The shape of the lesson
By the end of the lesson, learners can
- Expression, circuit and truth table: Choose an expression in the lab and watch it become a circuit, then switch the inputs to see the values travel through each gate.
- From a problem statement: Pick a problem in the lab, write your own expression first, then reveal the answer to check.
- From a truth table: Fill in truth tables in the lab until the output column comes easily.
- Expression to circuit: one gate per bracket, two inputs per gate (except NOT), label inputs and output.
- Truth table: rows 000 to 111 in order, a working column for each intermediate gate.
- Statement to expression: one letter per condition, brackets around conditions that belong together.
- Table to expression: one AND term per row with X = 1, joined with OR; check one row of each output.
One picture every idea returns to
A logic circuit connects gates so that the output of one is an input to the next, from the inputs on the left to the output on the right. A logic expression such as X = (A AND NOT B) OR (B NOR C) describes the same circuit: each bracket is one gate. To build a truth table, list every input combination in binary counting order and add a working column for each intermediate gate output. To get an expression from a truth table, take each row where X = 1, AND the inputs together (using NOT for each 0), then OR these terms: the sum of products. In 9618, every gate except NOT has exactly two inputs.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Circuit, table, expression.. A logic problem can be written in four ways: as a problem statement, a logic expression, a logic circuit or a truth table. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Expression, circuit and truth table | Choose an expression in the lab and watch it become a circuit, then switch the inputs to see the values travel through each gate. Project tab 2 of the lab (Expression to circuit) and let a student drive it while the class predicts. Model the worked example One row by hand on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: From a problem statement | Pick a problem in the lab, write your own expression first, then reveal the answer to check. Project tab 3 of the lab (Problem statements) and let a student drive it while the class predicts. Model the worked example An alarm on the board, one line at a time. Your turn: A pump P runs when the tank is not full (F = 0) and either the timer T is on or the manual switch M is on (or both). Write the expression and give the rows (F T M) where P = 1. | Pairs |
| 21 to 30 min | Teach and check: From a truth table | Fill in truth tables in the lab until the output column comes easily. Project tab 4 of the lab (Complete the truth table) and let a student drive it while the class predicts. Model the worked example Two rows give two terms on the board, one line at a time. Your turn: Look again at the two rows in the example: C is 1 in both, and A and B are different. Write a shorter expression with the same truth table. | Corners |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 20 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in From a truth table, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Corners
Put one answer or viewpoint in each corner of the room. Students move to the corner they agree with, justify the choice to a partner there, then hear one voice from each corner before the reveal.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Working columns earn marks and catch errors. | Examiners give credit for a correct output column; a missing or out-of-order row loses it, so always list the inputs 000 to 111. |
| Do not draw a three-input gate. | To combine three conditions with AND, use two two-input gates, for example (A AND B) AND C. |
| A sum-of-products answer is accepted even if it is not the simplest. | A shorter expression is only right if it gives the same output for all 8 rows: check it with a truth table. |
Logic circuits, truth tables and expressions: worksheet
6 questions, 20 marks. Written in the style of the papers; no past paper question is reproduced.
A logic circuit has a NAND gate with inputs A and B, and an OR gate with inputs B and C. The outputs of these two gates are the inputs of an XOR gate, whose output is X.
(a) Write the logic expression for the circuit.
(b) Complete the output column X for the inputs A B C = 000, 001, 010, 011, 100, 101, 110, 111.
A car's warning buzzer W sounds if the engine is running (E = 1) and the seat belt is not fastened (B = 0), or if the engine is running and a door is open (D = 1). Write a logic expression for W.
Complete the truth table for the car buzzer in the previous question, for the inputs E B D = 000 to 111.
A truth table for inputs A, B and C has the output X = 1 in two rows only: A B C = 001 and A B C = 110. Write a logic expression for X.
X = (A XOR B) NAND (NOT C). Find X for each set of inputs, showing your working: (a) A = 1, B = 1, C = 0; (b) A = 1, B = 0, C = 0; (c) A = 0, B = 1, C = 1.
A student draws the circuit for X = NOT (A AND B) OR (B XOR C) using a single three-input gate and no NOT gate. Describe a correct circuit and give the only input row for which X = 0.
Answer key and marking notes
- (a) X = (A NAND B) XOR (B OR C).
- (b) Working: A NAND B = 1, 1, 1, 1, 1, 1, 0, 0 and B OR C = 0, 1, 1, 1, 0, 1, 1, 1.
- Rows 000 to 011: X = 1, 0, 0, 0.
- Rows 100 to 111: X = 1, 0, 1, 1.
- (E AND NOT B) for the seat-belt condition.
- (E AND D) for the door condition.
- W = (E AND NOT B) OR (E AND D). (Equivalent: W = E AND (NOT B OR D).)
- Rows 000, 001, 010, 011 (engine off): W = 0, 0, 0, 0.
- Row 100: W = 1; row 101: W = 1.
- Row 110: W = 0 (belt fastened, door closed).
- Row 111: W = 1 (door open).
- Term for 001: (NOT A AND NOT B AND C).
- Term for 110: (A AND B AND NOT C).
- Terms joined by OR: X = (NOT A AND NOT B AND C) OR (A AND B AND NOT C).
- (a) A XOR B = 0, NOT C = 1, 0 NAND 1 = 1.
- (b) A XOR B = 1, NOT C = 1, 1 NAND 1 = 0.
- (c) A XOR B = 1, NOT C = 0, 1 NAND 0 = 1.
- An AND gate with inputs A and B, whose output goes into a NOT gate.
- An XOR gate with inputs B and C; the outputs of the NOT gate and the XOR gate are the two inputs of an OR gate, giving X.
- X = 0 only for A = 1, B = 1, C = 1 (NOT (1 AND 1) = 0 and 1 XOR 1 = 0).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
For X = (A OR B) AND NOT (B AND C), which set of inputs gives X = 1?
- AA = 1, B = 0, C = 1
- BA = 0, B = 1, C = 1
- CA = 1, B = 1, C = 1
- DA = 0, B = 0, C = 1
In a truth table with inputs A, B and C, X = 1 in the row A = 0, B = 1, C = 1. Which term does this row add to a sum-of-products expression?
- A(NOT A AND B AND C)
- B(A AND NOT B AND NOT C)
- C(NOT A OR B OR C)
- D(A OR NOT B OR NOT C)
A greenhouse fan F switches on if the temperature is high (T = 1) and the window is closed (W = 0), or if the humidity is high (H = 1). Which expression is correct?
- AF = (T AND NOT W) OR H
- BF = (T AND W) OR H
- CF = T AND (NOT W OR H)
- DF = (NOT T AND W) OR H
How many logic gates are needed to draw the circuit for X = (A AND NOT B) OR (B NOR C), using only the gates named in the expression?
- A4
- B3
- C5
- D6
For X = (A XOR B) AND (B OR C), in how many of the 8 rows of the truth table is X = 1?
- A3
- B4
- C2
- D6
A truth table for inputs A, B, C has X = 1 only in rows 001 and 111 (written ABC). Which expression gives exactly this truth table?
- AX = C AND NOT (A XOR B)
- BX = C AND (A XOR B)
- CX = C OR NOT (A XOR B)
- DX = NOT C AND NOT (A XOR B)
Quiz key
For 1, 0, 1: A OR B = 1 and B AND C = 0, so NOT (B AND C) = 1, giving X = 1 AND 1 = 1. For 0, 1, 1 and 1, 1, 1, B AND C = 1, so NOT (B AND C) = 0 and X = 0. For 0, 0, 1, A OR B = 0, so X = 0.
Each row with X = 1 gives a product term: AND all the inputs, with NOT on any input that is 0, so A = 0, B = 1, C = 1 gives NOT A AND B AND C. The second option puts NOT on the inputs that are 1, the wrong way round. The two OR terms use the wrong operator: a product term joins the inputs with AND, and the terms are then joined with OR.
The first condition needs T = 1 and W = 0, which is T AND NOT W; the second is H on its own; "or" joins them. (T AND W) misses that the window must be closed, so W must be inverted. T AND (NOT W OR H) brackets the wrong parts, so high humidity alone would not switch the fan on. (NOT T AND W) inverts the wrong inputs.
The gates are NOT (on B), AND, NOR and the OR that joins the two brackets: 4. 3 forgets the NOT gate. 5 adds an extra NOT after the NOR, but the inverted output is already part of the NOR gate. 6 counts each input as a gate.
A XOR B = 1 in rows 010, 011, 100 and 101. Of these, B OR C = 1 in 010, 011 and 101 but not in 100, so X = 1 in 3 rows. 4 counts every row where A XOR B = 1 and forgets the second bracket. 2 leaves out row 010, where C = 0 but B = 1 still makes B OR C = 1. 6 counts the rows where B OR C = 1 and ignores the XOR bracket.
In both rows C = 1, and A and B are equal (0, 0 and 1, 1), so NOT (A XOR B) = 1; combining with AND gives X = 1 in exactly 001 and 111. C AND (A XOR B) needs A and B to differ, giving rows 011 and 101. Using OR gives X = 1 in six rows. NOT C picks the rows with C = 0, giving 000 and 110.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.