Tag Archives: division

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

Q: Who needs polynomial division? A: In high school, NOBODY !

Polynomial division is a completely unnecessary procedure. It is not needed for partial fractions. It is not needed for finding factors, etcetera…

The same result can be obtained in a more logical and meaningful way, by considering the structure of polynomial and rational expressions.

Check this out :


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Filed under algebra

Long division, the explanation.

Since the kids have to explain everything in the new CCSS math standards,

they better have this under their belt, even if they have to learn it and parrot

it out in some test or other (careful, cynicism is not always just round the


So, here is a long division calculation for you, 32 divided into 2768, or if you

prefer the old fashioned, only used in schools notation,   2768 <the old

fashioned division sign, not on my keyboard>  32. (and < and > are not

representing inequalities at this point).

Division is at bottom repeated subtraction, so we do it:-

32)2768    100×32 = 3200 is too big
2560   so take a smaller multiple (in 10’s)
——   Choose from 90×32=2870, 80×32=2560 (OK!)
208   and subtract,leaving 208, and 10×32=320
192   So try 6×32=192 (OK!). 7×32 is too much.
—-   Subtract again, leaving 16,which is less
16    than 32 and so is the remainder.

So 2768 = 90×32 + 6×32 + remainder,
which is  96×32 + 16,
and so 2768 divided by 32 is 96  with remainder 16

The End

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Filed under arithmetic

CCSS rules!

Here are some gems from the CCSS math document.

CCSS grade 6
Apply and extend previous understandings of multiplication and
division to divide fractions by fractions.
How many 3/4-cup servings are in 2/3 of a cup of yogurt?

Well, I would say “None”

CCSS grade 7
Analyze proportional relationships and use them to solve real-world
and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios
of lengths, areas and other quantities measured in like or different
For example, if a person walks 1/2 mile in each 1/4 hour, compute
the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

And here we are asked to crack a nut with a sledgehammer.
Common sense to the rescue!
Half a mile in a quarter of an hour is one mile in half an hour and so 2 miles in one hour, or 2 miles per hour.

is a complex fraction …. really?  Only if fractions are not numbers!!!!!!!

More on fractions in a later post…………………………………….

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