Focus on Math

Helping children become mathematicians!

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,


Strategies for Student-Centered Secondary Mathematics February 24, 2011

Filed under: Middle School/Secondary School Math — Focus on Math @ 5:16 pm

I came across a blog this week that had an entry about secondary mathematics teaching. It tells a bit about Paul Bogdan’s journey from having a teacher-centered classroom to a student-centered one. The link below will take you to the page. Paul shares five specific strategies he uses to increase student engagement and to help students learn mathematics.

I hope you will check out Paul’s ideas and try them in your middle or secondary classroom!

Mathematically yours,


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,


Ten Frames for Learning Math: Basic Facts +5, =10 February 18, 2011

Ten frames (click to download), as discussed in the previous post, are powerful visual tools for helping students of all ages learn basic facts. Besides being helpful for facts that are +1, +2, +9, and +8, the ten frames are also great for +5 and =10 facts.

For the +5 facts, start with the 5 card. If you add any other number 1 to 5, when that card is laid beside (or partially on top of) the 5 card, together they look like one of the other cards 6-10. Since these cards are already familiar to the child, the answer to the fact can easily be visualized. If you add any number 6 to 9 to the five card, the cards can be placed so to see the column of 5 dots on the 5 card beside the column on 5 dots on the other card. Of course, the two columns of 5 together make 10, and then the remaining part of the second card is added to the 10. So, 5 + 8 becomes 5 + 5 + 3 or 10 + 3, a much easier fact to learn and remember.

The =10 facts are the ones apparent on every one of the ten frame cards. For instance, when looking at the 6 card, since there are 4 spaces without a dot, 4 must be added to make 10. The corresponding subtraction facts are useful here, as well. 10 – 4 = 6 is also clearly seen on the ten frame card for 6. The cards can be flipped over one at a time, and for each card the addition fact apparent from the dots and spaces can be practiced.
Remember, basic facts are basic! Have fun with them!
Mathematically yours,


Ten Frames for Learning Math: Basic Facts +1, +2, +9, +8 February 17, 2011

Basic facts are just that: basic. They are the foundation for a lot of other kinds of mathematical thinking including estimating, doing mental math, and doing other, more sophisticated kinds of computations.

Basic facts are generally defined as the set of addition and multiplication problems that are one digit +/x one digit, along with the “reversals”of those questions for subtraction and division. Thus 6 + 7 = 13 is a basic fact, as is 13 – 7 = 6. However, 13 – 2 = 11 is not, because its reversal, 11 + 2 = 13 begins with a double-digit number.

Ten frames (click for download) are a powerful tool for helping children learn the basic facts. They give children a way to visualize the numbers, to “see” their thinking with the card(s) until they are able to “see” the thinking in their minds. Once they own the images the empowerment happens.

So to start off, I recommend you download the 10 frames, cut the cards apart, and begin having your children learn the images. You can play games such as “war” or “concentration” to help them.

The facts that are +1/-1 and +2/-2 (which flow out of the “Big Ideas for Little People” workshop) can be visualized. I have often talked with kids about adding or removing dots on the cards with a “magic eye”. Thus 6 + 1 can be visualized by seeing the 6 card and adding another dot mentally. Children who have worked on the big ideas mentioned above will find these facts easy.

Several other kinds of addition and subtraction facts are supported by the ten frame visuals. Consider the +9 facts. 7 + 9 = 16 is one of the “hard” facts in the eyes of most children, but if they learn to visualize the ten frames, the fact can be quite easy. By looking at the 7 card and the 9 card, they can visually move a dot from the 7 card to fill the 9 card, turning it into 6 + 10 – 16, a much easier fact! Alternately, a child can look at 7 + 9 and think 7 + 10 is 17, but since I added one to the nine I must remove it (which is using the “zero principle” — adding and removing the same amount). Both strategies can be used on the +9 facts with great success! The +8 facts can be done in a similar manner, just movingĀ  two dots mentally or by adding 10 and then removing 2 from the result.

Most children learn their addition basic facts in the primary years (grades 1-3) but if yours did not master them early, it is never too late! In fact, if children are older and do not know them, it is usually because they are not good memorizers — and if no strategies for learning are taught, memorization is the only route. Older students can benefit from the ten frames, too! If you haven’t used ten frames with your children, now is a great time to start! Mathematically yours,



“What Do I Do When My Kids Don’t Get the Math?” February 15, 2011

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 9:26 pm
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This weekend is the annual parent conference here in Fort St John, BC, put on by School District #60 and some other community partners.

I have the privilege of presenting a session, the title of which is the title of this post. It is an important topic, one which is of concern for both parents and teachers alike. What do we do when our children or students are not “getting the math”? And what does it actually mean to “get the math”?

Because of the way that math as been traditionally taught (for many generations), most of us think that doing pages of problems all of which follow the same formula or algorithm is “doing math”. Indeed, we have little experience doing anything else in math. But once someone has correctly done their 50 problems (for instance, multiplying a 2-digit number by a 2-digit number), what can one see about the person’s learning? I propose that such a page of problems tell us only two things: first, that the person knows their basic facts, and second, that the person knows how to follow directions. There is nothing in a page of 50 problems to show that the learner understands anything about the concept of multiplication or knows when multiplying is the appropriate operation. The same can be said for any page of practice problems where there is no context given, and no questions which address the mathematical concepts which underlie the rule-based practice.

One way to change this is to ask students to do a word-based question through strategies which show their understanding of the math. Most of us, again because of our educational experiences, think there is only one “right way” to add, one “right way” to subtract, etc. In fact, there are many ways to do each of these operations, and almost all of the other non-traditional ways are more meaningful to students (and adults!).

If you will be in FSJ on Saturday, I encourage you to attend the parent conference (pre-register at the school district website and, of course, come see me!

Mathematically yours,


Problem Solving Question: Intermediate — Water Jugs

Filed under: Intermediate Math Ideas & Problems — Focus on Math @ 4:00 pm

I had a great time in Melissa’s grade 5/6 class yesterday doing a logic problem. Melissa had mentioned that she had done a few logic problems with her class and wanted me to facilitate one for her to observe.

I chose a classic “Water Jug” problem, one which, according to one story, was the inspiration for the 19th Century mathematician Simeon Denis Poisson’s pursuit of mathematics:

  • Two friends have and eight-litre jug of water and wish to share it evenly. They also have two empty jugs, one which holds exactly five litres and another which holds exactly three litres. How can they measure four litres using these jugs?

The students went to work on the problem, but clearly some were having more difficulty than others (as is often the case). After a short while we stopped to have a quick discussion about which operations were likely to be used in solving the problem (addition and subtraction, and not multiplication or division). A second discussion a bit later focused on how students might organize their thinking to keep track of the “pouring”. This is a perfect example of when making a chart is a useful problem-solving strategy. Some students got stuck in a “cycle” of pouring, where they would have a line on their charts exactly like a previous line (or stage in the pouring) and they could not figure out how to get out of the cycle. Still, a number of students were able to find solutions in the time we had — and there was a real excitement in the air.

There are many variations of this problem — different sized jugs, different numbers of jugs, different target measure — something to fit nearly every level! If you haven’t yet tried logic problems with your class, I encourage you to give it a go!

I’ll not post the solution here (it is always good for YOU to practice your problem solving skills, too!), but would love to hear what your students do with the problem.

Mathematically yours,