algebra 1 un arithmetic

8/11/2019 Algebra 1 Un Arithmetic

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The Game of Algebra

or

The Other Side ofArithmetic

2007 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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Lesson 1


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Unarithmetic

+-

2007 Herbert I. Grossnext


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What is Unarithmetic?

When young children are first taught to puton their shoes, they might refer to takingoff their shoes as unputting on theirshoes. In other words to unput on your

shoes might be a childs way of saying totake off your shoes. As awkward as thisphrase might seem, it does express the

relationship between putting on andtaking off shoes.

2007 Herbert I. Grossnextnext


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In a similar way to undo multiplication, a

child might have invented the wordunmultiply, which at the very least ismuch more suggestive than the word

division.

It is in the above context that we may beginour study of algebraby thinking of it asbeingunarithmetic.

Key Point

2007 Herbert I. Grossnextnext


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Lets keep in mind, whether weapprove or not, that calculators

and computers are nowhousehold items, and students

see nothing wrong in using them.

And, in fact, since theprerequisite for an algebracourse

is a knowledge of arithmetic;once this knowledge is assumed

there is nothing wrong withallowing students to use

calculators in an algebracourse. 2007 Herbert I. Gross

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In the language of calculators, wecall it an arithmetic problem, if an

answer to a computation problem canbe obtained by simply pressing keysin the order in which the operations

are introduced.

For Example

The sequence of steps Start with 6; multiply

by 5; and then add 4 would be called anarithmeticprocess or direct computation.Namely all we would have to do with a

calculator is enter the sequence of

key strokes 2007 Herbert I. Gross nextnext


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9 8 7 +

6 5 4 -

3 2 1

0 . =

6

5

=

30

+

4

=

34

6 5 + 4 =

34

The display window of the calculatordisplays 34 as the answer.

2007 Herbert I. Grossnextnextnext


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In terms of a computer analogy,think of 6 as being the input,

multiply by 5 and then add 4 as

being the program, and 34 as beingthe output. Putting this in

computer language, its arithmetic

(or a directcomputation) when theprogram and input are given, andthe output must be found.



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On the other hand, suppose wewanted to know the number we

had started with if the answerwas 59 after we first multiplied itby 5 and then added 4.

In this case, the output (59) is known,but the input must be determined.

Going back to our calculator, the sequenceof steps for this would have to be

? 5 + 4 = 59

But since the calculator doesnthave a ?

key, wecant

proceed. 2007 Herbert I. Gross nextnext


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In the above context, one of the ways we

define algebrais to say

Key Point


Algebrais the subject that allows usto paraphrase questions the calculatorcannot understand into equivalent

questions that the calculator canunderstand.

That is: algebraconverts an indirectcomputation (which we can think of asunarithmetic) into a directcomputation

(which we can think of as arithmetic). nextnext


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Pedagogy Note

2007 Herbert I. Gross next

Often students depend on a calculatorto do computations, but a calculator,alone, will nothelp them solve anyproblem that involves an indirectcomputation.


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Consider the fill in the blank question

that is designed to test whether thestudents know the number fact 2 + 3 = 5.

For Example

Form A 2 + 3 = __

If a student had no idea of what the meaningof + or = was, but had a calculator, hestill could get the correct answer by

pressing the following keys in order.

2 + 3 = 5

2007 Herbert I. Gross nextnext


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But suppose that, instead of Form A, thefill in the blank question was worded...

Form B 2 + __ = 5

This presents an obstacle. Namely, the

student can enter 2 and +, but now he isstymied by the blank. To be able to solve

this problem by using a calculator, the

student would have to be able toparaphrase Form Binto the equivalent form

52 = __.



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The above discussion is notlimitedto mathematics but rather exists

in any course that involves

fill in the blank questions.

Pedagogy Note


How well students will do on a fill-in-the-

blank type of question will often dependon how the question is worded.

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Suppose that students are tested

on whether they know Sacramentois the capital of California.

The question can be worded as...

For Example

____________ is the capital of California.or

Sacramento is the capital of __________.

Whether you use form (1) or form (2),the correct answer will be Sacramento

is the capital of California.

2007 Herbert I. Gross nextnextnext


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However, the number of students whoget the correct answer could very well

depend on whether form (1) or form (2)was used.


In particular, in (1) the proper noun is

California, and when thinking of California,the city name Sacramento may or may no t

come to mind.

On the other hand, in form (2) the onlyproper noun is Sacramento, and it is quite

likely Sacramento brings California tomind.

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The student might reason,Gee, I didnt know that

Sacramento was the capital ofanything, but knowing that its in

California, I think the correct

answer is probably California.

For Example



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How does this apply to the discussionabout arithmeticand algebra?

To give the multiply by 5 and then add4 a real-life interpretation, consider

The price of a box of candy in a catalogreads $5 per box plus $4 shipping andhandling. What is the cost of buying 6

boxes of candy?

Problem



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The thought process for solving thisproblem is rather straight-forward;

namely

Since each box costs $5, and you

want to buy 6 boxes

$Then, add an additional $4 for shipping(to the $30) to obtain the total cost, $34.


First, multiply $5 by 6, thus obtaining $30as the cost of the 6 boxes.

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If you didnt know how to perform theappropriate arithmetic, but you knew how

to use a calculator, you could enter thefollowing sequence of key strokes

And obtain 34 dollars as the answer.

6 5 + 4 = 34



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The previous sequence of key strokes isequivalent to what many textbooks refer to

as a function machine; and which isrepresented in a form similar to the one

shown below.

cost in dollars

OutputInput

number of boxes 5 + 4



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If we translate the diagram into plainEnglish, the following sequence of steps

is obtained.

Step 1: Start with the number of boxes

(in the present illustration; its 6).Step 2: Multiply by 5.

Step 3: Add 4 for shipping.Step 4: The answer is the cost in dollars

(34). 2007 Herbert I. Gross nextnext


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In essence, the calculator model,the function machine, and the plain

English model are equivalent.

However, our own belief is that theplain English model is the mostuser friendly, at least to those

students who may have vestiges ofmath anxiety.

Note



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To see the application of anindirect

computation (that is, unarithmetic),suppose were still buying from thesame candy catalog, but this time

weve decided to spend $59. Howmany boxes of candy could we buy

for that amount? Notice that to solve

this problem we have to know morethan just how to read a calculator.



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That is, the sequence of key strokeswould be

? 5 + 4 = 59

But, to compute the value of ?

we would have to do an indirectcomputation. In other words in this case,

we have defined the input implicitly

(rather than explicitly). That is:the input is that number which, when wemultiply it by 5 and then add 4, results in 59

being the output. 2007 Herbert I. Gross nextnext


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In terms of the functionmachine, the problem looks

like

cost in dollars

OutputInput =

number of boxes 5 + 4

? 59



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In terms of our plain English model theproblem would be...

Step 1: Start with the number of boxes (inthe present illustration; its ).

Step 2: Multiply by 5.

Step 3: Add 4 for shipping.

Step 4: The answer is the cost in dollars(59).


?


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Notice that the answer in Step 4(59) wasobtained after 4 was added. In other words

to get from Step 3to Step 4, the fill-in-the-blank question would have been

Form A ___ + 4 = 59

Form Atells us that 59 was obtainedafter 4 was added to the blank.

Therefore, to determine the number that isrepresented by the blank, we have tounadd 4 to 59 (that is, subtract 4 from 59).



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In other words, Form A ( i.e.___ + 4 = 59)is equivalent to

Form B 594 = ___

The difference between the two forms isthat the calculator can solve FormB, thus

making FormBa directcomputation

(arithmetic), but it cannot solve FormA(which is an indirectcomputation or

unarithmetic).



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It is in this context that we definealgebraas the subject that allows

us to paraphrase questions thatcannot be answered directlyby a calculator into equivalent

questions that canbe calculateddirectly.

Key Point



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Knowing that 55 (number of dollars)was the answer after we multiplied by 5,

we then unmultiplied (that is divided) by 5to determine that we had started with 11.

Program

Start with the number of boxes

Multiply by 5.

Add 4

Answers is the cost in dollars.

Answer is the number of boxes

Unmultiply (Divide) by 5.

Unadd (Subtract) 4

Start with the cost in dollars.

Undoing Program



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In terms of the function machinemodel: starting with an input of

11 boxes and obtaining anoutput of $59, as shown below, isconsidered an arithmeticproblem.

$59

Output

11

Input

number of boxes 5 + 4 cost in dollars

1155 59



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On the other hand, starting with thecost of $59 as being the input and

reversing the steps using theundoing process, as shown

below, is considered to be algebra.

11

Input

number of boxes

59

+ 45 cost in dollars

595511

Output


5 - 4 cost in dollarsnumber of boxes

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InputOutput


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A succinct way to emphasize what we just did,is to talk about formulas.

In essence, a formulais a well-defined rule

that tells how to deduce the value of anunknown quantity, by taking advantage ofknowing one (or more) related quantities.

Formulas as a Bridge betweenArithmeticandAlgebra


An elementary exampleis the rulethat tells us

the relationship between feet and inches.Since there are 12 inches in a foot: to convertfeet to inches, simply multiply the number of

feet by 12. next


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To write the relationship in the form ofa formula: let Fdenote the number of feet

andIthe number of inches.The formula would become

I = 12 F

If, for example, Fequals 5, the formulawould become

I = 12 5

and would thus be a directcomputation(arithmetic).



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On the other hand, if I equals 60, theformulawould become

60 = 12 F

In which case there would be an indirect

computation (algebra)

which byunmultiplying becomes the directcomputation.

60 12 = FHowever, keep in mind that the formula, in

itself, is neither arithmeticnor algebra.



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This is especially true in problems that

involve constant rates. For example,consider the following question

AppendixThe Corn Bread Model


Sometimes a picture is worth a thousand words

In a certain class, the ratio of boysto girls is 2:3. If there are 30 students inthe class, how many of them are boys?

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By a ratio of 2:3 (read as 2 to 3) we meanthat for every 2 boys in the class, there are

3 girls. Namely, a group consists of2 boys and 3 girls, so there are

5 students in each group.(In the language of common fractions,

this tells us 2/5 of the students are boys.)

ArithmeticSolution


And since there are 30 students in theclass, and since 2/5 of 30 is 12,there are 12 boys in the class.

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However, the above solution can bethreatening to students who come to

algebrastill uncomfortable with fractions.


Namely, draw a rectangle (which we like topersonify by referring to it as a corn bread).

This corn breadwill represent the

total number of students.

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Corn Bread

A simple way to make fractions easieronce and for all, is to make them visual.

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The fact that the ratio of boysto girlsis2:3 means that we can divide the rectangle

(corn bread) into 5 pieces of equal size.

Corn Bread

We then let 2 of the pieces (designated bythe letter B) represent the number of boys,

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B B G G G

and 3 of the pieces (designated by the letterG) represent the number of girls.

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Since the corn breadrepresents the totalnumber of students, and since there are 5

equally sized pieces and 30 students;each of the 5 pieces represents 30 5 or

6 students. That is

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B B G G G6 6 6 6 6

In summary

12 18

Boys Girls

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Th b d d l


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The corn breadmodeldoesnt depend on how

many students there are. If, for example,there are 1,000 students, still with a boy-to-girl ratio of 2:3, the corn breadwould still bedivided into 5 equally sized pieces.

But now, each of the 5 pieces represents1,000 5; that is, 200 students.

Note


Thus, there would be 400 boysand 600 girls.

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B B G G G200 200 200 200 200400 600

Boys Girls


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More generally: if we denote the total

number of students by T, then the numberof students in each of the 5 pieces is T 5.


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The corn bread model presents a niceintroduction to algebraicequations.

For example, we can let xrepresent thenumber of students in each of the 5 pieces.

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In that event, the picture translates into


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B B G G Gx x x x x

2x 3x

2x= the numberof boys. 3x= the numberof girls.

The total number of students would be

+

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Suppose now that the total number ofstudents is 150, and the ratio of boys to

girls is still 2 to 3. It follows that2x+ 3x = 150.


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2 x = the numberof boys.

3 x= the numberof girls.

5 5

nextnext

2 (30) = 60 3 (30) = 90next

5 x =

By dividing each side of the equation by 5

we obtain

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= 30150


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In between the abstractness of fractionsand the concreteness of the corn bread,one can always interject trial and error.One systematic approach to trial and

error is known as an input/output table.With respect to our original problem,

namely...

A Note on Bridging the Gap


In a certain class, the ratio of boys togirls is 2:3. If there are 30 students in the

class, how many of them are boys?


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We make a table in which we start with2 boysand 3 girlsand keep adding rows

that consist of 2 more boysand 3 moregirlsuntil we get to the row in which the

total number of students is 30.


Row Number of Boys Number of Girls Number of Students

1 2 3 5

2 4 6 10

3 6 9 15

4 8 12 20

5 10 15 25

6 12 18 3012 18

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30


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The chart offers the additional advantageof highlighting patterns.

For example, it makes it easy to see that

each time the number of boys increasesby 2, the number of girls increases by 3,

and that the total number of studentsincreases by 5. And this, in turn, is a

segue for helping students see a whole-number version of what 2/5 means.

Note


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For example, suppose there had been 60 boys

in the class, and we wanted to know how manystudents were in the class altogether.

Note on the Chart


Since each additional row adds 2 more boys

(and 3 more girls), the entry for 60 boys wouldoccur in the 30th row (60 2). It would be

cumbersome to extend such a chart to 30 rows.However, once we realize that every new row

shows 5 more students, we know that the entryin the 30th row has to be 60 boys (30 2),

90 girls (30 3) and a total of 30 5 (or 150)

students. next


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That is


Row Number of Boys Number of Girls Number of Students

1 2 3 5

2 4 6 10

3 6 9 15

4 8 12 205 10 15 25

6 12 18 30

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7 14 21 35

--- --- --- ---

30 30 2 30 3 30 560 90 150


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While the Corn breadmodel might seemrather simplistic, experience assures us that

the corn breadmodel can be used to goodadvantage throughout all school levels.

Applying the Corn BreadtoLesson 1


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F l ith t t li


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For example, with respect to our earlierproblem of buying 6 boxes of candy from a

catalog for $5 each, with $4 added to theorder to cover shipping; we can use thecorn breadmodel as representing the totalcost.

The corn breadwould be cut into 7 pieces.Namely 6 equal-sized pieces for the 6 boxesof candy costing $5 each; and then 1 smaller

piece for the $4 shipping.

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Corn Bread$5 $5 $5 $5 $5 $5 $4next

$5$10$15$20$25$30$34


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We have now begun our journey

from arithmeticto algebra,and we hope you areenjoying the trip.

Closing Note

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algebra 1 un arithmetic

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