## Tuesday, July 7, 2015

### 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?

mathwater said...

A question: What is 1010x1010?

Since 1010 = 101x10, we have:

101x10x101x10 = 101^2 x 10^2 = 10201 x 100 = 1020100.

This requires students to figure out how to square 101.

One way is: 101x101 = (100+1)101 = 10100 + 101 = 10201.

Back to the main story:

But this is: (1010 + 2)(1010 - 2) = 1010^2 - 2^2

= 1020100 - 4 = 1020096, as remarked.

(Take-home lesson: The product of two numbers with an even difference can always be re-written this way. When does it help? A nice example is 43x37...)

Dave Marain said...

Thanks "mathwater". Nice analysis. Some students might discover a gimmicky approach that can be verified algebraically.
Let's try 1009x1004.
Working with the 9 and 4 we first add, then multiply as follows: 1009+4=1013; 9x4=36
Put these together to obtain 1013036
Seems very contrived but it seems to work.
Let's try 1010x1010:1010+10=1020,10x10=100

Thus 1010x1010=1020100. Not as elegant as your methods but students might find it interesting and ask why it works! When we ask them to devise their own patterns strange things can happen! Our role is to enable this discovery and provide tools for them to verify their conjectures...