The shape of the lesson
By the end of the lesson, learners can
- Inequalities on a number line: Set an inequality in the lab and watch the number line: the circle at each end fills in or empties as you switch between \(<\) and \(\le\).
- Solving like an equation: (Extended) Add, subtract, multiply and divide on both sides just as you would for an equation.
- Double inequalities: (Extended) An inequality with three parts, such as \(-3 \le 3x - 2 < 7\), is solved by doing the same thing to ALL three parts at once until \(x\) is alone in the middle.
- Writing your own inequality: (Extended) Words such as 'at most', 'no more than' and 'cannot exceed' mean \(\le\); 'at least' and 'no fewer than' mean \(\ge\).
- Number line: open circle for \(<\) or \(>\); closed circle for \(\le\) or \(\ge\).
- Solve: do the same to every side, exactly as for an equation.
- Negative: multiplying or dividing by a negative number reverses the sign.
- Double inequality: work on all three parts at once.
- Integers: check each end point: is it included or not?
One picture every idea returns to
Solve an inequality like an equation: do the same thing to every side. The one extra rule: when you multiply or divide by a NEGATIVE number, reverse the inequality sign.
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, Not one answer, a range.. An equation pins \(x\) to a single value; an inequality allows a whole stretch of the number line. 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: Inequalities on a number line | Set an inequality in the lab and watch the number line: the circle at each end fills in or empties as you switch between \(<\) and \(\le\). Project tab 5 of the lab (Equations and inequalities lab, inequalities) and let a student drive it while the class predicts. Model the worked example Draw an inequality on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Solving like an equation | (Extended) Add, subtract, multiply and divide on both sides just as you would for an equation. Model the worked example One step at a time (Extended) on the board, one line at a time. Your turn: (Extended) Solve \(4(x + 1) > 2x - 6\). | Pairs |
| 17 to 23 min | Teach and check: Double inequalities | (Extended) An inequality with three parts, such as \(-3 \le 3x - 2 < 7\), is solved by doing the same thing to ALL three parts at once until \(x\) is alone in the middle. Model the worked example Three parts (Extended) on the board, one line at a time. Your turn: (Extended) Find the largest integer \(n\) such that \(7n - 3 < 40\). | Pairs |
| 23 to 30 min | Teach and check: Writing your own inequality | (Extended) Words such as 'at most', 'no more than' and 'cannot exceed' mean \(\le\); 'at least' and 'no fewer than' mean \(\ge\). Model the worked example Shopping budget (Extended) on the board, one line at a time. | Frayer Model |
| 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 Writing your own inequality, 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.
Frayer Model
Pairs fill a four-box grid for the key term: definition, characteristics, examples and non-examples. The non-examples box is where the misconceptions surface, so give it the most time.
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 |
|---|---|
| Open and closed circles are not decoration. | Drawing a closed circle for \(x < 3\) says 3 is allowed, and loses the mark. Strict sign, open circle. |
| Why the flip? \(2 -5\). | Forgetting to reverse the sign is the most common lost mark on this topic. You can avoid it by moving the \(x\) term to the side where it stays positive. |
| Do not split a double inequality and then only finish one half. | Whatever you do to the middle, do to both ends as well, or the answer range is wrong. |
| In context the answer is often a whole number. | Do not round 13.125 to the nearest integer by habit: think about which way keeps the inequality true. For 'at most' round down; for 'at least' round up. |
Linear inequalities and the number line: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Describe how to show \(-1 \le x < 3\) on a number line.
(b) A number line shows an open circle at 5 with an arrow pointing left. Write down the inequality.
List the integers \(n\) such that \(-4 \le n < 1\).
(Extended) Solve \(4x - 9 > x + 6\).
(Extended) Solve \(2 - 5x \ge 17\).
(Extended) Solve \(-7 < 2x - 3 \le 5\) and list the integer values of \(x\).
(Extended) A rectangle is \((x + 6)\) cm long and 4 cm wide.
Its perimeter must be at least 30 cm and its area must be less than 52 cm2.
(a) Write down two inequalities in \(x\) and solve them. (b) \(x\) is an integer. Find the possible values of \(x\).
Answer key and marking notes
- (a) Closed circle at \(-1\), open circle at 3, joined by a line.
- (b) \(x < 5\)
\(-4, -3, -2, -1, 0\)
- Subtract \(x\): \(3x - 9 > 6\).
- Add 9: \(3x > 15\), so \(x > 5\).
- Subtract 2: \(-5x \ge 15\).
- Divide by \(-5\) and reverse the sign: \(x \le -3\).
- Add 3 to all parts: \(-4 < 2x \le 8\).
- Divide by 2: \(-2 < x \le 4\).
- Integers: \(-1, 0, 1, 2, 3, 4\).
- Perimeter: \(2(x + 6) + 2 \times 4 = 2x + 20 \ge 30\), so \(2x \ge 10\) and \(x \ge 5\).
- Area: \(4(x + 6) < 52\), so \(x + 6 < 13\) and \(x < 7\).
- Together: \(5 \le x < 7\).
- (b) \(x = 5\) or \(x = 6\).
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.
Which inequality is shown on the number line?
- A\(-4 \le x < 2\)
- B\(-4 < x \le 2\)
- C\(-4 \le x \le 2\)
- D\(-4 < x < 2\)
List all the integers \(n\) such that \(-1 < n \le 4\).
- A\(0, 1, 2, 3, 4\)
- B\(-1, 0, 1, 2, 3, 4\)
- C\(0, 1, 2, 3\)
- D\(-1, 0, 1, 2, 3\)
What is the largest integer \(x\) such that \(x < -2.5\)?
- A\(-3\)
- B\(-2\)
- C\(-2.5\)
- D\(3\)
Solve \(8 - 3x > 2\).
- A\(x < 2\)
- B\(x > 2\)
- C\(x > -2\)
- D\(x < -2\)
List all the integers that satisfy \(-7 \le 3x + 2 < 11\).
- A\(-3, -2, -1, 0, 1, 2\)
- B\(-3, -2, -1, 0, 1, 2, 3\)
- C\(-2, -1, 0, 1, 2\)
- D\(-2, -1, 0, 1, 2, 3\)
A phone plan costs $12 a month plus $0.35 for each gigabyte (GB) of data. Mina can spend at most $30 in a month. What is the greatest whole number of GB she can use?
- A51
- B52
- C85
- D51.4
Quiz key
The closed circle at \(-4\) means \(-4\) is included, so \(-4 \le x\). The open circle at 2 means 2 is not included, so \(x < 2\). Together: \(-4 \le x < 2\). The answer \(-4 < x \le 2\) swaps the two circles round.
\(-1 < n\) is strict, so \(-1\) is NOT included; \(n \le 4\) includes 4. The integers are \(0, 1, 2, 3, 4\). Including \(-1\) treats the strict sign as if it were \(\le\).
Numbers less than \(-2.5\) are further left: \(-3, -4, -5, \ldots\) The largest of these integers is \(-3\). \(-2\) is tempting because 2 is less than 2.5, but \(-2\) lies to the right of \(-2.5\), so \(-2 > -2.5\). \(-2.5\) is not an integer and is not less than itself.
Subtract 8: \(-3x > -6\). Divide by \(-3\) and REVERSE the sign: \(x < 2\). Check with \(x = 0\): \(8 > 2\), true. \(x > 2\) comes from dividing by \(-3\) without reversing the sign; \(x = 3\) gives \(8 - 9 = -1\), which is not greater than 2.
Subtract 2 from all three parts: \(-9 \le 3x < 9\). Divide by 3: \(-3 \le x < 3\). So \(-3\) is included and 3 is not: \(-3, -2, -1, 0, 1, 2\). Including 3 ignores the strict \(<\) on the right.
\(12 + 0.35g \le 30\), so \(0.35g \le 18\) and \(g \le 51.43\). She cannot go over $30, so round DOWN to 51 GB (cost $29.85). 52 GB would cost $30.20, too much. 85 comes from \(30 \div 0.35\), forgetting the $12 fixed charge; 51.4 is not a whole number.
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.