Friday, January 19, 2007

Introducing Exponents (Middle School)

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

Here's an activity I recently used with my 9th grade class of students for whom math is a struggle. it is equally appropriate for middle school, modified as needed.
I gave them a copy of the famous "As I was going to St. Ives I met a man with 7 wives" nursery rhyme/riddle. This is well-known and many of you have probably used it. We modeled each stage of the poem using diagrams (tree model for example), multiplication in expanded form, exponent form and using a table to record the number of people or objects at each stage and the cumulative total. Pretty standard stuff...

I gave them two assessments.
1. Working with your partner, invent your own St. Ives poem. The first 2 lines had to rhyme and there had to be at least 4 levels as in the nursery rhyme. I received some interesting responses. Here are two:

K.G. and S.T. came up with:
Each girl had 5 designs.
Each design had 5 lines.
Each line had 5 colors.
Designs, lines, colors and girls.
How many all together?

K.C. and E.D. came up with:
Fuzzy Wuzzy was a bear.
Fuzzy Wuzzy had no hair.
Fuzzy did not care he had no hair.
So he went to town and found a crown.
Each crown had 5 points.
Each point had 5 diamonds.
Each diamond had 5 faces.
Faces, diamonds, points and crowns...
How many were going to town?

2. I gave a written individual assessment (pls excuse my lame attempt at poetry!)
As I was shopping in Value-Mart,
I found 9 special shopping carts.
In each cart I found 9 boxes.
In each box, I found 9 hats.
In each hat, I found 9 bats.
Bats, hats, boxes and carts,
How many did I find at Value-Mart?

They were expected to express the number of objects at each stage using both expanded and exponent notation. The calculator was allowed.
This is always a daily learning experience for me. I feel the need to be creative with this group to keep them involved and to change activities often. I know that any approach that engages their learning is worthwhile but I also know that activities do not easily transfer to traditional skills unless we enable that transfer. I also feel that the assessments for this activity were a critical piece to evaluate their learning and my instruction.

Any thoughts? See this as a waste of time? 'Fuzzy Wuzzy' math? Is this 'constructivist' according to someone's definition? Why not just define exponents and provide direct instruction? There are several other ways I get at the meaning of exponents and the ideas behind the 'laws' and 'rules' as well as zero and negative exponents. But that's another posting.


Unknown said...

There is also the story of rice on the chessboard that you could use to motivate the discussion: A king offers a reward to a person, who, instead of a pot of gold, asks for one grain of rice for the first square of the chessboard, and double the previous number of grains on any subsequent square. Finally, the number of grains needed exceed the total on earth.

Dave Marain said...

totally -- you are far from clueless!
The chessboard problem is a great one. I just like the St. Ives poem for some reason! Another powerful approach is to have students do about 20 calculator computations involving exponents and ask students to notice similarities among the results. This kind of dscovery lesson works. I just observed it yesterday. The key was careful selection of the examples and which groups were similar. Students then had to 'state a rule' based on these observations. This did not kill a full period. Students were lured into the subject by the calculator but some of the surprising results like -3^2 vs. (-3)^2 generated good dialogue. The teacher clearly explained the result using order of operations but there is something special when they see it on the display screen!

Anonymous said...

Do you tell them the answer to St. Ives?

I used that once a few years ago in an ESL Math class, just to introduce counting. You know what the had trouble with? Understanding that St. Ives was a place. They read so slowly and so carefully that most correctly answered "How many going to St. Ives?" Very very wierd.

Dave Marain said...

we worked out the details of St. Ives ; I had prepared a worksheet with the steps laid out and they had to fill in the details; I'm sure some might have thought St. Ives was a brand of shampoo!
The issue that did arise was whether the man should have been included since it appeared he was headed that way too! This led to more careful reading of the phrase: "Kits, cats, sacks and wives, how many were going to St. Ives?"
ESL, huh?!? I feel like an ESL teacher EVERY day! (or should I say 'EFL'!).

Anonymous said...


it was a content class, math of course, taught in English, but to a class of students of mixed origin, and who had not passed a (State? City?) exam that would make them no longer ELL's. I got a handout of suggestions, but then sought out some teachers who sat with me and gave me a better sense of how I should modify my teaching for this group (most of the techniques were useful with all my classes.)

I remember most of their names, and it was 6 years ago. Eduart, Arjan, Besmira and Brunilda were twins, Rithita, Sabah, Maimounah, Edvinas, Sokhom, Mohammed, Agnes, Meron, Jose, Eunice.. there were 25, and I can see more faces, but now I will be trying to recall their names for the next week...

So, you do know only 1 was going to St. Ives?

Dave Marain said...

interesting stuff about those youngsters and your accommodations. similar to when i teach skills classes - i have to be so much more creative and use methods that help me elsewhere!
you're so right about st ives - only ONE is going for sure, namely 'I'! However, the question is qualified by the phrase 'kits, cats, sacks and wives.' So technically we cannot be certain that any of THAT group was going so i guess the answer is 'cannot be determined' or ZERO!' I never saw the 'official' solution to this riddle but unless it was intended to be that tricky (and it probably was), I still like using it for the math... Now in which direction were they really going, hmm...
Either way, I'm chagrined!