The shape of the lesson
By the end of the lesson, learners can
- Three ways to write one number: In the lab, set the numerator and the denominator and read the same value as a fraction in its simplest form, a decimal and a percentage.
- Mixed numbers and improper fractions: Calculations with fractions are easier with improper fractions; answers usually look better as mixed numbers.
- Recurring decimals: Extended.
- Fraction to decimal: top \(\div\) bottom.
- Decimal to percentage: \(\times 100\); percentage to decimal: \(\div 100\).
- Decimal to fraction: place value (tenths, hundredths, ...), then cancel by the HCF.
- Mixed to improper: whole \(\times\) bottom \(+\) top, over the same bottom.
- Recurring (Extended): let \(x\) = the decimal, multiply by a power of 10 to line up the repeats, subtract.
One picture every idea returns to
Percent means 'out of 100'. To convert, go through the decimal: fraction to decimal by dividing top by bottom, decimal to percentage by multiplying by 100. Always cancel a fraction to its simplest form by dividing top and bottom by their HCF.
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, One number, three costumes.. \(\dfrac{3}{8}\), 0.375 and 37.5% are the same number written three ways. 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: Three ways to write one number | In the lab, set the numerator and the denominator and read the same value as a fraction in its simplest form, a decimal and a percentage. Project tab 4 of the lab (Number lab, fractions, decimals and percentages) and let a student drive it while the class predicts. Model the worked example Fraction to decimal and percentage on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Mixed numbers and improper fractions | Calculations with fractions are easier with improper fractions; answers usually look better as mixed numbers. Model the worked example Both directions on the board, one line at a time. Your turn: Write 165% as a mixed number in its simplest form. | Pairs |
| 21 to 30 min | Teach and check: Recurring decimals | Extended. Model the worked example One repeating digit (Extended) on the board, one line at a time. Your turn: (Extended) Write \(0.\dot{4}\) as a fraction. | Round Table |
| 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 Recurring decimals, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| \(0.4\) is \(40\%\), not \(4\%\). | Multiplying by 100 moves the digits TWO places. \(4\% = 0.04\). |
| A percentage over 100 is fine: 165% is more than one whole. | Do not 'correct' it to 16.5% or 1.65%; as a decimal it is 1.65. |
| \(0.1\dot{7}\) (0.1777...) and \(0.\dot{1}\dot{7}\) (0.171 717...) are different numbers. | Read the dots carefully before choosing what to multiply by, and show the subtraction: 'show that' questions give no marks for a calculator answer. |
Fractions, decimals and percentages: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
Write 0.65 as (a) a fraction in its simplest form (b) a percentage.
(a) Write \(4\dfrac{5}{7}\) as an improper fraction. (b) Write \(\dfrac{29}{4}\) as a mixed number.
Without a calculator, write \(\dfrac{7}{8}\) as (a) a decimal (b) a percentage.
(Extended) Write \(\dfrac{7}{12}\) as a recurring decimal, using dot notation.
(Extended) Show that \(0.3\dot{6} = \dfrac{11}{30}\).
240 people chose their favourite colour. 0.35 of them chose red, \(\dfrac{3}{8}\) chose blue, 15% chose green and the rest chose yellow.
(a) How many chose yellow? (b) What fraction of the people chose yellow? Give your answer in its simplest form.
Answer key and marking notes
- (a) \(\dfrac{65}{100} = \dfrac{13}{20}\)
- (b) 65%
- (a) \(\dfrac{4 \times 7 + 5}{7} = \dfrac{33}{7}\)
- (b) \(29 \div 4 = 7\) r 1, so \(7\dfrac{1}{4}\)
- (a) \(7 \div 8 = 0.875\)
- (b) \(0.875 \times 100 = 87.5\%\)
- \(7 \div 12 = 0.58333\dots\)
- Only the 3 repeats: \(0.58\dot{3}\)
- Let \(x = 0.3666\dots\)
- \(100x = 36.666\dots\) and \(10x = 3.666\dots\)
- \(100x - 10x = 33\), so \(90x = 33\)
- \(x = \dfrac{33}{90} = \dfrac{11}{30}\)
- Red: \(0.35 \times 240 = 84\). Blue: \(\dfrac{3}{8} \times 240 = 90\). Green: \(0.15 \times 240 = 36\).
- (a) Yellow: \(240 - 84 - 90 - 36 = 30\)
- (b) \(\dfrac{30}{240} = \dfrac{1}{8}\)
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.
Write \(\dfrac{7}{8}\) as a percentage.
- A87.5%
- B78%
- C8.75%
- D0.875%
Write 0.36 as a fraction in its simplest form.
- A\(\dfrac{9}{25}\)
- B\(\dfrac{36}{100}\)
- C\(\dfrac{18}{50}\)
- D\(\dfrac{36}{10}\)
Write 165% as a decimal.
- A1.65
- B16.5
- C0.165
- D165
In a survey of 80 students, 12 said they walk to school. What percentage of the students walk to school?
- A15%
- B12%
- C85%
- D6.67%
Write \(0.\dot{7}\dot{2}\) as a fraction in its simplest form.
- A\(\dfrac{8}{11}\)
- B\(\dfrac{18}{25}\)
- C\(\dfrac{7}{9}\)
- D\(\dfrac{4}{5}\)
Write \(0.2\dot{6}\) as a fraction in its simplest form.
- A\(\dfrac{4}{15}\)
- B\(\dfrac{26}{99}\)
- C\(\dfrac{13}{50}\)
- D\(\dfrac{13}{45}\)
Quiz key
\(7 \div 8 = 0.875\), and \(0.875 \times 100 = 87.5\%\). 0.875% forgets to multiply the decimal by 100, and 8.75% moves the digits only one place instead of two. 78% just writes the digits 7 and 8 side by side.
\(0.36 = \dfrac{36}{100}\). The HCF of 36 and 100 is 4, so \(\dfrac{36}{100} = \dfrac{9}{25}\). \(\dfrac{36}{100}\) and \(\dfrac{18}{50}\) are equal to 0.36 but are not in simplest form (both can still be divided further), so they would lose the mark. \(\dfrac{36}{10} = 3.6\), the wrong place value.
\(165\% = 165 \div 100 = 1.65\). A percentage over 100 is more than one whole, so the decimal is more than 1. 16.5 divides by only 10 and 0.165 divides by 1000. 165 forgets to divide at all.
\(\dfrac{12}{80} = 0.15 = 15\%\). 12% treats the 12 students as a percentage, but there are 80 students, not 100. 85% is the percentage who do NOT walk, and 6.67% comes from \(80 \div 12\), dividing the wrong way round.
Let \(x = 0.727272\ldots\). Then \(100x = 72.727272\ldots\). Subtracting, \(99x = 72\), so \(x = \dfrac{72}{99} = \dfrac{8}{11}\). The tempting \(\dfrac{18}{25} = 0.72\) ignores the dots: it is the terminating decimal 0.72. \(\dfrac{7}{9} = 0.\dot{7}\) repeats only the 7, and \(\dfrac{4}{5} = \dfrac{72}{90}\) uses 90 instead of 99.
Let \(x = 0.2666\ldots\). Then \(100x = 26.666\ldots\) and \(10x = 2.666\ldots\). Subtracting, \(90x = 24\), so \(x = \dfrac{24}{90} = \dfrac{4}{15}\). The tempting \(\dfrac{26}{99}\) is \(0.\dot{2}\dot{6}\), where both digits repeat. \(\dfrac{13}{50} = 0.26\) exactly, and \(\dfrac{13}{45} = \dfrac{26}{90}\) forgets to subtract the 2.
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.