algebra 1 and arithmetic

The Game of Algebraor

The Other Side of Arithmetic

© 2007 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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Lesson 1

Unarithmetic+ -

× ÷© 2007 Herbert I. Gross

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What is Unarithmetic?When young children are first taught to put on their shoes, they might refer to taking off their shoes as “unputting on” their shoes. In other words “to unput on your shoes” might be a child’s way of saying “to take off your shoes”. As awkward as this phrase might seem, it does express the relationship between “putting on” and “taking off” shoes.

© 2007 Herbert I. Grossnextnext

In a similar way to undo multiplication, a child might have invented the word

“unmultiply”, which at the very least is much more suggestive than the word

division.

It is in the above context that we may begin our study of algebra by thinking of it as

being “unarithmetic”.

Key Point


Let’s keep in mind, whether we approve or not, that calculators

and computers are now household items, and students

see nothing wrong in using them. And, in fact, since the

prerequisite for an algebra course is a knowledge of arithmetic;

once this knowledge is assumed there is nothing wrong with

allowing students to use calculators in an algebra course.

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In the language of calculators, we call it an arithmetic problem, if an

answer to a computation problem can be obtained by simply pressing keys in 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 an arithmetic process or direct computation.

Namely all we would have to do with a calculator is enter the sequence of

key strokes…© 2007 Herbert I. Grossnextnext

9 8 7 +6 5 4 -3 2 1 ×0 . = ÷

6×

5

=

30

+4

=

34

6 × 5 + 4 =

34

The display window of the calculator displays 34 as the answer.

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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 being

the output. Putting this in computer language, it’s arithmetic (or a direct computation) when the program and input are given, and

the output must be found.


On the other hand, suppose we wanted to know the number we had started with if the answer was 59 after we first multiplied it by 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 sequence of steps for this would have to be…

?? × 5 + 4 = 59 But since the calculator doesn’t have a ??

key, we can’t proceed.© 2007 Herbert I. Gross

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In the above context, one of the ways we define algebra is to say…

Key Point

© 2007 Herbert I. Gross

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Algebra is the subject that allows us to paraphrase questions the calculator

cannot “understand” into equivalent questions that the calculator can

“understand”. That is: algebra converts an indirect computation (which we can think of as “unarithmetic”) into a direct computation (which we can think of as arithmetic). nextnext

Pedagogy Note


Often students depend on a calculator to do computations, but a calculator, alone, will not help them solve any problem that involves an indirect computation.

Consider the “fill in the blank” question that is designed to test whether the

students know the number fact 2 + 3 = 5.

For Example

Form A 2 + 3 = __

If a student had no idea of what the meaning of “+” or “=” was, but had a calculator, he still could get the correct answer by pressing the following keys in order.

2 + 3 = 5© 2007 Herbert I. Gross

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But suppose that, instead of Form A, the “fill in the blank” question was worded...

Form B 2 + __ = 5 This presents an obstacle. Namely, the student can enter 2 and +, but now he is

stymied by the “blank”. To be able to solve this problem by using a calculator, the

student would have to be able to paraphrase Form B into the equivalent form

5 – 2 = __.


The above discussion is not limited to 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 depend

on how the question is worded.

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Suppose that students are tested on whether they know Sacramento

is 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.


However, the number of students who get 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 not

come to mind. On the other hand, in form (2) the only

proper noun is Sacramento, and it is quite likely Sacramento brings California to

mind. nextnext

The student might reason, “Gee, I didn’t know that

Sacramento was the capital of anything, but knowing that it’s in

California, I think the correct answer is probably California.”

For Example


How does this apply to the discussion about arithmetic and algebra?

To give the “multiply by 5 and then add 4” a “real-life” interpretation, consider…

The price of a box of candy in a catalog reads “$5 per box plus $4 shipping and handling”. What is the cost of buying 6

boxes of candy?

Problem


The thought process for solving this problem 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 $30 as the cost of the 6 boxes.

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If you didn’t know how to perform the appropriate arithmetic, but you knew how

to use a calculator, you could enter the following sequence of key strokes…

And obtain 34 dollars as the answer.

6 × 5 + 4 = 34


The previous sequence of key strokes is equivalent to what many textbooks refer to

as a “function machine”; and which is represented in a form similar to the one

shown below.

cost in dollars

OutputInput

number of boxes × 5 + 4


If we translate the diagram into “plain English”, the following sequence of steps

is obtained.

Step 1: Start with the number of boxes (in the present illustration; it’s 6).

Step 2: Multiply by 5.

Step 3: Add 4 for shipping.

Step 4: The answer is the cost in dollars (34).


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

English” model are equivalent. However, our own belief is that the “plain English” model is the most “user friendly”, at least to those

students who may have vestiges of “math anxiety”.

Note


To see the application of an indirect computation (that is, “unarithmetic”), suppose we’re still buying from the same candy catalog, but this time we’ve decided to spend $59. How

many boxes of candy could we buy for that amount? Notice that to solve this problem we have to know more than just how to read a calculator.


That is, the sequence of key strokes would be…

?? × 5 + 4 = 59

But, to compute the value of ?? we would have to do an indirect

computation. In other words in this case, we have defined the input implicitly

(rather than explicitly). That is: the input is that number which, when we

multiply it by 5 and then add 4, results in 59 being the output.


In terms of the “function machine”, the problem looks

like…

cost in dollars

OutputInput =

number of boxes × 5 + 4?? 59


In terms of our “plain English” model the problem would be...

Step 1: Start with the number of boxes (in the present illustration; it’s ).

Step 2: Multiply by 5.

Step 3: Add 4 for shipping.

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


??

Notice that the answer in Step 4 (59) was obtained after 4 was added. In other words to get from Step 3 to Step 4, the fill-in-the-

blank question would have been…Form A ___ + 4 = 59

Form A tells us that 59 was obtained after 4 was added to the “blank”.

Therefore, to determine the number that is represented by the blank, we have to

“unadd” 4 to 59 (that is, subtract 4 from 59).


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

Form B 59 – 4 = ___

The difference between the two forms is that the calculator can solve Form B, thus

making Form B a direct computation (arithmetic), but it cannot solve Form A

(which is an indirect computation or “unarithmetic”).


It is in this context that we define algebra as the subject that allows us to paraphrase questions that

cannot be answered directly by a calculator into equivalent

questions that can be calculated directly.

Key Point


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

we then unmultiplied (that is divided) by 5 to 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


In terms of the function machine model: starting with an input of

11 boxes and obtaining an output of $59, as shown below, is considered an arithmetic problem.

$59

Output

11Input

number of boxes × 5 + 4 cost in dollars

11 55 59


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

reversing the steps using the “undoing” process, as shown

below, is considered to be algebra.

11Input

number of boxes

59

+ 4× 5 cost in dollars

595511Output


÷ 5 - 4 cost in dollarsnumber of boxes

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InputOutput

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

In essence, a formula is a well-defined rule that tells how to deduce the value of an

unknown quantity, by taking advantage of knowing one (or more) related quantities.

Formulas as a Bridge between Arithmetic and Algebra


An elementary example is the rule that tells us the relationship between feet and inches.

Since there are 12 inches in a foot: to convert feet to inches, simply multiply the number of

feet by 12. next

To write the relationship in the form of a formula: let F denote the number of feet and I the number of inches. The formula would become…

I = 12 × FIf, for example, F equals 5, the formula would become…

I = 12 × 5

and would thus be a direct computation (arithmetic).


On the other hand, if I equals 60, the formula would become…

60 = 12 × FIn which case there would be an indirect

computation (algebra) which by “unmultiplying” becomes the direct

computation.

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

itself, is neither arithmetic nor algebra.© 2007 Herbert I. Gross

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This is especially true in problems that involve constant rates. For example,

consider the following question…

Appendix The “Corn Bread” Model


Sometimes a picture is worth a thousand words…

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

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

3 girls. Namely, a group consists of 2 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.)

Arithmetic Solution


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

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However, the above solution can be “threatening” to students who come to

algebra still uncomfortable with fractions.


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

This corn bread will represent the total number of students.

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Corn Bread

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

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The fact that the ratio of boys to girls is 2:3 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 by the letter B) represent the number of boys,

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B B G G G

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

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Since the corn bread represents the total number 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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The corn bread model doesn’t 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 bread would still be divided into 5 equally sized pieces.

But now, each of the 5 pieces represents 1,000 ÷ 5; that is, 200 students.

Note


Thus, there would be 400 boys and 600 girls.next

B B G G G200 200 200 200 200400 600

Boys Girls

More generally: if we denote the total number of students by T, then the number of students in each of the 5 pieces is T ÷ 5.


The “corn bread” model presents a nice introduction to algebraic equations.

For example, we can let x represent the number of students in each of the 5 pieces.

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In that event, the picture translates into…


B B G G Gx x x x x

2x 3x

2x = the number of boys.

3x = the number of girls.

The total number of students would be…

+

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Suppose now that the total number of students is 150, and the ratio of boys to girls is still 2 to 3. It follows that…

2x + 3x = 150.


2 x = the number of boys.

3 x = the number of girls.

5

5

nextnext2 (30) = 60 3 (30) = 90 next

5 x =

By dividing each side of the equation by 5 we obtain…

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= 30 150

In between the abstractness of fractions and 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 to girls is 2:3. If there are 30 students in the

class, how many of them are boys?

We make a table in which we start with 2 boys and 3 girls and keep adding rows that consist of 2 more boys and 3 more girls until 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 52 4 6 103 6 9 154 8 12 205 10 15 256 12 18 3012 18

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30

The chart offers the additional advantage of highlighting patterns.

For example, it makes it easy to see that each time the number of boys increases by 2, the number of girls increases by 3,

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

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

Note


For example, suppose there had been 60 boys in the class, and we wanted to know how many

students were in the class altogether.

Note on the Chart

© 2007 Herbert I. Gross next

Since each additional row adds 2 more boys (and 3 more girls), the entry for 60 boys would

occur 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 entry in the 30th row has to be 60 boys (30 × 2),

90 girls (30 × 3) and a total of 30 × 5 (or 150) students. next

That is…


Row Number of Boys Number of Girls Number of Students

1 2 3 52 4 6 103 6 9 154 8 12 205 10 15 256 12 18 30

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7 14 21 35--- --- --- ---30 30 × 2 30 × 3 30 × 560 90 150

While the Corn bread model might seem rather simplistic, experience assures us that the corn bread model can be used to good advantage throughout all school levels.

Applying the Corn Bread to Lesson 1



For example, with respect to our earlier problem of buying 6 boxes of candy from a catalog for $5 each, with $4 added to the order to cover shipping; we can use the corn bread model as representing the total cost. The corn bread would be cut into 7 pieces. Namely… 6 equal-sized pieces for the 6 boxes of candy costing $5 each; and then 1 smaller piece for the $4 shipping.

nextCorn Bread$5 $5 $5 $5 $5 $5 $4

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

We have now begun our journey from arithmetic to algebra,

and we hope you are enjoying the trip.

Closing Note


algebra 1 and arithmetic

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