The shape of the lesson
By the end of the lesson, learners can
- Line them up: In the lab, enter each fraction as numerator and denominator and read off its decimal.
- Inequality symbols: The symbol always opens towards the larger number: the wide end faces the bigger side.
- Roots, pi and close calls: When values are very close, a calculator decides it.
- Convert everything to decimals (or to one unit).
- Pad to the same number of decimal places, 4 if values are close.
- Negatives: further from zero means smaller.
- Symbols: the wide end of \(<\) or \(>\) faces the larger number; \(\le\) and \(\ge\) include the end value.
- Answer in the original forms, in the order the question asks.
One picture every idea returns to
To order mixed quantities, convert every one into the same form (usually a decimal to at least 3 or 4 places, or the same unit), compare, then write the answer using the ORIGINAL forms in the order asked for.
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, Same form, then compare.. Fractions, decimals, percentages, negatives, roots and measurements in different units can all be put in order once they are written the same way. 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: Line them up | In the lab, enter each fraction as numerator and denominator and read off its decimal. 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 Fractions, decimals and percentages on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Inequality symbols | The symbol always opens towards the larger number: the wide end faces the bigger side. Model the worked example Choosing the symbol on the board, one line at a time. Your turn: Write as an inequality: \(x\) is at least 4 but less than 9. Then list the integer values of \(x\). | Pairs |
| 21 to 30 min | Teach and check: Roots, pi and close calls | When values are very close, a calculator decides it. Model the worked example Very close values on the board, one line at a time. | Mix-Pair-Share |
| 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, 14 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 Roots, pi and close calls, 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.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
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 |
|---|---|
| More digits does not mean bigger: \(0.405 < 0.45 < 0.5\). | Compare place by place from the left (or pad with zeros: 0.405, 0.450, 0.500). |
| \(-5 > -2\) is FALSE. | With negatives, the number that looks bigger is smaller: \(-5\) is 3 below \(-2\). |
| Read the instruction: 'starting with the largest' reverses the order. | And always give the final list in the original forms, not as your decimals. |
Ordering quantities and inequality symbols: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
Write \(<\), \(>\) or \(=\) in each gap.
(a) \(-6 \ \_\_\ -9\) (b) \(0.3 \ \_\_\ \dfrac{1}{3}\) (c) \(25\% \ \_\_\ \dfrac{1}{4}\)
Write in order, smallest first: \(\ -1.5,\ 0.2,\ -2,\ -0.25,\ 1\).
Write in order, smallest first: \(\ \dfrac{4}{9},\ 0.45,\ 44\%,\ \dfrac{7}{16}\).
Use your calculator to write in order, smallest first: \(\ \sqrt{17},\ 4.1,\ \dfrac{33}{8},\ \sqrt[3]{70}\).
(a) List the integers \(n\) such that \(-2 \le n < 3\).
(b) The temperature \(t\) °C at a ski resort was above \(-5\) °C and at most 12 °C. Write this as an inequality.
Four sprinters' times for a race are: Ana 12.08 s, Ben 12.8 s, Chen \(12\dfrac{3}{25}\) s and Dev 12.085 s.
(a) List the sprinters from fastest to slowest. (b) Find the difference between the fastest and slowest times.
Answer key and marking notes
- (a) \(-6 > -9\)
- (b) \(\dfrac{1}{3} = 0.333\dots\), so \(0.3 < \dfrac{1}{3}\)
- (c) \(25\% = \dfrac{1}{4}\)
\(-2,\ -1.5,\ -0.25,\ 0.2,\ 1\)
- \(\dfrac{4}{9} = 0.4444\), \(0.45 = 0.4500\), \(44\% = 0.4400\), \(\dfrac{7}{16} = 0.4375\)
- \(\dfrac{7}{16},\ 44\%,\ \dfrac{4}{9},\ 0.45\)
- \(\sqrt{17} = 4.1231\), \(\dfrac{33}{8} = 4.1250\), \(\sqrt[3]{70} = 4.1213\), \(4.1 = 4.1000\)
- \(4.1,\ \sqrt[3]{70},\ \sqrt{17},\ \dfrac{33}{8}\)
- (a) \(-2,\ -1,\ 0,\ 1,\ 2\)
- (b) \(-5 < t \le 12\)
- \(12\dfrac{3}{25} = 12.12\). In 3 d.p.: Ana 12.080, Ben 12.800, Chen 12.120, Dev 12.085.
- (a) Fastest is the smallest time: Ana, Dev, Chen, Ben
- (b) \(12.8 - 12.08 = 0.72\) s
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 these in order of size, starting with the smallest.
\(0.6 \quad \dfrac{5}{8} \quad 62\% \quad 0.06\)
- A\(0.06,\ 0.6,\ 62\%,\ \dfrac{5}{8}\)
- B\(0.06,\ \dfrac{5}{8},\ 0.6,\ 62\%\)
- C\(0.6,\ 0.06,\ 62\%,\ \dfrac{5}{8}\)
- D\(\dfrac{5}{8},\ 62\%,\ 0.6,\ 0.06\)
Which statement is true?
- A\(-7 < -3\)
- B\(-2 < -5\)
- C\(0 < -1\)
- D\(-4 > -1\)
Write these masses in order, starting with the smallest.
1.2 kg, 1250 g, 980 g, 0.9 kg
- A0.9 kg, 980 g, 1.2 kg, 1250 g
- B1.2 kg, 0.9 kg, 980 g, 1250 g
- C980 g, 0.9 kg, 1250 g, 1.2 kg
- D0.9 kg, 1.2 kg, 980 g, 1250 g
\(n\) is an integer and \(-3 < n \le 2\). How many possible values of \(n\) are there?
- A5
- B6
- C4
- D2
Write these in order of size, starting with the largest.
\(\dfrac{3}{7} \quad 0.43 \quad 42.8\% \quad 0.4\dot{2}\)
- A\(0.43,\ \dfrac{3}{7},\ 42.8\%,\ 0.4\dot{2}\)
- B\(0.4\dot{2},\ 42.8\%,\ \dfrac{3}{7},\ 0.43\)
- C\(\dfrac{3}{7},\ 0.43,\ 42.8\%,\ 0.4\dot{2}\)
- D\(0.4\dot{2},\ 0.43,\ \dfrac{3}{7},\ 42.8\%\)
Which symbol makes this statement true?
\(\sqrt{50} \ \square \ \dfrac{50}{7}\)
- A\(<\)
- B\(>\)
- C\(=\)
- D\(\ge\)
Quiz key
As decimals: \(0.6 = 0.600\), \(\dfrac{5}{8} = 0.625\), \(62\% = 0.620\), \(0.06 = 0.060\). Smallest first: 0.060, 0.600, 0.620, 0.625, so the order is \(0.06,\ 0.6,\ 62\%,\ \dfrac{5}{8}\). Putting \(\dfrac{5}{8}\) second reads it as 0.58; it is \(5 \div 8 = 0.625\). The last option is the right order but reversed (largest first), and putting 0.6 before 0.06 assumes more digits means bigger.
\(-7\) is 4 places to the left of \(-3\) on the number line, so \(-7 < -3\) is true. \(-2 < -5\) and \(-4 > -1\) both make the mistake of comparing the digits without the signs: with negatives, the one that looks bigger is smaller. \(0 < -1\) is false because every negative number is less than 0.
In grams: 1.2 kg = 1200 g and 0.9 kg = 900 g. Smallest first: 900, 980, 1200, 1250, so the order is 0.9 kg, 980 g, 1.2 kg, 1250 g. Ordering the bare numbers 0.9, 1.2, 980, 1250 without converting gives '1.2 kg, 0.9 kg...' or '0.9 kg, 1.2 kg, 980 g...', which ignore the units. Putting 980 g before 0.9 kg misreads 0.9 kg as 90 g.
\(-3 < n\) means \(-3\) is NOT included, and \(n \le 2\) means 2 IS included. The integers are \(-2, -1, 0, 1, 2\): 5 values. The tempting 6 includes \(-3\), but the symbol \(<\) excludes it. 4 leaves out 2 as well, and 2 counts only the positive integers 1 and 2.
\(\dfrac{3}{7} = 0.4285\ldots\), \(0.43 = 0.4300\), \(42.8\% = 0.4280\) and \(0.4\dot{2} = 0.4222\ldots\). Largest first: 0.4300, 0.4285, 0.4280, 0.4222, giving \(0.43,\ \dfrac{3}{7},\ 42.8\%,\ 0.4\dot{2}\). Putting \(\dfrac{3}{7}\) first comes from rounding it to 0.43 too early, which hides the difference. The second option is the correct order reversed: the question asks for the largest first. The last option assumes the recurring decimal is biggest because its digits go on for ever.
\(\sqrt{50} = 7.071\ldots\) and \(\dfrac{50}{7} = 7.142\ldots\), so \(\sqrt{50} < \dfrac{50}{7}\). Both round to 7.1 to 2 s.f., which is why \(=\) is tempting, but they are not equal. \(>\) and \(\ge\) both claim the square root is at least as big, which is false.
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.