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 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.

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Driving the Integer Road

Posted by mark schwartz on April 27, 2016



A lot of texts these days use colored chips or other manipulatives to introduce students to the operations with signed numbers. This article goes one step further. I was using a “chip” concept but then realized that even the chip idea is somewhat abstract. So I visualized something that turned out to be effective … and fun.

The Story

Operations with signed numbers seem to escape a lot of students. This is particularly so when they are confronted with all the rules for the four basic operations. The one they seem to remember is “a minus and a minus is a plus” but they tend not to remember what operation is involved. This rule tends to get applied whenever they see two negatives in proximity to each other in a problem.

.Rather than presenting the rules to the class, they were provided a visual and kinesthetic activity which “generated” the rules. They were shown one way in which the rules were embedded in a daily activity. In essence, any rule or formula is a statement of a relationship between quantifiable activities and in our daily lives, one was found that could be used.

So looking at activities and playing with them might lead to the realization that there are quantifiable components, which ultimately may lead to or actually be the rule. It’s a matter of observation and a matter of determining which symbolizing system to use.

For example, in the following demonstration, letters and numbers will be used to indicate an action. The action will be precisely prescribed. Students will be asked to do a specified series of actions. These actions are based on something that they (well, at least most of them) do.

Here we Go …

The activity that most students do is drive. Their path is a road, but in this activity their path will be a number line. Certain conditions must be set: (1)  a car on the number line (the road) can face forward ( FF, toward the positive end of the number line) or face back ( FB, toward the negative end of the number line) and (2)  the car can move forward (MF, put it in drive) or move back (MB, put it in reverse).

A car (I use Tonka toys: small cars and trucks) can be put on the number line either facing forward or facing back. Once it’s on the number line¸ it can move either forward or back. At this point, it has to be clarified and emphasized that facing is with reference to the number line and moving is with reference to the car. I believe this to be a critical distinction because in our math symbolizing system, we use “ – “ for the action of subtraction but also the indicator for the sign of a number (negative). We use “ + “ to indicate the action of addition, but also the indicator of the sign of the number (positive). This is a confusing element for a lot of students. The notation needs to be discussed, examined, and understood. The exercise with the car makes this point very apparent, by having students begin with the FF, FB, MF, MB symbolizing system and ultimately showing how it is comparable to the use of “ + “,“ – “, and parentheses.

A demonstration of how the system works is given by drawing a number line on the board and using a picture of a car to face and move. I drew one on a piece of cardboard so that facing forward or facing back can be on opposite sides of the cardboard. The number line need only go to plus and minus five. The exercise always begins at zero. The direction to follow is stated as FFMF2. This means “face forward, move forward two”.  But combinations of facing and movement can be strung together as a “command line”. For example, FFMF2, FBMF3, FBMB1, FFMB3 would be done in sequence, beginning at zero with the second, third, and fourth command done from the previous end point.

Using an arrow to indicate where the car starts and which way the car is facing, and using a ● to indicate where the car would finish, the sequence FFMF2, FBMF3, FBMB1, FFMB3 would look like this.


-5     -4     -3     -2     -1     0     1     2     3     4     5

FFMF2                                →           


-5     -4     -3     -2     -1     0     1     2     3     4     5

FBMF3                        ●                  


-5     -4     -3     -2     -1     0     1     2     3     4     5

FBMB1                         ←   


-5     -4     -3     -2     -1     0     1     2     3     4     5

FFMB3        ●                      

After having students practice a bunch of these, a shift to conventional symbols can be made. But don’t hurry the class through this because it takes a while for them to get comfortable with the idea that facing is with respect to the line and moving is with respect to the car.

For example, FF and MF are +; FB and MB are −, and here’s a substitution to show how it works (BUT THIS IS NOT TO BE SHOWN TO STUDENTS UNTIL LATER).

FFMF2,    FBMF3,    FBMB1,    FFMB3

+ ( +2 )  – ( +3)   – ( –1 )  + ( –3 )

This standard notation can be further simplified but this relationship between facing and moving and the standard notation needs to be discussed, examined and understood before proceeding.

There are a few side benefits with this. Doing the facing and movement slows them down, engages cognition, sight, hand movement, decision making, all of which might be put in the context of problem solving. Further, when they are working in groups (in all my classes, students work in groups), they learn how to accept and give questioning and correction from peers, evaluate others’ actions as well as their own, and be deliberate. It also provides a kinesthetic and visual representation that aids the transition from positive-number-only operations to operations with signed numbers.

The key outcome, however, is that the “code” of facing and moving has a parallel pattern with the traditional notation. The pattern can be generated by asking the following for each command: Ask the class: for FFMFn (“n” represents any number), in what direction do you move? For FFMFn, you move in a “forward” direction. Asking the question for FFMB, FBMF and FBMB; for FFMBn, you move in a “back” direction; for FBMFn, you move in a “back” direction and for FBMBn, you move in a “forward” direction. As the answer to each of these questions is given, write down the following pattern:

Command       direction moved

FFMFn            forward

FFMBn           back

FBMFn           back

FBMBn           forward

This pattern contains most of the rules for operations with signed numbers, but this “code” has to be translated to the traditional notation. This is done by talking to the students about the slight difference in notation, including the use of parentheses as an indicator of multiplication. We earlier established (and it’s still on the number line on the board) that forward is “+” and back is “─”. Also remind students that facing and moving are two separate events.

The traditional notation can now be substituted in the above “coded” pattern. The “command” code list is the traditional notation “action’ list and the “direction moved” code is the traditional “outcome” list.

……  code   ……                                   …  traditional notation …

Command       direction moved             action           outcome

FFMFn                forward                       + (+n)              + n

FFMBn               back                             + (─n)              ─ n

FBMFn                back                            ─ (+n)              ─ n

FBMBn                forward                       ─ (─n)             + n

For example, take the problem 2 + ( ─ 3 ) + 4 ─ ( ─ 5 ) ─ ( +2 ). The first step is to look at the problem and see if any substitutions can be done based on the patterns. In talking about substitution, stress to students that this is, in fact, multiplication.

Look at the substitutions that can be made:  2 + (─ 3) + 4 ─ (─ 5) ─ (+2):

+ (─ 3) becomes ─ 3 ….. ─ (─ 5) becomes + 5 ….. ─ (+ 2) becomes ─ 2

So, the problem now is 2 ─ 3 + 4 + 5 ─ 2.

I recommend that at this point, students practice only the substitution idea with at least 10 more problems. I spend a lot of time at this point and don’t talk about what to do with this string of numbers. Finishing this problem after the substitution can be done a variety of ways. I prefer to have this string of numbers seen as a string of signed numbers to be added, employing the commutative and associate properties as needed. Again, there are multiple approaches but the focus of this article is to get to the string of numbers, not how to handle them, so the way to handle the numbers should come from experiences which you have found successful.


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