Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Vedic Version of a Line From Two Points

Posted by mark schwartz on September 25, 2016

In Vedic Mathematics (revised edition, 1992) a very interesting algorithm is presented. It allows one to find the equation of a line in standard form by visually examining the values of the two points, doing a little mental calculation, and writing down the equation! One need not use the slope-intercept or the point-slope formula.

Given two points (a,b) and (c,d), the vedic version (pg. 343) is: x(b-d) – y(a-c) = bc – ad

A slight notation change gives the standard form (ax+by =c), thus (b-d)x – (a-c)y= bc – ad

For example, using the vedic version with (9,7) and (5,2) the equation is:

(7 – 2)x – (9 – 5)y = 7⦁5 – 9⦁2, giving 5x – 4y = 17.

I was curious about this because it looked familiar; basically, the difference in the y-values is the x-coefficient and the difference in the x-values is the y-coefficient. The constant is the ‘inner’ minus the ‘outer’, if you are familiar with FOIL. As I played with this, I realized that the vedic algorithm could be derived from combining the slope-intercept and the point-slope formulae. Starting with the point-slope formula, one gets:

(y – y1) = m(x – x1

(y – y1) = ((y2 ̶ y1)/(x2 ̶ x1)) (x – x1)

(x2 – x1) (y – y1) = (y2 – y1)(x – x1)

(x2 – x1)y – (x2 – x1)y1 = (y2 – y1)x – (y2 – y1)x1

– (y2 – y1)x + (x2 – x1)y = (x2 – x1)y1 – (y2 – y1)x1

– (y2 – y1)x + (x2 – x1)y = x2y1 – x1y1 – x1y2 + x1y1

 – (y2 – y1)x + (x2 – x1)y = x2y1 – x1y2

 (y2 – y1)x – (x2 – x1)y = x1y2 – x2y1

 -1(y1 – y2)x – (-1)(x1 – x2)y = (-1)(x2y1 – x1y2)

(y1 – y2)x – (x1 – x2)y = x2y1 – x1y2

This form (y1 – y2)x – (x1 – x2)y = x2y1 – x1y2 is the vedic form (b-d)x – (a-c)y = bc – ad.

Furthermore, this vedic form allows one to generate the equation of the line if given the slope and a point, or a point with a line perpendicular or parallel to a given line because a second point can be found from the given point and the slope.

Using the same example as above, if presented the point (9,7) and the slope 5/4, the second point is (9 + 4, 7 + 5), or (13,12), as well as (9 – 4, 7 – 5), or (5,2). Consider using this vedic version.

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In 1877, Mr. Ray Reasons with Fractions

Posted by mark schwartz on September 8, 2016

Introduction

In Mr. Ray’s 1877 Ray’s New Intellectual Arithmetic, an elementary school text, he presents some of the problems with their solution. A sample of these are worth looking at because in every case he shows a solution method which is based on fractions and knowing how to handle a sequence of fractions. But it’s not only the sequence of fraction operations but also the logic of these fraction operations that elementary school children had to follow. This required them to think about the relationships in the problem. I’d like to further note that this method of solution for all 7 problems presented here is seen in many of the texts of that era. It really required students to understand fractions! I’m not proposing that we use this “fractional” method in lieu of solving them by either proportions – the first 4 problems – or simple equations, the last 3 problems.

The Story

All these problems are from his text. Read the solutions slowly to really enjoy the subtlety of the method.

  1. A yard of cloth costs $6, what would 2/3 of a yard cost?  (Pg. 48, # 3)

Solution: 1/3 of a yard would cost 1/3 of $6, which is $2; then, 2/3 of a yard would cost 2 times $2, which are $4.

  1. If 3 oranges are worth 15 cents, what are 2 oranges worth?  (Pg. 49, #19)

Solution: 1 orange is worth 1/3 of 15, or 5 cents; then 2 oranges are worth 2 times 5 cents, which are 10 cents.

  1. At $2/3 a yard, how much cloth can be purchased for $3/4?  (Pg. 75, # 5)

Solution: For $1/3, 1/2 a yard can be purchased, and for $1, 3/2 of a yard; then, for $1/4, 1/4 of 3/2, or 5/8 of a yard can be purchased, and for $3/4, 9/8 = 1 and 1/8.

  1. If 2/3 of a yard o cloth costs $5, what will 3/4 of a yard cost?  (Pg. 101, # 2)

Solution: The cost of 1/3 of a yard will be 1/2 of $5 = $5/2; and a yard will cost 3 times $5/2 = $15/2; then, 1/4 of a yard will cost 1/4 of $15/2 = $15/8; and 3/4 of a yard will cost 3 times $15/8 = $5 and 5/8.

Note that these 4 problems lend themselves well to being solved using proportions. What follows now are 3 more problems, which if presented in today’s texts would likely be solved with simple equations, but again Mr. Ray’s solutions are a sequence of fraction operations.

  1. If you have 8 cents and 3/4 of your money equals 2/3 of mine, how many cents have I? (Pg. 52, #17)

Solution: ¾ of 8 cents = 6 cents; then 2/3 of my money = 6 cents, 1/3 of my money is 1/2 of 6 cents = 3 cents, and all my money is 3 times 3 cents = 9 cents.

  1. Divide 15 into two parts, so that the less part may be 2/3 of the greater.  (Pg. 106, #1)

Solution: 3/3 + 2/3 = 5/3; 5/3 of the greater part = 15; then, 1/3 of the greater part is 1/5 of 15 = 3, and the greater part is 3 times 3 = 9; the less part is 15 ̶ 9 = 6.

  1. A and B mow a field in 4 days; B can mow it alone in 12 days: in what time can A mow it?  (Pg. 110, #14)

Solution: A can mow 1/4 ̶ 1/12 = 1/6 of the field in 1 day; then he can mow the whole field in 6 days.

I hope you appreciate what elementary school students had to do at that time. Since it was elementary school, they weren’t taught proportions and simple equations but they were “exercised” with fractions in a way that I believe could benefit today’s students understanding of fractions.

Posted in algebra, basic math operations, fractions, Historical Math, math instruction, mathematics, proportion, Proportions, remedial/developmental math | Tagged: , , , , , , | Leave a Comment »

Math Stories

Posted by mark schwartz on August 30, 2016

We all believe were good teachers, yet also we all believe we can improve. Our training and experience, and occasional validation by our peers helps us feel good at what we do. Students also give us comforting feedback, directly through their improved performance and “thank you” and indirectly as we hear from other students, friends and colleagues. We continuously share our ideas with our peers, discuss successes and failures, ask for guidance and help with a pedagogical idea. We teach with inspiration and concern, and treat students with respect, care and a nurturing attitude.

So, what’s the problem? How come we are distressed at the 4-5 students in a class of 20-some who don’t seem motivated, don’t do the work, sometimes don’t even show up for class? Are we to parent them? Are we to act as shepherds of their lives as well as math instructors? Are we to be motivational speakers to grab their attention and rouse them to dynamic learning?

Perhaps we believe, at the core, that everyone is capable of learning math, if only the learning environment were appropriate. If only we were able to be more adept at unraveling the history of their prior experiences and build from the math rabble they possess. And, paying attention to the importance of learning styles, we tend to adopt a host of teaching methods until we find the one that’s just right for each student. Well, at least, we try.

So, what’s the problem?

Is it them or us? This short question is the problem. Once ‘them’ and ‘us’ are separate entities, once we see ourselves as separate, the problem begins. Really? After all, we are individuals and it is pure happenstance that brings us together in the same classroom.

And that’s the point. We are a unit and theoretically, I’m the instructor and they the learners. There is no point to my teaching if there is no audience and there is no point of their sitting in a classroom if no activity is to occur. So, we are a unit; an educational unit.

And what is the dynamic of this unit? One way to view this is to allow that the instructor is to tell stories to the students. The stories will be about relationships between quantities. Usually these collective and cumulative stories will be called math; there are prescribed relationships that have been identified through the ages as essential and core to mankind, and these relationships are bundled together in a text and it is called mathematics.

Some of these relationships – some of these stories – don’t seem to make sense to the students because they have heard these stories before and the stories don’t seem to make sense or don’t seem to have any meaning, or further, don’t seem to have any meaning to that person’s unique story. Is this a critical event? Must all math – or for that matter – any content area have immediate and purposeful meaning to a person before they feel engaged with it and feel impelled to learn it because it has a visible connection to the daily lives of the student?

What about history, English, biology, psychology? Do these disciplines have value to the learner? Can the learner see how these can be used, integrated into their daily lives and become valued learning? What about math?

If math is taught as it traditionally is, then the likely answer is no, there is no apparent real-life immediate utility. This, of course, is based on the assumption that if the utility were made visible, the students would then be more engaged in learning it. But let’s return to the idea of ‘traditional’ for a moment. Traditional means that sequence of information presented in texts which math curriculum developers and teachers believe to be important for students to learn.

In basic math the four basic operations are learned, and then the order of operations is learned because a person confronted with a problem in which all the operations occur must have a ‘rule’ for which operations to do first. Why? So that not only will they get consistent answers but also so that all the people doing to problem will arrive at the same answer.

Is there something sacred about the order of operations? Not particularly, but it has been made to be so. Violate the rules and you will err. For example, in class the other day the problem was (2/5)^3x(5/8)^2.

The solution in the text showed (2/5)(2/5)(2/5)(5/8)(5/8) and then showed some canceling of common factors, resulting in an answer of 1/40. However, notice that in order to do this, the order of operations has been ignored. If one were to literally apply the order of operations the exponents would have been the first operation to do, giving 8/125 x 25/64. The next step would have been to do the multiplication of the fractions, and finally the answer would be simplified completely. Somewhere in the text, by the way, the way this problem was worked was consistent with what was described as a ‘shortcut’, meaning when you have multiplication of fractions you should look at the fractions to see if they can first be reduced, and then do the multiplication. This is a valuable and time-saving strategy, but in the example as worked out in this case, reference to the ‘shortcut’ wasn’t made.

The issue here is the story of how to do this kind of a problem. Is there more than one story that can be told? Yes. Be literal and follow the order of operations or alternatively, use the ‘shortcut’. Which one is correct? Both are correct. Which one should a student use? Whichever story resonates with them. What? Isn’t math a little more linear and logical than that? Allowing students to craft their own method of solution?

Well, isn’t crafting an individual and unique method for problem solution what students do in everyday life? Why not in math class?

A lot of students have the perception that math is very linear, very logical, and has one and only one path to the answer. Starting with the very basic idea, for example, that 2 + 2 always and only equals 4, this concept scaffolds into other math operations, like the above noted order of operations. Until of course, one has to do algebra, where the numbers don’t exist and letters representing them exist.

For example, what happens to the order of operations in algebra vs. in arithmetic? In arithmetic, students are taught to do the work inside any grouping symbols first, if there is work to do. So, 2(5 + 6), is done by adding 5 and 6 = 11 and then multiplying by 2 = 22. What happens if a student knows the distributive law and uses it? Is that student to be corrected?

But back to algebra. In algebra a statement like 2(5 + 6) can be seen as a(b + c) and of course, the order of operations still applies. ‘Do the work inside the grouping symbol first’, so the student is to add ‘b’ and ‘c’., Well, algebraically, it can’t be done. All one can do is indicate it by leaving the expression as written, b + c. Then multiply this expression by ‘a’. Is this consistent with order of operations? Absolutely, so why does this bedevil some students?

There are some students who contend that b + c = bc. We now have to address issues of notation. In algebra, ‘bc’ means to multiply ‘b’ times ‘c’, once we have actually substituted numerical values for ‘b’ and ‘c’. But if we were to be very literal, if ‘b’ = 7 and ‘c’ = 8, then what would be written is 78. But this isn’t done. By the way, 78 means 7(10) + 8(1), a whole other notation issue, and certainly doesn’t mean 7 times 8.

But when we write 78 because of having substituted ‘b’ = 7 and ‘c’ = 8, different notation is employed. To keep the arithmetic consistent with the algebra; we use one of the available notations for multiplication such as (7)(8). Why not use (b)(c)? Because b + c doesn’t result in (b)(c).

I think you get the point. Math stories seem to have inconsistencies. We seem to lurch around and come to realize that some of the arithmetic rules work with algebra and some of them don’t. And, what’s more, we have to know both domains, because while doing algebra, the arithmetic rules still apply, but the converse isn’t necessarily true. So, although 2 + 2 = 4 is true in arithmetic and in algebra, students must understand that although the order of operations is the same in both cases, it ‘feels’ different.

So, why don’t we as math instructors, teach students the algebraic story of grouping symbols to bring consistency? Why not allow that if one sees 2(3 + 4), use the distributive law? Is it because of exponents? A student would have to see that 2(3 + 4) and 2(3 + 4)2 have to be handled differently. And, by the way, look at what happens – and it happens regularly with some students – when a(b+c) and a(b+c)2 are presented. Have you ever seen (3 + 4)2 = 9 + 16?

Part of the issue here is that sometimes, the rules that apply when the values or expressions have an exponent of 1 are different from the rules that apply when a value or an expression has an exponent greater than 1. Wow, what a story.

For example, when trying to make the point of the difference between (-4)2 and –42, one very direct way is to say ‘follow the order of operations’. In (-4)2, a student would do the exponent first, thus 16. If the student were to do –42 according to the order of operations, the ‘-‘ has to be seen as subtraction, not the sign of the number, and the order of operations says to do exponents before subtraction, so the outcome is –16. Is this too subtle for students to see? Should this understanding of it be presented? And, why-oh-why does math allow the same symbol to be used for two different things: ‘+’ can mean addition or be the sign of the number and the same for the symbol ‘-‘.

So, look at (-4)2 and –42 again. Using grouping symbols makes it clear that the student is to apply the exponent to everything inside the grouping symbol. This works readily. But what if students had been taught to read the ‘-‘ not as subtraction but as the sign of the number? Or, that the symbol ‘-‘ can mean either? Given this understanding of ‘-‘, then (-4)2 and –42 could mean the same thing. The symbol ‘-‘ is the sign of the number, therefore understand this to mean ‘negative four times negative four’.

Here again, however, the idea of an exponent of 1 versus an exponent greater than 1 can come into play. If a value has an exponent of 1, the ‘-‘ can be either the sign of the number or the operation of subtraction, but if the exponent is greater than 1, the ‘-‘ means subtraction and not the sign of the number. Given this, should it ever be taught that ‘-‘ can mean both, or more critically, knowing that this distinction between the forms (-4)2 and –42 forces different meanings, perhaps the point should be made about how the rules differ when exponents are equal to 1 or greater than 1. Or, ignoring all this, simply point out that the order of operations applies.

However, texts present the following: “When subtracting, add the opposite” and demonstrate this algebraically as a-b = a + (-b). So, in a problem such as 25 – 42, if a student were to follow the subtraction algorithm, the outcome would be 25 + (-4)2 and thus again showing that (-4)2 and –42 mean the same thing. And it is again true that if a student were to be literal about the order of operations, the 42 has to be done before the subtraction. But where is it ever designated that changing the notation form isn’t allowed? After all, changing the notation form is not doing the operation. But, is rewriting 25 – 42 as 25 + (-4)2 actually doing the operation of subtraction or simply changing notation? Again, if it’s understood to be doing the operation, then again the order of operations prevails, and thus a student couldn’t apply the subtraction algorithm before doing the exponentiation. But, how is a student to know this? Well, maybe again, this defaults to distinguishing between the operations if the exponent is 1 versus greater than 1. If it’s greater than one, suspend transforming subtraction to addition. So, it seems that raising numbers to exponents greater than one has a subtle but critical meaning which isn’t addressed in texts or by instructors.

Perhaps all the above is irrelevant since the rules and algorithms that we present are presented in a very direct, concise, logical way. We need not presume that any student would ever get caught up in the tangle of symbols, words and relationships as written above.

But these little inconsistencies, which never reach consciousness necessarily, can generate cognitive dissonance. It may well be that some part of the students ‘math brain’ is saying, “Huh…this doesn’t make sense. We were taught that 6-2 means 6 + (-2) is true in one circumstance, but not true in another circumstance, 6-22 ≠ 6 + (-2)2.”

Perhaps this gives too much importance to the thinking students bring to math classes. Have they ever been bedazzled by other such seeming anomalies in math notation and math meaning? Have they every sensed something about the rules not always being applicable and no instructor has ever pointed this out? Should I, a student, ask about this if I see it?

Fundamentally, I believe teaching math is telling stories and listening to students’ math stories, the stories that they carry around in their heads, perhaps fantasies, hardly realities of math.

Getting people to reflect on what they know – actually, reflect on the fragments that they carry around that they call math – is important to the story telling. Again, these are the student’s stories. For example, writing on the board that 2/4 = and asking them what the answer is usually gets a collective ½. If it is then asked “What math operations did you do?” the answers will vary but be things like, cancelled, or divided, or reduced, simplified, got rid of. The point is that students know how to do, metaphorically, math. The words they use aren’t math operations but symbolically represent several math operations. In order to get 2/4 to 1/2 involves several math operations. These are sequential events, algorithmically organized, compartmentalized, and further trivialized by summarizing and compressing them into a word like ‘cancel’.

Getting students to reflect on what they know gives them a starting point from which understanding, not just doing, math can occur. Puzzling them, disabling their usual routes to answers, and making them appear as deer in the headlights. But not doing it in a threatening or punitive environment.

For example, we were talking about dividing by powers of 10, and used the example of 300 divided by 40. I then wrote it as a fraction, and brought it back to the same notation form as 2/4. I got a wow response, which was nice but also something which I could get only if students had been misperceiving what was really being expressed. How did they generate this misperception? They’ve been in math classes before mine and have come ‘armed’ with these fragments and pieces of math-like words.

I do believe that if they can gain an understanding of the basics, then they can always reconstruct in their own way with their own logic a pathway from the problem given to the set of possible solutions available.

Keeping it basic and visible is helpful. For example, most textbooks give one page to exponents and square root. It is written in ‘mathese’, dry presentation of what the notation is and what it ‘means’. However, what it means can also be demonstrated.

Put on the board the following: (2)(2)(2)(2)(2)(2)(2)(2)(2)

(3)(3)(3)(3)(3)

(5)(5)(5)

(7)(7)(7)(7)(7)(7)(7)

Then talk about notation; no mystery here, simply a notation issue.

Try this. Put on the board a cluster of 9 “postits” and a separate cluster of 16 “postits”. Ask for a volunteer to come to the board and arrange these in a square. Once done, they will be able to see that the square made by 9 is 3 on a side, and 4 on a side for the 16. Then ask for a volunteer to put them together in a square, and this gives a square of 5 on a side for the total of 25. This demonstrates square root.

Students have been trained to believe that if you do math well, you must also do it fast. This I find to be one of the perceptions that students bring which interferes with their being able to learn math. The part that isn’t seen is the thinking, the reflection, the connecting the ‘new’ stuff with stuff they already know. Their stories of math typically aren’t complete. Rather than identifying what they do as wrong, I try to identify what they do as incomplete. Sometimes I prompt them with the idea that if they have a strategy that works, they need to know if it works all the time. If it doesn’t, let’s talk about it and find out how to make it work all the time.

To be continued … meaning there is always more stories to tell – a lot of stories by a great number of teacher and students. I would hope that framing teaching math as a way that teachers and students can share math stories will reframe some of the classroom conversation.

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Is it ̶ 3 or is it ̶ 3?

Posted by mark schwartz on August 27, 2016

Introduction

I know. The title “Is it -3 or is it -3?” looks weird but it’s not a typographical error. It’s a way to bring attention to algebraic notation. The question is: how did you read -3? Did you say “minus 3” or did you say “negative 3”? Does it make a difference?

In Day’s 1853 An Introduction to Algebra, he writes 5 pages on the topic – yes, 5 whole pages of words discussing negative quantities. He wants to make sure that students understand that the 4 basic operations in arithmetic are different from the 4 basic operations in algebra because of the introduction of negative quantities in algebra. In lengthy discussions he cites how negative quantities appear in profits of trade, ascent and decent from earth, progress of a ship relative to a latitude, and of course money. Clearly he’s conveying what I would call the algebraic trip-wire – how to handle negative quantities. This kind of lengthy discussion isn’t presented in today’s texts but rather students are presented with diagrams and number lines and visual aids to help them understand the rules. An instructor can supplement the text with their own creative explanations and demonstrations. But Day’s emphasis on this point may well be what is needed in today’s texts – a core understanding of the rationale behind the rules.

The Story

So, back to “is it  − 3 or is it – 3?”

Day’s writing prompted me to recall a question from a student. We were working with operations with signed numbers. Typically I am very careful to reference any “ ̶ “ in a problem or an answer as a negative or as a minus, depending on its use in the problem. Knowing, for example, that + ( ̶ 3) gives the same result as ̶ (+3), in the former the “ ̶ “ is understood as negative 3 but in the latter it’s understood as minus 3. As noted, it ultimately makes no difference, but a student stopped me during a discussion and pointed out that in the same problem I had referred to a term as both and it didn’t seem right to him … and in a most technical sense, he was right. I asked if he were the only one bothered by this and other students felt as he did.

I admitted to my sloppy use of the terms and we got back to discussing operations with signed numbers and then again, this student stopped me. He asked “what about – and in his words – minus a minus 5” – how come it’s plus 5?” I wrote ̶ ( ̶ 5) the board and asked him if this is what he meant and he said yes. I asked him then what operation is being indicated and he said that it indicated to subtract a negative. So, the sign inside the parenthesis isn’t a minus, rather it’s a negative sign, a sign of the number. The class was muttering about this somewhat lengthy Socratic discussion – and they participated too – which really was a very positive result of the initial question … what some might call an unintended consequence … but a good one.

And of course, there was the question of “does it make a difference what I call it if I get the right answer?” So, we played language games with various examples until there was consensus that there was a difference between “minus” as the operation of subtraction and “negative” as the sign of the number. But, for most of the class, this difference didn’t make a difference as long as they understood what the notation in the problem was asking. So, I asked them to think about this:

Don’t do this problem yet but within your group, discuss the “ ̶ “ signs in the problem 4 ̶ 6 + 2 ̶ 3 ̶ 5 + 7. Signs of the number of signs of the operation? It was fun to roam the room and listen to the within-group discussions. As expected, there were disagreements, yet those that disagreed came to understand that both were correct! It was a matter of what procedure made each person feel most comfortable.

After allowing for discussions, I asked for volunteers to go to the board and demonstrate their solution. There were two primary solutions: first, just use the order of operations and do the indicated operations from left to right, although there was some stumbling to explain how to handle “2 ̶ 3 ̶ 5”. The language used in explaining the whole problem was interesting. For example, “4 ̶ 6” equals minus 2 (not negative 2) and minus 2 and plus 2 is zero (adding two operation not two values). Then zero minus 3 (the “ ̶ “ is the sign of the operation) gave “minus 3” and the next operation was expressed as “a minus 3 and a minus 5 equals negative 8”. Think about that. Technically, the 3 and the 5 were expressed as adding two subtractions (minus wasn’t seen as an operation) yet the answer of negative 8 was correct notation. But the real thing to notice is that the answer is correct independent of technically incorrect labelling of the values.

As much as I believe in the importance of carefully using either minus or negative correctly, it clearly seems that – at least for this student and his group – knowing how to handle the negative is more important.

The second solution was given with a preface. This student rewrote the problem as 4 + ( ̶ 6) + (+2) + ( ̶ 3) + ( ̶ 5) + ( +7). She pointed out that her group saw all the signs as signs of the numbers and therefore they just added them all together. Neat.

Of course there are more ways to handle this problem but these two examples show that as long as students understand the basic rules and relationships with signed numbers, the right answer will be found. We talked about these two solutions and how to handle the signs and operations.

I then asked if all the talk we had about the difference between negative 3 and minus 3 made a difference for them. The consensus was yes and that it showed up when they were talking about the problem in their group. Apparently, it provided a clearer understanding of the difference.

There was also the comment that allowing them to challenge me (I pointed out it wasn’t challenging me but rather challenging the math content) gave them a sense that the “rules” and labels weren’t arbitrary – that there really was sense to it.

Finally, I’d like to note that hearing a student’s question as a real interest in knowing rather than a hostile kind of “whatever”, opened the door to the discussion which further opened the door for their better understanding – again an unintended positive consequence. If you have time, try it.

Posted in algebra, basic math operations, Historical Math, math instruction, remedial/developmental math | Tagged: , , , , , | Leave a Comment »

The Importance of a Clearly Stated Algorithm

Posted by mark schwartz on August 22, 2016

Introduction

I posted a piece earlier in this blog titled Sheldon’s Compound Proportions. It describes what Sheldon labels the “cause and effect” method for solving compound proportions, which as far as I can tell, aren’t in todays’ texts. His work was in 1886. You might want to take a look at his idea because this posting talks about other compound proportion procedures at that time and I did it to emphasize the importance of a clearly stated procedure for doing an operation.

The Story

I strolled through my collection of old texts and in quite a few of them found the same prescription for solving compound proportions not using cause and effect. I picked 5 which cover about a 20 year span from 1864 to 1883. They all have the same procedure and what I suspect is that it was the established and accepted solution method at that time. As in todays’ texts, it was just a simple matter of “borrowing” a basic algorithm from someone else’s work. There are other texts of that era which reference Sheldon’s cause and effect method and a few of them introduce it along with the procedure I’ll cite below.

The point is that his method is a much clearer statement of how to handle the information in a compound proportion problem. Further, what I’m suggesting is that we should carefully examine some of our current traditional algorithms to see if the reason students have trouble with them is because of the way they are worded and presented. For example, finding the lowest common denominator (LCD) in order to add/subtract fractions doesn’t require the extended way it’s been typically taught. In fact, I have seen some texts introducing a method which doesn’t require finding an LCD at all. Certain mixture problems can more readily be solved with an 1864 method Mixing it up with Alligation, posted earlier in this blog.

By the way, the 5 texts in which I found this procedure are all arithmetic texts, which indicates to me that this somewhat sophisticated idea of compound proportion was taught in elementary school. I’ll give you example problems from an old text to indicate that, in my view, it was a very handy procedure for the real world experience at that time. Today we call these “application” problems.

Here’s the rule as stated in Greenleaf’s 1881 The Complete Arithmetic, page 235 (the other 4 books are cited below and present the same rule).

Rule for Compound Proportions

“Make that number which is like the answer the third term. Form a ratio of each pair of the remaining numbers of the same kind according to the rule for simple proportion, as if the answer depended on them alone. Divide the product of the means by the product of the given extreme, and the quotient is the fourth term, or answer.”

Embedded in this is reference to “…the rule for simple proportion …” which Greenleaf provides on page 233 and it is:

Rule for Simple Proportions

“Make that number which is of the same kind as the answer the third term. If from the nature of the question the answer is to be larger than the third term, make the larger of the remaining numbers the second and the smaller the first term; but if the answer is to be smaller than the third term, make the second term smaller than the first. Divide the product of the means by the given extreme, and the quotient is the fourth term, or answer.”

Students had to be able to apply this latter rule for simple proportion before being presented compound proportion. There is no conflict between the two rules; in fact, there is some overlap. For simple proportions, the rule directs the student to understand “the nature of the question …” and use that to determine which values go in which of the 4 places in the proportion. The students had to be able to assess and estimate if the answer was going to be larger or smaller and place the correct terms in the first and second places. Wow! There is a lot of estimating and juggling of values and basically it seems that all of this effort is aimed at what we would say today as determining whether it’s a direct or inverse proportion. With problems with simple values, this is a somewhat manageable issue.

For example, a problem from the text is “If a man travel 319 miles in 11 days, how far will he travel in 47 days?” Using the rule for simple proportion, the setup would be:

11/47 = 319/x    (the rule doesn’t use “x”, but I did for demonstration purposes)

The solution is (47×319) ÷ 11 = 1363

However, in today’s approach to simple proportion, the setup (in most cases) simply follows from the order of the information in the problem, giving:

319/11 = x/47

This gives the same answer but notice that the rule states “Divide the product of the means by the given extreme …” and that doesn’t apply here. So, the 1881 rule is quite constraining when it comes to writing the proportion, when indeed there are several ways to set up the proportion for the problem.

Again, there is nothing wrong about the simple or compound proportion rules as provided by Greenleaf. The issue is that the rules are somewhat convoluted and constraining. If a student doesn’t learn this algorithm and follow it precisely, the likelihood is that the correct answer won’t be found. There are a lot of words referring to the terms and judgements that a student must make about which terms go where in the proportion. Further, look at what happens with a compound proportion problem, again from Greenleaf (#67, page 236):

“If 12 men in 15 days can build a wall 30 feet long, 6 feet high, and 3 feet thick, working 12 hours a day, in what time will 30 men build a wall 300 feet long, 8 feet high, and 6 feet thick, working 8 hours a day?”

Now, where does a student begin sorting through all this information if they use the rule above for simple proportion? What’s the “nature of the question”? For example, the rule states “…make the larger of the remaining numbers …” and how is a student to know which number is to be selected? I can visualize the instructor explaining in excruciating terms how all this works. Again, it’s not impossible to apply the rules as stated in 1881 but I urge you to look at Sheldon’s Compound Proportions in this blog and see how much more direct the rule is by framing information as cause and effect.

Briefly, Sheldon’s 1886 statement of the procedure:

“The solution of every example in proportion proceeds on the assumption that effects are in the same ratio as the causes that produce them. Every proportion is the comparison of two causes and two effects. In the method known as Cause and Effect, the causes form one ratio, and the effects the other. The first cause and the first effect are antecedents; the second cause and second effect consequents.”

Notice the simplicity of identifying cause and effect and then the causes forming one ratio and the effects the other. The words” antecedents” and “consequents” could be updated to 1st and 3rd term and 2nd and 4th term, respectively.

Taking the above compound problem the 1st causes are 12 men, 15 days 12 hours a day and the 1st effect is to build the wall 30 feet long, 6 feet high, and 3 feet thick. The 2nd causes are 30 men working 8 hours a day and the 2nd effect is to build a wall 300 feet long, 8 feet high, and 6 feet thick. You are to find “…in what time…” which is a 2nd cause. There is a shortcut that can be used but let me show you – in what I call slow-motion-math – one way to make sure the terms get placed correctly. I typically use the labels and then replace it with the values (for a lot of different types of problems, not just compound proportions). The proportion following Sheldon’s procedure is:

Causes                     Effects

1st       men, days, hours         length, height, thickness

2nd       men, x, hours               length, height, thickness

I used “x” for days in the second cause. If the numbers are substituted, we have:

12•15•12 = 30•6•3
30•x•8     300•8•6

Cross-multiply and divide, solving for x and the answer is 240.

Again, a detailed description of the “cause and effect” is in Sheldon’s Compound Proportions in this blog.

The essence of this posting is to demonstrate the importance of a well thought-out procedure expressed in easily understood language. If you are an instructor, you likely have done this kind of “simplifying” of the algorithm because as stated in the text, it seemed too fussy for students to follow. Not every algorithm can be simplified but I believe it’s an instructor’s responsibility to make math more accessible to students by removing the fog of awkwardly phrased rules and algorithms. Give it a try.

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Commentary: Algebra – yes or no?

Posted by mark schwartz on August 17, 2016

Commentary: Algebra – yes or no?

Andrew Hacker is proposing not teaching algebra but rather teaching math in a real-world context. He has created a conversation about the utility of algebra and proposes in his book “the math myth” that algebra is a cause of the loss of talent because many students can’t get past the algebra filter. I propose that it’s not the algebra content but rather the bad teaching of algebra. Consider this.

What does a student see when presented with 2(3x ̶ 1) = 10? As they have been taught, they will start rummaging through the rules for solving equations – order of operations, distributive law, operations with signed numbers, basic addition and subtraction – and anything else needed to solve for “x”.

This is sad. Algebra is an aid to help us see patterns and relationships and the equation presented above contains those things, yet that’s not what students have been taught to see. Poor teaching of algebra is the issue, not the algebra content itself. The argument about when will students use algebra in the future or in their daily lives can be asked also of history and poetry. If the premise is utility in the future, then many topics need not be taught at all. I’m going to shy away from the philosophical framework of the purpose of education, outside of stating it’s a way of introducing people to the social, economic, religious, literate and cultural concepts in which they will live their lives.

As humans, we tend to learn not individual facts but rather how these facts aggregate to patterns and relationships. Using math as the example, students in elementary school learn arithmetic – the four basic operations and maybe a few extended procedures, for example what is called long division. But what is really being learned? Are there patterns and relationships from which the four basic operations emerge?

Consider addition. The “new” concepts being presented in the common core curriculum have students thinking about numbers differently. In essence, they are taught that it is ok to rephrase the problem. They are presented with a specific procedure on how to do this but what I suspect is that they aren’t taught that they have some alternatives which also work. For example, 14 ̶ 9 can be rephrased as 5 + 9 ̶ 9, giving the answer 5. There are alternatives, which I have seen students create and use. One example is a student who realized that in the problem 14 ̶ 9 if 1 is added to both numbers, you see 15 ̶ 10, giving 5. This is a sound “theory” and if extended leads to a different conception of subtraction. This students can’t “prove” (or demonstrate) why this procedure works, but he knows it does consistently.

This type of thinking happens at the algebraic level as well. Hacker spends a lot time focusing on the idea of “rigor” and provides examples how rigor is factor which keeps students from passing algebra. Hacker misses what this student has done; Immanuel Kant stated that Math requires two things: imagination and rigorous logic and this student has employed both of these aspects, while hacker focused on only one.

I’ve strayed from the basic theme here. To repeat, it’s not algebra it’s the way it is taught. What’s the purpose of teaching math beyond the basics of arithmetic? Simplistically, our society needs to identify students with math capability who really will be needed as our society develops and grows. Mathematical modeling has become a core element across a wide spectrum of our lives.

Beyond this identification purpose, math allows us to examine a cluster of information and toss out the distractors, those elements that have no serious relationship to the core relationships in that set of information. This is what happens in what are called application problems. This idea alone is a very essential life value. Again, given a flurry of information, we tend to look for the essential pattern and relationship. This happens when driving. There’s a heap of stuff we see and hear but we pay attention only to those things that help us drive safely. Is this algebra? I believe some of it is.

Allow me the license of calling algebraic math something that we do automatically at incredible speed. The stuff we see in texts and the formulae and procedures are nothing more than slowed-down and recorded versions of what we do automatically. For example, I will start a math class by asking if there are any softball, baseball or basketball players and usually there’s more than one in the class. I ask one of them to stand and I toss them an eraser. They catch it and throw it back. I then ask if there was any math involved. As the discussion develops, it turns out that the class identifies a bunch of quantitative judgements involved. For example, how much speed must be used to get the eraser from the thrower to the catcher; how is the trajectory of the eraser tracked; how does the catcher determine where to place a hand to intercept the eraser’s path; how much energy must be applied to grip and hold the incoming eraser? There are other issues related to these core activities but the overall effect is that students start realizing that they employ “math” all the time. What’s more, they offer many more examples of activities that involve making quantitative judgements – driving, even walking up and down steps.

So, in my judgement, algebra is a tool we use to help us identify, connect and summarize quantifiable relationships and if taught this way, I tend to believe people would actually enjoy it.

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An 8th Grade Final Exam: Salina , KS – 1895

Posted by mark schwartz on August 16, 2016

Introduction

In the story below is an 8th Grade Final Exam given in Salina, KS in 1895. After you look at the problems, I’ve posed some questions and commented on a few things.

The Story

Here are the problems.

Name and define the Fundamental Rules of Arithmetic.

  1.  Name and define the Fundamental Rules of Arithmetic.
  2. A wagon box is 2 ft. Deep, 10 feet long, and 3 ft. Wide. How many bushels of wheat will it hold?
  3.  If a load of wheat weighs 3,942 lbs., what is it worth at 50cts/bushel, deducting 1,050 lbs. for tare?
  4. District No 33 has a valuation of $35,000.. What is the necessary levy to carry on a school seven months at $50 per month, and have $104 for incidentals
  5. Find the cost of 6,720 lbs. Coal at $6.00 per ton.
  6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.
  7. What is the cost of 40 boards 12 inches wide and 16 ft. long at $20 per metre?
  8. Find bank discount on $300 for 90 days (no grace) at 10 percent.
  9. What is the cost of a square farm at $15 per acre, the distance of which is 640 rods

That’s it; just 9 questions. I guess the belief was that a student can demonstrate what they know in 9 questions, rather than 20 or so as we seem to do in today’s examination mode. In essence, either they learned it or they didn’t.

Let’s consider these problems one at a time in the context of what an 8th grader had to know. It is most likely that the information they had to know to answer these problems was, at some point, presented and discussed in class. Basically, it’s a set of memorized information. Think about this in the pedagogy at that time compared to what students today are expected to know. What they needed to know then is in the context of their daily lives and the business of the day.

In question 1 students had to know the Fundamental Rules of Arithmetic. Do we teach this today? Would it be of value for students to know? What are they? They are the four basic operations – addition, subtraction, multiplication, division which we teach but don’t identify as the fundamental rules.

In question 2 students have to be able to calculate volume but the critical thing they have to know – because it’s not given in the problem – is the volume of a bushel of wheat. Quite likely, since this is an example in Kansas, knowing the volume of a bushel is a very handy piece of agricultural information. As it turns out, there are two possibilities and we have to assume that the teacher at that time made it clear what was being asked in the question. A bushel of wheat has a volume of about 1.2445 cubic feet. There is also a heaped bushel, which is 27.8% (sometimes 25%) larger than a regular bushel. The regular bushel is also called struck measure to indicate that the bushels have been struck, or leveled, rather than heaped. And by the way, I didn’t know any of this and had to look it up. I didn’t grow up in an agricultural area. One more thing – given 4 decimal places in the volume and the 27.8% (sometimes 25%), we have to again assume that the teacher gave precise directions on how to handle these two values, most likely – my guess – by rounding 1.2445 to 1.

Question 3 looks like a straight forward calculation problem. The only possible issues are students (1) knowing how to handle the 1050 lbs. – do I subtract the 1050 before or after calculating the worth or (2) mishandling the decimal point in the 50cts.

Question 4 is again a straight forward calculation problem. What got my attention here is carrying on school for 7 months. It made me wonder if school was a 7 month period or if they simply used 7 because it fit the other numbers well for the calculation? It seems plausible that there could be a 7 month school year because it’s an agricultural area and families worked together to get the farm work done. Just a thought.

In question 5, students need to know how many pounds in a ton, which we assume was talked about in class at some point.

Question 6 seems to be a rather sophisticated problem for the 8th grade, but again in the context of life at that time, it seems reasonable that an 8th grader might be involved in the family’s business and would use this kind of calculation. I suspect it was a simple plug-these-values into the formula they learned. At that time, rote knowledge was highly prized. Note several things: how to use percent in decimal form and also realize that this was all paper and pencil calculations; no calculators then.

Question 7 has inches, feet, and metre as measurements so the student is being tested on measurement conversions. Once all the conversions are done, it again becomes a straight forward calculation. The issue, since 1 meter is equal to 3 feet and 3.37 inches, is what decimal value was used or were they taught to use just 39 inches. But again, it’s something that they presumably were presented and were expected to know. It’s sort of cute to use 12 inches, simplifying the calculation.

Question 8 is like question 6 in the sense that it’s sophisticated for the 8th grade, but again something very useful if a 14-15 year old was helping out with a family business. I again suspect this was a simple plug-these-values into the formula they learned. And as in question 6, students had to know how to write 10 percent as a decimal and further, again, there were no calculators then; all pencil and paper calculation. And what, exactly, is a bank discount?

Question 9 presents conversion issues between acre and rod – common, useful agricultural measures at that time which I again will presume were covered in class and students had to know. But they still had to do the calculation with paper and pencil. Technically and precisely both acre and rod had decimal values but I suspect it’s possible it was rounded off when talking about it and when doing calculation. The rod was a measure of 5 and a half feet, so the students had to again know how to handle decimal calculation if that was the value used. It’s interesting that they noted a “square” farm, since acre is a measure of area and the farm could be any shape. Perhaps it was just the teacher tinkering with the students, as some teachers – even today – are wont to do.

When you look across these 9 questions, it becomes apparent that they are all what we call today “application” or real life problems. These questions also involved sophisticated calculations based on formulae they clearly had to learn. It seems that the information that 8th graders needed to know was normal for that time, yet the pencil and paper calculations they needed to do were demanding. In this day and age, would there be an outcry if students had to do similar problems, having had to memorize the formulae and had to do the calculations without a calculator? I’m not going to venture a guess.

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Unequations Buzz

Posted by mark schwartz on August 11, 2016

Introduction

Had a thought. Simple one-variable 1st degree equations, by definition, state that there is a bunch of stuff “here” that equals a bunch of stuff “there”. For example, 2(3x ̶ 1) = 5(x + 1). What is meant by “equal”? Looking at this equation, obviously the two bunches of stuff are not equal! What this statement means is that if you can find the value of the variable “x”, replace the “x’ with that value in both sides of the equation and evaluate both sides, the value on both sides of the equation sign will be equal. Thus, that’s why one solves for the value of “x”.

The fundamental rule for solving equations is “whatever you do to one side of the equation, you do to the other side.” This, in essence, maintains the equality. My thought was that rather than start with an equality and burp out the rules, start with an unequation and have students play with it to find out how to make it an equation. However, we won’t use paper and pencil; we’ll use poker chips.

The Story

In order to solve an equation of this order, students need to know a lot of stuff – identification of terms, order of operations, distributive law, the four basic operations with signed numbers and to verify their answer, substitution of a value for the unknown and of course the basic rule of “whatever you do to one side of the equation, you do to the other side.”

Solving unequations is simpler and is a kinesthetic, visual way to have students play with all those things which, in my view, expands their conception of equations. In many instances, I’ve seen students who know all the elements but somehow can’t blend them together to solve equations. Here’s how unequations work.

Each group of students (2 or 3 to a group) gets a handful of white poker chips and each chip has a positive on one side and a negative on the other. You can use other markers if you choose.

I ask them to put 1 to 5 chips in each pile but the total value in each pile can’t be the same. Two questions that always comes up are (1) can we put positives in one pile and negatives in the other and (2) can we put positives and negatives in the same pile? So, right away, they’re thinking about this exercise; they’re engaged. We have a discussion about this and although they don’t yet know what to do with these 2 piles (although some guess they’re equations), I let them determine what is allowable. So again, right away they “own” this exercise because they have determined what’s allowable. By the way, the discussion about what is allowable has many branches and typically includes a lot of “what if” banter. I just listen.

Once this is resolved, I then ask them to label the pile on the left “A” and the pile on the right “B”. This also is fun because there typically is someone who stacks the piles vertically rather than horizontally, so I simply say the pile furthest from you is A and the pile closest is B.

When everyone is ready I then ask them to do something to their pile A such that the total value in both piles is equal. This is also a fun point in the exercise for classes that allow positives in one pile and negatives in the other, but overall the buzz within each group again is one of the goals of this exercise. When this is done, I ask them to return to their original piles and then I ask them to do something to their pile B such that the total value in both piles is equal.

In both cases, I ask them if there was only 1 way to make the piles equal. Buzz, buzz again and the consensus was yes.

The next question to them was do something to both piles at the same time such that the total value in both piles is equal. This really generates buzz and questions to me, which I say I’ll answer later. The reason I won’t answer is that I want them to explore how this works. What they discover is that there is an unlimited number of ways to do this. For example, if A = 2 and B = 4, add 5 to A and 3 to B and both piles equal 7. There usually is an “aha” moment when they realize that as long as the difference between the two numbers added to A and B is 2, the total value will always be equal. Some also discover that unequal amounts can be subtracted from both piles and further that two numbers differing by 2 can result in an equal value in both piles. And there’s another “aha” moment – the total value in both piles can be negative if both were positive at first! And what’s more, zero is a valid value!

So, we played with these 3 options for a while and there was discussion all along about not only what was allowable but also the range of answers under the different conditions. Then we moved to equal piles to begin the exercise.

I ask them to adjust their piles so that there is an equal number in both piles. This then brings up the issue of their rule allowing positives in one pile and negatives in the other, if they allowed this. They realize they have to rule it out. But I then ask if they can have an equal value in both piles while having positives and negatives in the same pile. Can the total in both piles be positive or negative? Buzz, buzz and the conclusion is that it’s ok but this comes after a lot of discussion and this really gets them going about signed numbers. For example, if they are to have 3 positives in both piles to begin, they could put 4 positives and 1 negative, or 6 positives and 3 negatives or … here it goes again with an unlimited number of both as long as the total is 3.

So, I ask them to consider there beginning equal value in both piles and typically they make it simple – either all positive or all negative and they do this partly – they tell me – because they don’t know what I’m going to ask them to do. At this point, the equation question arises and I have to admit that we’re headed in that direction. After playing with this for a while, the class concludes (again) that there is an unlimited number of values that can be added or subtracted to maintain the inequality.

The next step is to give each group a few blue chips. What the group is asked to do is have one person look away of shut their eyes while the others in the group do two things: (1) set up two piles with an equal number of chips in both and (2) remove a certain number of chips from one of the piles and place a blue chip in that pile. In essence, create a simple equation. When they are done setting it up, the closed-eye person is to look at what they’ve done and answer the question: what must you replace the blue chip with in order to make the piles have equal value?

Do each of these exercises until the class seems comfortable with all the ideas that got buzzed about.

At this point, if you’d like to extend this 2-pile concept to work with introducing work with equations, see Chipping Away at Equations in this blog. It links up with this posting and together it gives students a different view of equations.

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Walk the Clock: It’s Fractions

Posted by mark schwartz on August 3, 2016

Introduction

For some reason, or perhaps reasons, fractions don’t make sense to many students. Despite the visual representations in text and/or the use of manipulatives such as Cuisenaire rods, fractions seem to remain a mystery to students. One day I asked all my basic math classes “What makes fractions so hard?” The overwhelming response focused on remembering the steps of the 4 basic operations. For them, operations with fractions seemed nothing more than trying to remember the steps to get the answer. Somehow, math instruction throughout elementary and secondary education led students to think not about what fractions mean and what they represent but rather to think about how to “solve the problem”. So, I played with an idea which seems to have provided a way for students to “see” fractions a little differently.

The Story

DON’T TELL STUDENTS THIS IS FRACTIONS! If someone asks if this is fractions, tell them it will be discussed after the activities are done. I’ve provided an idea on how to do this in the discussion section following the demonstration of the activity.

The students preferably will work in groups of 3 (or 2, depending on the size of the class). The minimum grouping is 2. Each group gets a magic marker and 12 paper plates. The students are to number the plates 1 to 12. The plates are to be placed on the ground as a clock face. This activity is best done outdoors but if not, move the desks and chairs to allow for each group to have enough space for one person to walk inside and one person to walk outside a clock face circle. If neither of these spaces are available, the plates can be cut down in size and placed on a table top. If it’s a rainy cold day and going outside is a bad idea, and if the curriculum allows and time allows, make it a “review” day and hope for sunny and warm tomorrow. This activity works indoors and on the desk top but outside is best; it’s more fun. If done on the desk top, 2 markers per group will be needed. These markers will be the “walkers” in the activity (this will be explained below).

Here’s how it works. Lay out the plates as a clock face. One of the people in the group will walk outside the circle (call this person the outsider); one person will walk inside the circle (call this person the insider); the third person will be the reader/recorder (call this person reader). Give each group a copy of the activities (a sample is below) which states what the insider, outsider and reader are to do. Once the groups have figured out who will do what, give a demonstration of what they are to do, using the 1st activity.

Using the first activity and using one group to demonstrate, note that both walkers will walk twice. Both walkers start at “12”. In each activity, the insider walks first and then the outsider. The first walk is done when the insider reaches “12”. The second walk for both starts where their first walk ended. The reader is to watch and verify that each walker takes the right number of steps (others in that group can help verify).

1st Activity: on the first walk, the insider walks 2 units while the outsider walks 1 unit. On the second walk, the insider walks 3 units while the outsider walks 1 unit. The reader will note “outsider.insider”. In this activity, the record should show 10.12.

If there is confusion about the walking and/or the recording, just repeat the first activity. When everyone’s ready, move on to the next activities.

2nd Activity: on the first walk, the “insider” walks 4 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 6 units while the “outsider” walks 1 unit. The reader should note 5.12.

3rd Activity: on the first walk, the “insider” walks 2 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 3 units while the “outsider” walks 2 unit. The reader should note 14.12. (There should be questions on how to record this. Show students “military” time.

It’s important that if more activities are to be done, don’t allow students to do it. The reason: activities provided by students may result in a very time consuming set of walks and more critically, present a new issue to handle. For example, although subtraction of fractions can be done this way, I suggest not doing it. You could get a negative answer and you might want to avoid this. Just stick with one concept at a time; adding fractions (although they may not realize it). Given this, you might want to prepare and walk through a bunch of activities and be careful that none of them take too much time, yet enough time for the students to play and enjoy it.

Again, do not say anything about fractions at this point, but what has happened is that the problem 1/2 + 1/3 has been done. The record “outsider.insider” is 10.12 , or in reduced fractional notation is 5/6. Most likely, someone has noticed that the insider always has a value of 12. You sort of have to weasel your way around this and don’t yet call it a common denominator.

A Little Discussion. After these activities, you can transition to presenting fractions as you usually do. But, here’s one idea to consider in talking to students about how this activity demonstrates addition of fractions. What is seen and used but not referenced is the common denominator of 12. This explains why the insider’s walking the line twice isn’t counted twice. In the problem 2/3 + 3/4 , the denominator could be any multiple of 12 but in this case since it is 12 and you know it, don’t count it twice. Students may balk at this idea but it can be explained further. The insider always walks the line twice but always restarts the 2nd walk at “12”, while the outsider restarts the 2nd walk where the first walk ended so the insider’s walks aren’t added, rather they simply repeat.

Also not seen is the addition, but it occurs in the outside walk when the second walk starts where the first walk ended. The outside walker’s position at the end of the first walk is added to the beginning of the second walk. Please note that using this method for a problem such as 1/2 + 1/3 would give the answer 10/12, not 5/6, so clearly reducing fractions has to be addressed before this activity. Further, you might question how to get from this activity to the “rules” for addition and subtraction, but that’s not the point, although it can be seen because both fractions in this example, were converted to equivalent fractions with a denominator of 12, although in this case and others, it wouldn’t necessarily be the lowest common denominator. This again, could create a teaching moment, discussing the issue of common denominator versus lowest common denominator.

I suggest that different sets of students get a chance to walk the line. In fact, teams of students could do it; two walk and the others verify that their walking is accurate. Further, the point at which the transition from this activity to the traditional fraction work is to be made is a matter of how the class is collectively responding. In some instances, students caught on and realized that this was adding fractions. But even if they caught on, I still had them walk through all the activities. In several classes, students wanted more exercises. I think it was because it was a nice warm day. It’s a judgment call.

One more thing. Recall that the purpose of this activity is to give a visual and kinesthetic sense to the “rules” and it does seem to have a positive effect on students. When we got to the traditional rules and procedures, I heard students talking about how it “matched up” with what they were doing outside. Play with it.

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Metric to Metric Conversion: Ultimately, it’s a Proportion!

Posted by mark schwartz on July 27, 2016

Introduction

This discussion came about because one student in one class simply asked “Why does this work”? He was referring to the procedure for converting metric units to other metric units, for example, “how many centimeters are there in 10 kilometers?” He could see that the “moving the decimal point” procedure worked but he kept insisting that there must be more to it; that somehow someone had figured this out and he wanted to know how it had been figured out. I had no clear answer to this and told him (and the class) that I would think about it. What I came up with isn’t necessarily the reality of the derivation of that procedure, but it did start with something that we had already discussed in class – proportions – and he was willing to accept this as a demonstration of why it works but wasn’t about to consider it a true explanation. Loved this guy!

The Story

In Colaw and Ellwood’s 1900 School Arithmetic: Advanced Book (page 252) is a discussion of the metric system. Among other interesting things, they note that kilo hecta and deka are Greek, while deci, centi, milli are Latin. In reading through their discussion, I got to thinking about how they, as we do today, convert one metric unit to another: a 7-point scale and simply “move the decimal point” … but some of their commentary made me think about how and why this 7-point scale works.

If asked how many decimeters are in .04 kilometers, one has a variety of strategies to use. If the 7-point scale (kilo hecta, deka, unit, deci, centi, milli) is known, one can write .04 at the kilo point on the scale and then visualize moving from kilo to deci, which would give a move of 4 places to the right. If the decimal point is moved four places to the right, this shows that .04 kilos is 400 decis. Typically, students are accepting of this ‘shortcut’ because it is much more manageable than other systems. But, the question was “why does it work?”

I believe the underpinning for the move-the-decimal method is to do the problem by first converting all the units to the amount at each point on the scale that equals one unit. This by no means is a rigid mathematical derivation but rather a way of demonstrating the relationships using a previously studied math relationship, namely proportions.

The traditional 7-point scale looks like this:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1000             100                  10                    1                      1/10                 1/100               1/1000

This scale shows the number of units in a named place-value. “Kilo” means 1000 units; “deci” means 1/10 of a unit, etc.. But let’s ask the question from the point of view of the unit: how many kilos would it take to make a unit? How many decis would it take to make a unit, etc.?

Here’s how the “unitized” 7-point scale would look:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1/1000          1/100                 1/10                 1                      10                    100                  1000

It appears as though the scale has been reversed, and it has because we are viewing the information from the perspective of what it takes to make one unit. For example, it can be read as “1/1000 of a kilo equals 1 unit” or “10 decis equals 1 unit”, etc. The point of this is that all of the place-value names are now on the same scale and having them on the same scale permits one to establish proportions.

For example, on this scale 1/1000 of a kilo equals 10 decis because they both equal 1 unit. Another way of stating this relationship is to state that “1/1000 kilos is to 10 decis”, which is a phrase describing the first rate of two rates that would make up a proportion. What would be the second rate? The original problem was “how many decimeters are in .04 kilometers?”

In this case, being consistent with the idea in proportions that the numerators are all the same type of units and the denominators are all the same type of units, what is seen is the relationship of kilos to decis, is:

kilo  1/1000  =  .04  = 400 decis
deci    10        x

It is this proportional relationship which provides the basis for conversions from one metric unit to another, as long as the units used are those that “equate” them to 1. Students must be comfortable knowing how many ‘dekas’ it takes to make one unit (since a ‘deka’ is 10 units, it takes 1/10 of a ‘deka’ to equal a unit). This may seem counterintuitive since we typically say, for example, that a kilo is a thousand units, which is true but the focus here is with how many kilos it takes to make a unit.

Given this discussion of the two methods, it seems most likely that students would tend toward the ‘move the decimal point’ system. It doesn’t require any computation. But the point of presenting both of these it to bring out the reality that the ‘easier’ system is based on a proportional system. Just another example of the power of proportions based on an interested student’s insightful inquiry.

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