In the last blog, NUMBER SENSE STARTS WITH COUNTING, we have discussed the first pillar of Number Sense: Counting. Let’s talk about the second pillar now: Wholes & Parts – another type of math problems a student encounters from Kindergarten to Grade 12. A strong understanding of the concept of Whole and Parts will make a difference in understanding problem-solving: do I add or do I subtract? Do I multiply or do I divide?

What is “Wholes and Parts”?

The whole equals the sum of its parts: Whole = Part + Part + … + Part. 

The whole thing is equal to the total of all the little things that make it up. This is one essential thread that binds together many of the diverse elements of mathematics, because Wholes and Parts, and the relationship between them, unite various aspects of math.

To understand what is going on in math, it is necessary to see the Big Picture of the subject at hand, the problem on which you’re now working. Seeing this Big Picture nearly always involves determining the whole, its parts, and the relationship between them.

By seeing how parts come together to form a whole, and how the whole is broken-down into meaningful parts, many aspects of mathematical problem solving can be learned very quickly. 

The Law of “Wholes and Parts” is the basis of solving strategies of many diverse branches of math

When two parts form a whole, each part is called the complement of the other with respect to the whole. When three or more parts are considered, each part individually is the complement of the group of all other parts.

Here are the two most common tasks in math – with basic problem examples:

1. Given the parts, find the whole:  There are 18 boys and 12 girls in class. How many students are there altogether? 18 + 12 = 30. (Addition).
2. Given the whole and one or more part(s) find the other part: A box contains 40 marbles. 10 are white, 20 are black, and the rest are green. How many marbles are green? 40 – 10 – 20. (Subtraction).

When your child is confused in solving a problem-solving question, and you see that it’s about wholes and parts like the above example, ask them: read carefully, is it about addition or subtraction?  

Finding the whole and finding the missing part are found in all branches of math. For example, finding the supplementary or complementary angles in Geometry, or writing equations and solving it in Algebra. Another example is solving probability problems in Statistics e.g. the probability of winning plus the probability of losing = 1. In Calculus: the whole area under a curve equals the sum of the areas of all the thin little rectangles, parts, that can be arranged to most completely cover that area. 

Starts Early!

As in other aspects of learning such as sports, language, music, etc. the earlier you learn a new skill, the stronger and the more fluent you are going to be.

Introducing this concept should be done early to make it possible for their brain to grow with “problem-solving brain wiring”. With a kindergartener (as we did with our model kindergartener in the picture),  start with very basic problems such as 10 + 2 = ? and 10 + ? = 12. To make it easy for them to visualize the problem, we could ask: if you have 10 candies and your brother gives you another two candies, how many do you have now? (Finding the whole). If you have 10 candies, how many more should your brother give you to make it 12? (Finding the parts). It’s ok if they don’t get it yet; they may not have mastered this skill yet, but at least we have brought awareness to them. With repetition and patience (and of course some fun), they will master it one day.

We have a grade 4 student who has been with us for a little more than 2 years, and we notice that he is now very savvy in identifying wholes and parts and solving it all mentally. This is so rewarding to us and of course his parents, considering he was not able to do basic math when he first came in.

In the next blog, we will be talking about The Birth of Fractions, because Fractions are a natural extension of seeing things in terms of wholes and parts. So stay tuned!