Category Archives: fractions

Fractions, at home and away

In the morning Johnny’s mom
Said “Here’s six candies for your break.
“Give your sister half of them”.
Now Johnny’s brain is on the make.
He gives her one, and then another.
Little sister stamps her feet!
“And the last one!” says his mother.
“Damn” thinks Johnny, “I can’t cheat!”.

Later that day

“Johnny, what is half of six?”.
“I dunno”.
“Well, go get out six lego bricks
“And make a row.
“Now break the row right in the middle.
“That’s half the row.
“Just split the half and count the bricks”.
“I got three”.
“So now you see, three’s half of six”.
But does he know?

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Filed under arithmetic, fractions, humor, language in math, teaching, verse

Adding fractions

pizza fraction 2

When adding two fractions
Take care, delay your actions.
You must allow the whole
To exercise its role.

(Possibly and unwittingly owing something to Ogden Nash)


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Read this!

This is a plug for Jose Vilson, in particular this post:

The comments at the end of this post about the futile nature of current math education by  Ted Dintersmith  should be posted on the side of the Empire State Building.

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Filed under abstract, algebra, arithmetic, education, fractions, geometry, teaching, Uncategorized

Let’s actually use the number line

Let’s have a number line. We can count up or down by moving to the right or the left, or actually up or down if the number line is drawn vertically.
But what else ?????

The following pictures show how points on the number line which represent fractions can be found exactly by simple geometrical construction, and then how results of multiplication and division of fractions can be found exactly as points on the number line (sorry, the numerical values are however not found).

This arose from a statement in the CCSS math document that the fraction 1/6 could be represented by a point one sixth of the way from zero to one, BUT NOWHERE DOES IT SAY HOW TO FIND THAT POINT.

numberline locating fractions

numberline multiply 1

numberline multiply frac

numberline frac division


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Filed under arithmetic, fractions, geometry

HCF and LCM – why?

Here’s another thing that is long past its expiry date.

What are Highest Common Factor and Least Common Multiple actually for? And “It makes adding fractions simpler” is just nonsense.  It may be that much later on, for those with a real interest in pursuing math, there is something useful there, but such people at that stage would be able to pick up the ideas and apply them in about 10 minutes.

If I want to add two fractions with different denominators then the simple and fairly obvious rule: “multiply top and bottom of each fraction by the denominator of the other one” is perfectly adequate, and if the student spots a common factor in the resulting fraction it can be removed then.

A separate but equally pointless activity is adding fractions with seriously different denominators. If you really want the answer then use a calculator. And get a clue about checking the result for realism.

I am sure there is more stuff in math courses which could be sent on its way, leaving time to do some interesting stuff.



Filed under arithmetic, fractions, Uncategorized

Angles and the CCSS – oops!

Check this out:


Grade 4 content
Geometric measurement: understand concepts of angle and measure angles.
5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.


My comments:
1. Well, this is the first occasion that the word “ray” is used, in the whole document, and as expected by now there is no definition!
2. There is an attempt to define measurement of angle, which really confuses me.  Which circle ? Any circle ? What is “the circular arc” ? Since when did angles turn ? And I have this vision of making a one degree angle out of card and measuring angles with it.
3. An this is from the Fractions section : “note 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.”
So what about 1/360 ?

Am I nitpicking again?

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Filed under fractions, geometry, language in math

Pythagoras converse, proof from scratch

There is a website with 100 proofs of the famous theorem of Pythagoras, but when I trawled the net looking for a proof of the converse, they all assume the basic theorem.

Here’s how to do it from scratch, which is considerably more satisfying, and also a simple application of similar triangles and basic algebra:

pythag converse diagram pythag converse text

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July 2, 2014 · 7:21 pm

Fractions are parts of the same whole, part 3

When is a whole not a whole? (again)

When it’s two wholes (or more) :-

John eats 1/2 of his pizza, Mary eats 3/4 of her pizza. So between them they ate 1/2 + 3/4 of a pizza, or 5/4 of a pizza.

So which whole are we referring to ? John’s pizza ……. No.   Mary’s pizza ……. No.    Both pizzas …….. No.    John’s pizza and  Mary’s pizza and  both pizzas …….. No.

Conclusion: What we are referring to as “the same whole” is an abstract unit of one pizza, and the fractions are measurements using this unit. Wouldn’t it be a good idea to start off like this, with fractions as measurements, and avoid years of misunderstanding, stress and confusion.

Is this so different from adding whole(adjective!) numbers , as when  adding two numbers they have to be counts of the same thing (or whole(!) before it is chopped up).?

Fun arithmetic:        3 apples + 4 bananas = 7 applanas

Desperately fun arithmetic :  1/2 of my money + 1/2 of your money = 1/2 of our money


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Fractions are parts of the same whole, part 2

When is a whole not a whole ?

When it’s a hole.

(which half of the hole shall we fill first, the top half or the bottom half?)

Besides, I thought whole was an adjective.

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Fractions are parts of the same whole

Just a bit of math fun :


For each triangle, what fraction of the triangle is the (partially) shaded bit ?


Filed under arithmetic, fractions, geometry