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.