What exactly is Base 10 arithmetic ?

Teacher: “Now we’re going to learn about base 10 arithmetic”.
Wise guy: “Is that where 3 + 4 = 12, or is it where 3 x 4 = 12 ?”.

I did a search on the net and found the term “base 10” all over the place. What does it mean?

An apparently annoying question:
“Does the 1 in 10 stand for the number 10’s in 10?”.

The interpretation of 10 in the system described as “Base 10” depends on the base of the system, so what is it? How do I find out?

We have here a logical problem. The term “Base 10” as a definition is self referential. It is more subtle than this definition of a straight line:

“A straight line is a line which is straight”.

The problem arises from the almost universal confusion between the two things:
1: The name of a number, in this case “ten” is supposedly implied
2: The symbols representing a number, in this case 10 in the base ten system”

So the answers to the questions “What is it? How do I find out?” above are “Unknown” and “You can’t”

Writing “Base 10” when you mean “Base ten” is probably the first step in making math meaningless.

 

How to murder algebra

Idly poking around the net I found this example:

Mathway example
x+4x=9
Solve for x
Answer
Move all terms that don’t contain x to the right-hand side and solve.
x=9/5

I cried !

Read more from mathway.com

Smarter Balanced: Lacking Smarts; Precariously Balanced

It’s a bit long, but it sure takes the lid off the CCSS. Read it now.

deutsch29: Mercedes Schneider's Blog

In this time of  “public-education-targeted boldness,” the Common Core State Standards (CCSS) has made the American public one whopper of a “bold” promise:

The standards were created to ensure that all students graduate from high school with the skills and knowledge necessary to succeed in college, career, and life, regardless of where they live. [Emphasis added.]

There is neither now nor never has been any empirical investigation to substantiate this “bold” claim.

Indeed, CCSS has not been around long enough to have been thoroughly tested. Instead, the above statement–which amounts to little more than oft-repeated advertising– serves as its own evidence.

However, if it’s on the *official* CCSS website, and if CCSS proponents repeat it constantly, that must make it true… right?

Keep clicking your heels, Dorothy.

Now, it is one issue to declare that CCSS works. It is quite another to attempt to anchor CCSS assessments to the above cotton…

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Commutative, distributive, illustrative-ly

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

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)

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 !

Fractions, at home and away.

So one day recently I was bored, and then the following rushed onto the page:

Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.

Is this 3 hours or 1/4 of a pizza?

How many hours equals 1/4 of a pizza?

Apologies to those for whom a clock face is a historical artifact.

How do I know it’s a pizza and not a cake?

Does half a day include half the night?

There are four feet in our yard, mine and my sister’s.

Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?

Are ratios numbers or fractions (or neither) ?

Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)

You cut, I choose !

This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.

2/3 and 4/6 are equivalent fractions. Equivalent to what ?

The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.

The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.

Back to the heavy stuff next time !

Fractions as parts of a whole, again !

This was found on “talking math with your kids” as an example of the “strange” stuff that kids bring home and cause mystification in their parents.

“The whole is 8. One part is 8. What is the other part ?”.

Just what exactly is this supposed to mean?
That the whole always consists of two parts?
Since when did numbers have parts?
What is the definition of “part”?
Even if we are talking about 8 things, then the natural AND logical answer is “There isn’t another part”.
If I want to see ways of creating 8, using adding, then what is wrong with
8=1+7 8=2+6 8=3+5 … 8=7+1 and 8=8+0 for completeness’ sake.

To call zero a part of 8 is going to lay the groundwork for a feeling that math hasn’t got a lot to do with real life, which is a crying shame. This feeling can arise at any stage, we should give reality a chance at this level. To conclude “What a stupid question!”.

Double negatives, or the meaning of -(-2)

In the extended number system of signed numbers, that is, the positive and negative numbers I see a lot of heart searching over the meaning of -(-2). This can be put to rest in one or both of two quite satisfactory ways:

1: Signed numbers are directed numbers, used for position, temperature, voltage etcetera. The basic question is “How far apart are the two numbers  A  and  B ?”, or more useful in a practical situation “How far is it from  A  to  B ?”.

This is a subtraction problem with direction and the answer is  B – A
For A=3 and B=7 we get
Distance from A to B = B – A = 7 – 3 = 4
For A=-3 and B=7 we get
Distance from A to B = B – A = 7 – (-3) = ???????????????
But a quick look at a number line shows that the distance is 10
So 7 – (-3) = 10
But 7 + 3 = 10 as well
Conclusion: -(-3) = +3

2: A simple and more abstract approach:
Starting with 7 – (-3) = ??????????????? we give a name to the unknown answer. Call it D.
Then using the basic fact that 12 – 4 = 8 is equivalent to 12 = 8 + 4 we have
7 – (-3) = D is equivalent to 7 = D + (-3)
7 = D + (-3) is equivalent to 7 = D – 3
7 = D – 3 is equivalent to 7 + 3 = D
which says that D = 10
So subtracting -3 is the same as adding 3

A meaningful example is as follows:
My friend from Anchorage calls me and says “It’s cold here this morning, -5 degrees”.
Down here in Puerto Rico it’s 68 degrees this morning.
How much warmer is it here than in Alaska?
 

 

Common sense versus logic and math: Congruence again

I thought I would write a computer routine to check if two figures were congruent by the CCSS definition (rigid motions). One day I will post it.

The most important thing was to be specific as to what is a geometrical figure. You can read the CCSS document from front to back, back to front, upside down and more, but NO DEFINITION of a geometrical figure. For the computer program I decided that a geometrical figure was simply a set of points. My diagram may show some of them joined, but any two points describe a line segment (or a line). So a line segment “exists” for any pair of points.

The question is “Are the two figures shown below congruent or not?

I rest my case…..