Focus on Math

Helping children become mathematicians!

Dot Cards Revisited November 19, 2014

Filed under: General Math — Focus on Math @ 2:29 pm

Screen Shot 2014-11-19 at 1.17.10 PMNumber sense is an important thing for students to develop. Some do it naturally, but most do so as the result of activities and lessons that are designed to have students think about numbers and their relationships with each other.

I recently had an email from a teacher asking how she might use dot cards to help the students in her grade 2/3 split class develop number sense, so this post is specifically to address that question of Amanda’s.

I use dot cards extensively in K and grade 1 classrooms, but they can be used for grade 2 and 3 students as well, especially if those students have not been exposed to many visual representations of numbers. If students are rather unfamiliar with the visuals, going back to the “basics” and working with them regularly for short periods of time is profitable. There are many ideas for such activities in the handout from my Dot Plate Workshop. (Here is a link for my other posts that refer to using dot cards.)

Here are a few ways to use the dot cards for grade 2/3 students:

Estimation: have the dot cards in a pile, a basket, or other way to easily access them. Have students draw three or more cards and then estimate the sum. It is important to have students discuss why they made the estimation that they did. It is also important to not force them to get really close to the actual sum. Rather, the skill is to practice the thinking about the numbers to get a reasonable sum. There are many ways students might do this: they might look for pairs of cards close to 10; they might decide that every card is either over or under 5 by a bit, and thus just add 5 for every card; they might realize that no card has more than 10 dots, so the answer cannot be more than the number of cards x 10; they might use a combination of methods to get as close to the answer as they can. Again, it is the process of thinking that is important here.

Mental Math: again, have the dot cards ready to be accessed, but also have some “decade cards” marked with 20, 30, 40, 50, 60, 70, 80. 90, 100 (and higher if you wish, depending on the ability of the children.) Have the students draw a dot card and a decade card, and subtract the dot card from the other number: e.g., if a student draws the 60 card and a 4 dot card, he would do the question 60 – 4 = 5. We have students practice subtracting in basic facts (including 10 – x) but often we do not have them practice the mental math of subtracting from other two-digit numbers. This is a great place to begin that practice.

Match and Add: place the dot cards face up. Have students scan the cards for two that have the same visual someplace on the card. For instance, a student may pick two different cards that each have “5-on-a dice” on them, and then add the two amounts (In the illustration, the student would be adding 9 + 7 but may see the strategy 5 + 5 + 4 + 2).

Once students are quite familiar with the dot card/dot plate visuals, it is fine to do the exercises with numerals cards.

Mathematically yours,


Responding to a FB Post… October 29, 2014

Filed under: General Math — Focus on Math @ 10:40 pm

Screen Shot 2014-10-29 at 10.25.01 PMI just finished posting an opinion to a thread on FB that had this graphic with it. It began here:

Do you know how to do Common Core math? Confusion over the standards has some calling for their removal. NBC’s Rehema Ellis reports tonight.

One reader had addressed me specifically in that thread, asking for my opinion, but as I responded to her under my personal FB account and not my Focus on Math account, I decided to copy and past my thoughts here on the blog (and thus to my FoM FB account) so they are officially “on the record”. Here is my post:

Louise Cook, This is about learning to think about numbers deeply. Most adults, if asked to add or subtract two two-digit numbers in their heads, cannot do it. They have to to the “carrying” or “borrowing” formula (on paper usually, or with great difficulty in their heads), and have no alternative ways to think about the numbers. If children are truly exploring numbers regularly it is AMAZING how they learn to manipulate numbers. More importantly, they understand deeply about the number system and why things work the way they do. The true mathematics is not just in memorizing methods that spit out an answer, but understanding the concepts that under lie those methods. Adults may be able to divide fractions (say, divide 1/2 by 3/8 using the “invert and multiply” method, but very, very few understand and can explain WHY they get an answer of 1 1/3. They think it just “magically” appears as an answer, but they don’t know why. I think that moving numbers around in operations (adding, subtracting, multiplying, dividing) without understanding why the methods work is like allowing a student to “read” without having any comprehension of what the words said. In fact, is it really reading without comprehension? Part of the problem is that we have been teaching math is the “spit out an answer way” for so long, that we believe it is right just because it feels comfortable. (I will also add that many teachers have been asked to teach for deeper understanding, such as with multiple strategies, without having developed an understanding for themselves. That is rather like trying to teach a foreign language that you don’t speak yourself — a difficult task to say the least. No wonder is does not work!) It was George Polya, a pioneer in problem solving, who said, “better to solve one problem five ways that five problems one way.” (end of that post)

I believe deeply in letting children really think about math, not spitting out a page of fifty numerical problems on a page without a context, ones they are expected to solve in a prescribed manner. If this method was doing such a great job we would have all of Canada and the US loving math and feeling confident about it. Instead, the vast majority of folks will say they hated math, were not (and are not) good at it, etc. Math usually wins as the “least-liked subject in school”. Oh, dear, I am still fired up about this and could go on and on, but it is late and I must stop. I am happy to continue chatting about this.

Mathematically yours,



What’s Important to Have in a Grade 1 Classroom? October 2, 2014

Screen Shot 2014-10-02 at 10.02.55 AMI was recently contacted by a former colleague, Dawn, regarding what manipulatives a grade one classroom might need to have on hand to support effective learning math. It seems a friend of Dawn’s is in a classroom which really has nothing for the children to use for hands-on math learning and they were wondering what was needed.

First off the classroom needs counters — counters in different shapes, sizes, etc. They can be purchased ones (such as mini plastic teddy bears) or ones gathered from home (such as bread tags, but†ons, etc.). But the need to be abundant and available.

Students need a way to count efficiently, especially in tens and ones. Egg cartons cut down to 10 holes, blank 10-frames printed on paper or card stock, or commercially produced 10-frames can all be used. I even like using cookie sheets (non-aluminum) and marking them with coloured tape as a giant 10-frame for use with magnets.

Base 10 blocks are also great for young students. These a generally in the form of small 1 cm cubes for “ones”, sticks for the “tens”, and flats for the “100’s”. I do want to make a critical point here: students may be engaged in a game of trading 10 cubes for a stick, or 10 sticks for a flat with every appearance of understanding the “ten-ness” of our base-10 number system. But be careful here. Student can be following your rule of trading 10 for 1 without that understanding. They might be just as happy to trade 8 for 1 or 12 for 1. The manipulatives give a opportunity for students to develop that important base-10 understanding, but moving blocks around correctly does not necessarily indicate that the understanding has been built in the student’s mind.

I think a grade one classroom needs “pop cubes” (multi-link cubes) — those blocks about 1inch in each dimension that can be attached together. I like to store them sticks of 5. If students need a particular amount for an activity, say 18, we discuss how many sticks each student will need, and then go get them. I also use these in many quick number-sense building activities. If I have students hold up a certain number of blocks, I want them to do so to model a ten frame. If I ask for 9 blocks and a student were to hold up a single stick of 9, I, as the teacher, cannot tell from a distance if the student is holding 8, 9, 10, or 11 blocks. But if he holds up a five stick beside a four stick, I can tell at a glance that he has the correct number. Pop cubes can be used in a multitude of math activities and should be on-hand for regular use.

Another must-have in my book are pattern blocks. They are particularly great for patterning activities for exploring symmetry, not to mention the creativity factor! I love them!

There are a number of things that I think should be in the classroom that are “make-able” such as dot cards, dot plates, printed ten-frames, even printed dominoes (click for more info on these)— all useful in exploring numbers, in building number sense, and  in helping students develop the skill of subtilizing. Students need to SEE the numbers in math, and these materials can help develop that “seeing” in the children.

Of course there are many other things that are fun to have in the math classroom, such as dice, dominoes, blocks, playing cards, geoboards, plastic coins, bingo chips, square tiles, Cuisenaire rods, and two-colour counters, to name a few. But lots of math learning can take place with some thoughtfully crafted lessons and activities and just the basics.

I hope you will focus on the math understanding with whatever materials you have at your disposal!

Mathematically yours,


Add Math to Your Open House September 18, 2014

Filed under: General Math — Focus on Math @ 10:09 am

Screen Shot 2014-09-17 at 9.55.22 PMBack-to-school usually brings with it an open house for parents and guardians to come in meet the teacher and see the classroom. As a classroom teacher I always encouraged the students to come to the open house along with their parents*, giving the students the opportunity to act as docents or guides for the occasion, taking their parents through the various stations (usually 5 or 6) set up around the room.**

Before the event we practiced the proper way to conduct introductions, and the students would put that into practice at the open house. Introducing their parents to me, the teacher, was the last station to complete for the students to complete.

I always included one or more math components in the event, most regularly a graph. Earlier in the day or week the students and I would make a graph of an opinion question (e.g., favourite pasta, sport, or singer). Once the graph was created we would use math to investigate the data (math applicable to the grade level of the students: from simple “How many more…?” and “Which was preferred most?”  kinds of questions for younger students to “What percentage of students preferred…?” kinds of questions for older students.)

The graph and what we learned from the data would be displayed at the open house, and the students were to discuss it with their parents. Additionally we asked the parents to participate in creating a new graph — we put the same question put to them. I always asked the students to predict ahead how they thought the parents’ graph would be similar to the students’ graph, and how they thought the two would be different. The day after the open house we would investigate the data again and compare our predictions.

*note: All my experience as a classroom teacher was in schools designated “inner city” with a transient student population. I always invited my students to come to the open house on their own, even if their parents/guardians could not come. I also made sure a few of my own friends were there that evening, near the classroom, so that any student who had no adult with them could still have the experience of acting as docent to one of my friends.

** note: Other stations set up for the evening included things like these: explaining our classroom behaviour contract; discussing our goals for the year; demonstrating some of the math “thinker tools” (manipulatives); demonstrating a simple science experiment.

I hope you will try making your open house an interactive experience!
Mathematically yours,


Welcome to SUCCESS! July 31, 2014

Screen Shot 2014-07-31 at 12.36.30 PMAs I write this the summer is half over — at least for students and teachers in BC, Canada. Schools are generally out for July and August, and then begin after Labour Day in September. I know in other places school will resume in mid- or late-August.

Whenever it begins for you, my question is this: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

For me it was simply this: WELCOME TO SUCCESS!  I had cut the letters for that saying out of construction paper 12 inches high (one letter for each page) and I stapled the message above the chalk board at the front of the classroom.

I talked about student success many times each day for the first few weeks. I basically inundated the students with the message that they would succeed in my classroom because I would not let them fail. I would do whatever it takes to work with them to be successful throughout the year. Failure was NOT an option — this was a classroom of successful students! I even went so far as to tell them that it was their lucky year getting  me for a teacher! Oh, there would be work involved along with lots of learning, talking, thinking, wondering, solving, thinking, testing, proving, thinking, recording, demonstrating, thinking… But we would be working together as a class and each and every student would be successful.

I was especially vocal about success in mathematics. I was teaching grade 6/7 in the early years of my “WELCOME TO SUCCESS” campaign, and it was clearly evident that a large proportion of the class came to me telling me they did not like math and that they were not good at it. I knew the real story was that they did not UNDERSTAND the math and they were not good at remembering all the rules. My plan was to work continually with the concepts in the mathematics knowing that once they understood they would get better and be more confident. My promise was to help them be successful even in an area of study they thought they could not be successful in.

The wonderful things was, of course, that my statement of declaration proved true year after year. All of the students WERE successful! They believed me when I declared it (I guess I said it and they read it so many times that they could not help but believe it!) and ultimately their personal belief regarding their personal success was a turning point for them.

I will ask my question again: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

Mathematically yours,


Math in Nature July 15, 2014

Filed under: General Math,Parents — Focus on Math @ 3:03 pm
Tags: ,

Pinecone fib pic 1When you are spending time in the outdoors, one of the mathematical things you can look for is the connection in nature to the Fibonacci number sequence. Of course, one must know the sequence in order to recognize when it shows up in nature, and it is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34…
Do you see the pattern? It begins, with 0 and 1, and after that each successive number is the sum of the two previous numbers. Thus the number after 34 is 55, derived by 21 + 34 = 55.

The sequence shows up in nature in a variety of ways. For many flowers, the number of petals that they have is a Fibonacci number. Looking at the bottom of pinecones, the number of spirals formed, whether left spirals or right spirals, is a Fibonacci number. This is true for pineapples and sunflowers as well. The number of seeds found in fruit is often a Fibonacci number. The next time you are eating an apple or orange, or squeezing lemons for lemonade, stop and count the seeds! Even when looking at how plants branch off a stalk or grow their leaves we can see the Fibonacci sequence.

Click here for one great site for delving into this further— there are many others out there, too!
Happy number hunting!
Mathematically yours,

Pinecone fib pic 2  tree branches Fib pic


GPS: It’s All Based on Math! June 12, 2014

Filed under: General Math — Focus on Math @ 12:55 pm

Ever wonder about how a GPS works? It is all based on math — lots of math, in fact! Here is a delightful explanation of how it works that was shared with me recently. I enjoyed watching it — I hope you do to!

Mathematically yours,



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