Pre-Algebra and My Dear Aunt Sally

Will you Please Excuse My Dear Aunt Sally? It seems she had indigestion and burped. Can you find where she is? This is how I had fun publishing my Unit on Pre-Algebra! This unit includes in-depth lessons on:

1. Order of Operations
2. Order of Operations with Exponents
Order of Operations with Integers
3. Writing AlgebraicExpressions
4. Writing AlgebraicEquations
5. Practice Exercises
6. Challenge Exercises

So can you find my Dear Aunt Sally? And will you excuse her?

factor tree calculator

I Love Me Some ExamView

I spent the last two days on school business building a set of common formative assessments in ExamView that are aligned to my department's skills list for our algebra and pre-algebra classes. Many of the questions are dynamic which means that for any given question type, multiple versions of the question can be created by the click of a button. As I was looking through the question banks, I realized that there were some skills that didn't have questions that fit what we will be trying to assess. Solution? Build it yourself.

The process is a bit tedious at first, but I was able to use previously made questions to figure out how to build a simple question regarding slope.

Step 1: Define your algorithms.

Step 2 : Create a graph using pre-defined variables.

Step 3: Define the distractors.

Yeah, gotta say I love this feature. Finding questions with good distractors is tough at best. But if you decide on an algorithm based on common misconceptions, you're good to go. In this problem, I d! ecided the common misconceptions for students calculating slop! e based on a graph were:

• Using Run/Rise instead of Rise/Run
• Confusing negative with positive slope
• Confusing the y-intercept with the slope

Of course, there may be some discussion on what the best misconceptions would be, but for the sake of practicing, those are what I went with.

Step 4: Decide how many of these type of question you want in your assessment.

I'm glad that I don't have to build all of these from scratch, but I like the fact that filling in the holes isn't too tough.

finding the vertex of a parabola

"Weather icons"

Set of some nice vector weather icons. Enjoy.
Author unknown. Only for personal use.
1 AI : 4,8 MB

Download

free vector

Probability that mangles the mind

Generally speaking I'm a bit so-so about recreational mathematics. I can't get very excited about polyominoes or tiling, for instance. But when the field strays into probability I get fascinated - and the mind gets boggled. Take the little probability problem mentioned in the New Scientist article I've linked to there. It gets rather lost in the article, and they don't describe it particularly well. Let's take a look.

The problem statement is simple. I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? But to get a grip on this problem we need first to take a step back and look at a more basic problem. I have two children. One is a boy. What is the probability I have two boys?

A knee-jerk reaction to this is to think 'One's a boy - the other can either be a boy or a girl. So there's a 50:50 chance that the other is a boy. The probability that there are two boys is 50%.' Unfortunately that's wrong.

You can see why with this handy diagram. The first blobs are the older child. It's a boy or a girl, 50:50. Then in each case we've a 50:50 chance of a boy or girl for the second child. So each of the combinations has a 1 in 4 (or 25%) chance of occuring.

All the combinations except Girl-Girl fit our statement 'I have two children. One is a boy.' So we've got three equally like possibilities, of which only one has two boys. So there's a 1 in 3 chance that there are two boys.

If this sounds surprising, it's because the statement 'One is a boy' doesn't tell us which of the two children it's referring to. If we say 'The eldest one is a boy', then our 'common sense' assessment of probability applies. If the eldest is a boy, there are only two options with equal probability - second child is a boy or second child is a girl. So it's 50:50.

Now we're equipped to move on to the full version of the problem. I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

Again, gut feel says 'The extra information provided can't make any difference. It must still be 1 in 3.' But startlingly, the probability is now 13 in 27 - pretty close to 50:50.

To explain this I should draw another diagram, but I can't be bothered, you'll have to imagine it. In this diagram there are 14 children in the first column. First boy born on a Sunday, First boy born on a Monday, First boy born on Tuesday... First girl born on a Sunday... through to First girl born on a Saturday.

Each of these fourteen first children has fourteen second children options. Second boy born on a Sunday... etc.

That's 196 combinations, but luckily we can eliminate most of them. Either the first or second boy must be born on a Tuesday. So the combinations were interested in are the fourteen that spread out from 'First boy born on a Tuesday' plus the thirteen that start from one of the other first children and are linked to 'Second boy born on a Tuesday.' So there are 27 combinations in all. How many of these involve two boys? Half of the first fourteen do - one for a second boy born on each day of the week. And for those thirteen with links on the right to 'second boy born on a Tuesday' six of them will have a boy as the first child (because we don't include 'First boy born on a Tuesday.') So thats 7+6 i.e. 13 combinations that provide us with two boys. So the chances of having two boys is 13 in 27.

Common sense really revolts at this. By simply saying what day of the week a boy was born on, we increase the probability of the other child being a boy. But we could have said any day of the week, so how can this possibly work? The only way I can think to describe what's happening is to say that by limiting the boy we know about to a certain birth day, we cut out a lot of the options. We are, in effect, bringing it closer to the sort of effect we get by saying 'the oldest child is a boy'. We are adding information to the picture.

The probabilites work. You can model this in a computer if you like and it's correct. But what's going on mangles the mind. Don't you just love probability?

(I ought to say, by the way, that this isn't quite a match to reality. It assumes there is an equal chance that either child is a boy or a girl, and that there is an equal chances of being born on each day of the week. In reality neither of these is quite true, but that doesn't matter for the purposes of the exercise.)

probability coins

Least Common Multiples, modern and ancient

Three 2001 Russian encyclopedia entries, a 2008 blog, and a 2008 Wikipedia entry point out Ahmes' 1650 BCE red auxiliary numbers as a central least common multiple (LCM) theme. The LCM theme reports modern and ancient terms of a unifying theme of ancient Egyptian arithmetic, rather than a secondary fragmented bit of information. Ahmes' LCMs had been under read by 1920s math historians. After 2002 close reviews of 2/n tables and other information, Ahmes has been shown to have selected optimized, but not always optimal, LCMs to write out 2/n table and calculate other optimized Egyptian fraction series. The Russian encyclopedia entries are published on-line by Springer with the first entry being:

http://eom.springer.de/a/a011920.htm

The first entry suggests that a LCM form may have been known to Ahmes by writing out

3/11 = ! 1/6 + 1/11 + 1/66

It is highly likely that Ahmes shorthand included:

3/11 times (6/6) = (18/66) = (11 + 6+ 1)/66 = 1/6 + 1/11+ 1/66

As many readers of Egyptian math history are aware Ahmes' red auxiliary numbers was an idea that has long been associated with LCMs. The precise nature of LCMs has come to light.

The second Russian entry only mentions modern LCMs per:

http://eom.springer.de/F/f041200.htm

and therefore the entry did not hypothetically offer an ancient leap back in time to decode the 2/n table. Had a formal LCM decoding hypothesis of the RMP 2/n table been offered, academic discussions may have been established within an interdisciplinary context. Following the modern logic offered by the Russian encyclopedia entries, formal proofs, or refutations could be discussed.

Four ways to di! scuss formal Egyptian proofs, taken from

http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html ,


that Russians had not considered are summarized by:

1. Read/translte RMP 38, that solved the problem:

a. 10 hekat equals 3200 ro times 7/22 equals 101 9/11

b. proving 101 9/11 times 22/7 equals 3200 ro

c. with a footnote that 35/11 times 1/10 = 35/110 = 7/22

2. Read/translate RMP 47, that solved the problem

a. 100 hekat equals 6400/64 hekat divided by 70

b. proving (91/64)hekat + 150/70 ro was the answer

3. Read/translate RMP 66 problem

a. 10 hekat times 32o ro equals 3200 ro, divided by 365, the number of days per year

b. 3200/365 equals 8 + 280/365 with

c. a duplation proof for the quotient 8, by writing

365 1
730 2
1460 4
2920 8

and the remainder (3200 - 2920) ! equals 280 by

(243 1/3 + 36 1/2 + 1/6)/365 = 280/365

4. RMP 82 an began with a hekat unity, (64/64), divided by 29 divisors n in the range 1/64 < n < 64, as RMP 83 divided (64/64), divided by 6, 20 , and 40.


A final Russian entry generally used an ancient aliquot fraction or ratio idea (Russian terms for red auxiliary numbers) without identifying a hypothetical use of ancient LCMs per:

http://eom.springer.de/A/a013260.htm

Note the straight forward modern and ancient arithmetic that easily fills in Ahmes use of red auxiliary LCMs per:

http://rmprectotable.blogspot.com/

and,

http://en.wikipedia.org/wiki/Red_auxiliary_numbers

least common multiple lcm

Calculating percent with mental math

Would you say that students' understanding of percent is sometimes - or often - hazy?

Find the number of which 79.5% is 101.

Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.

Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:

Y × 79.5% = 101.

0.795 Y = 101

Y = 101/0.795 = 127.044025157

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I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?

So while this idea is great and works, it is also necessary for students to understand the concept of percent well.

In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:

"79.5% of that number is 101"... so 0.795 Y = 101.


Ideas for using MENTAL math for calculating percent problems

I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.

1. Find 10% of some example numbers (by dividing by 10).
   
   
2. Find 1% of some example numbers (by dividing by 100).
   
   
3. Find 20%, 30%, 40% etc. of these numbers.
   FIRST find 10% of the number, then multiply by 2, 3, 4, etc.
   For example, find 20% of 18. Find 40% of $44. Find 80% of 120.
   
   I know you can teach the student to go 0.2 × 18, 0.4 × 0.44, and 0.8 × 120 - however when using mental math, the above method seems to me to be more natural.
   
   
4. Find 3%, 4%, 6% etc. of these numbers.
   FIRST find 1% of the number, then multiply.
   
   
5. Find 15% of some numbers.
   First find 10%, halve that to find 5%, and add the two results.
   
   
6. Calculate some s! imple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
   
   
7. "40% of a number is 56. What is the number?" - types of problems.
   
   You can do this mentally, too: First FIND 10% and then multiply that result by 10, to find 100% of the number (which is the number itself).
   
   If 40% is 56, then 10% is 14. So 100% of the number is 140. This result is reasonable, because 40% of this number was 56, so the actual number (140) needs to be more than double that.
   
   
8. "34% of a number is 129. What is the number?" (Now you need a calculator.)
   
   You don't need to write an equation. You could also first find 1% of this number, and then find 100% of the number.
   
   If 34% of a number is 129, then 1% of that number is 129/34. Find that, and multiply the result by 100.
   

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I recently got this sort of homework question sent to me:

I have a problem and I ! don't know how to solve it so here is the problem: At a popula! r clothi ng store clothes are on sale when they have hung on the rack too long. When an item is first put on sale, the store marks the prices down 30% off. If some shoes are regular-priced at $50.00, how much will they cost after the discount?

You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!

(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)

Calculate percentages

playing in bourgeois ideology's sandbox -- pedagogical reflections

Fellow UMass Econ blogger Mark Silverman discusses the important features of capitalism in this post. His premise is a capitalist propaganda video from 1948.

If you watch the video, you'll see that it consists of a (mock) portrait of a terribly earnest and engaged discussion among high school students about the definition of "capitalism." They use, as an example, a visit to Mr. Brown's grocery store to buy "weenies" for the class weenie roast. After arguing vigorously (and letting us see their visit to his store) they come up with the following list of defining features of "capitalism":
(1) Private Property
(2) Profit motive
(3) Competition
(4) Freedom of contract
(5) Gover! nment-enacted laws granting rights (including certain Constitutional rights) to items #1 and #4

And, they conclude, that (1)-(5) adds up to (6) "Free Enterprise" (which, of course, is a much more attractive sounding term than "capitalism.") (Incidentally, #5 is a rather sophisticated observation-- at least relative to what most economists generally discuss. Certainly they failed to recall it when administering so-called "shock therapy" to the former Soviet Union.)

After playing this video in my class, I listed these features on the blackboard. I asked my students: Is there anything else you'd add? Or does this seem like an extensive, and exhaustive, list?

I confine my comments to Mark's use of this video in teaching his history of thought class.

I have purposely never addressed the issue of defining capitalism in my teaching. You may think that's odd given that I've taught a course in American ! economic history, but in that course I found it much more help! ful to i nstead pose different accounts of the rise or success of capitalism and then debate the rigor of the various arguments. In this way, for example, we were able to poke holes in both leftist accounts of proletarianization, as well as rightist accounts of the rise of a liberal democratic society in the U.S. by the early 1800s.

Basically, in the quote above I think Mark goes too far in assuming some of the traits presented in the video as given. Competition, for example, is not a cornerstone of all "free enterprise systems," as Schumpeter pointed out and of course many before him (Schumpeter's is the account I am most familiar with in detail). But, whatever -- debating whether capitalist institutions really are efficient is not the point of my argument. I assert that there is a much larger point to be made about assumptions, other than the apparent inconsistency between what ideology says is capitalism, and what it really is. The larger point t! o be made is in how Mark relates propaganda's discussion of capitalism to how capitalism is defined by others.

We need to consider two related points. First, bourgeois ideology has its own definition of capitalism. Second, many academics (left and right) also have their own definition of capitalism. What does it all mean for teaching what capitalism is? By playing in bourgeois ideology's sandbox, Mark's "lesson on capitalism" implicitly accepts the first point and disregards the second, which compromises the strength of the lesson learned. Surely the students still get the main point that the propaganda is "biased" by not discussing wage labor as a central institution of capitalism -- but biased against what? The simple inclusion of wage labor, alongside the other 5 properties of capitalism or a free enterprise system? Is that the model Mark adopts? Mark does not tell us, but it seems implied that this reference point is precisely th! e one chosen by him.

I argue that ! the fail ure to coherently present an alternative model when attacking the mainstream one is one of the biggest weaknesses of heterodox teaching. I might be going too far in grilling Mark here -- and I hope he calls me out for doing so -- but as teachers, it seems like the best we can do is get out of bourgeois ideology's sandbox and start levelling critiques from the standpoint of the rigor of all definitions and arguments concerning capitalism. Doing so will poke holes in all the arguments, but it will also allow the students to gain familiarity with various perspectives, allowing students to judge for themselves.

This is not a "bias-free" method -- each instructor will nevertheless be more difficult on certain positions that are against his or her political leanings. But I think that recognizing that there are many different definitions of capitalism is an important first step at deconstructing "dominant" paradigms.

One final ! point. It appears that toward the end of his post, Mark does begin to discuss some of the historical origins of wage labor, and how unnatural it is. He remarks that Polanyi discusses wage labor as an absurd condition of modern society. Is this the alternative model of capitalism Mark has in mind?

homogeneous function of degree zero,