Focus on Math

Helping children become mathematicians!

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,


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.