Last session ended with a diagram that looked ridiculous. Nobody in the room proposed the real fix, because everybody assumed there has to be one way of writing a number.
There does not. The human's notation and the machine's notation are allowed to differ, as long as something translates at the boundary.
Ninety seconds on this. The resolution is not the hour, the translator is.
before we start, one promise
Today: whole numbers. Nothing else.
No fractions. No negative numbers.
Not because they are difficult. Because each of them breaks something different, and each one is worth an hour on its own.
So everything you see today will work perfectly, and I am telling you in advance that the perfection is because of what we left out.
Naming the restriction now stops it feeling like an omission later, and it makes Class 5 feel like a promise being kept rather than a complication.
start somewhere you already know
What does this actually mean?
47
Not what is it. What does the writing mean. Say it out loud.
Wait for somebody to say "four tens and seven ones". Do not accept "forty seven". Make them unpack the notation.
you just stated the rule yourselves
Digits, times powers of the base.
Pick any base b. You get b symbols, from 0 up to b−1.
The columns are worth 1, b, b², b³ and onwards, right to left.
This is Babylon from two sessions ago, written as a formula. Nothing here is new.
Say the callback explicitly. They invented positional notation in Class 2 and are now writing down what they invented.
so put a two in it
Two symbols. Same rule.
click the bits · the weights are just powers of two
ADDS UP TO0
Reading binary is nothing more than adding up the columns that are switched on.
Get a student to call out a pattern and read it. Nobody finds this direction hard, which is exactly the setup for the next slide.
and yet
Reading binary took you four minutes.
You learned no new rule. You used the one you already had, with a different number in it.
So why does going the other way feel like a trick you have to be taught?
Here is 47. Give me the bits.
Let them try it in their heads for thirty seconds. Most will do greedy subtraction without knowing it has a name.
the way you probably just did it
Take the biggest column that fits.
Find the largest power of two that is not bigger than what is left. Put a one there. Subtract it.
Repeat until nothing is left. Every column you skipped gets a zero.
watch the number get eaten
Type in a number a student shouts out. It works for anything, which matters more than the example.
the way the textbook does it
Divide by two. Keep the remainder.
Divide, write down the remainder, divide the quotient again, and keep going until you reach zero.
Then read the remainders upwards. That is your answer.
Which looks exactly like a rule you are expected to memorise.
the division ladder
Deliberately present this as arbitrary. The whole point of the next slide is that it is not.
look at what each line is actually asking
Every one of those is the same question.
Is this number odd?
The remainder is a one when the answer is yes and a zero when it is no. That is all a remainder after dividing by two can ever be.
So the bits of a number are the answers to "is it odd" asked over and over, each time after halving.
same ladder, one extra column
There was never a rule to memorise. Nobody ever needs to look this up again.
This is the slide of the hour. If they leave with one thing, it is this. Slow right down.
now go back and look again
Point at the step that needed it to be two.
Divide. Write the remainder. Divide again. Read upwards.
There isn't one.
Ask it as a real question and wait. Somebody will start to answer and then stop, which is the moment you want.
divide by b, remainders upwards
The same procedure, six different answers.
Click any base above to run the ladder in it.
NUMBER
the ladder in base 2
Change nothing but the dial. The algorithm on screen never changes, only the number you divide by.
so what did we actually build
It was never a binary translator.
It is a base translator. Binary is one setting on a dial that has no special position.
The machine picked two for reasons that had nothing to do with mathematics, and mathematics did not object.
Which raises the obvious next question. Writing numbers down is one thing. Can you still compute with them?
Hold the room here. Everything so far has been notation. The rest of the hour is arithmetic.
the same two quantities, three times
Twenty three plus twelve.
Find the difference between these three. There is exactly one, and it is not the answer.
Do not explain. Let them stare. Somebody will say "the carries happen in different places", which is nearly right and is the door to the next slide.
here it is
You carry when a column reaches the base.
In base ten you carry at ten. In base five you carry at five. In base two you carry at two.
Line up the columns, add each one, carry when it fills. That is the procedure, and it is word for word the same in every base.
add anything, in any base
+
Same dial as three slides ago, on purpose. The control is the argument: the base is a setting, not a subject.
and now the part nobody warns you about
This is what you memorised as a child.
BASE TEN · 100 ENTRIES
BASE TWO · 4 ENTRIES
The left one took you about a year. The right one took you the length of this sentence.
Let the two tables sit side by side in silence for a few seconds. The size difference does the arguing.
and the same thing again
Multiplication is worse in decimal, and trivial here.
BASE TEN TIMES TABLE
BASE TWO TIMES TABLE
Multiplying by a binary digit is either copy it or write nothing. There is no third case.
The right hand table has no multiplication in it at all. Every entry is a decision about whether to copy.
so long multiplication collapses
Eleven times five, in binary.
Every row is either a copy of the top number or a row of nothing, shifted along.
No products to remember. No times table to recall. Just shift and add.
A machine that can shift and add can multiply.
Work it on the board in twenty seconds while they watch. Also flag that shift and add is literally what you will build in Class 24.
be honest about what this means
Binary arithmetic is easier than decimal.
Fewer symbols means smaller tables, simpler carries and shorter rules. Genuinely easier, not merely different.
And fewer symbols is exactly why it is unreadable to you.
The same property causes both. It is short for a machine and long for a person, and there is no version of this where you get both.
This is the honest centre of the trade. Neither side is a flaw. It is one property seen from two directions.
so let us settle it
Forty seven times nineteen.
Three different notations. Three different sets of marks on the page. One quantity.
Have somebody check the base five and base two answers convert back to 893. Do not assert it, verify it in front of them.
remember this
You could not do XLVII × XIX at all.
Two sessions ago you had two minutes and got nowhere, and it was not because Roman numerals are hard.
It was because they have no columns, so there is nothing to carry into, so there is no procedure to run.
Every positional base just did it. Base two, base five, base ten, all of them.
The base was never what made the method work. Position was.
Worth naming: they have now proved something about Class 2 using tools built in Class 4. That is what the arc is for.
everything today, in one line
Carry when a column reaches the base.
That is the only difference the procedure ever notices. Not the symbols, not the size of the tables, not which base a machine happens to prefer.
Put this on the physical board. It comes back in Class 6 when they run the same algorithms in base three and base seven.
so where does that leave last session
The machine choosing two costs us nothing.
Mathematically, nothing at all. Every procedure survives the change of base untouched.
It costs readability, and readability is fixed by a translator that fits on one slide and runs on a question a child can answer.
Those two red arrows are not waste. They are the price of letting each side be good at its own job.
Bring back the Class 3 pipeline image on the board if you can. Same picture, and it no longer looks absurd.
Every number in this hour has been a whole one.
Next session, try writing down half of something.
Stop here. Say nothing else. Do not hint at what goes wrong, and do not mention one tenth. They should walk in next time expecting it to be easy.