Focus on Math

Helping children become mathematicians!

Communication: Recording a Problem Solving Discussion June 29, 2011

One of the things I learned this year was the advantage of recording student ideas as they shared strategies and solutions after solving an “rich” math problem [and I will restate my personal definition of a “rich” problem as one with a) many solutions; b) one solution but many strategies for finding it; or c) both many solutions and many possible strategies] .

For most of the school year as I would be working with students, I would record the students’ strategies and solutions on the chalkboard. My general rule of thumb is to have students tell me what they did, and I would do the recording. This was done very specifically so that the students had to practice verbalizing their thoughts — they usually found it easier to write things down with pictures and symbols that to tell me how to write things down.

I carefully recorded all that the students would tell me, but then, before my next class of students would come, I had to erase the board and get ready for another round. It occurred to me during the year that I was missing out on an important scaffolding step for students: if I were to record the work on large chart paper rather than the chalkboard, then the work could be hung for all to see could be referred to in later classes.

I should point out that in the particular case depicted in the photo, the T-chart on the right was done after the students had shared various solutions. It is important that, toward the end of the discussion, the teacher pose questions that can help the students move to a “bigger picture” — in this case I was moving toward an “n-rule” with the class.

So, that is how I began recording the discussion. I even got to the point where I printed off a large copy of the problem the students were working on. (NOTE: Thanks to Sharlene K’s brilliant idea, I now always write the student problems on the computer and copy and paste to fill a page. I usually only have to print off a few sheets, then use a paper cutter to cut them apart. The students begin each session by gluing the question strip into their exercise books — no one has to take the time to copy out a question, while it ensures that there is a copy of the question on the page.) Making a large copy of the problem and gluing it onto the recording sheet was easy since I was typing out the question anyway. I hung the chart paper on the side wall where it could be viewed. Each week, on Wednesdays, as I worked with the students, I taped the new work for a class on top of that class’ previous week’s work.

It was soon apparent that recording the discussions of solutions and strategies was a good idea. Students referred to solving previous problems (knowing the solution was still visible) making connections between one problem and another. I also referred to the previous problems, reminding them of strategies they could not clearly recall.

Even though I was doing this in an elementary setting, the principle of recording student solutions would work at ANY grade level. I HIGHLY recommend recording the class discussions on chart paper! I think you, too, will find it valuable for students and yourself.

Mathematically yours,

PS:The problem on the page, used for a grade 3 class, is this:

Jacob, Charlie, Kara, and Heather shared a bag of Skittles.
They each ate the same amount. There were 2 left over, and
they gave those to Jacob’s little sister. How many Skittles could
have been in the bag?


Learning and the Brain: UPSHIFT! June 27, 2011

It is nearly the end of the school year for our district, with Wednesday being the last 1/2 day for students and we teachers working through Thursday. That being said, I am well into planning for the fall! I will be doing three days of workshops at the end of August I am calling “Math Camp”, with each day focused on a particular grade band.

As I have been preparing for these days, I have been compelled to add in a bit on learning and the brain. There is so much out there concerning that connection that one could do a whole day’s workshop on that topic, and indeed, there are those who present such workshops regularly.

While each day will be devoted to math, I will be tying in a small piece about the brain, in particular, about student engagement, because if there is no engagement, there can be no learning.

The Reticular Activating System or RAS is a densely packed bundle of nerve cells in the central core of the brain stem. Roughly the size of a little finger, it it said to contain about 70% of all the brain’s cells.

The RAS is the “attention” centre of the brain. It is the key to “turning on your brain” and is the centre of motivation. It acts like the brain’s “secretary” or “gatekeeper” monitoring what gets in. It allows only two kinds of information:
–Info that is deemed valuable right now
–Any alert to a threat or danger

Now, once any threat or danger has been perceived and the brain alerted, it sends chemicals into the bloodstream: adrenalin for an immediate response, and cortisol if the threat is ongoing. Both of these chemicals shut down the learning of the brain — they short-circuit the learning, as it were. There is an immediate DOWHSHIFT in the brain’s state, and as long as the downshift is in place, learning is blocked.

To re-engage the learning there must be an UP-SHIFT in the brain’s state, a reactivation of the RAS to something other than a threat or danger. There upshift is not immediate. It may take seconds or even minutes to move the brain back into a state for learning. But it is critical that it be done. In the down-shifted state no learning will take place!

So, what kinds of things can we do to help students up-shift, to
activate the RAS toward positive learning outcomes?
— Get kids moving — and tie learning to this movement
— Ask questions: “What if…?” or “What do you think about…?”
— Engage their curiosity
— Help them achieve a state of relaxed alertness (relaxed but up-shifted)
— Play Baroque music (dendrites dance to the music!)–Mozart’s is good, too!
— Give them choices
— Shout, “Yes! Yes! Yes!” — sounds silly, but it really works! “Woohoo!” works quite nicely, too! I tend to do that one quite often in math 🙂

Now, personally, I try to get all those suggestions in during a math class with the exception of the music. I only remember to play it sometimes. All the other ideas I put there are easily achieved during a problem solving lesson! — So that’s another great reason to give your students rich problems to “noodle” over 🙂

I talk to my students about the two brain states I described here. I have even made little posters (click here to download the pdf file) that I hang in the classroom to remind them that a down-shift hinders learning, but an up-shift allows the brain to engage in learning.

If you are stressed right now about anything, try letting out a good “Yes!” shout or a loud, “Woohoo!” and upshift your brain. Repeat as needed!!

Mathematically yours,


Mental Math: Adding and Subtracting on the 100 Chart June 17, 2011

In keeping with the theme of mental math, I would like to propose that students in grades 2+ be challenged with learning to add and subtract any two digit numbers. This can easily be supported by using the 100 chart until the number relationships become second nature to the students. After that students can either just do the mental calculations, or they can close their eyes, “see” the 100 chart in their minds, and calculate from the visual image they produce.

It is easiest to begin by adding/subtracting 1 and 10 from a given number. If students have done the 100 chart tic-tac-toe, this will be an extension of that. From there it is easy to move on to adding/subtracting more than one (2 to 9) and multiples of 10.

The answer to 38 + 10 is 48, which is immediately below 48. The answer to 38 + 20 is 58, found two rows below 38, which is the same as adding 10 twice. The answer to 38 – 1 is 37, found to the immediate left of 38 on the 100 chart. The answer to 38 – 5, 33, is found five spaces to the left of 38.

When it comes to adding and subtracting other two-digit numbers, the above procedures can be combined. To add 54 + 23 a student can move down two rows (adding 20) and then move right three squares (adding 3) to end up at 77.

Strategies can be developed for adding or subtracting numbers that would “wrap around” ends of the 100 chart. When adding 54 + 29 rather than move down two rows and try to count 9 to the right (which moves down to the next row) students can devise strategies that make use of “nice” or “friendly” numbers. In the above example it is much easier to add 30 to 54 and then subtract one for a total addition of the required 29.

My personal belief is that every student above grade 2 and every adult should be able to add and subtract two-digit numbers mentally with ease. If the skill is not there, it is only because it has not been developed and practiced.

Spending a few minutes every math class on mental math helps develop life-long skills. Most of us, as adults, do much more mental math and estimation than we do with paper and pencil for exact amounts. We figure time, mileage, money, etc. daily in our heads.

Help your students develop their mental math skills by planning these kinds of practice sessions into your lessons.

Mathematically yours,


Mental Math: Finding Compatible Numbers June 14, 2011

Using pairs of compatible numbers is a great way to do mental math. Learning about compatible numbers can begin in the early grades. Students who use 10 frames (see an earlier post for copies of those) to learn about numbers to 10 can visualize the number compatible for 10 easily. For instance, looking at the 6 card, it is clearly apparent that there are 4 “empty” spots on the card, and thus 6 and 4 are compatible for 10.

As in the last post, it is easy to work on finding compatible numbers for 100 using the 100-dot array. First, using whole rows, students make the same “to ten” connection as for small numbers, but using full rows of tens. Thus 60 (or 6 full rows of 10) can be seen to be compatible with 40 (or 4 full rows of 10). From there the 100-dot array can be use for pairs of compatible numbers: 55 and 45 become compatible, etc.

Older students can work on compatible pairs for 1000. The same principle of ten works for hundreds (600 and 400 are compatible, 630 and 370 are compatible, as are 639 and 361). It is helpful for students if they examine and discuss the pattens that appear: the one’s place digits add to 10, all other place value digits add to 9).

A great way to practice finding compatible numbers is to display on the chalkboard or overhead groups of numbers, say 10 or 12, in which there are compatible pairs. You can write the numbers so that every number has a compatible partner, or have some “distractors” in the group that have no match. Students can find pairs of compatible numbers and display these on response boards.

Once students have practiced finding compatible numbers and become comfortable with that process, the skill can be used for other mental math. Consider adding 78 + 33. If a student recognizes that 78 + 22 = 100, and 33 is 11 more than 22, then 78 + 33 = 111. Using larger numbers, when doing 880 + 250, a student can split apart the second number into 2 smaller numbers, one of which is the compatible number. So 880 + 120 + 130 = 1130.

Using the same principle, students can practice finding compatible decimal numbers (e.g., 0.6 and 0.4 are compatible) use those numbers for mental computation as well.

Students should also play with fractions that are compatible (e.g., 3/8 and 5/8 are compatible, as are 13/16 and 3/16) and do some mental computation with those at the appropriate level.

There is some definite benefits in being able to find and use pairs of numbers compatible to multiple of ten (and for fractions, compatible to a whole number). I hope you will consider spending a few minutes throughout your math week working on some mental math skills with your students.

Mathematically yours,


Mental Math: “How Many to 100?” on the 100-Dot Array June 12, 2011

I mentioned in an earlier post that one of my favourite tools to use with students is the 100-dot array. I want to share one way that I use it to help students develop some mental math skills.

I have used this with students as young as grade two, but if older students have developed few mental math skills, this is a great way to add to their strategies. The fact that this mental math is grounded in the use of a visual tool allows every student to have success with the method.

I have traditionally done this strategy using an overhead and a transparency of the 100-chart, but if you have access to a document camera, you can easily use that. The other thing I usually do while doing this activity with students is have them write their responses on some kind of response boards (e.g., small chalk boards, small white boards, etc). It is a great way to incorporate formative assessment in the math classroom. As students write their responses, you can see at a glance who is “getting it” quickly and who needs a bit more help. I should mention, too, that the first time I use the 100-dot array with any group of students I always spend 10-15 minutes having the kids notice things about the array and talk about it. We pay attention to the 10’s, the 5’s the 25’s and the 50’s that are displayed on the chart. It is an important step if you are going to use the tool for any activity.

So, now to the actual activity. I begin by displaying the 100-dot array on the overhead, the I cover some rows with a piece of paper. I ask two questions: How many do you see? and How many to 100? I look for the answers in two forms: the number of tens the students see, and the number of dots they see. For instance, if I have covered 3 rows, I want students to say they see 7 rows of 10 dots, or 70 dots. The “how many to 100?” is answered by 3 rows of 10 dots or 30 dots. Each answer uses the relationship of 10 as an anchor, which is one of the foundational number relationships that students need to develop. Depending on the age and quickness of the students, I spend the first number of days doing this part of the activity.

The second stage is to use two pieces of paper to cover dots. With the first, I cover full rows as before. With the second I cover part of the last exposed row. So, from the previous example, if I were covering 3 full rows, I would go on and cover part of the next row, say 8 more dots. Now when I ask, “How many do you see?” the answer is 62 (6 full rows and 2 more dots). When I ask, “How many to 100?” students must “complete” two types of tens: in the individual row and in the number of rows. So, completing the row in the example there are 8, and then there are 3 full rows covered, for a total of 38 needed to make 100.

The year I taught grade 2 I did this activity regularly throughout the year. By late spring I could just say to my students (without showing the 100-dot array), if you start at 57, how many do you need to make 100? Across the whole classroom, almost as one, the eyes of every student would close as they would visualize the array. Heads would swivel and bob as students were completing the 10’s, and they all could find the needed amount to make 100.

It’s an easy way to build some mental math skills with your students. I recommend doing it fairly regularly as a warm-up activity. Build these skills, and then build some more!

Below  are some links to download some 100 dot arrays in different sizes. (Students use the small ones all the time in my class. When they are problem solving, if their strategy makes use of the array, they grab a small one from the basket at the back of the classroom and glue it onto the sheet they so they can represent their thinking. Choose a size that works for your students!)

one large 100 dot array

4 small 100 dot arrays

6 small 100 dot arrays

12 small 100 dot arrays

Mathematically yours,

PS The link for the 100 dot arrays was not working for a while (thanks, Pat, for pointing out the glitch!), but I think I have it fixed now! — at least it downloaded OK for me. Let me know if you have more trouble.


Communication: Using a Frayer Model to Develop Math Vocabulary June 10, 2011

Inherent in the study and learning of mathematics is rich language, and we, as teachers, need to help students make sense of the vocabulary involved. One way to help with this is by using the Frayer Model to have students thoughtfully consider individual vocabulary terms.

This graphic organizer begins in the centre, with the particular term written there. Then around the term are sections in which the student writes the definition (in his own words); facts about and/or characteristics of the term; examples of the term; and non-examples of the term. The box at the bottom of the page allows the student to write any questions he has about the term (and the answers to those questions, once they are discovered). I have used reduced-sized pages of this to have students make personal vocabulary booklets for particular units or strands (such as for fractions, for angles, for measurement, etc.). Having to pay attention to terms is especially useful for ESL or struggling students, but, in fact, all students can benefit from this activity.

Mathematically yours,

PS: Download the full-sized .doc version of the graphic organizer “Frayer Model” here.


100 Chart Tic-Tac-Toe June 9, 2011

One of my favorite activities with young mathematicians is to explore the 100 chart and all of its patterns. It is something I do with every class, not just the early learners! There is power in really knowing and seeing the patterns that are there, and they are many!

With young learners, one of the things I like to do is 100-chart tic-tac-toe. Now, I must confess that it is not really a game as is suggested by the name (although one could make a version of the game– feel free to adapt this!) I use the name because of the grid set-up. Practically every child knows how to draw the two vertical and two horizontal lines to set up the game of tic-tac-toe, and that is what starts us off, too.

Once kids draw the grid, I always begin by choosing a number to place in the middle. In the example shown you see the number 28 as the start number. We work together to place the other numbers. “What number is one less/one more than 28?” or “What number comes right before/after 28?” are the kinds of prompts to use. During the first number of times I use this with a class, I only go on to the numbers immediately above and below the centre number. Again, the questioning/talking is important as it is not the positional relationship we want to emphasize, but rather the 10 more/10 less relationships of those other numbers. Once students are familiar with this (and we keep looking at an actual 100 chart during this process), it is fine to extend the +10/-10 relationships to the “one before” and “one after” numbers on the sides.

The illustration also show two things that can be done to allow students, even the weakest ones, to fill in their grids independently. One way is to use a cut-out that fits the particular 100 chart the students are working with. By cutting a 5 x 5 square grid, and then removing the 3 x 3 centre, a window is created for showcasing the particular 3×3 section of the 100 chart that the students are trying to fill in on their grids. If you are doing this, it is best to use cardstock for durability. The second way is to have the students use a highlighter on the 100 chart (or a washable marker on a laminated 100 chart) to mark off the 3 x 3 grid.

You will notice that in two of the tic-tac-toe grids in the illustration that the starter number is not in the centre of the grid. This is where I go with students once they are successful completing the grids from a centre start. Of course, the starter number can be placed in any of the outside blocks, and the students enjoy the challenge a new spot gives.

A further extension is to start with a three-digit number, say 128, in the centre. For students dealing with numbers 100-999, this is a way to help them understand the one more/one less and ten more/ten less relationships.

Once students know how to do this, it can be done quickly as a warm-up/refresher. At this point a 100 chart on the classroom wall is often the only visual needed as a reference.

Give it a try with your students and let me know how it goes!
Mathematically yours,