter Math Challenges

Free solutions sent to your inbox?
Only 3 wks left...

1. Circle has center (0,0), radius 2. Line y=1 intersects circle at P and Q. PQ=?
(A)1 (B)√3 (C)2 (D)2√2 (E)2√3

(a) On a number line, how many positive integers are "closer" to 500 than to 1000?
b) Express the condition in (a) as an inequality using absolute values.

3. If the avg of 2 #'s is n, n>0, and the smaller # is 25% of the larger, the smaller # is what % of n?

4. For how many numbers x is the square of x equal to the opposite of x?

5. For how many integer values of x is |50-x| < 10?

6. A set of integers, T, has the property that if x is in T then x^2-1 is also in T. What is the least possible number of #'s in T?

7. Three-eighths of a candle's material remains after burning for 8 hrs. At this rate how many hrs for the rest to melt?

Monday, September 8, 2014

MathNotations Passionate View of the Common Core Controversy

Despite all the backlash against the Common Core, I am deeply committed to raising the bar for all students. The Common Core is an important step in this direction. I may have deep concerns about implementation and assessment, but I have no significant concerns about the quality, appropriateness and importance of the Core math standards and, in particular, the Eight Mathematical Practices.

Those states that are now defending their pre-CCSS standards are forgetting that the Common Core was developed by these states and approved by these states, not the Federal Government. The Common Core was a response to the fact that our students have been falling behind a majority of industrialized nations, particularly in math and science. The future of our children is at stake.

We need to stop the political rhetoric and do what's right for our children. Raise concerns about assessments and the timetable. You should. But remember this. On every international comparison our students have been performing at a mediocre level. That's reality. If there were no problem we wouldn't need to fix it. But there is a problem...

My commitment to our children and our teachers has been demonstrated by giving away hundreds of free higher-order math problems on this blog and on twitter for years now.

But implementation of CCSS will take the commitment of everyone. Or like every other educational initiative it will fail. But this time, if this fails, we will be failing our children. Hyperbole? Just check our nation's performance on international comparisons...


Circle has center (0,0), radius 2. Line y=1 intersects circle at P and Q. PQ=?

(A)1 (B)√3 (C)2 (D)2√2 (E)2√3

I'm sorry to keep reminding my readers that free detailed solutions to all current Twitter problems can be sent to your inbox if you request it via Blogger Contact Form.

As any of my followers know, I've posted hundred of free problems on my blog and on Twitter for a very long time. However I have omitted answers and solutions to many of these.

For the past month I have offered to send free detailed solutions of these to your inbox if requested.  I will continue to do so until the end of this month. After that I will continue to post problems but solutions will be available only via paid subscription.

In addition to solutions those on the mailing list can confirm that I'm also including alternate methods and specific references to the Mathematical Practices of the Common Core and strategies for implementing CCSS. This is my commitment to our teachers and our children.

Saturday, September 6, 2014

Free Twitter CCSS/SAT Problems/Solutions ending on 9-30

Right now it's still free to have these sent to your inbox. Blogger Contact Form must be filled out with all requested info. See

After that date by subscription only...

More Recent Twitter Problems (

1. Right circ cylinder: base circumf 16π. If greatest distance between bases=20, volume =kπ, k=?

2. #ImplementTheCore

Median of 1st 1000 pos even int's?
Name of strategy one could use?

Free soln?

3. If area between 2 concentric circles=area of smaller circle then Larger radius:Smaller radius=?
Ans: √2:1
Free soln to inbox?

2^100-1 must be div by
I. 2^10-1
II. 2^50-1
III. 2^50+1

Sometimes "free" actually means free!

Thursday, September 4, 2014

Update on Free Solutions to Problems

Based on response thus far I am restricting emails to Solutions of the Twitter problems I post @dmarain.

The opportunity to be added to the mail list and have free solutions sent to you is limited.  Pls include all requested info - see link below...

IMPLEMENT THE CORE-SIX Twitter/CCSS/SATPREP Math Questions to Start the Year

From Twitter (@dmarain)...

Want free answers solutions/discussion sent to your inbox?

1. Reg price:$N
40% discount $32 better than successive disc of 20%,20%
Want ans/soln/details/discussion sent to inbox?

2^100-1 must be div by
I. 2^10-1
II. 2^50-1
III. 2^50+1

Need  quarters&dimes for 35¢ bus fare. Asked bank for most I could get from $10. How many dimes did I get?

(3/a)•(3/(a-12)) = -1

Relate this equation to a coord problem involving a right angle!

Want answer/solution/details?

Mork's age
O-yrs  46   90  
E-yrs   1     10
When O-age:E-age=5:1, E-age=? (Assume linear)
Soln sent to inbox?

6. Inscribe a circle of rad=1 in a reg hexagon and inscribe a 2nd reg hex in the circle. Show that the avg of the areas of the hex's = (7√3)/4.
What does this have to do with π?