Exploding Dots

1.4 Arithmetic in a \(1 \leftarrow 10\) Machine

Lesson materials located below the video overview.

We humans have a predilection for the number ten. We base our system of arithmetic on this number, and its powers, most likely because of our physiology: we are born with ten fingers on our hands.


Comment: There are many cultures that work with numbers systems of base \(20\). Why the number twenty do you think?

And we have vestiges if base \(20\)  in our western system. For example, how does the Gettysburg address begin? (What’s a “score”?) How do you say \(87\) in French? (Translate it, literally, word for word!)


The \(1 \leftarrow 10\)  machine expresses all numbers in base ten. For example, here is the number \(273\) in this base-ten system:


Here is the number \(512\):


If we add these we obtain clearly get seven dots in the hundreds box, eight in the tens box, and five in the units box:


This is the number \(785\)  and we’ve just computed the sum:


And, again, this reads:

Two hundreds plus five hundreds gives \(7\) hundreds

Seven ones plus one ten gives \(8\) tens

Three ones plus 2 units gives \(5\) units


Question 20:  Draw the \(1 \leftarrow 10\) dots and boxes picture for the number \(3704\). Add to this the picture for \(2214\). What is \(3704+2214\)?


Let’s do another one. Consider \(163+489\).


And this is absolutely mathematically correct:


One hundred plus four hundreds does give \(5\) hundreds

Six tens plus eight tens does give \(14\) tens

Three ones plus nine ones does give \(12\) ones.


The answer is \(5 | 14 | 12\), which we might try to pronounce as “five hundred and fourteeny-tenty twelve”! (Oh my!) The trouble with this answer – though correct – is that most of the rest of the world wouldn’t understand what we are talking about!

Since this is a \(1 \leftarrow 10\) system we can do some explosions.


The answer is “six hundred forty twelve”! Still correct, but let’s do another explosion:


The answer is “six hundred fifty two.” Okay, the world can understand this one!


Question 21:  Solve the following problems thinking about the dots and boxes. (You don’t have to draw the pictures; just do it!) And then translate the answer into something the rest of the world can understand.ex14010


Recall that ALL solutions appear in the COMPANION GUIDE to this EXPLODING DOTS course.




Let’s go back to the example \(163+489\). Some teachers don’t like writing:


They prefer to teach their students to start with the 3 and 9 at the end and sum those to get 12. This is of course correct – we got 12 as well.


But they don’t want students to write or think “twelve” and so they have them explode ten dots:


and they have their students write:


which seems mysterious at first.  But it makes sense to us, because have put that “1” in the tens place which is exactly what an explosion does.

Now we carry on with the problem and add the tens:


and students are taught to write:


And now we finish the problem.


and we write:


Teachers like to teach their students to do all their explosions as they go along. This means that students must start to the right of the problem and work towards the left “carrying” digits that come from the explosions. (This can be seen as odd since we are taught to read left to right!)


In the dots and boxes method one adds in any direction or order one likes and performs the explosions at the end.


WHY DO TEACHERS LIKE THEIR METHOD? Because it is efficient.

WHY DO I LIKE THE DOTS AND BOXES METHOD? Because it is easy to understand.


Question 22: Redo the problems of question 21 the teachers’ way. You will see that it is quicker.


Question 23: HARD CHALLENGE. Here is an addition problem in a \(1 \leftarrow 5\) system. (THIS IS NOT \(1 \leftarrow 10\) ).ex14019


a)    What is the \(1 \leftarrow 5\) answer?


b)    If this were an addition problem in a \(1 \leftarrow 10\) system, what would the answer be?


Question 24: Jenny was asked to compute \(243192 \times 4\). She wrote:ex14020

a)    What was she thinking? Why is this a mathematically correct answer?


b)    Translate the answer into a number that the rest of the world can understand.


c)     Find the answers to these multiplication problems:




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