Life Among the Elms

3.2 million

Last Monday I began what I thought was going to be a quick math lesson to explain a statistic that popped up in class, "80% of Minnesota's 3.2 million registered voters are expected to vote."  But wait a second...when I went to unpack all of the math involved this is what we ended up doing:

First, a review of big numbers and the base ten system.  You have to know what a million is and what a tenth of a million is.  Second, rounding.  Obviously, there weren't exactly 3,200,000 registered voters in Minnesota.  How do you create an appropriately precise number for your purpose?  Third, decimals.  Here we have three wholes and two tenths.  Students are often used to seeing decimal points but don't really know what they mean for anything other than money.  Even in money, their understanding is shaky -- very few can articulate "tenths" and "hundredths" at the beginning of fourth grade.  Fifth, using a place value chart to turn "3 whole millions and two tenths of a million" into a standard number.

All of that was for 3.2!  We moved on to 80%.  First I explained that "per" meant "for every" or "for each" and "cent" was Latin for 100.  Once students know that "percent" means "for every hundred" they begin to be able to see its connection to decimals and fractions.  We used base 10 blocks to further illustrate what 80% meant and then I very quickly exposed them to the procedure for finding out what 80% of 3.2 million is.

Whew.  All of that for 7 words of text.  It certainly reminded me of how much is involved in being numerically literate.

Candy Consultants, Wrestlemania and Greek Sieves

We are wrapping up a short unit on multiplication.  It began with an unexpected debate.  We had been doing a lesson on how to add, subtract and multiply with multiples of ten when the commutative property of addition came up.  Students knew that they could "flip flop" the numbers in an addition problem and, after reflection, they agreed that they could not do the same for subtraction.  Could you flip multiplication?  About a third of the class wasn't sure.  "Three groups of two is different than two groups of three."  The pro-commutative group tried to convince us but struggled until we looked at multiplication as an array, specifically, a rectangular candy bar.  Was the candy bar with three rows of two squares the same as the candy bar with two rows of three squares.  Everyone agreed that it was.

Sensing a rich vein, I became the owner of a candy company who wanted students to design the best box to hold 24 cube chocolates.  This gave students the opportunity to experiment with arrays and multiples.  Many chose to build their boxes out of graph paper - a great spacial exercise.

When everyone had a chance to present their ideas to the shareholders.  I explained that we now were facing a new challenge - we needed to find the most flexible number of candies so that we could change our box often.  Consumers like things to be "new," afterall, so we wanted to know the number of chocolates that would give us the most boxing flexibility.  The table the students created allowed us to explore primes and square numbers as well as see which numbers had the most factors.

Our multiplication minds warmed up, I introduced multiplication wrestling (which you may recognize from your algebra days as the F.O.I.L. method).  This is a multiplcation algorithm that emphasizes the "groups of" nature of multiplication.  Students can see how quantities can be broken done into smaller quantities that are easier to multiply and then reassembled to find the final product.  We used base ten blocks to create a concrete representation of what we were doing with the numbers on the board.  I was excited to see several students setting up their own problems with the blocks when we had open work later in the day.

Today we explored the Sieve of Eratosthenes, an ancient tool for finding prime numbers.  By crossing out the multiples of a prime, you begin to winnow down the remaining integers, slowly revealing only prime numbers.  (The link above is a slick interactive web version.)  Don't worry, we only went to 100 or 200, not the two million plus digit number that is the largest prime number discovered yet.

I hope this overview gives you an idea of how we are attempting to weave more concrete, computational math into authentic, engaging experiences.  By following students' ideas and needs, we help them construct the understanding that "math makes sense."