Sunday, October 4, 2015
Wednesday, August 26, 2015
Click on graph to see the activity!
Set viewing window: -5<=x<=5
Also, it's better to manually animate the tangent lines by dragging the point or the slider!
Just a simple demo of the power of Desmos to build interactive Investigations for our math students.
Posted by Dave Marain at 3:41 PM
Thursday, July 30, 2015
***BALANCING*** PROCEDURAL LEARNING and CONCEPTUAL UNDERSTANDING.
Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....
Posted by Dave Marain at 7:53 PM
Tuesday, July 28, 2015
Estimated population after 10 yrs?
Should some background be developed BEFORE exploring with technology or AFTER or something in between?
Posted by Dave Marain at 9:20 PM
Saturday, July 25, 2015
Posted by Dave Marain at 6:31 AM
Monday, July 20, 2015
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).
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
Saturday, July 18, 2015
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?
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
Sunday, July 12, 2015
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!
Posted by Dave Marain at 10:02 AM
Thursday, July 9, 2015
Find x such that f(x)=
5 [no soln]
1 [no soln]
Fairly traditional rational function question in precalculus? Normally we'd ask students to analyze the function. What was my focus here?
Posted by Dave Marain at 6:54 AM
Tuesday, July 7, 2015
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?
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
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!
Posted by Dave Marain at 6:48 AM
Sunday, July 5, 2015
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.
Posted by Dave Marain at 1:18 PM
Saturday, July 4, 2015
Friday, July 3, 2015
Posted on twitter.com/dmarain@gmail...
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)!
Posted by Dave Marain at 8:45 AM
Posted by Dave Marain at 7:19 AM
Monday, June 29, 2015
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".
---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!
Posted by Dave Marain at 1:23 PM
Tuesday, June 9, 2015
Tuesday, June 2, 2015
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!
Posted by Dave Marain at 1:03 PM
Thursday, May 28, 2015
Wednesday, May 27, 2015
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!
Posted by Dave Marain at 8:43 AM
Thursday, April 30, 2015
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?
Posted by Dave Marain at 8:33 AM
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...
Posted by Dave Marain at 7:23 PM
Wednesday, February 11, 2015
Child will either draw a diagram, be given grid paper (1"x1") or use a bucket of at least 30 unit tiles.
DIRECTIONS TO CHILD
(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!
Posted by Dave Marain at 5:21 PM
Saturday, February 7, 2015
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?
Posted by Dave Marain at 3:25 PM
Wednesday, January 21, 2015
As posted on twitter.com/dmarain ...
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'être for teaching. Idealistic perhaps but when that is lost, what's left?
Posted by Dave Marain at 4:52 PM
Monday, January 12, 2015
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!!
Posted by Dave Marain at 5:12 PM
Friday, January 9, 2015
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!
Posted by Dave Marain at 10:55 AM