Why binary: logic gates

Students try every combination of two switches in a circuit to build AND and OR truth tables, use NOT, then add two bits with a half adder, to explain why digital systems are built on binary (AC9TDI8K04 node 3).

Data representation · Binary numbers · about 40 minutes · Open the student lesson

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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Worksheet Download Print the student copy; the Teacher edition has the answers.

Students will be able to

I can give the output of AND, OR and NOT gates.
I can explain why computers use binary.
I can show how gates add two bits.

Timing

Predict and Try 7 min · Worked example 10 min · Vary 10 min · Practise and Make 13 min

Before class

Try the boards yourself. A torch, a battery and two real switches (or a breadboard) make the Try concrete in a minute.

The one idea (the kit's chain for this node): electronic components have two reliable states → those states are 0 and 1 → gates combine them → so binary is the foundation of all computing. The half adder in Make shows gates doing the binary addition students know from lesson 5.

Simplifications: real gates are made of transistors, not mechanical switches; the circuit view is an analogy that holds for AND and OR.

Running the lesson

Predict: students who chose "yes" picture parallel wiring. Ask them to trace the current.
Try: have students fill a paper truth table as the board's fills.
Worked example: before the OR line, ask what changes when the switches sit side by side.
Vary: ask why AND has only one row that gives 1.
Make: connect to 1 + 1 = 10 in binary and to carrying in base 10.

The hard part (on purpose)

Students read OR as everyday English (one or the other, not both). Point at the circuit: with both switches on, is there still a path for the current?

When students get stuck

What you will see	A question to ask	The fix
A student writes 1 OR 1 = 0 (Sam's mistake).	With both switches on side by side, can the current reach the lamp?	Yes: OR means at least one. One but not both is XOR.
A student thinks gates add (1 AND 1 = 2).	How many states can the output wire have?	One bit: 0 or 1. Adding takes several gates, as in the half adder.

Answers to the checks

two switches in a line: A battery, two switches and a lamp are wired in one loop, one switch after the other. Switch A is **on** and switch B is **off**. Does the lamp light?
No. With the switches in a line, the current has to pass through both, so B being off breaks the loop. The lamp lights only when A AND B are on.
Try: every combination: Click the switches to try all four combinations of A and B. When does the lamp light?

Only when both switches are on: 1 AND 1 = 1. Every other row gives 0. That is an AND gate.
Truth tables: Two switches side by side (an **OR** gate). A = 1 and B = 0. What is the output?
1
OR gives 1 when A or B is 1 (or both). A is 1, so the output is 1.
Find the mistake: Sam fills in the OR gate's truth table. Click the line where it first goes wrong.
line 5
Line 5: OR gives 1 when at least one input is 1, so 1 OR 1 = 1. "One or the other but not both" is a different gate, XOR.
Why binary?: Why are computers built on binary, not on ten levels for the digits 0 to 9?
Switches have two states that are easy to tell apart, and gates can combine them to do any calculation
Two states are reliable even with a noisy signal, and switches wired as gates (AND, OR, NOT) can be combined to add, compare and store anything.
OR with one on: Switch the gate to OR. Turn on only A. What is the output?

OR needs only one input on: 1 OR 0 = 1.
OR with both on: Keep OR and turn B on too.

1 OR 1 = 1: both on still counts as at least one.
NOT: Switch to NOT. Turn A off. What comes out?

NOT flips its one input: NOT 0 = 1.
AND's one row: Switch back to AND and make the output 1.

Only one row gives 1: A = 1 and B = 1.
Does B matter?: An AND gate has A = 0. Does it matter what B is?
No: the output is 0 whatever B is
AND needs both. With A = 0, the output is 0 for either B.
AND: What is **1 AND 0**?
0
AND gives 1 only when both are 1.
NOT: What is **NOT 1**?
0
NOT flips 1 to 0.
Alarm: A car alarm should sound if the door is opened **or** the window is broken. Which gate?
OR
Either input should set it off, so OR.
Machine guard: A machine should run only when the guard is closed **and** the start button is pressed. Which gate?
AND
Both must be true: AND.
Review: binary: Review: what number is **110** in binary?
6
110 = 4 + 2 = 6.
add two bits: This circuit adds two bits: the sum is A XOR B and the carry is A AND B. Set the switches so the answer is 10 in binary (2).

1 + 1 = 10 in binary: the sum bit is 0 and the carry is 1, just like 5 + 5 = 10 carries a 1 in base 10. Chains of these gates add numbers of any size.