# Focus on Math

## Helping children become mathematicians!

### Metacognition using an Addition Strategy Math MatFebruary 28, 2013

Filed under: Basic Facts,General Math,Parents,Primary Math Ideas & Problems — Focus on Math @ 9:14 am

Recently a teacher shared with me a math mat she had created listing a huge variety of addition strategies (thanks, Doreen M.!) Now, the idea of such a strategy mat is not brand new. Indeed, many such mats are circulating on Pintrest and other sites.

What delighted me about Doreen and her mat was not the uniqueness of the mat, but how she was using it. Doreen told the story of having given the mats to her students after they, as a class, had talked about each of the strategies. The math problem-solving lesson was structured with the students being given a block of time to solve the day’s problem using as many strategies as possible. Doreen hoped the students would use the strategy mat to prompt their thinking as they were solving word problems.

However, Doreen observed that the students rarely referred to the mats during their actual problem solving work. The students basically ignored the mats, even those who particularly needed the scaffolding.

Her response to this was to have the students do some metacognition regarding the strategies they had actually used in their personal solutions – but to do this once they were done solving the problem. Each student already had a laminated copy of the mat, but now she gave each student a marking pen. Students were asked to look over their work, and each time they noticed that they had used a particular strategy on the mat (e.g., breaking down a number into smaller parts or using doubles), they marked it on their own mat.

This, then, become the “norm” when doing problem solving. Over time, the students in the class became much more aware of which strategies they were using as well as the ones they weren’t using. This metacognitive thinking provided a great starting place for discussions when sharing solutions for the particular problem of the day.

So I invite you to try the adding strategy mat with your students. But more than that, I hope you will also try Doreen’s method of having your students do some “thinking about their thinking.” There is great power in metacognition!

Mathematically yours,
Carollee

### I Have No Cue! Subtraction without the “Cuing” wordsFebruary 21, 2013

A while ago I gave my grade 2 and 3 math classes a whole-part-part subtraction question, which is a different format from what is often given to students. Many times word problems for addition and/or subtraction are the kind where the initial amount is given, the amount of change is given, and the sum or difference is what is missing. Those problems go something like this (using “small” numbers):
• There were 2 birds in a tree (initial amount). 3 more landed in the tree (change). How many were then in the tree (missing sum)?
• There were 5 birds in a tree (initial amount). 2 flew away (change). How many birds were left in the tree (missing difference)?

In both of those questions, students hear “cueing” words (“more” and “away” respectively) and figure out from those cues which operation to perform.

Problems, however, can be written in ways that do not give such obvious cues, thus providing situations for students to think more deeply about the problems. Using a situation involving a whole and two comprising parts (whole-part-part) is one way to write such questions. Solving such a word problem may be a bit more difficult for students as nothing is actually added to or subtracted from the initial amount.

In a certain city there were 93 children enrolled in soccer. 55 of them were girls and the rest were boys. How many boys were enrolled in the program? (Incidentally, we did have to have a conversation in each class about the meaning of the word ‘enrolled’.)

The grade 3 classes were given this version of the question:
In a certain city there were 321 children enrolled in soccer. 148 of them were girls and the rest were boys. How many boys were enrolled in the program?

At least one student in each of the 5 classes that did this problem came up to me immediately after the class had started to work and asked what he should do. “Should I add these numbers?” “Should I subtract the two numbers?” I expected that this would happen given the wording of the problem.

The goal, as always, was for the students to make sense of the problem and then to solve it using more than one strategy. In the end I was quite delighted in the various strategies that the students used to solve the problem.

LOOKING AT THE GRADE 2 STRATEGIES:
Mrs. Sequin’s grade 2 class shared several strategies. A number of students made use of the 100 dot array, most of them first colouring 93 dots to represent all of the children, then crossing out 55 of those as representing the girls. The remaining dots, not crossed out, were counted to tell the number of boys playing soccer.

One boy showed his numerical solution where he started at 55 and added up (mainly by 5’s and 10’s) until he reached 93.

We had been playing with open number lines in the previous weeks, and three different students shared strategies for solving the problem with this tool. Using such tools allows students great flexibility as they think about the numbers in a problem.

Notice the first number line on the chart paper. The student worked backwards towards 55 (really doing this: 93 – ? = 55), starting at 93, jumping back 20 to 73, jumping back 20 more to 53, then realized that that was 2 too far back, so changed the second jump backwards to 18 (for a total of 38 back).

The second number line records a student who used addition to solve the problem (55 + ? = 93). She began with the 55 girls, then jumped by 10’s up toward 93, passing it, landing on 95. She solves the problem then by adding her 4 ten jumps and subtracting the 2.

The third number line also shows subtraction like the first method: again the student is looking for the missing amount that will leave 55. The student begins by jumping just 3 backwards to get to the “friendly” number 90, jumps 20 back once, twice, then recognized he has “over shot” 55 and changes the last jump to 15.

LOOKING AT THE GRADE 3 STRATEGIES:
Mrs. Ranger’s grade 3 class shared 4 strategies in the time we had for our discussion. First, one girl shared how she arrived at the answer using the traditional algorithm for subtraction. After that a boy shared his strategy using an open number line. He used the “think addition” method (148 + ? = 321).

I was quite impressed with the third strategy shared. My students have used 100 dot arrays for thinking about numbers, but the boy did not want to have to use 4 arrays on his page. Instead, he drew his own dots, explaining that each dot represented 10 players. Thus each row of 10 dots represented 100 players. He had two more dots for the 20 players, and a single tally mark for the last player (300 + 20 + 1). From that point he crosses out 14 dots (representing 140 players), crossed out one more dot but puts a small 2 beside it to show that not all 10 were removed, just 8 leaving 2. Pretty cool thinking, eh?!

In the final method shared the student had used sketches to represent base-10 blocks. She began with 321 and shows step by step how 100, 40, and 8 are removed.

I hope you see how wonderful problem solving can be with students. When we arm them with tools and strategies for thinking, they can do wonderful things!

Mathematically yours,
Carollee

### SD#60’s 6th Annual Parent ConferenceFebruary 18, 2013

School District #60’s Annual Parent Conference is just around the corner. I am delighted to again be presenting two sessions at the conference.

First, for parents of primary-aged children (K-3) there is the session Numbers, Numbers Everywhere! This session offers visual tools and strategies geared towards a new wayof thinking about mathematics! We will explore early numeracy concepts and ways to support your child’s learning in math.

For parents of intermediate-aged  children (grades 4-7) there is the session Understanding Math in the Intermediate Years. Parents often wonder why their child’s math looks different from the math the parent did in school. This session will offer visual tools and strategies geared towards a new way of thinking about mathematics! We  will explore mathematical operations (addition, subtraction, multiplication, and division) and mathematical concepts (place value, fractions) in ways that help parents and students make sense of the math.

I am sure you won’t be disappointed in any of the conference sessions, but I especially hope you will join me for math!

Mathematically yours,

Caollee

### How Many More to Make 30?February 12, 2013

This is an activity I created to use with two grade 2 classes that I work with at Charlie Lake Elementary. In BC, grade two students work extensively with numbers to 100. The activity, like “How many more to make 20?” (see post from Feb. 5, 2013),  is based on one of the foundational number relationships which is, for numbers 1 to 10, anchoring each number to 10.

30 was chosen as the focal point for this activity since multiples of 10 are also important anchoring numbers.

Once again I was delighted to put some special dice to use, in this case 30-sided dice.** Each child rolled the dice and then, using a set of 10 frames, created the number rolled at the top of the sheet, right over the blank ten frames there. Thus, if 14 were rolled, the child placed a full ten frame and a one showing four on the paper, and then recorded the number 14 in the roll column of the T-chart. Then he looked to see how many would be needed to make 30, in this case 6 to fill the partial ten frame and one more full ten. 16 was  recorded beside 14 on the T-chart (see picture).

Similarly, if 7 were rolled, the child placed a ten frame showing seven on the paper, and then recorded the number 7 in the roll column of the T-chart. He could see that to make 30 he would need 3 more to fill the partial ten frame and two more full 10 frames, and thus 23 was recorded on the T-chart (see picture).

As in the “How many more to make 20?” activity, some of the children stopped making the numbers with their ten frames soon into the activity. Clearly they could imagine the anchoring relationship in their minds and did not need to manipulate the cards to “see” the numbers. Other children needed the support for every roll, but they were still able to be successful because of the scaffolding the ten frames provided.

I hope you will try the activity with your students!

Mathematically yours,

Carollee

**If you do not have 30-sided dice, having students draw numbers from a bag or spinning numbers on a spinner will do nicely. You could even give students the page with the first column already filled in with numbers of your choice.

### How many more to make 20?February 5, 2013

This is an activity I created to use in a grade one classroom here my school district. (In BC grade one students work extensively with numbers to 20.) It is based on one of the foundational number relationships for numbers 1 to 10: anchoring each number to 10. A set of ten frames is a fabulous tool to help build this relational understanding with young children. The ten frames provide a visual representation of each number and clearly show how far away each number is from 10.

Along with 10 being an anchoring number, multiples of 10 are also important
anchors. With this in mind, I felt it was important to give grade one children the opportunity to practice anchoring numbers to 20.

I had some 20-sided dice that were perfect for the activity**. Each child rolled a die and then, using a set of 10 frames, created the number rolled at the top of the sheet, right over the blank ten frames there. Thus, if 14 were rolled, the child placed a full ten frame and a one showing four on the paper, and then recorded the number 14 in the roll column of the T-chart. Then he looked to see how many would be needed to make 20, in this case 6, and recorded it beside 14 on the T-chart.

Similarly, if 7 were rolled, the child placed a partially filled ten frame showing seven on the paper, and then recorded the number 7 in the roll column of the T-chart. He could see that to make 20 he would need 3 more to fill the partial ten frame as well as a full ten more, and thus recorded 13 on the T-chart.

Some of the children stopped making the numbers with their ten frames soon into the activity. Clearly they could imagine the anchoring relationship in their minds and did not need to manipulate the cards to “see” the numbers. Other children needed the support for every roll, but they were still able to be successful because of the scaffolding the ten frames provided.

I hope you will try the activity with your students!