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".
COREFLECTIONS
---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

A COORDINATE SAT/COMMON CORE PROBLEM

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

25²=625=6|25,2•3=6
35²=1225=12|25,3•4=12
45²=2025=20|25,4•5=20

COREFLECTIONS...
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?
(10t+5)²=100t²+100t+25
=100t(t+1)+25
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

PLAYING BY THE RULES

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.

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!