Focus on Math

Helping children become mathematicians!

Seeing Dots: NCTM 2014 New Orleans Presentation April 11, 2014

Screen shot 100 dot arrayI am excited to be here in New Orleans at the 2014 NCTM conference. Yesterday was a great day of sessions for me, and I am delighted to be presenting a session in just a couple of hours! “Seeing Dots: Using Arrays to Add, Subtract, Multiply and Divide” will focus on all the different ways the 100 dot array can be used to help students visualize and represent numbers — something which leads to a deeper understanding of numbers.

I am posting the handout from the workshop as well as links to 100 dot arrays is the different sizes.

I hope you try using the 100 dot array in your elementary classroom!

Download the conference handout here.

Download a 100 dot large array here.

Download 4 arrays on a page here.

Download 6 arrays on a page here.

Download 12 arrays on a page here.

Mathematically yours,

Carollee

 

12 Days of Christmas: Canadian Style December 11, 2013

Screen shot 2013-12-11 at 9.57.17 AMIf you are looking for a great Christmas-themed book for the young (or young at heart), you will want to check out A Porcupine in a Pine Tree: A Canadian 12 Days of Christmas (written by Helaine Becker, illustrated by Werner Zimmerman). Of course, the math connection is in the counting and in the adding up of all the items being given during the 12 days – all totaled it adds up to quite a tidy sum.

Certainly, any book with numbers tends to make me smile, but this one more than most. It is seriously delightful! It offers Mounties frolicking, squirrels curling, moose calling, hockey players a-leaping, and more.

The hardcover version is available from Amazon.ca (not Amazon.com) for the bargain price of $12.26 (price valid as of date of posting). Actually, for just a bit more, you can get the gift set version with the hardcover book plus an adorable little plush porcupine (sounds like an oxymoron doesn’t it!)

I am not affiliated with the author or publisher – I was given a copy of this book last year for Chrismas and I love it! I know you will, too.

May your days be merry and bright!

Mathematically yours,

Carollee

 

Strike it Out: a Primary Game from NRICH March 5, 2013

Filed under: Basic Facts,General Math,Parents,Primary Math Ideas & Problems — Focus on Math @ 12:06 pm
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Screen shot 2013-03-05 at 8.43.25 AM One thing that I came across recently on the NRICH math website is an quick primary students’ game for practicing addition and subtraction with numbers to 20. On the particular webpage for “Strike it Out”, they offer a poster (a picture of which is posted here), a short video clip of the game, and a power point file all which give the instructions for the game.

I had each of my grade two math classes playing this game recently as a warm-up activity, and they loved it! The games go quickly for the most part – of course, some of the pairs of students were slower at the game, but those students were still engaged and trying their best.

The rules, simply, are these:
• Using a number line marked 0-20, one student begins by creating and recording an addition or subtraction equation, e.g., 4 + 10 = 14. On the number line he crosses out the 4 and 10 and circles the 14. His turn is over.
• The partner must now create a new addition or subtraction equation, but it must use the number 14 as one of the first two numbers, e.g., 14 – 6 = 8. She would crosses out the circled 14, crosses out the 6, and circles the 8. Her turn is over.
• The 8 must be used now by the first partner in his new equation, with the recording and crossing out and circling continuing.
• Play continues until one of the partners cannot make a correct number sentence, and the player who made the last correct equation wins.

Although there are many possibilities for equations near the beginning of the game, there are fewer possibilities as the game progresses. I watched students doing a lot of mental math trying to come up with appropriate equations. The students who needed support had a set of ten frames on the table to use to help the visualize and calculate.

I am including for download the game board page I made for students to use. (Cut the page in half to use.) Two students play on a single game board at a time.

There are lots of other great ideas on the NRICH math site for many different levels. I hope you will take some time and explore what is there!

Mathematically yours,
Carollee

Link to game on the NRICH math site here

Download the game board here.

 

Addition and Subtraction: A Really Big Idea October 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!

Whatever the grade you teach, whatever the level of mathematics, I encourage you to help your students understand this big idea about addition and subtraction: We ALWAYS add or subtract LIKE THINGS!
Mathematically yours,
Carollee

 

A Patterning Problem: Finding Sums of Consecutive Numbers October 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

 

Basic Facts: the Last Addition Facts February 25, 2011

In three previous entries I have discussed the learning of basic facts with the visual tool of ten frames. Using such visual tools can help children (especially the ones who are not good “memorizers”) use strategies to get get from things they know to things they don’t know — which in this case are the basic facts.

There are, of course, the +0 facts. For young children it is not trivial — they must make the connection that you can add zero things to other things, but that the number of other things does not change. So although we often generalize that adding numbers together gets a bigger number, this is not the case with adding zero (that generalization fall short elsewhere, too, such as with adding integers — it is, in fact, a false generalization!)

So,we have looked at strategies for +0, +1, +2, +9, +8, +5, =10, doubles, and near doubles, and, amazingly enough, every fact of the 0 to 9 addition grid has now been addressed with the exception of four facts (see inserted grid): 3 + 6; its pair 6 + 3; 7 + 4; and its pair 4 + 7.

Both of these pairs of facts can be tied to strategies that children already know. Each can be tied to an =10 fact: if 6 + 4 = 10, then 6 + 3 must equal only 9 (if adding one less, the answer must be one less). Similarly, if 7 + 3 = 10, then 7 + 4 must equal 11 (if adding one more, the answer must be one more). The 6 + 3 = 9 may also be tied to the visual of three rows of three dots, as is on a domino: two rows of three is 6, another row of three totals nine.

It is important to remember that all facts can be learned with efficient, mental strategies. Counting on fingers or with pencil taps is NOT an efficient strategy or a mental strategy, and we should strongly discourage children from using these methods. Drill of addition facts should only take place once the facts and/or strategies are in place. Drilling does not help a child learn the facts if he does not already know them.

Again, basic facts are truly basic. It is very important for children to learn them, and a strategy approach is useful. Here’s hoping things all “add up” for you and your children!

Mathematically yours,
Carollee

 

Ten Frames for Learning Math: Basic Facts – Doubles and Near Doubles February 22, 2011

The basic facts which are the doubles are those facts which add two of the same: 3 + 3, 7 + 7, and so forth. They have many connections to things in real life, and these should be explored. Many things on the human body come in pairs of 2; some things come pairs of 5. The legs on insects come in pairs of 3, while spider legs come in pairs of 4. Talking about such doubles is a great way to start with young children.

Skip counting by two’s is also a great connection for the doubles. Many children are quite fascinated when they realize that the answers to all of the doubles questions lie in the skip counting sequence.

The doubles can be tied to ten frames, especially those larger than 5 + 5. If a child looks at two six cards, each has a full five on it. Together these can be put together as a full 10, leaving only the two other single dots. Thus 6 + 6 becomes 5 + 5 + 1 + 1 or 10 +2. Similarly, 8 + 8 can become 10 + 3 + 3. Such strategies offer ways for children to eventually close their eyes, see the needed 10 frames, and answer the questions.

Once the doubles are learned, then the near doubles can be addressed. We want children to recognize that 6 + 7 can be thought of as 6 + 6 + 1 (or 7 + 7 – 1, as some children want to double the larger number). Even the “two-aways” can be learned in this matter. 6 + 8 can be 6 + 6 + 2. That fact can also be addressed by compensation: take one from the 8 and move it over to the 6, thus changing 6 + 8 to an actual double 7 + 7.

Although the ten frames provide a visual/pictorial tool, younger children can use actual counter to go through the motions of these kinds of strategies. Egg cartons are a wonderful tool for this! Just cut two of the “cups” off one end of a carton leaving 10 cups in the same formation of a 10 frame. Children can then put 6 counters in one egg carton, 8 in another and then physically move one from the 8 to the 6, revealing the 7 + 7.

Mathematically yours,
Carollee