Focus on Math

Helping children become mathematicians!

A Combining Problem: Correct Postage September 29, 2011

Filed under: General Math,Primary Math Ideas & Problems — Focus on Math @ 11:29 am

Yesterday I did problem solving with a number of classes at Charlie Lake School. Since it is early in the year and students have not yet had a chance to develop a range of strategies for problem solving, I used a problem that I could specifically base around manipulatives.

The problem, originally created by someone at the University of Wisconson (and I apologize that I cannot locate a specific name, group, or link), was this:

Jeff is sending a small package to Kelsi, who lives in another city. Jeff has to put 18 cents worth of stamps on the package. Jeff has 10-cent stamps, 4-cent stamps, and 2-cent stamps that he can use. Show different ways that Jeff could make 18 cents using his stamps.

I used 18 cents as a total for the grade 2’s, but gave the grade 3’s the same question with a total of 24 cents (thus allowing for more possible combinations).

Each student had 18(or 24) counters which could be moved into groups of 10, 4 or 2 to represent the value of each stamp. Because it was a hands-on approach, every student was able to be successful.

We shared our solutions and I recorded them on chart paper.  Some of the students realized that the repeated addition of similar stamps could be combined using “groups of” the value, so these ideas were recorded also. The students all worked hard and had a great time — many of them saying so when going out the door! Isn’t that what you want to hear?!

Mathematically yours,


Supporting Angela September 27, 2011

Filed under: General Math — Focus on Math @ 3:47 pm

I am delighted to report that one teacher from our district has posted work to the NCTM facebook page in response to the contest “Illuminations: Do You Notice Sum-Thing?” In fact, I am so delighted that I am going to campaign a little on Angela’s behalf and ask my readers to go to the page and vote for Angela by clicking “like” at her photos. This link will take you directly to her photos (from the main page, given in the previous post, you have to go under “photos” at the left side to find her entry).

Way to go Angela!! If anyone else enters the contest let me know and I will campaign for you, too!


Win Sum-Thing from NCTM Illuminations! September 20, 2011

I subscribe to a list serve from the National Council of Teachers of Mathematics‘ (NCTM) website Illuminations. If you have not visited there, it is a great source of lesson ideas for mathematics classes of all levels.

I am copying part of one of their recent emails where they are announcing a contest based on trying a particular posted lesson which explores patterns on a 100 chart. Even if this particular lesson does not “tickle your fancy” or you feel it is not at an appropriate grade level for your class, it is well worth your time to take a look at the Illuminations site. There are lots and lots of good lesson ideas there. Here is the information that came to me:

Win Sum-Thing!

Win a classroom set of A+ Tiles simply by trying your hand at an Illuminations lesson. In the lesson Do You Notice Sum-Thing?, students are asked to consider patterns that occur when various tiles are placed on a hundreds board. For a chance to win a full classroom set of the tiles or an individual set of A+ Tiles, try the lesson with some students and then do one of the following:

* Submit a picture (or link to a gallery of photos) of students participating in the lesson.

* Share a link of student work from the lesson.

* Post a link to a video of students participating in the lesson.

* Submit a write-up of things your students discovered during the lesson.

* Create an extension for the lesson.

Share your picture, video link, or write-up on our Facebook page, “NCTM Illuminations.” The link or photo that receives the most “likes” by 5pm ET on Thursday, October 6, will receive a classroom set of hundreds boards provided by A+ Compass. Additionally, ten other randomly selected entries will win an individual set of A+ Tiles.

Entries should be uploaded to our Facebook page, NCTM Illuminations. Or, you can submit to Christa Koskosky,, who will post any emailed entries to the Facebook page.

Mathematically yours,


Upcoming “Toolkit” Workshop September 19, 2011

Filed under: General Math — Focus on Math @ 4:31 pm

For those of you in the Fort St. John, BC area, I will be doing a workshop here at the SD#60 Board Office on October 13. Registration is free to SD#60 employees (teachers, TOC’s. EA’s, ASW’s); for others there is a cost of $25 for the session.

Creating toolkits with students is a wonderful way for them to develop a mindset of using concrete and visual things to help them understand and talk about mathematical concepts. Manipulatives and tools are less effective if  brought out by the teacher only for the occasional lesson. It is much more powerful if tools are available to the students every day for solving problems, part of the norm, not an “add on”.

I hope you get a chance to join us for the session and build a sample toolkit.
Mathematically yours,


Ten Frames for Solving Problems September 14, 2011

Earlier I wrote about using 10 frames to help students learn basic facts. Using those pre-made ten frames, students used strategies to help them solve equations such as 9 + 7 = ?

Blank ten frames can also be useful in solving problems, those which involve numbers in the “basic facts” category as well as those which involve larger numbers. In the first case, students might use large blank ten frames and put blocks or other counters on them to work out the problem. Egg cartons can also be cut down to replicate a 10 frame and used with blocks or counters.

When students do problem solving with me, and am usually interested in having them document their thinking using pictures, numbers, and/or words. I want students of any age to learn how to record the mathematical thinking they used in solving a problem.

One of the ways I facilitate this recording is to provide mini versions of visual tools we use in the classroom. These sit in small baskets in the room available for students to come and take and to then glue into their exercise book. Mini blank 10 frames (27 per page)  (40 per page) are useful in such situations, particularly for primary students. If a student has used larger 10 frames (or egg cartons) with blocks, he can record what he has done by gluing on however many little blank ten frames he needs, and then drawing circles on them (or colouring in the squares) to record the solutions. Some students are happy to not use the larger version of the 10 frames, and just use the mini version to work out the solution to the problem.

I use mini 100 dot arrays and mini 100 charts in this same way. Baskets of each of these sit at the back of the classroom (cut apart and ready for the student to “grab and glue”). Students who have solved a particular problem in more than one way may have used several of these tools (or the same tool but with different thinking strategies shown).

Thanks to Charlene K. for sharing these mini 10 frames with me. She made the original sheet and has allowed me to share it with others.

Remember, students even in Kindergarten and grade 1 can learn to represent their mathematical thinking, and providing tools for them can make it easier.

Mathematically yours,


“Each Orange Had 8 Slices”: Exploring Multiplication September 7, 2011

This is a great book to help students develop an understanding of multiplication. It is important that students internalize that multiplication and its inverse operation of division are always, always, always about groups of things. One of the factors in the multiplication problem must name the grouping mechanism.

When I used this with my grade two and thee classes last school year, I read the book to them. After that I set the kids the task of using the pattern of the book to create their own grouping page.

Here are several examples of the students’ work. I created the template for them to use (with prompts beneath the blanks to help them make their statement properly). As a bonus, the finished pieces made a great hall bulletin board! I was always looking for interesting math to post for public viewing. Note: the “teacher” line is because I was doing this with students from 4 different classroom teachers and needed to keep classes straight!

It is an easy lesson to do, but it helps develop the concept of multiplication. Give it a try!

Mathematically yours,


Math Camp (gr 2-5) 2011 September 2, 2011

Once again I am delighted to say that math camp was a success! We had a productive day exploring learning and the brain, mental math, strategies for all 4 operations (addition, subtraction, multiplication, an division), and equality.

We discussed that both cortisol and adrenalin, when present in the blood stream, tend to “shut down” thinking and memory in the brain. Adrenalin induces the “fight or flight” reaction, while cortisol induces a state of stress when other things are more important than learning. If we want students to learn well in our classrooms, we must take the time to build a safe learning community and do all we can to reduce students’ stress.

The mental math we talked about was similar to what was presented in the Grade 6-8 math camp, so I will direct the readers there for some notes about that topic.

As for the basic operations, it is critical that students do these in ways that are meaningful for them. We shared many strategies for addition and subtraction, some based on numbers only and some based on tools (e.g., 100 dot arrays, blank number lines, base to blocks, etc.). One of the most important things that students should know about the operations of multiplication and division is that they are always, always, always about groups. In representing multiplication, the area model is very effective and can help students understand multiplication beyond basic facts. (Base 10 grid paper is useful for this).

Equality is an important for students to develop in the elementary years. Studies show that when students see the equal sign in an equation, they do not think of equality. Rather, they think it means, ‘put the answer here’ or ‘now do what the sign says to do’. I have had students tell me, when I wrote an equation such as 8 = 2 + 6 on the board, that I wrote it “backwards”. Students expect the single number to be on the left because that is how they always see it! Not only should they see equations written “backwards”, but they should explore equalities with “multiple parts” on both sides, such as 5 + 3 = 2 + 6. Here is a “balance scale” which can be useful in exploring equalities.

I wish you all a wonderful school year!
Mathematically yours,

PS: I just remembered that I told all of you at the workshop to write on your 100-dot arrays because there would be clean copies available! So here are the links to the 100-dot arrays:  large 4 small, 6 small, 12 small.