Tag Archives: arithmetic

The Future

“How’s your Mary doing?”.

“She’s doing well. She’s 8 now. She’s in Grade 3. She really enjoys the Pre-Algebra and the Pre-Textual Analysis.”.

3 Comments

Filed under algebra, education, language in math, teaching

Memorization versus Number Sense

This is a link to a post by WatsonMath about Stanford Professor Jo Boaler and her thoughts, opinions and research backed statements of the counterproductive combination of “learn your tables” and “take this (yet another !) timed test on them”.

Definitely worth reading, and worth passing on as well.

http://www.watsonmath.com/2015/02/07/jo-boaler-fluency-without-fear/

1 Comment

Filed under arithmetic, education, math, teaching

Commutative, distributive, illustrative-ly

Here they all are, apart from associative, as it belongs to algebra.

gif commutative add

gif commutative mult

gif distributive law

5 Comments

Filed under abstract, arithmetic, language in math, operations, teaching

The Distributive Law, again !

The formal statement of the distributive law should read as follows:

If a, b, c and d are numbers, or algebraic expressions (same thing really) and b = c + d then ab = ac + ad

It is a by-product of the law that it tells you how to expand an expression with a bracketed factor.

In any case, what’s the big deal ?

gif distributive law

2 Comments

Filed under abstract, algebra, arithmetic, language in math, teaching

More bad language in math

Here is another horror which I found recently:

The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

distributive property really

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Example:
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and here it is:

set b – c equal to w (why not!)
then b = c + w
multiply both sides by a
ab = a(c + w)
expand the right hand side by the distributive law
ab = ac + aw
subtract ac from both sides
ab – ac = aw
replace w by b – c, and then
ab – ac = a(b – c)
done !

9 Comments

Filed under abstract, arithmetic, language in math, teaching

Common Core math testing – oh dear!

You should all read this, from the Washington Post October 2013.

“Why are some kids crying when they do homework these days? Here’s why, from award-winning Principal Carol Burris of South Side High School in New York”.

http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/10/31/a-ridiculous-common-core-test-for-first-graders/

Here is the actual test paper (for 5-year-olds), to save you time:

the-math-test NY grade 1 Pearson

1 Comment

Filed under arithmetic, education, language in math, teaching

Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

Leave a comment

Filed under abstract, algebra, arithmetic, education, language in math, teaching

Number lines, or lines with numbers on them

Image

It sure is a number line, and it works perfectly well with the whole or natural numbers.

The question is “How did the number line become straight, with equally spaced numbers, when the ideas of length and measurement have not yet been developed?”.  This is the math version of the “what came first, the chicken or the egg?” question.

And, with zero not there no-one can take my last cupcake.

Leave a comment

Filed under arithmetic, humor