A Middle School Mental Math 'Practical' Problem

Look here for links to MathNotations π-Day Resources, Activities and Investigations!


You're on a superhighway in the middle of nowhere at 10 PM, your fuel tank shows about a quarter of a tank remaining, your trip odometer shows you've gone about 180 miles since the last fill-up and you just passed a fuel rest stop. The sign reads "Next Rest Area and Gas - 58 miles." Are you in trouble or will you make it?

Comments
(1) Unrealistic scenario? Rest areas rarely that far apart on the Interstate? Why didn't the driver realize that he was low on gas and just stop! Why not just get off the Interstate and look for a local station or call AAA or Onstar if needed or use a navigation system to locate another gas station? (Do all of us have access to these technologies?). Does the student have to know that one still has some fuel remaining even when the gauge points to empty? Should a discussion of these kinds of practical issues be an integral part of 'real-world' problem-solving?

(2) Is this type of question fairly common in the texts you're using?

(3) What prerequisite skills and conceptual understandings are needed for this problem? Should most 7th or 8th graders be able to do this?

(4) Why do you think I suggested a mental math approach? Should we eliminate the mental math/estimation aspect altogether or use it as a starting point for further discussion?

(5) How do you think your students would fare on this problem? How many students would recognize the 1:3 ratio between gas remaining to gas consumed? Is this part:part construct something stressed in texts and in our classrooms? Should it be? Is it worthwhile discussing several approaches to this problem?

(6) Do you have a favorite visual model for this kind of ratio problem? Your thoughts?

A Much Harder Mental Math 'Trick'?? - Algebra May Not Be Optional!

Update: For more background on the investigation below, I strongly encourage readers to visit Sol's wonderful blog wildaboutmath and, specifically, the post he wrote about the algebra behind multiplying 2- and 3-digit numbers. Further, he developed a video (mathcast) to demonstrate the method for 'mentally' mutiplying any two 3-digit numbers. Sol inspired me to develop this activity!


Target audience: Strong Algebra 1 or Algebra 2 students

Ok, I hope you enjoyed challenging yourself or your students to describe a step-by-step procedure for squaring three-digit numbers with a middle digit of zero. I believe many middle school students could devise a method using only operations on the digits, without the need to use algebraic representation or proof. I really believe that recording and organizing the necessary data and the search for patterns make that a valuable activity. Perhaps, even more importantly, the verbalization of the strategy or trick may be the most important benefit for our students! Finally, I strongly believe these kinds of investigations deepen student understanding of place-value and develop number sense.

But now we will move on to a much more significant challenge. I highly doubt that any of your students will be able to find the right combinations of digits to square any 3-digit number! One will still have to record the data and attempt to see a pattern, but, this time, a non-algebraic approach may be too formidable. I am fully aware that there are classical mental math methods for multiplying numbers (Vedic Maths and Trachtenberg to name a couple), but I didn't research those when writing this activity. I simply applied algebraic methods and looked for an algorithm that made sense to me. Thus my method may be similar to those others or not!

This challenge should be frustrating for some and many of you will question whether there is any benefit in looking for some strategy for squaring such numbers by some artificial-looking or convoluted combinations of digits. If the object here were to develop an easy mental math trick, you'd be right. BUT that is not my objective - the idea is to use algebra to motivate an alternate approach to multiplying 3-digit numbers. And, yes, there is some mental math involved, although, I suspect one will need to record intermediate results and do some paper and pencil arithmetic along the way (sorry, no calculator allowed other than to check your answer!). In the end, students should have a much deeper understanding of the meaning of digits, place value that is.

We begin with the algebra prerequisite for this problem, then move on to the real challenge!

READER/STUDENT CHALLENGE/ACTIVITY

(1) Show that
(p+q+r)² =
p² + q² + r² + 2pq + 2pr +2qr

Ok, here goes...
(2) Using the algebraic formula above, devise a method/strategy/algorithm/procedure/'trick' using combinations of digits to square ANY THREE-DIGIT NUMBER 'HTU'.
Note: Again, I reiterate, others have probably invented much cleverer methods for multiplication than this. The key here is to use the above algebraic representation!

In the end, you should be able to do an example 'mentally' like

463² = 214,369 (or should I write it as 21-4-3-6-9).
Lots of 'carrying' here or 'regrouping' as we say these days! If I give you the intermediate step, it may give it away. If I don't, you may give up! Let's see what happens...

508^2 = 258,064: Invent your own Mental Math 'Trick' and Prove It!

UPDATE: READ THE COMMENTS FOR A DISCUSSION OF NOT ONLY A POSSIBLE SOLUTION BUT ALSO THE PEDAGOGICAL IMPLICATIONS OF SUCH AN ACTIVITY.


What child (ok, adults too!) is not mesmerized by a magician. The Hindu-Arabic place value system allows for endless possibilities for mathemagic, aka, mental math strategies. Adults and children alike are fascinated by these as is apparent from the re-discovery of Vedic Maths. There are several excellent resources for this. You may want to check out The Vedic Maths Forum India Blog, which includes an interesting video showing youngsters demonstrating one of these methods in a dance routine! It's also no wonder that Sol's Impress your friends with mental Math tricks post over at wildaboutmath went viral! Sol discusses many of these wonderful mental math strategies and the response has been overwhelming.

I've personally always wanted to understand the basis for many of these mental math 'tricks' that have been around for centuries. Since a significant part of teaching is performing, math educators are often interested in any mathematical sleight of hand that provokes wonder in the child or older student. We would hope the student would want to know the WHY behind the trick but most are more interested in just performing the feat. What child doesn't come home and want to challenge their parents to square 45 in their heads or multiply 68 by 62 in 5 seconds or less! Hey, anything that turns kids on to mathematics is alright in my book. However, the purpose of this post is to have the reader discover a mental math method and use algebra to delve beneath the surface. Magic? Perhaps that is what teaching is all about...

STUDENT/READER ACTIVITY/CHALLENGE

Study the following:
101² = 10201
401² = 160801
508² = 258064
306² = 93636
909² = 826281


1. From studying the above examples, discover a mental math strategy which would enable you to square any 3-digit number whose middle digit is zero. Write the precise steps of the method. Be careful here - make sure your method works for all 5 of the above examples! Note: That last one is a bit harder to do mentally but give it your best.

2. Demonstrate (perform!) your method by having someone challenge you to square such a number! You need to be able to perform the trick in less than 5 seconds (ok, maybe 10 for old people like me!). Repeat your amazing performance at least 5 times!
Note: You may need to practice this for awhile before going on American MathIdol!

3. Ok, now PROVE IT ALGEBRAICALLY!
(Gee, that takes all the fun out of it.)

A couple of additional comments...

(i) Did you ever notice that all of these multiplication tricks actually require that the child knows the basic facts cold! Perhaps, these tricks could even serve as motivation to learn them!

(ii) When I discovered this trick, I was naturally excited and of course wanted to share it with my wife who has a droll way of putting my geekiness in perspective. After she picked a random 3-digit number and I did the math correctly, she seemed unimpressed and replied, "But can you cook a chicken?" I was tempted to suggest that I could square a chicken if it had a hole in the middle, but I knew she would not be amused....