Focus on Math

Helping children become mathematicians!

Reasoning About Fractions Using Benchmarks January 6, 2012

Filed under: General Math — Focus on Math @ 2:28 pm

lg fr pocket chartI first came across the idea of using benchmarks with fractions in John Van de Walle’s book Elementary and Middle School Mathematics: Teaching Developmentally (Pearson). I loved Van de Walle’s idea that students could quickly compare any proper fraction to the benchmarks of 0, 1/2, and 1 and decide which value the fraction would be closest to, thus giving a quick estimation of the fraction. By actually placing the fraction on a number line showing 0 to 1, students have the opportunity to develop the understanding that every fraction has a particular place on the number line (though it might share the spot with other fractions). Students generally have had real life experiences with fractions such as 1/2, 1/4, and 3/4 and thus I have found that all students can be successful thinking about fractions in this way.

Given the fraction 15/16, it is clear that having 15 of the 16 parts means that it is nearly the full amount, and thus is close to 1. Similarly for 19/24. A student might say here that there are 9 pieces more than half, and only 5 pieces away from being the whole amount, thus 19/24 is close to 1. (A word of caution here — make sure that students give sound reasoning. In the example of 19/24 to allow a student to say “it is only missing 5 pieces so it is close to 1” is not sound, for 3/8 is also missing 5 pieces but is close to 1/2. It is the relative amount missing that is important. Similarly 123/1000 has many fractional parts, but because the number of parts there is small relative to the number of parts in the whole, this fraction is closer to 0.)

Non traditional reasoning can be used when benchmarking fractions such as 4/9. A student might think that since 4 1/2 is exactly one half of 9, 4/9 must be close to 1/2. Elementary and middle school students are not usually exposed to fractions within fractions, but this reasoning is mathematically sound and is useful.

Once students are comfortable benchmarking fractions, they can move on to comparing fractions. For most of us, our experiences with fractions were such that if asked to compare 3/10 to 11/15, we would have changed the denominators to the common multiple of 30 and then compared the fractions. However, using benchmarks it is east to see that 3/10 is less than 1/2 and 11/15 is greater than 1/2, thus making 11/15 greater than 3/10. There is much to be gained by taking the time to explore fractions using benchmarks.

I have created pocket charts to go along with this activity, a large one to be used at the front of the class as well as smaller individual ones students can use. Having done this with a number of classes, including recently with my grade 4 and grade 4/5 classes at Charlie Lake Elementary, I have found that students enjoy using the little pocket charts. One teacher, after I had made and used pocket charts with her class, emailed me later that day to tell me the students were begging to take the pocket charts home so they could show them to their parents! Seriously, when was the last time your class begged to go home and do fractions with their parents?

I explored this topic more fully in an article published in the Summer 2011 issue of Vector (the journal of the BC Association of Mathematics Teachers). That issue was a special elementary edition of Vector and was sent out to all public elementary schools in BC. If you are in the province, hopefully you have access to that article!

You can download directions for making the pocket chart in both sizes here.

I hope you try using fraction pocket charts with your students. Let me know how it goes!
Mathematically yours,


Calgary City Teachers’ Conf 2014 February 17, 2014

Screen shot 2014-02-17 at 9.04.21 AMIt was wonderful to share the Friday morning session at the Calgary City Teachers’ Conference with so many new friends! I hope you walked away with some ideas for helping your students understand mathematics is a deeper way. Congratulations to Shannon Muir who won the math coaching session in the draw!

If you remember one idea from the morning, I hope it is one about building understanding in math. Students need to make sense of the concepts using first concrete materials, then with pictorial representations, and then with symbolic (or numeric) representation. Rules for manipulating numbers are not remembered well if they are not based in meaning. Caine and Caine report from their brain research, “The brain resists meaninglessness.”

As promised, I am posting here the tools we used and referred to for your easy access.

100 dot arrays (1 large)

100 dot arrays (6 per page)

100 dot arrays (12 per page)

ten frames (teacher size)

ten frames (mini blank ones, 40 per page)

base 10 grid paper (enlarge as needed)

fraction & percent circles

fraction pocket chart    (link here for more discussion about these)

I think that is everything. If I have missed something let me know. And I would love to hear how this all makes a difference for your students!

Mathematically yours,


Screen shot 2014-02-17 at 11.27.57 AM


Twain Sullivan Elementary Revisited! November 28, 2013

Screen shot 2013-11-28 at 2.12.06 PMA big shout out to all the participants of the workshop last week at Twain Sullivan Elementary in Houston, BC! I appreciated your participation and your enthusiasm as we delved into the wonderful world of mathematics!

A focus for the day was on sense-making in mathematics — that it is important for students to be making meaning in what they do each day. We talked about the importance of number relationships, particularly the whole-part-part relationship. The more ways a student knows to pull apart a number, the stronger his mathematical thinking! We played with mental math in halving and doubling (even using Russian Peasant Multiplication) and in using number strings.

We used a variety of visual tools for making sense in mathematics:§    Fraction pocket charts (link here for a blog post on those)

I trust you had ideas and tools to use right away in your classrooms with your students. Let me know how it goes!

Mathematically yours,



Math Camp 2013 Reflections… August 28, 2013

Screen shot 2013-08-27 at 7.13.49 PMWow! Math Camp 2013 was a resounding success! The focus each day was on how we can structure routine activities for our students that will allow them to build number sense. We also talked about Carol Dweck’s research about mindsets and looked at how we could help our students build a ‘growth mindse’t in mathematics and not be stuck in a ‘fixed mindset’. (If you have not read Dweck’s book Mindset, I encourage you to get a copy asap!)

We looked at visual routines, counting routines, and routines involving number quantity, and discussed how each of these can be utilized for learning.

Our visual routines involved using 10 frames, dot cards, dot plates, 100 dot arrays, fraction pocket charts, percent circles, base-10 grid paper, and number lines (I always have students draw these rather than use ones that are pre-drawn and pre-marked). See end of post to download the various tools.

Our counting routines involved choral counting, counting around the circle, and stop and start counting, and counting up and back.

Our routines for number quantity involved mental math, number strings, “hanging balances”, and decomposing numbers.

It would take too long to write here in one post about how best to use/do each of these ideas, but over time I will get to them. Are you interested in something in particular? Email me and let me know and I’ll get to that one right away!

All of the “math campers” went away with lots of ideas that can be implemented in the classroom right away. I’ll be excited to hear from them how it goes it their classrooms.

I’ll leave you with my favourite definition of number sense: “Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Hilde Howden, Arithmetic Teacher, Feb., 1989, p.11).

There is much food for thought in that quote alone!

Mathematically yours,


Click to download: student 10 frames , teacher 10 frames; student dot cardslarge 100 dot array, 12 small 100 dot arrays, 6 small 100 dot arrays, 4 small 100 dot arrays, teacher dot cards set 1, set 2, set 3; template for making dot platesbase-10 grid paper, percent circles; directions for making fraction pocket charts;


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,


Math Camp (gr 6-8) 2011 August 31, 2011

What a great group of participants we had at yesterday’s gr 6-8 math camp! I was delighted with all of the group sharing that we were able to facilitate — there is always so much that teachers can learn from each other.

The workshop yesterday focused on several things: mental math, integers, fractions, and algebra. These areas, I believe, are important for students to master if they are to be successful in subsequent levels of mathematics.

Mental math is a skill which is only developed if practiced, and we discussed some particularly useful strategies that might be incorporated into regular practice sessions. First, mental math begins with basic facts! From there we can have students practice thins such as these:

  • adding multiples of tens (MOT’s) (20 + 50)
  • subtracting single digits from MOT’s (50 – 8 )
  • adding single digits to non-MOT’s (39 + 6)
  • adding to get 100 or – from 100
  • adding any two-digit #’s (47 + 39)
  • doubling numbers
  • halving numbers
  • multiplying by 10, 100
  • dividing by 10, 100
  • multiplying by 20 (by doubling and then multiplying by 10)
  • multiplying by breaking up numbers (using “nice” numbers)

Response boards are a great way to do immediate full-class assessment during mental math practice.

Our focus on integers was in using “chips” to have students learn about positive and negative numbers in a very visual way. It is helpful for students if they understand the power of zero in adding and subtracting integers, and zeros can be visualized by an equal quantity of positive and negative chips. Using the chips to solve addition, subtraction, multiplication, and division problems involving integers allows students to build a conceptual understanding of integers which goes far beyond memorized rules.

For fractions, we discussed the need for students to develop number sense regarding them. Using benchmarks to estimate fractions is one way to help facilitate this. We talked about making pocket charts for both the teacher and the students to use in practicing this. (Pocket chart directions.)

As for algebra, once again, visualization was the key to helping students make sense of this generally abstract area. By having students display and manipulate equations in a concrete way on a “balance scale”, they have the opportunity to learn what are acceptable or “legal” moves in solving algebraic equations. (Balance scale.) A hands-on, visual approach to algebra allows every student to be successful in this area!

One of the teachers at the workshop is going to email me a rubric that she used in her math classes last year, and I will up-date this post with that rubric once I have received it. (later) Here is the link for the rubric. When we discussed this, the “traffic light” part was really important. Remember that students can self-assess their understanding and record it as red (“I am totally stuck.), yellow (“I am able to work some on the problem but not I am not really sure about it.), or green (“I understand this well enough that I could teach someone else.)

In the meantime, I hope you will think about how you might better teach these areas of mathematics that are critical for students in these grades.

Mathematically yours,