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.

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!

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.

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.

POWER OF THE FOURTH!

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?

A "Fitting" Celebration of the Fourth!

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)!

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. 

HAPPY FOURTH!

(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!

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...

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!