The shape of the lesson
By the end of the lesson, learners can
- Negative numbers: In the lab, set a starting number and a second number and step along the number line: adding a negative moves left, subtracting a negative moves right.
- Order of operations: Brackets, Indices, Division and Multiplication, Addition and Subtraction.
- Fractions: all four operations: Change mixed numbers to improper fractions first.
- Decimals without a calculator: Line up the decimal points to add and subtract.
- Order: brackets, indices, \(\times \div\) left to right, \(+ -\) left to right.
- Signs: subtracting a negative adds; same signs multiply or divide to a positive.
- Mixed numbers: make them improper first.
- Add/subtract fractions: common denominator. Divide: multiply by the reciprocal.
- Decimals: count decimal places to multiply; scale both numbers to divide by a whole number.
One picture every idea returns to
Brackets first, then indices, then multiply and divide (left to right), then add and subtract (left to right). For fractions, change mixed numbers to improper fractions first; to divide, multiply by the reciprocal.
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, Right answer, right order.. Add, subtract, multiply and divide with negative numbers, fractions, mixed numbers and decimals, in the correct order, mostly without a calculator. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Negative numbers | In the lab, set a starting number and a second number and step along the number line: adding a negative moves left, subtracting a negative moves right. Project tab 5 of the lab (Number lab, fractions and negatives) and let a student drive it while the class predicts. Model the worked example Temperature changes on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Order of operations | Brackets, Indices, Division and Multiplication, Addition and Subtraction. Model the worked example Brackets change everything on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Fractions: all four operations | Change mixed numbers to improper fractions first. Model the worked example Add and subtract mixed numbers on the board, one line at a time. Your turn: Without a calculator, work out \(3\dfrac{3}{4} \div 1\dfrac{1}{4}\). | Pairs |
| 23 to 30 min | Teach and check: Decimals without a calculator | Line up the decimal points to add and subtract. Model the worked example Multiply and divide decimals on the board, one line at a time. | Numbered Heads Together |
| 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, 15 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 Decimals without a calculator, 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.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
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 |
|---|---|
| \(-3^{2} = -9\) because the square is done before the minus sign, but \((-3)^{2} = 9\). | On a calculator, type the brackets. |
| \(20 - 12 + 2\) is 10, not 6. | Addition does not come before subtraction: work left to right. |
| \(\dfrac{2}{5} + \dfrac{1}{3}\) is NOT \(\dfrac{3}{8}\). | Never add the bottoms: use a common denominator, \(\dfrac{6}{15} + \dfrac{5}{15} = \dfrac{11}{15}\). |
| Dividing by a number less than 1 makes the answer BIGGER: \(4.8 \div 0.06 = 80\), not 0.8. | Check the size of your answer makes sense. |
The four operations: fractions, negatives and order of operations: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
Work out (a) \(7 - (-5)\) (b) \((-3) \times 4 \times (-2)\).
Without a calculator, work out \(2\dfrac{2}{5} + 1\dfrac{3}{4}\). Give your answer as a mixed number.
Without a calculator, work out \(3\dfrac{1}{3} \div 1\dfrac{7}{9}\). Give your answer as a mixed number in its simplest form.
Insert two pairs of brackets to make this statement true: \(\ 3 + 5 \times 6 - 2 = 32\)
Without a calculator, work out (a) \(0.36 \div 0.004\) (b) \(2.5 \times 0.12\).
A water tank holds 240 litres when full. It is \(\dfrac{5}{6}\) full. Sam uses \(\dfrac{2}{5}\) of the water in the tank, and then 35 litres are added.
(a) How many litres are now in the tank? (b) What fraction of the tank is now full? Give your answer in its simplest form.
Answer key and marking notes
- (a) \(7 + 5 = 12\)
- (b) \(-12 \times -2 = 24\)
- \(\dfrac{12}{5} + \dfrac{7}{4} = \dfrac{48}{20} + \dfrac{35}{20} = \dfrac{83}{20}\)
- \(= 4\dfrac{3}{20}\)
- \(\dfrac{10}{3} \div \dfrac{16}{9} = \dfrac{10}{3} \times \dfrac{9}{16}\)
- \(= \dfrac{90}{48} = \dfrac{15}{8}\)
- \(= 1\dfrac{7}{8}\)
\((3 + 5) \times (6 - 2) = 8 \times 4 = 32\)
- (a) Multiply both by 1000: \(360 \div 4 = 90\)
- (b) \(25 \times 12 = 300\); 3 decimal places gives \(0.300 = 0.3\)
- Water at the start: \(\dfrac{5}{6} \times 240 = 200\) litres.
- Sam uses \(\dfrac{2}{5} \times 200 = 80\) litres, leaving 120 litres.
- (a) \(120 + 35 = 155\) litres
- (b) \(\dfrac{155}{240} = \dfrac{31}{48}\)
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.
Work out \(3 + 4 \times 5 - 2^{3}\).
- A15
- B27
- C17
- D29
At midnight the temperature is \(-6^{\circ}\text{C}\). By midday it has risen by \(9^{\circ}\text{C}\). By the next midnight it has fallen by \(15^{\circ}\text{C}\) from the midday value. What is the temperature at the next midnight?
- A\(-12^{\circ}\text{C}\)
- B\(-30^{\circ}\text{C}\)
- C\(0^{\circ}\text{C}\)
- D\(12^{\circ}\text{C}\)
Work out \(2\dfrac{1}{3} + 1\dfrac{3}{4}\).
- A\(4\dfrac{1}{12}\)
- B\(3\dfrac{4}{7}\)
- C\(3\dfrac{1}{12}\)
- D\(4\dfrac{1}{7}\)
Work out \(2\dfrac{1}{4} \div \dfrac{3}{8}\).
- A6
- B\(\dfrac{27}{32}\)
- C\(\dfrac{1}{6}\)
- D\(2\dfrac{2}{3}\)
Without a calculator, work out \(0.36 \times 0.4 \div 0.012\).
- A12
- B1.2
- C120
- D0.12
Which placing of brackets makes the statement true?
\(12 - 4 \times 2 + 1 = 24\)
- A\((12 - 4) \times (2 + 1)\)
- B\((12 - 4) \times 2 + 1\)
- C\(12 - 4 \times (2 + 1)\)
- D\(12 - (4 \times 2) + 1\)
Quiz key
Indices: \(2^{3} = 8\). Multiply: \(4 \times 5 = 20\). Then \(3 + 20 - 8 = 15\). The tempting 27 works from left to right, \((3 + 4) \times 5 - 8\), but multiplication comes before addition. 17 treats \(2^{3}\) as \(2 \times 3 = 6\), and 29 makes both mistakes.
\(-6 + 9 = 3^{\circ}\text{C}\) at midday, then \(3 - 15 = -12^{\circ}\text{C}\). \(-30\) subtracts the rise as well as the fall (\(-6 - 9 - 15\)). 0 adds the fall instead of subtracting it (\(-6 - 9 + 15\)), and \(+12\) gets the size right but drops the minus sign.
Wholes: \(2 + 1 = 3\). Fractions: \(\dfrac{1}{3} + \dfrac{3}{4} = \dfrac{4}{12} + \dfrac{9}{12} = \dfrac{13}{12} = 1\dfrac{1}{12}\). Total \(3 + 1\dfrac{1}{12} = 4\dfrac{1}{12}\). \(3\dfrac{4}{7}\) adds the tops and the bottoms, which is never allowed. \(3\dfrac{1}{12}\) forgets to carry the extra whole one from \(\dfrac{13}{12}\).
\(2\dfrac{1}{4} = \dfrac{9}{4}\). Then \(\dfrac{9}{4} \div \dfrac{3}{8} = \dfrac{9}{4} \times \dfrac{8}{3} = \dfrac{72}{12} = 6\). The tempting \(\dfrac{27}{32}\) multiplies by \(\dfrac{3}{8}\) without turning it over. \(\dfrac{1}{6}\) divides the wrong way round, and \(2\dfrac{2}{3}\) divides only the \(\dfrac{1}{4}\), leaving the 2 untouched.
\(0.36 \times 0.4 = 0.144\) (\(36 \times 4 = 144\), with 3 decimal places). Then \(0.144 \div 0.012 = 144 \div 12 = 12\) (multiply both by 1000). 1.2 and 0.12 put the decimal point in the wrong place: dividing by a number less than 1 makes the answer BIGGER than 0.144. 120 multiplies one number by 1000 but the other by only 100.
\((12 - 4) \times (2 + 1) = 8 \times 3 = 24\). The tempting \((12 - 4) \times 2 + 1 = 16 + 1 = 17\), because without a second pair of brackets the \(+ 1\) is done last. \(12 - 4 \times (2 + 1) = 12 - 12 = 0\), and \(12 - (4 \times 2) + 1 = 5\), the same as having no brackets at all.
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.