I was working with some kids who wanted to boost their arithmetic skills and so together we thought through the following project.
We wanted to beef up skills in addition, subtraction, multiplication and division. One way to do it was to teach the computer how to make up an exercise, then ask the user to solve the problem, and check the response with the actual answer.
To analyze the problem we wrote out some sample exercises such as
6 + x = 12, 13 - x = 6, 5 * x = 25, 30 / x = 6
To make up one of these problems the computer has to generate two of the numbers and compute the third.
We decided to start with addition.
Logo is equipped with a random number generator. It takes one input and then selects a number from 0 up to the input.
make “number1 random 11
:number1 is then 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10
and then we make the second number.
make “number2 random 11
Add the two numbers and save the result.
make “result :num1 + :num2
So we make a procedure that prepares the three numbers and then states the problem, asks for answer, checks it against the correct one, and keeps asking the same question until the user responds with the right answer.
In fooling around we found that two procedures were needed. One procedure makes the three numbers. It then turns the job over to an assistant (subprocedure) to ask the question and insist on getting the right answer.
to add make "num1 random 11 make "num2 random 11 make "result :num1 + :num2 add2 end to add2 question (se :num1 [+ x =] :result char 13 [What is x?]) if answer = :num2 [stop] add2 end
Try it, it works.
Have you ever thought of the relationship between addition and multiplication? Well, in writing these procedures we saw that the procedure for multiplying was very similar to addition. The real difference lay in how the procedure computed the result.
to multiply make “num1 random 11 make “num2 random 11 make "result :num1 * :num2 multiply2 end to multiply2 question (se :num1 [* x =] :result char 13 [What is x?]) if answer = :num2 [stop] multiply2 end
Try it, it works.
But wait a minute! we noticed that add2 and multiply2 do similar
jobs. From a programming point of view the only difference is whether
a + sign or a * sign is printed. Is it possible to make one
procedure work for both add and multiply? We could then use that
procedure for subtracting and dividing.
So we make a new procedure which takes an input. The input is either
+ x or * x.
to check :phrase question (se :num1 :phrase :result char 13 [What is x?]) if answer = :num2 [stop] check :phrase end
and edit add and multiply.
to add make "num1 random 11 make "num2 random 11 make "result :num1 + :num2 check [ + x =] end to multiply make “num1 random 11 make “num2 random 11 make "result :num1 * :num2 check [* x =] end
They work but I now see another fix I’d like.
We want a superprocedure which calls on add and multiply and eventually subtract and divide.
to math add multiply end
So now on to subtraction.
In this case we wanted to make sure x is always positive. There are different ways to do this. I chose is to get num1 by adding result and num2. Thus subtracting num1.
To subtract make “result random 11 make “num2 random 11 make "num1 :result + :num2 check [ - x =] end
Try it.
Dividing uses the same technique as the subtract procedure. The divisor is computed from the other two numbers. The difference here is that we wanted to avoid the possibility of 0 being any of the numbers. This is accomplished by adding 1 to the random number.
to divide make “result 1 + random 10 make “num2 1 + random 10 make “num1 :result * :num2 check [/ x =] end
The basics check out. But the poor user is forced to keep trying until arriving at the right answer. Limits were needed. I chose to move on after the third incorrect answer.
To accomplish this we had to set up a counter initially set to 1.
make “times 1
The check procedure would then have the job of keeping tabs on times and then either stopping the procedure or adding 1 to times and asking again for the answer.
Here is the new check procedure.
to check :phrase question (se :num1 :phrase :result char 13 [What is x?]) if answer = :num2 [stop] if :times > 2 [stop] make “times :times + 1 check :phrase end
For visuals we, of course, use turtles. We played around and made several shapes or costumes including the words yes, no, answer and =. To display this much information we used three turtles. Next, we wanted to show the correct answer and so I drew the ten digits.
As we were debugging, we decided to plan for answers being as large as
Numbers are words and we can take them apart using first and butfirst etc. By putting the digits in shape cells 1 through 10 and placing 0 in cell 10 we could say setsh first :number. Of course, we would have to see if first :number were 0. Then the command would be setsh 10.
showit displays the answer digits from left to right. Here is Showit
with comments.
to showit :number :places ; :places is the list of the four answer turtles. ask [t2 t3][ st] ; shows answer = if empty? :number [wait 20 ask [t1 t2 t3 t4 t5 t6 t7] [ht] stop] ; display the answer for 2 seconds, then hide all tto first :places ;display a digit ifelse 0 = first :number [setsh 10 st][setsh first :number st] showit bf :number bf :places ;do the same thing for the next digit end
to math setup Add Subtract Multiply divide end to add make "times 1 make "num1 random 11 make "num2 random 11 make "result :num1 + :num2 check [ + x =] end to multiply make "times 1 make “num1 random 11 make “num2 random 11 make "result :num1 * :num2 check [* x =] end to subtract make "times 1 make “result random 11 make “num2 random 11 make "num1 :result + :num2 check [ - x =] end to divide :result :num2 make "times 1 make “result 1 + random 10 make “num2 1 + random 10 make "num1 :result * :num2 check[ / x =] end
The check procedure looks complicated because the if instructions are long. I put yes in cell 17 and no in cell 18.
to check :phrase question (se :num1 :phrase :result char 13 [What is x?]) if empty? Answer [stop] if :num2 = answer [t1, setsh 17 st showit :num2 [t4 t5 t6 t7] stop] if :times > 2 [t1, setsh 18 st showit :num2 [t4 t5 t6 t7] wait 20 stop] make "times :times + 1 t1, setsh 18 st wait 10 ht check :phrase end to showit :number :places ask [t2 t3][ st] if empty? :number [wait 20 ask [t1 t2 t3 t4 t5 t6 t7] [ht] stop] tto first :places ifelse 0 = first :number [setsh 10 st][setsh first :number st] showit bf :number bf :places end
The setup procedure is only to restore the seven turtles to their original positions and looks.
to setup ask [t1 t2 t3 t4 t5 t6 t7][ht sety 130] t1, setx -250 t2, setx 50 setsh 19 t3, setx 85 setsh 20 t4, setx 170 t5, setx 200 t6, setx 230 t7, setx 260 end
Sometimes display the math sentence.
6 + x = 12, 13 - x = 6, 5 * x = 25, 30 / x = 6