Posted in Elementary School Math, Math videos, Number Sense

Why PEMDAS is ‘morally’ wrong

Here’s a video that explains why you need to memorise PEMDAS (or BODMAS, BIDMAS, depending where you are in the world) order of operation and why you don’t need to. Minute Physics who made this video made a mistake in assuming that PEMDAS is taught in schools without emphasising that multiplication and division should be done whichever comes first from left to right. But he does explain the ‘why’ behind the rule plus the importance of knowing fundamental ideas such as the distributive property and the associative property. This is what makes the video worth viewing.

You want to test your PEMDAS skill try this problem.

Posted in Elementary School Math, Number Sense

Why do we ‘invert’ the divisor in division of fractions

The following sequence of tasks shows how we can teach a procedure for dividing fractions, conceptually. The technique involves the same idea used for finding equivalent fraction or proportions – multiplying the upper and lower number of a proportion by the same number preserves the proportion. This is something that they learned before. The task of division of fractions becomes a simple extension of finding an equivalent fraction.

Teaching Sequence on Division of Fractions
Task 1 – Find fractions equivalent to the fraction 5/8.

This should be easy for learners as all they needed to do is to multiply the same number in the numerator and denominator. This is more of a revision for them.

Task 2 – Find fractions equivalent to

fraction divided by wholeHere they will apply the same idea used in task 1: If you multiply, the same number (except 0) to numerator and denominator, you produce equivalent fractions. You can ask the students to classify the fractions they made. One group I’m sure will have a fraction for numerator and whole number for denominator; another group will have fractions for both numerator and denominator; and, another will have whole numbers for numerator and denominator. The last group is what you want. This fraction is in simplest form. They should be ready for Task 3 after this.

Task 3 – Find the fraction in simplest form equivalent to

fraction divided by a fraction

From here you can ask the students to express the fraction as a division (this is one of the meaning of fraction – an indicated division) and rework their solution. It should be something like this:

dividing by fraction

You can challenge your students to find the shortest possible solution of getting the correct answer. It will involve the same idea of multiplying the dividend and the divisor by the same number. I’m sure that after doing the tasks above, they will be able to figure out the following solution which now leads to the the procedure ‘when dividing fractions, just multiply it by the reciprocal of the divisor’:

dividing by fraction

Procedural fluency does not mean doing calculation with speed and accuracy even without understanding. Remember that procedure is only powerful and useful in problem solving when students understand what it means and why the procedure is such. I suggest you also read my post on what it means to understand fractions and math knowledge needed by teachers to teach fractions and decimals.

Note:

The above lesson is not just about division of fractions. I made it in such a way that weaved in the lesson are the ideas of equivalent fractions, proportion, the property that when you multiply same number to the numerator (or dividend) and to the denominator (divisor) it does not change the value of the quotient, division by 1, etc. Working with the tasks engages students to the same process/technique they will be applying when they work with rational algebraic expressions. The main point is to use the lesson on division of fractions as context to make connections and to teach important ideas in mathematics. I think this is how we should teach mathematics.

Posted in Elementary School Math, Geometry

A triangle is a fish

Why is it that students find it easier to calculate the area of triangle ABC but will have difficulty calculating the area of triangle DEF? Middle school students even believe that it’s impossible to find the area of DEF because the triangle has no base and height!right triangles

That knowing the invariant properties that makes a triangle a triangle (or any geometrical shape for that matter), is not an easy concept to learn is illustrated by this conversation I had with my 4-year old niece who proudly announced she can name any shape. The teacher in me has to assess.

Thinking about how a four-year old could possibly think of these meaning of the shapes made me ask: If four-year olds are capable of thinking this way then why do we think that there are students who can’t do math or doubt the idea that algebra is for all

Posted in Elementary School Math, Number Sense

Bob is learning calculation

Bob is an elementary school student. He is learning to calculate. He just learned about addition and multiplication but there are some things that he doesn’t understand. For example, how come 1+3 = 3 + 1? How can it be the same thought Bob? Every morning I have 1 piece of bread for breakfast while Dad has 3 pieces. If I have 3 pieces while Dad has 1 piece, I will be too full and Dad will be hungry?

When they added three numbers, Bob did not understand (1+2) + 1 = 1 + (2+1). Usually I like to drink 1 cup of coffee with 2 spoons of milk then afterwards have a piece of bread. I would not feel well if I first drink a cup of coffee then afterwards drink 2 spoons of milk while having 1 piece of bread. How come they are the same, thought Bob.

The most confusing part was after the lesson on fraction. Bob learned that 1/2 = 2/4. So when he got back home he tried to share 6 apples with his sister Linda. He divided the 6 apples into two groups – 2 apples in one group and 4 apples in another group.

apples, dividing apples

From the set of two apples he gave 1 to Linda. That’s 1/2. From the set of four apples, he took 2, that’s 2/4. It is equal he said. But Linda did not agree with him because she got 1 apple less that he. Bob thought, how can this be? Why would 1/2 = 2/4 not work for apples!

The next day, the teacher asked Bob to add 1/2 and 2/4? Bob wrote 1/2 + 2/4 = 3/6 because taking 1 apple from 2 apples then 2 apples from 4 apples, he must have taken a total of 3 apples from 6 apples!

This story is adapted from A Framework of Mathematical Knowledge for Teaching by J. Li, X. Fan, and Y Zhui at the EARCOME5 2010 conference.

Point for reflection:

What has Bob missed about the meaning of addition of natural numbers? the meaning of fraction?

You may want to read the following posts about math knowledge for teaching: