Thursday, July 30, 2015

Balanced Learning is not Blended or Flipped

Haven't been up to one my passionate rants in a long time so buckle up...
Technology has enabled educators to reimagine the  traditional classroom, how students learn and how we facilitate this process, aka teach. Incredible new opportunities to empower students to take more control over their own learning in a "space-time continuum" sort of way. Not to mention providing powerful tools to analyze data to individualize and maximize learning. Are there any buzzwords I omitted!
We have strayed from NCTM'S central message from over 25 years ago:
Technology changes the landscape in a fundamental way but the best source code cannot quite replace the critical dialog and face-to-face interaction that is needed to accomplish the above goal. The spontaneous give-and-take of questions and ideas. Interaction vs interactive...

Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....
The hexagon/triangle problem in the diagram above can be approached using dynamic software like Geogebra & Desmos. You could develop an extraordinary exploration with carefully crafted questions enabling the student to discover relationships in the figures. I love doing that. I used to do this in the classroom long before it was fashionable. Anyone who follows me knows I'm a techie geek at heart.
BUT I came to realize that there was something missing. If you believe I'm not knowledgeable enough of how these new tools can accomplish the BALANCING I speak of, then challenge my premise! Let the New Math Wars begin...
And I haven't even addressed the myriad of approaches to "solving" this multifaceted geometry problem. Most students/groups will find their own solution paths but it is human nature to CHOOSE THE METHOD THAT FITS YOUR OWN WAY OF THINKING.
To develop the deeper ideas of geometry - symmetry, transformations, dissecting, combining and rearranging pieces of a puzzle, students need to be TAKEN OUT OF THEIR COMFORT ZONE and experience others' ideas and we need to fill in the gaps. That is part of teaching, yes?
And, oh yes, there certainly are algebraic/geometric approaches here with lots of nice formulas like (x²√3)/4...

Tuesday, July 28, 2015

Modeling - What Algebra Looks Like on the New SAT/PSAT and the Common Core

A bear population, P(t), after t yrs,  is modeled by
P(t)=M-k(t-20)², 0≤t≤20.
Initial population:356
Max pop'n:500
Estimated population after 10 yrs?
Answer: 464
Is this the "new" algebra? Students given a function with PARAMETERS which "models" real world data? Questions like this have appeared on SATs for a few years now and, based on the sample new SAT/PSATs released by the College Board, they will become even more common. Students will be asked to analyze the function and use it in application.
The Common Core also emphasizes algebra models - "using" algebra to solve applied problems.
Middle and secondary math educators are not surprised by any of this as these changes have been around for a while but textbooks may need to include even more examples and homework problems of this type.
The real challenge, IMO, is to find that proverbial BALANCE between traditional algebra skills and  applications.
How much knowledge of quadratic functions is needed for this question? Will most students relate the form of the model to f(x) = a(x-h)²+k? Will they immediately recognize that M must be 500 since (20,500) is the vertex or maximum point? Try it and let me know!
Students should be allowed to explore this function using powerful software like Desmos and Geogebra. Sliders in Desmos allow for considerable analysis when parameters like M and k are given. 
BUT they also need to develop a fundamental knowledge of quadratic functions.
A key question for me is:
Should some background be developed BEFORE exploring with technology or AFTER or something in between?
I included a screenshot from Desmos for the bear population problem but this does NOT show how to IMPLEMENT this powerful tool in the classroom. I'll leave that for the real experts like John Golden! (@mathhombre).

Saturday, July 25, 2015

37 not 42 the Answer to The Meaning of Life? A Common Core Investigation

A Middle School Common Core Investigation
Is 37 an "interesting" #?
37x4=148; 4-1="3",8-1="7"
37x5=185; 8-5="3",8-1="7"
How far can you extend the pattern?
And is 37 patriotic (apologies to AK&HI)??
And my favorite ...
Is it all because 37x3=111?
First we engage, then illuminate...

Monday, July 20, 2015

Parabolas, NEW PSAT/SAT and the Common Core

As posted on

SHOW: The line with slope 1 intersecting y=-(x-h)²+k at its vertex also intersects at (h-1,k-1).

How would you modify this to make a grid-in or multiple choice question? A question similar to this appears on the published practice NEW PSAT. It is one of the last 3-4 questions on the grid-in with calculator section and was rated "medium" difficulty. I would rate it as more difficult! I recently tweeted the link for this practice test but easy to find on the College Board website.

Do the parameters h,k discourage use of graphing software?

Does the student need the equation of the line to solve the linear-quadratic system? Why does (h-1,k-1) have to be on the line? Then what?

What will be your source of questions like this for your students?

Saturday, July 18, 2015

Median = Geometric Mean? A Common Core Investigation

As tweeted on

J noticed that for an arithmetic sequence like 3,7,11,15,19 the median equals the arithmetic mean. In this case, the median and "mean" are both 11. She found this was well-known and not too difficult to prove.

She wondered if there was an analogous rule for geometric sequences like 2,4,8,16,32. Instead of the arithmetic mean she tried the geometric mean:
(2•4•8•16•32)^(1/5) which equals 8, the median. VERIFY THIS WITHOUT A CALCULATOR!

Unfortunately her conjecture failed for a geometric sequence with an even number of terms like 2,4,8,16 in which the median equals 6 while the GM = 4√2.

(a) Test her conjectures with at least 4 other finite geometric sequences, some with an odd number of terms, some with an even #.

(b) PROVE her conjecture for an odd number of terms.
Hint: If n is odd then a,ar,ar²,...,ar^(n-1) would have an odd # of terms. Why?

(c) How would the definition of median have to be modified for an even # of terms?

How much arithmetic/algebraic background is needed here?

Arithmetic sequences more than enough for middle schoolers to explore? Geometric too ambitious?

PROOF too sophisticated for middle schoolers? How would you adapt it? We are trying to raise the bar, right?

Sunday, July 12, 2015

Tangrams Forever...

Posted today on

Math educators K-14 have used tangrams for creative activities and to make learning "fun" but the underlying mathematics is rich. Whether you cut out the 7 pieces and rearrange to re-form the original square or a cat or a swan it's all math! Enjoy!

Thursday, July 9, 2015

Dys-Functional but Rational

As posted on
Find x such that f(x)=
0 [-2]
5 [no soln]
1 [no soln]
Answers in brackets
Fairly traditional rational function question in precalculus? Normally we'd ask students to analyze the function. What was my focus here?
Of course graphing software and CAS systems can be used but students need to walk before they run. The technology helps visualization but, more importantly it allows DIFFERENT, MORE PROBING QUESTIONS to be asked! I'm sure many of you might provide the graph and some key points and ask students to construct a rational function that fits the graph!
So why didn't I simply ask for the zeros, asymptotes, "holes"??

Tuesday, July 7, 2015

0,1,2,3,x If mean=median, x=?

As posted on today...

Let's get the "answer" out of the way first.
x can = -1,1.5 or 4. Not much more to say about this, right?


If this were an SAT-type question, it might be a "grid-in" asking for a possible value of x.

So what is needed to be successful with this type of problem? A basic understanding of mean and median for sure but there are the intangibles of problem-solving here. This question requires clear thinking/reasoning. Confident risk-taking is very important also. When one seems blocked, not knowing how to start, some students just jump in anywhere and see where it goes. Insight enables a student to move in the right direction more quickly.

Many students intuitively suspect that the median could be 1 or 2 or something in between. Even if they can't precisely justify this, they should be encouraged to run with their ideas. "Guessing" the median first seems easier than guessing a mean! One can always test conjectures.

Recognizing that there are THREE cases to consider is critical here. In retrospect, this will make sense for most but they have to make that sense of it for themselves!

So why not just give a nice clean efficient solution here? Because problem-solving for most of us is not clean at all! When the student is GIVEN the solution it may help them to grasp the essence of the problem but more often it shuts down thinking and doesn't help the student learn to overcome frustration. Yes, we can provide a model solution but how will that lead to solving a similar but different problem. We learn when we construct a solution for ourselves or reconstruct other's solutions in our own way.

Annoyed yet? If you solved it, you're fine. If not, frustration sets in quickly for some. If everyone in the class is stumped we can always give a hint. I think I already did!

1,012×1,008=1,020,096 A Mental Math Shortcut for MS!

Calculators and other technology enable students to "see" possible patterns/relationships without being discouraged by arduous calculations. The above multiplication is a well-known type of example to engage students in the mystery, magic and beauty of our subject.

Would you expect groups of middle schoolers to devise a rule or observe and describe a pattern based on this one example?

Would you start with simpler 3-digit examples like 102x103=10506 first to make relationships easier to see and formulate or does that depend on the group?

What do you find are the greatest challenges when implementing these kinds of activities? Is helping them express ideas in verbal and symbolic form one of them?

How important is "testing hypotheses" in this discovery/problem-solving process. Some students are naturally more patient and careful about "jumping to conclusions", a quality we should cultivate. But the risk-takers are necessary to move forward. The " testers" and skeptics are cautious and equally necessary, n'est-ce pas?

I don't expect many comments but if you have the opportunity to share this with children, pls share your experiences!

Sunday, July 5, 2015

If (a-3)x+(b+2)=0 for at least 2 values of x then...

Many conclusions here but would you want your students to know why 'a' MUST EQUAL 3 and 'b' MUST EQUAL -2.

So what's the BIG IDEA here? Is this really "Fundamental"? Where is it in the Common Core?

So if a polynomial equation of "degree" n has more than n solutions, what exactly does that imply? Any restrictions on the coefficients? And what does this have to do with an identity?

For me, it's critical that we don't see these problems as curiosities or challenges designed for only the accelerated groups or the mathletes.

Saturday, July 4, 2015


As just posted on

Using only "mental math" explain why 173^4+179^4+183 is not prime.

Should 5th graders be expected to understand this?

Friday, July 3, 2015

A "Fitting" Celebration of the Fourth!

Posted on

A "Fitting" Celebration of the Fourth
The next term could be 40. Explain using the quoted hint!

Students, as most people do, tend to look for simple arithmetic patterns like "subtract 3, add 8" but this problem  can be "fitted" into a quadratic pattern. Common Core and STEM strongly recommend that math educators include Least Squares methods into our curriculum using appropriate technology. But algebra teachers can seize the opportunity as well to fit a parabola thru the points (1,7), (2,4) and (3,15)!

Catch A Few "Rays" for July 4th

Not going to add much to the diagram above but STEM is all about APPLYING math and science, yes?
OK, so the expression for the angle labeled y is y=90-x!
The graphic isn't great so I hope you can read that angle! I left the diagram as open-ended as possible so students and educators can make conjectures and "assumptions". Feel free to comments or send me a direct email via the Blogger Contact Form. 

Monday, June 29, 2015

(0,0),(2,3) opp vert of a rectangle. Max area? And it's not 6!

Just posted this problem on Twitter. Might not get seen by too many educators/students who are on summer break here in US but when an idea comes to me I have to publish it or it will perish!
My passion is writing questions which promote divergent thinking and dialogue while developing conceptual understanding of the Big Ideas of Math. And of course encouraging all 8 Mathematical Practices simultaneously!
Since most texts have a dearth of these nonroutine questions I found myself creating my own when I was in the classroom. Now I share them with my online "family".
---So the answer is 6.5. But to me the EXPLANATION is always part of the"answer"!
---Would you give this problem or a version of it to 6th graders? Earlier? Only students in a geometry class? Only accelerated/honors students? My belief is it's appropriate for many "levels" but how we PRESENT it will change!
---Of course students need to sketch or graph it but is there benefit from both hand graphing and use of software like Geogebra? I believe the software can open vistas and promote inquiry not possible with just a manual sketch but a balance is still important. Learning HOW to use interactive geometry software is an aim here but it's not an END!
---Can you predict how many of your students would consider rectangles other than the obvious one whose sides are parallel to the axes? Should asking for the  "maximum" area suggest there is more than one possible rectangle, in fact infinitely many? Would you give them the "answer", 6.5, and have them justify it?
---How exactly would you want them to draw and consider other rectangles? This is not an obvious issue at all in my opinion.
---Would it be too much of a reach to expect a DEMONSTRATION of WHY the square is the rectangle of maximum area with a given diagonal? Would you relate this to the important idea that the triangle of max area with 2 given sides is a right triangle?
---Do you think discussion in class would lead students to a deeper understanding of the diagonal properties of a rectangle and the square as a special case? It isn't necessary for us to anticipate ALL the BIG IDEAS which emanate from problem-solving. What do you see as the main ones here?
---I depend on your comments here otherwise I'm writing in a vacuum. Your thoughts and constructive suggestions are not only welcomed but strongly encouraged!

Tuesday, June 9, 2015

A June 9th 2015 Riddle

At ~11:41 am (EDT/ "GM"T-4), today's date will "mean" something! Explain! Three embedded hints may help if you know what I "mean".

Comment with your solution or email it to me via the  Blogger Contact Form in sidebar...

Tuesday, June 2, 2015

Riddle of The El--- Wa-- and The Re---------- St---

DEA---- HA----- RIDDLE
Show that the length of the El--- Wa-- is THREE times the radius of the Re---------- St---!

I know I can't be the only Potterphile on the Math Blogosphere! Maybe your students will want to join the club!

Thursday, May 28, 2015


Note that the figure is not drawn to scale and point C(x,y) is on line L. Hopefully image file will upload and be viewable. You may have to click on IMG to enlarge.

Wednesday, May 27, 2015

Perspectives on Math Ed Tech

Discussed this often but worth revisiting in light of some outstanding new online apps...

My fundamental belief is that tech enhances and supplements instruction. Most students can not learn concepts without effective instructors who understand the variety of ways in which children learn. The best adaptive software can not replace human interaction. However, ed tech has come a long way in enabling a skilled instructor to help children better understand essential ideas thru visualization and interactivity. But there is often too much emphasis on creating "pretty" graphs or real-world activities, not enough on helping children grasp the "big ideas" of math. You can always tell when a company has or does not have strong professional math consultants on board. That is where most products are still lacking. But there is hope and I plan to review some of the best I've seen. I'll mention 3 for now, well just the 1st letter of each. I'm sure you can fill in the gaps!
1) G
2) D
3) W

Thursday, April 30, 2015

Patterns ending in 25...

Middle schooler playing on calculator observes


1) What can we do to make this a teachable moment?
2) To which of the Mathematical Practices does this relate?
3) Do you think it's important for students to describe the pattern both verbally and symbolically?
3) Algebraic derivation?
4) Extensions? Does the pattern continue into 3 digits e.g., 105²=11025=110|25, 235²=55225=552|25, 23•24=552
Does the pattern eventually break down?

Sunday, April 26, 2015


Not exactly a math story but...

My 7-yr old grandson was at bat today. The next pitch appeared to graze him and the umpire told him to take 1st base. He turned around and said that the ball hit his bat and not him. The shocked ump told him to get back in. Two pitches later, he walked anyway. There's a moral here somewhere...

Wednesday, February 11, 2015

Open-Ended 2nd-3rd grade PARCC-Type Challenge Activity

Child will either draw a diagram, be given grid paper (1"x1") or use a bucket of at least 30 unit tiles.

(a) Make the largest square you can from 18 small equal squares. Use the grid paper to show this.
(c) Any left over? If yes, how many? (d) Which of the following multiplication problems is your big square most like?
3x3? 4x4? 5x5?
(e) How many more squares do you need to make the next larger square? Draw it or use tiles.

This is just a springboard for your own ideas. Your reaction to this?
How would YOU word this type of question? Share!

Saturday, February 7, 2015

MathNotations Survey - Best Ed Reforms Over Past 25 Years

My tweet @dmarain on 2-6-15 has generated hundreds of views but no opportunity to freely exchange ideas. I thought about setting up a twitter chat but, for now,I've opted for my blog as the vehicle.

So here's the question which is trending ...

What changes in education over the past 25 yrs do you think have most impacted how children learn?

Since this blog focuses mostly on issues in *math* ed
I'll kick this off by suggesting

1) Increased communication in the math classroom. I favor a balanced approach of direct instruction and posing open-ended nonroutine problems requiring team effort. NCTM promoted the importance of greater student-student and student-teacher interaction in 1989 and this is far more evident in our classrooms today.

2) Technology of course but specifically which technologies have had the greatest effect on learning math skills and math concepts. As a math educator, the access to and strategic use of powerful graphing calculators has enabled students to solve traditional problems in a variety of ways (multiple representation), explore topics in greater depth and model data more efficiently.

Anyone remember great software like Green Globs from the early 90's? We've come a long way since then with interactive geometry and algebra software and apps and we're still just at the toddler stage!

Haven't even mentioned the many curriculum projects, Common Core, and assessments but opinions on their benefits should vary widely.

OK, your turn. I hope this prompt will encourage comments but that's up to you. If you'd rather use twitter or other social network sites let me know. A chat on twitter might be the way to go in 2015.

So what is your Top 5 List of most significant changes?

Wednesday, January 21, 2015

When is a rectangle an equilateral triangle?

As posted on ...

Diagonal of a rectangle has length 6 and makes a 30° angle with a side.
(a) Area of rectangle=?
(b) If diagonal has length d, area=?


(1) A moderate difficulty problem for SATs? Appropriate or too hard for a PARCC assessment with both parts?

(2) Should diagram be given or is drawing part of what's being assessed?

(3) Will some students recognize that the expression in terms of 'd' is the formula for the area of an equilateral triangle of side length d? If no one does then is it our responsibility to model and facilitate "connection-making"? Uh, yes ...

More interestingly, will some students realize that the rectangle divides into two 30-60-90 triangles which can be rearranged to form an equilateral triangle? Hence, the title of this post!

If we create this kind of environment in our classes it may happen. I think we're all conditioned to thinking that's only for the top honors groups,  and only for a few students. But, for me, helping ALL children discover and uncover the beauty of mathematics was my raison d'ĂȘtre for teaching. Idealistic perhaps but when that is lost, what's left?

Monday, January 12, 2015

How one 2nd grader knows his 7 Times Table!

As posted on twitter @dmarain today...

Question to 2nd grader: 7×6
How did you know that?
Easy --- 6 touchdowns! I know all my 7's!

1) So do you think this is about a real 7-yr old?

2) Would this be useful to many or just for girls/boys who watch a lot of football? OR

Is there a bigger issue here re the individual ways in which children learn? I think there are some HUGE implications here for teaching/learning in the Common Core and beyond...

3) All these "strategies" turning you off? Yearning for the good old days -- having children write their facts 10 times each or flash cards and memorization?

I have mixed emotions since I'm probably older than most of my readers but the anecdote above is real and it did work for this particular child! Further the child said, "I know some of my sixes too!" Missed PATs?? Maybe field goals will help with the 3s!!

Your thoughts?

Friday, January 9, 2015

Implement The Core: Quadratic Function SAT-Type Assessment

As posted on Twitter @dmarain...

The graph of f(x)=-(x-k)^2+h has one x-int and a y-int=-16. Coordinates of all possible vertices? Sketch graph(s).


(1) How do you feel about the "h,k switch" on an assessment? Would you revise it or leave it alone?
(2) Level of difficulty here? How do you think your students would perform? Let me know if you use it!
(3) Are you finding more of these types of questions in current texts? If not, what resources do you use to raise the bar?
(4) What if the question had asked for the PRODUCT of all possible x-intercepts? Better question for standardized assessments? Since the answer to this revised question is -16 do you think some of your students would ask if that's a coincidence? Why not ask them to check that -- Then GENERALIZE!

Thursday, January 8, 2015

BREAK THE CODE: 12-91-1305

As tweeted  on Twitter @dmarain today:

Break the code:12-91-1305
Then multiply these #'s by 4/3...
And you'll get my OBJective here!

Use contact form in sidebar to send me your answer/thoughts or leave a "hint" or question in Comments!