Free Fractions Pamphlet

I just want to promote in this post James Tanton’s latest pamphlet on fractions. It’s FREE for download. Just click Pamphlet on Fractions. Tanton writes:

If fractions are pieces of pie, then what does the multiplication of fractions mean? (You can’t multiply pie!)
If fractions are proportions, then what are their units? Amount of pie per student (and not just pie)?
If fractions are points on the number line, then what does half a pie mean?

Fractions are slippery and tricky and, in the end, abstract. It is actually unfair to expect students to have a good grasp of fractions during their middle-school and high-school years. This pamphlet explains why, and offers the means to have an honest conversation with students as to why this is the case. Their confusion and haziness about them is well founded!

What do you say about the statement I highlighted above? What a relief to know that? 🙂

I highly recommend that you also checkout his collection of past MATH ESSAYS.

You may also want to read a couple of my posts on fractions:

  1. Why do we ‘invert’ the divisor in division of fractions
  2. What are fractions and what does it mean to understand them?
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 Algebra

Which is easier to teach and understand – fractions or negative numbers?

Which concept is easier for students to understand and perform operations on, numbers in fraction form or negative numbers? I think fractions may be harder to work with, but people understand what it is; at least, as an expression to describe a quantity that is a part of a whole. Like the counting numbers, fractions came into being because we needed to describe a quantity that is part of a whole or a part of a set. The fraction notation later became powerful also in denoting comparison between quantities (ratio) and even as an operator. See What are fractions and what does it mean to understand them?  And negative numbers? Do we also use them as frequently like we would fractions? I think not. People would rather say ‘I’m 100 bucks short’ than ‘I have -100 bucks’.

How did negative numbers come into being? As early as 200 BCE the Chinese number rod system represented positive numbers in Red and Negative numbers in black. There was no notion of negative numbers as numbers, yet. The Chinese just use them to denote opposites. There was no record of calculation involving negative numbers.  Although the first set of rules for dealing with negative numbers was stated in the 7th century by the Indian mathematician Brahmagupta, it was only in the middle of the 19th century, when mathematicians began to work on the ‘logic’of arithmetic and algebra that a clearer definition of negative numbers and the nature of the operations on them began to emerge (you may want to read the brief history of negative numbers). It was not easy for many mathematicians before that time to accept negative numbers as ‘legitimate’ numbers. Why did it take that long? In her article Negative numbers: obstacles in their evolution from intuitive to intellectual constructs, Lisa Hefendehl-Hebeker (1991) identified the hurdles in the acceptance of negative numbers:

  1. There was no notion of a uniform number line.The English mathematician, John Wallis (1616 – 1703) is yet to invent the number line which helps give meaning to the negative numbers. Note that it did not make learning operations easy.The preferred model was that of two distinct oppositely oriented half lines. This reinforced the stubborn insistence on the qualitative difference between positive and negative numbers. In other words, these numbers were not viewed as “relative numbers.”  You may want to read Historical objections against the number line.
  2. A related and long-lasting view was that of zero as absolute zero with nothing “below” it. The transition to zero as origin selected arbitrarily on an oriented axis was yet to come. There was attachment to a concrete viewpoint, that is, attempts were made to assign to numbers and to operations on them a “concrete sense.”
  3. In particular, one felt the need to introduce a single model that would give a satisfactory explanation of all rules of computation with negative numbers. The well-known credit-debit model can “play an explanatory but not a self-explanatory role”.  [Until now, no such model exists. More and more math education researchers are saying that you need several models to teach operations on integers]
  4. But the key problem was the elimination of the Aristotelian notion of number that subordinated the notion of number to that of magnitude.

Lisa Hefendehl-Hebeker #4 statement is very important for teachers to understand. If you keep on teaching the concept of negative numbers like you did with the whole numbers and fractions which naturally describes magnitude, the longer and harder it would take the students to understand and perform operations on negative numbers. The notion of negative numbers as representing a real-life situation say, a debt, becomes a cognitive obstacle when they now do operations on these numbers. I am not of course saying you should not introduce negative numbers this way. You just don’t over emphasize it to the point that students won’t be able to think of negative numbers as an abstract object. I would even suggest that when you teach the operation on negative numbers, make sure the introduction of it as representation of a real-life situation has been done a year earlier. Here’s one way of doing it – Introducing negative numbers.

Here’s Brahmagupta (598 – 670) rules for calculating negative and positive numbers. See how confusing the rules of operations are if  students think of negative numbers as representing magnitude.

rules of operation on integers

 Image from Nrich.

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: