Focus on Math

Helping children become mathematicians!

Equality: Balancing the Scale January 19, 2012

Many elementary children have serious misconceptions about the meaning of the equal sign (“Fostering Relational Thinking while Negotiating the Meaning of the Equals Sign”, Molina & Ambrose, Teaching Children Mathematics, Sept. 2006). Most of them think it means “put the answer here” or “do the adding” or whatever operation was involved in the equation. I personally have had students, when I wrote something like 5 = 2 + 3 on the board, say, “Mrs. Norris, you wrote it wrong!” The comment is not surprising since they almost always see equations with the answer on the right.

Students tend to be uncomfortable when their notions of the equal sign are challenged. “Backwards” equations (as was 5 = 2 + 3) or ones such as 2 + 3 = 4 + 1 put students’ understanding at a disequilibrium as they struggle to make sense of what is being said in the equation.

I had the opportunity this week to work with five different classes around the concept of equality. The school had purchased (at my request!) a class set of student balances, along with a larger, demonstration-sized balance, which we used to represent equations. We began with some “regular” ones, and then moved on to showing 5 = 5, 12 = 12, etc.. Equalities like that, with no operation symbol at all, were a bit startling to most of the students, but they quickly understood the logic of such statements as we represented them on the balance scale. Since the scale only goes to 10 on each side, we explored how to represent double-digit numbers greater than ten using base-10 representation. Thus 12 was put on the balance as 10 and 2, 20 was put on as two 10, etc.

We represented and recorded many “backwards” equations, and then moved on to ones that had two numbers on both sides of the equal sign, or multiple numbers on each side. We explored multiplication by hanging multiple weights on a single number (e.g. 4 groups of 3 balanced with 10 and 2).

The drawback of using the scales is that you cannot represent subtraction. In most cases the children used the tool, then wrote the equation they had created. One girl wrote her equation first, 9 – 1 = 6 + 2 but then could not represent that on these simple balances. We will explore such extensions in further sessions.

Because the class was hands-on and very interactive, every student was engaged. There were many comments made about the balance system being “cool” and many questions about when we would use the scales again. And, seriously, don’t we want them eager to come back for more math?!

Children need many experiences with equations that are not in “regular” form if they are to build an understanding of the true meaning of equality. I encourage you to find ways to explore this concept, one that is a critical component of algebra, with your students.

Mathematically yours,


Fibonacci Numbers: A Fascinating Sequence January 10, 2012

I was recently given the gift of this delightful interactive book written by Emily Gravett. Although it appears to be a children’s book, The Rabbit Problem can be appreciated on the adult level as well. This tale of Lonely Rabbit and Chalk Rabbit is actually a retelling of a scenario that, according to Wikipedia, first appeared in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considered the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
•    At the end of the first month, they mate, but there is still only 1 pair.
•    At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
•    At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
•    At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The question, of course, is how do we know how many pairs of rabbits there will be at the end of any given month??***
The answer lies in the Fibonacci sequence of numbers, a fascinating set of numbers that keep popping up in nature in amazing ways. The sequence begins with 0 (or should I say “can begin”?), then add 1, and from there the next number in the sequence is always derived from adding the two previous numbers. So the third number is 0 + 1 or another 1, the fourth number is 1 + 1 or 2, then 1 + 2 or 3, then 2 + 3 or 5 and so forth, giving this sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 …
I recommend this short  video by Vi Hart to jumpstart your thinking about these numbers (it’s about 6 minutes long). She builds the topic of Fibonacci numbers off the topic of spirals, so be patient and the number part will come. As you watch, keep in mind the kinds of explorations that you and your class can pursue with pinecones, pineapples, artichokes, cactus fruit, flowers, and such. Often the number of seeds that show up in fruit and vegetables is a Fibonacci number. Try counting the seeds of the next apple or orange you eat!
This website has a whole host of information about connections to the Fibonacci series. Note especially the section about plants. You might be able to do some of your explorations from the photos included here is real fruit, pine cones, etc are not readily available.
Exploring Fibonacci numbers can be a great “hook” to grab students’ interest about numbers and mathematics.
Mathematically yours,
***By the way, the answer the the rabbit question is this: at the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.


Reasoning About Fractions Using Benchmarks January 6, 2012

Filed under: General Math — Focus on Math @ 2:28 pm

lg fr pocket chartI first came across the idea of using benchmarks with fractions in John Van de Walle’s book Elementary and Middle School Mathematics: Teaching Developmentally (Pearson). I loved Van de Walle’s idea that students could quickly compare any proper fraction to the benchmarks of 0, 1/2, and 1 and decide which value the fraction would be closest to, thus giving a quick estimation of the fraction. By actually placing the fraction on a number line showing 0 to 1, students have the opportunity to develop the understanding that every fraction has a particular place on the number line (though it might share the spot with other fractions). Students generally have had real life experiences with fractions such as 1/2, 1/4, and 3/4 and thus I have found that all students can be successful thinking about fractions in this way.

Given the fraction 15/16, it is clear that having 15 of the 16 parts means that it is nearly the full amount, and thus is close to 1. Similarly for 19/24. A student might say here that there are 9 pieces more than half, and only 5 pieces away from being the whole amount, thus 19/24 is close to 1. (A word of caution here — make sure that students give sound reasoning. In the example of 19/24 to allow a student to say “it is only missing 5 pieces so it is close to 1” is not sound, for 3/8 is also missing 5 pieces but is close to 1/2. It is the relative amount missing that is important. Similarly 123/1000 has many fractional parts, but because the number of parts there is small relative to the number of parts in the whole, this fraction is closer to 0.)

Non traditional reasoning can be used when benchmarking fractions such as 4/9. A student might think that since 4 1/2 is exactly one half of 9, 4/9 must be close to 1/2. Elementary and middle school students are not usually exposed to fractions within fractions, but this reasoning is mathematically sound and is useful.

Once students are comfortable benchmarking fractions, they can move on to comparing fractions. For most of us, our experiences with fractions were such that if asked to compare 3/10 to 11/15, we would have changed the denominators to the common multiple of 30 and then compared the fractions. However, using benchmarks it is east to see that 3/10 is less than 1/2 and 11/15 is greater than 1/2, thus making 11/15 greater than 3/10. There is much to be gained by taking the time to explore fractions using benchmarks.

I have created pocket charts to go along with this activity, a large one to be used at the front of the class as well as smaller individual ones students can use. Having done this with a number of classes, including recently with my grade 4 and grade 4/5 classes at Charlie Lake Elementary, I have found that students enjoy using the little pocket charts. One teacher, after I had made and used pocket charts with her class, emailed me later that day to tell me the students were begging to take the pocket charts home so they could show them to their parents! Seriously, when was the last time your class begged to go home and do fractions with their parents?

I explored this topic more fully in an article published in the Summer 2011 issue of Vector (the journal of the BC Association of Mathematics Teachers). That issue was a special elementary edition of Vector and was sent out to all public elementary schools in BC. If you are in the province, hopefully you have access to that article!

You can download directions for making the pocket chart in both sizes here.

I hope you try using fraction pocket charts with your students. Let me know how it goes!
Mathematically yours,


Questions Regarding Snowmen: gr 2-4 January 5, 2012

Yesterday five of my seven math classes at Charlie Lake School did questions that had to do with building snowmen. The two grade 2 classes did this version:

Some children at Charlie Lake School are having a contest to see who can make the best snowman. Each snowman is to be made with three big snowballs and then decorated. If there are 16 snowmen being built at the school, how many snowballs have to be rolled?

I was delighted with the thinking and figuring of the students. One grade 2 girl used two mini 10-frames to solve this. She began to skip count by threes, writing a number in each of the spaces until she had written 16 numbers.  Other students used the 100-dot array, while others added 3 16’s explaining that there would be 16 snowmen heads, 16 “middles” and 16 “bottoms”. Many students drew pictures of all 16 snowmen and use their pictures to count the number of snowballs. As I have stated before, I like giving word problems such as this that are open-ended regarding the strategies that students can use.

The other three classes (grade 2/3, grade 3, grade 3/4) did this version of the snowmen question:

Some grade 2, 3 and 4 classes at Charlie Lake School are building snowmen. Grade 2’s will use two snowballs, grade 3’s will use three snowballs, and grade 4’s will use 4 snowballs. If the grade 2’s are building 9 snowmen, the grade 3’s are building 16 snowmen, and the grade 4’s are building 15 snowmen, how many snowballs in all have to be rolled?

This question had more numbers in it than any other question I had given to date and thus was a bit more complex. In the grade 2/3 class we created a chart showing for each grade in the question what their snowmen would look like and how many were to be built. I suggested to those students that they work on the grade 2 and 3 snowmen and than go on to the grade 4 snowmen as they had time. In all three of the classes doing this version, students wanted to solve the problem by adding 9 + 16 + 15, which would give the total number of snowmen but not the total number of snowballs. In many cases students had to draw pictures to sort out all the numbers. As always, students who solved the problem before the working time was up were to find other strategies/methods for solving the problem.

In all it was a great day of solving snowmen problems. Too bad that the weather did not cooperate so the children could go outside and build some real ones!

Mathematically yours,