# Focus on Math

## Helping children become mathematicians!

### I Have No Cue! Subtraction without the “Cuing” words February 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