At a glance. One lesson of about 60 minutes. The four symbols, open and filled circles, and solving one-step and two-step inequalities with a positive multiplier.
Calculator: no calculator. The numbers are small on purpose: the work is in the symbol and the circle.
What students will learn
Know
- <, >, ≤ and ≥ and what each means.
- An open circle leaves the boundary out; a filled circle includes it.
- Solving by balancing keeps the sign when the multiplier is positive.
Understand
- An inequality has a range of solutions, not a single answer.
- Each solving step keeps the same solutions.
Do
- Draw an inequality on a number line.
- Solve one-step and two-step inequalities.
- Check a solution by substituting values on and either side of the boundary.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving and graphing (questions 2 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to say what does not change from step to step and why.
- Criterion C. Communicating: correct use of the four symbols and the circle, and the exit ticket's justification.
- Criterion D. Applying mathematics in real-life contexts: height limits and phone plans written as inequalities (question 7).
The one thing to do with the figure
On the first figure, set x < 3, then x ≤ 3, and ask what changed. Only the circle at 3. Then ask: “Is 3 a solution of each?”
On the second figure, ask students to look at the pattern of dots before each press. It never changes, and that is why each step is allowed.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2. |
|---|---|
| 10 min | Hook: the height rule. At least against taller than. |
| 10 min | The first figure; the key idea. |
| 10 min | The second figure; worked examples. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 6; −3; 4. |
|---|---|
| Worked examples | x ≥ 8; x > 4; x ≥ −2. |
| Faded example | x ≤ 3. |
| 1. Reading a graph | A. B uses the circle for >; C and D point the arrow towards the smaller numbers. |
| 2. Strict inequality | 8. |
| 3. Counting solutions | 4. 1, 2, 3, 4. |
| 4. One step | 3. x < 4. |
| 5. Two steps | A, x ≥ 5. B leaves out 5; C stops at 3x ≥ 15; D subtracts the 3. |
| 6. Matching solutions | C. A has the wrong direction; B and D have boundary 4. |
| 7. Context | 30 texts. (75 − 15) ÷ 2. |
| 8. x on both sides | 6. x > 5. |
| 9. Solve and draw | B. A uses an open circle; C subtracts the 5; D points the wrong way. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Wrong circle, or the arrow pointing the wrong way. | Ask whether the boundary is a solution, then which side is bigger. |
| 7 or 6 in question 2 | Included the boundary, or went the wrong way. | Substitute 7 into x > 7. |
| 3 or 5 in question 3 | Left out 4, or counted 0. | List the values on a number line. |
| 4 or 16 in question 4 | Included the boundary, or added. | Substitute 4 into x + 6 < 10. |
| B, C or D in question 5 | Left out the boundary, stopped at 3x, or subtracted 3. | Substitute x = 5 into 3x − 3 ≥ 12. |
| A, B or D in question 6 | Wrong direction or wrong boundary. | Solve each and compare. |
| 37, 60 or 45 in question 7 | Ignored the fee, stopped at 2m, or added the fee. | Check: 15 + 2 × their answer, against 75. |
| 5 or 10 in question 8 | Included the boundary, or stopped at 2x. | At x = 5 both sides are 27. |
Questions worth asking
- Is 3 a solution of x < 3? Of x ≤ 3? No for the first; yes for the second.
- Why does the sign stay when you divide by 3? Dividing every number by a positive number keeps their order.
- How many solutions does x > 5 have? Infinitely many: every number above 5, not just whole numbers.
Differentiation
A way in for anyone stuck
- Use a physical number line and a counter for each whole number.
- Read each inequality aloud in words before drawing it.
- Give a table: symbol, in words, circle, arrow direction.
Stretch
- Write −3 ≤ x < 2 in words and draw it. (x is at least −3 and less than 2: a filled circle at −3, an open circle at 2, and the whole numbers −3, −2, −1, 0 and 1.)
- Solve 3(x − 2) < 2x + 5. (x < 11.)
Students learning in English as an additional language
At least, at most, more than and less than are often mixed up. Pair each with its symbol, a gesture and a circle.
| Word | Plain meaning | Example |
|---|---|---|
| inequality | a comparison using <, >, ≤ or ≥ | x > 4 |
| at least | equal to or more than | at least 120 cm means x ≥ 120 |
| at most | equal to or less than | at most 5 means x ≤ 5 |
| boundary | the number where the range starts | 4 in x > 4 |
| solution set | all the numbers that make it true | every number above 4 |
Sentence starters for explaining aloud or in writing:
- x > 4 means x is …
- The circle is open / filled because …
- The arrow points left / right because …
Two things not to say
“The alligator eats the bigger number.” It does not explain at least or at most. Read the sign in words.
“Treat it as an equation, then put the sign back.” It works until the multiplier is negative, in the next lesson.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- DrawDescribe x ≤ −1 on a number line.
Filled circle at −1; arrow pointing left. - SolveSolve x − 9 > 2.
x > 11.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA lift in a Nairobi tower is safe for at most 600 kg. A trolley weighs 80 kg and each crate weighs 40 kg. What is the largest number of crates?
13 crates. 40n ≤ 520. - Earlier work, equationsSolve 6x + 2 = 4x + 14.
x = 6.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningx is a whole number and 2 < x ≤ 6. List the values.
3, 4, 5, 6. - Earlier work, integersOrder from smallest: −7, 2, −1, −12.
−12, −7, −1, 2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each, “Show me the steps” gives the same states in words.
No device? A number line drawn on the floor with tape, and students stand on solutions.