Focus on Math

Helping children become mathematicians!

NCTM 2013 Denver Presentation April 18, 2013

NCTM workshop picI am looking forward to a great hands-on session tomorrow as we are “Packing a Powerful Punch with Patterns” (presentation #500, located in the Hyatt Regency, Centennial Ballroom E beginning at 1:00). We will be focusing on how to help students make the transition from basic patterning skills to algebraic thinking, uncovering the deeper math that is embedded in patterning. Our vehicle will be growth patterns that we make out of pattern blocks. If you are here in Denver this week, I hope you are able to join us for the session.

The handouts given out in the session were a truncated version of the PowerPoint presentation, and as promised I am making the full version of the handout available here. If you use these in your classroom, I would love to hear from you about the lessons and even see some samples of student work, too.

Mathematically yours,

grow pat chart pic


North Central Zone Conference Sessions March 9, 2012

Thanks to all of you who participated in my two math sessions here in Prince George, BC, today as part of the Zone Conference. In spite of my sore throat, we covered a lot of ground about fractions, decimals, percents, integers, and algebra.

I promised some “clean copies” of the handouts, so here they are, ready for downloading:
balance scale (for doing algebra equations concretely)
percent circles
percent grids and blanks (see activity below)
mini 10 frames (for representing decimal tenths in adding, multiplying, etc.) 27 per page40 per page

Remember, the BC curricular documents stress over and over that students should demonstrate their mathematical understanding “concretely, pictorially, and symbolically”. We tend to not do enough of the concrete and pictorial parts, but hopefully after today’s session(s) you have a few more ideas of how you can fit those in to your lessons.

We did not do anything with the percent grids and blanks, so let me share an  activity you can do with those. Before photocopying to give to students, partially shade several of the “blank” squares. Don’t worry about shading in a particular manner, just draw a random closed curve in each square and shade it in. Ask the students to estimate the percentage of the square that is shaded, and then give each of them one of the little 100 grids copied onto acetate and cut out individually. Students can lay the acetate grid over the partially shaded square and count the number of little grid squares that are completely shaded. Then have them count all the little squares that are partially shaded (for instance, the curve goes through the square leaving part of the grid square shaded and part not shaded) and divide by two to average out the ones mostly shaded and mostly not shaded.

For a recap on what we talked about using the fraction pocket charts, check out the blog post called “Reasoning About Fractions Using Benchmarks”. That post goes over what we talked about today and includes directions for making the pocket charts as well.

If I have forgotten something, let me know and I will edit this post and add to it! I hope you try something from the workshop right away with your students!

Mathematically yours,


Equality: Balancing the Scale January 19, 2012

Many elementary children have serious misconceptions about the meaning of the equal sign (“Fostering Relational Thinking while Negotiating the Meaning of the Equals Sign”, Molina & Ambrose, Teaching Children Mathematics, Sept. 2006). Most of them think it means “put the answer here” or “do the adding” or whatever operation was involved in the equation. I personally have had students, when I wrote something like 5 = 2 + 3 on the board, say, “Mrs. Norris, you wrote it wrong!” The comment is not surprising since they almost always see equations with the answer on the right.

Students tend to be uncomfortable when their notions of the equal sign are challenged. “Backwards” equations (as was 5 = 2 + 3) or ones such as 2 + 3 = 4 + 1 put students’ understanding at a disequilibrium as they struggle to make sense of what is being said in the equation.

I had the opportunity this week to work with five different classes around the concept of equality. The school had purchased (at my request!) a class set of student balances, along with a larger, demonstration-sized balance, which we used to represent equations. We began with some “regular” ones, and then moved on to showing 5 = 5, 12 = 12, etc.. Equalities like that, with no operation symbol at all, were a bit startling to most of the students, but they quickly understood the logic of such statements as we represented them on the balance scale. Since the scale only goes to 10 on each side, we explored how to represent double-digit numbers greater than ten using base-10 representation. Thus 12 was put on the balance as 10 and 2, 20 was put on as two 10, etc.

We represented and recorded many “backwards” equations, and then moved on to ones that had two numbers on both sides of the equal sign, or multiple numbers on each side. We explored multiplication by hanging multiple weights on a single number (e.g. 4 groups of 3 balanced with 10 and 2).

The drawback of using the scales is that you cannot represent subtraction. In most cases the children used the tool, then wrote the equation they had created. One girl wrote her equation first, 9 – 1 = 6 + 2 but then could not represent that on these simple balances. We will explore such extensions in further sessions.

Because the class was hands-on and very interactive, every student was engaged. There were many comments made about the balance system being “cool” and many questions about when we would use the scales again. And, seriously, don’t we want them eager to come back for more math?!

Children need many experiences with equations that are not in “regular” form if they are to build an understanding of the true meaning of equality. I encourage you to find ways to explore this concept, one that is a critical component of algebra, with your students.

Mathematically yours,


The Meaning of Equality April 23, 2011

I was in Indianapolis, Indiana recently at the National Council of Teachers of Mathematics (NCTM) National Conference. First of all, let me say that it was a fabulous meeting of minds! I am full of new ideas to try with my students and to share with teachers in my district.

I was honoured to get to present at the conference. I did my workshop “Packing a Powerful Punch with Patterns” which is about using patterns (especially pictorial ones) to help younger students build algebraic thinking. One of the important points in the workshop is about equality. We, as teachers, need to make sure we offer lots of opportunities for students to build an understanding of this concept. We use the term frequently in our classrooms, saying things like “two plus three equals five” referring to the symbolic notation of that: 2 + 3 = 5. Day after day, week after week, month after month, year after year we use the term “equals”, but seldom do we stop and check to see if the students understand the term as we mean it. It is most likely that they do not. Studies have shown that students then to think that “equals” means “put the answer here”, or “now do the operation”. They tend to have no reference point for the idea that whatever is on one side of the equal sign has the same value as what is on the other side.

Teachers can help students with this by doing a few simple things. First, try writing equations with the “answer” on the left: 5 = 2 + 3. When children see that for the first time, they tend to tell you that you have written it incorrectly! Another great thing to try is to write equations such as 2 + 3 = 4 + 1. You can also leave out any one part of that equation and have students solve it. Again, they are likely to have a misunderstanding and tell you that 2 + 3 does NOT equal 4 since that is where they are used to stopping in their thinking. Using actual balance scales with small blocks or other regularly-sized manipulatives is a great way to help students develop this important concept.

So, the next time you are using equations in a lesson, take some time and find out how well your students understand equality. It’s worth putting some time into this topic!

Mathematically yours,