***BALANCING*** PROCEDURAL LEARNING and CONCEPTUAL UNDERSTANDING.
Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....
Look for fully developed K-14 math investigations, math challenges, and standardized test practice both for SATs and Common Core assessments. The emphasis will always be on maintaining a balance between skills proficiency and developing conceptual understanding in mathematics. There will also be dialogue on key issues in mathematics education. Use Blogger Contact Form at right for personal communication. See top sidebar to subscribe to solutions of Daily Twitter Problems.
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As posted on twitter.com/dmarain...
SHOW: The line with slope 1 intersecting y=-(x-h)²+k at its vertex also intersects at (h-1,k-1).
COREFLECTIONS
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?
Posted by Dave Marain at 1:11 PM 2 comments Links to this post
As tweeted on twitter.com/dmarain...
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?
COREFLECTIONS
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?
Posted by Dave Marain at 7:12 PM 2 comments Links to this post
Posted today on twitter.com/dmarain...
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!
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As posted on twitter.com/dmarain 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?
COREFLECTIONS
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!
Posted by Dave Marain at 5:31 PM 0 comments Links to this post
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.
COREFLECTIONS
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!
Posted by Dave Marain at 6:48 AM 2 comments Links to this post
Many conclusions here but would you want your students to know why 'a' MUST EQUAL 3 and 'b' MUST EQUAL -2.
COREFLECTIONS
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.
Posted by Dave Marain at 1:18 PM 2 comments Links to this post
As just posted on twitter.com/dmarain@gmail...
Using only "mental math" explain why 173^4+179^4+183 is not prime.
Should 5th graders be expected to understand this?
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Posted on twitter.com/dmarain@gmail...
A "Fitting" Celebration of the Fourth
7,4,15,?
The next term could be 40. Explain using the quoted hint!
COREFLECTIONS
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)!
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