# Focus on Math

## Helping children become mathematicians!

### Addition and Subtraction: A Really Big IdeaOctober 25, 2011

Much of what students do in mathematics across the years involves four basic operations: addition, subtraction, multiplication, and division. Within these four operations we really can consider two pairs of operations since

• addition and subtraction are inverse operations (each “undoing” the other)
• multiplication and division are inverse operations.

As students’ mathematics skills increase, so does what we ask them to do in the operations. (NOTE: As I write, I shall concentrate on the process of addition, but please keep in mind that what I am saying applies equally to subtraction since it is the inverse operation of addition.)

Young children begin by adding whole numbers such as 2 + 5. Put into a context, it might be a question such as this: There were two birds in a tree. Five more birds came and perched in the same tree. Then how many birds were in the tree?

Later children begin adding multi-digit whole numbers. They might be given a problem such as this: 257 + 48. If children write this problem in the typical “stacked” format, we teach them to line up the numbers, beginning on the right.

Decimal addition follows soon after. This time the problem may look like this: 2.57 + 4.8. Again, such a problem is typically “stacked”, and we teach students a rule that has them lining up the decimal points.

Adding fractions brings about a new situation, and typically a new rule. If students are given a question such as 2/5 + 1/2, we teach them about finding a common denominator, changing one or both fractions (depending on the particular question) into equivalent fractions with the same denominators, and then proceeding to add the numerators. In our example we find the common denominator of 10, change the problem to be 4/10 + 5/10, then add to get 9/10. The common denominator rule must be followed if students are to arrive at a correct solution.

Students soon encounter algebra and its accompanying variables. Now they see problems such as 3x + 4y + 2x and we teach another rule: you always add “like terms”. Thus the example can be simplified to 5x + 4y, but these two terms cannot be combined because they are not alike.

I propose that the idea of adding “like terms” or “like things” is not new to algebra. In fact, it is exactly the same rule we have been using in every situation. We just have not been calling it that, but maybe we should!! How different would students’ understanding be of all of the addition and subtraction situations if we continually pointed out that we always, always, always add and subtract like things.

For young children, we must add things like apples to apples, birds to birds, pencils to pencils, etc.. I once worked with a young girl and asked her to make up a word problem for 2 + 5. She looked out the window from where we were sitting and she saw some birds in an apple tree. She began her story, “There were  2 birds in a tree and 5 apples…”.Then she paused, looked at me, and stated: “That doesn’t make sense!” I asked her what would make sense, and she responded, ” I think I should have apples and apples.” She was right. It only makes sense to add things that are the same. Even when we add 12 boys in a class to 14 girls, we end up saying there are 26 children or students. We actually had to find a “same name” or a “like term” for them if we were going to add them.

When it comes to adding multi-digit whole numbers, why is it important to “line up the right side”? Because in writing the problem in that manner we are setting it up so that we can add ones to ones, tens to tens, hundreds to hundreds, etc. This principle is the same for why we “line up the decimal points” when we add decimal numbers. We are ensuring we add ones to ones, tenths to tenths, hundredths to hundredths, etc. In our place-value-based number system, we must add like values, or like terms.

Students are often confused when doing an operation on fractions. They have trouble remembering when they have to change the denominators and when they can go “straight across”. I feel there would be less confusion regarding fractions if they knew the big idea about addition and subtraction: namely, that you always add and subtract like things.

By the time students begin doing algebra, it should be easy understanding about adding “like terms” since that is what they have been doing the whole time in addition, if only we would help them see that!

Mathematically yours,
Carollee

### Combined (“Split”) Grade Math Learning Outcomes: BC/WNCPOctober 20, 2011

Filed under: General Math — Focus on Math @ 11:56 am

Tomorrow morning I am doing a professional development day session about teaching math in “split” or combined grades. It is helpful for teachers of such “split” grades to have side-by-side learning outcomes for their particular grades.

Teachers from the Vancouver IslandNet worked to prepare tables with the learning outcomes for different grades “matched up” in that manner. In other words, for the two (or three) specific grades, all of the outcomes are listed, each beside the outcome(s) that correlate between those grades. If an outcome in one grade does not specifically correlate to the other, then it is suggested that students in the other grade may either explore or review the concept. Such a chart is a very handy tool to have when planning!

This link will take you the the Vancouver IslandNet At-A-Glance Project. The learning outcomes for British Columbia (and any other province or territory using the Western and Northern Canadian Protocol [WNCP] math outcomes) are available here for single grades, combined grades (two), or mulitgrades (three).

Our thanks to the IslandNet group for providing this valuable resource!
Mathematically yours,
Carollee

### The Importance of Speaking Clearly

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 10:10 am

The toolkit workshop last week was a success overall. We looked (and “played with”) a variety of items that students can use as tools for solving problems.

Along the way, however, I learned a valuable lesson about speaking clearly. One of the tools we used at the workshop was the 100-dot array. Now, as it turned out, the participants had a list of possible toolkit items, and we were exploring a number of items on the list, but I was not introducing the items in any particular order. When I began to speak of the 100-dot array and ask the participants to find the samples provided on the tables, quite a few folks looked at me rather strangely. I asked them again to each get a sample of the 100-dot array to use, and finally several folks stopped me to ask me what, exactly, did I want them to use!

I tend to talk quickly in a workshop (there is always SOOOOO much math to talk about, and I am trying to squeeze in as much as I can to any given time period!) and in speaking quickly I was not speaking clearly. In fact, one participant handed me a note at the end of the session with these words written on it:

• 100 dotter, eh?
• 100 daughter a?
• 100 daughter ray?
• 100 dotteray?

Clearly, in knowing for myself what I was referring to when I said “100-dot array” I assumed that everyone else was hearing what I was saying. They were not!

In the end we sorted it out, I pronounced the tool’s name slowly and clearly, and we went on with the session. But it made me wonder about other workshops, other sessions, other classes of students, other teachers with other classes of students. Do others hear clearly what we are trying to say? And what about the students who assume that if they aren’t “getting it” that it is something wrong with them, and not the speaker? There are so many math terms. It is important that we speak them clearly.

I hope I have learned my lesson from the “!00-dot array”. Maybe the next workshop will provide the evidence.

Mathematically yours,
Carollee

### A Patterning Problem: Finding Sums of Consecutive NumbersOctober 6, 2011

This problem is one of my favourites! I first came across this problem about the sums of consecutive numbers in Marilyn Burns‘ book About Teaching Mathematics: A K-8 Resource (Sausalito, California: Math Solutions) and I thought it was an intriguing question. Burns presents the problem to be solved this way:
“Ask the students, in their groups, to find all the ways to write the numbers from one to twenty-five as the sum of consecutive numbers. (For younger children, finding the sums for the numbers from one to fifteen may be sufficient.) Tell them that some of the numbers are impossible; challenge them to see if they can find the pattern of those numbers. Direct them to search for other patterns as well.” p. 58

I have done this problem over the last few years with quite a few classes in a range of grades (usually somewhere in the grade 3-8 span). For the older grades I have extended the problem, asking students to write sums for numbers up to 35.) I feel that it is not a problem to hurry the students through, and I often take more than one day with the task. What I like most about the problem is that it is full of patterns, and that, in finding the patterns, one “unlocks” the problem. It becomes so much easier to find and predict  the various sums when one notices the patterns that are produced.

I also like this problem because we tend to teach patterning to students isolated from problems, and I think that there is something quite powerful about a problem that uses patterns to solve it!

The task of finding sums of consecutive numbers provides a good opportunity for students to develop some problem solving strategies.  Burns suggests a list of useful problem-solving strategies, similar to lists proposed by other math authors, naming the major strategies useful to untangle problems:

• look for a pattern
• construct a table (chart)
• make an organized list
• draw a picture
• use objects
• guess and check
• work backward
• write an equation
• solve a simpler (or similar) problem
• make a model

I have found that students doing this problem tend to make use of  a number of the above strategies including these: look for a pattern, make an organized list, use objets, guess and check, and work backward. Any time that students are engaged in problem solving It is important to discuss with them  both the specific strategies they use to solve the problem and why those strategies were (or were not) effective choices. Additionally, this problem is simple enough on its most basic level that everyone has the chance to delve in and come up with some of the consecutive number sums. At the same time it is quite sophisticated and offers a challenge to the bright students in the class.

Burns offers ideas for extensions for this activity, too. For instance, she suggests students try to predict how many ways thirty-six can be written as the sum of consecutive numbers. Going further, she asks if a prediction can be made for any particular number.

So, I encourage you to give this one a try with your students — and it is OK if YOU do not have it all figured out ahead of time. Let your students know that you are solving the problem along with them.

Mathematically yours,
Carollee

### Upcoming Math ConferenceOctober 3, 2011

The British Columbia Association of Mathematics Teaacher’s annual conference will be taking place October 21 in Burnaby, BC. Here is the schedule for the day (at least as posted presently — some changes/additions will likely yet be made) and the link for registering for the conference. There are a wide range of speakers, both for grade levels and for topics. I know you will  find sessions that interest you and I encourage you to go to the conference if you can!