At a glance. One lesson of about 60 minutes. It assumes absolute value from the Standard course and linear equations. Students solve |x − a| = r and < r, equal absolute values, and compare an error in a length with the error in an area or volume.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students who have a GDC the abs( function and why both brackets must be closed.
What students will learn
Know
- |x − a| is the distance from x to a.
- |A| = |B| gives A = B or A = −B.
- |a + b| ≤ |a| + |b|, with equality only when a and b have the same sign.
Understand
- An inequality with absolute value describes a set of points within a distance, not a single bound.
- Errors from separate measurements can reinforce each other, so bounds add.
- A small percentage error multiplies by the power when a length is squared or cubed.
Do
- Solve absolute value equations and inequalities and show them on a number line.
- Decide whether a statement about absolute values is always, sometimes or never true.
- Find the percentage error in an area or volume from the error in a length.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving equations and inequalities with absolute value and finding errors (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure and the what-if. Students find the factor between a length's percentage error and the area's, and generalize to volume.
- Criterion C. Communicating: writing a solution to |A| = |B| in both cases and checking each in the original (worked example 2 and question 4).
- Criterion D. Applying mathematics in real-life contexts: bottling tolerance and a surveyed plot, where a small length error produces a larger area error (the hook and question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With a = 1 and r = 3, ask: “is x = 4 inside? Is x = 5?” before students move the point. Ask why x = −2 counts, and whether it would still count if the sign were < instead of ≤.
In the second figure, set the error to +1 and then −1 and ask which gives the larger area error, and why.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the bottling plant. Collect predictions. |
| 12 min | Both figures with predictions; the key idea. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or D, return to the first figure, set a = −5 and read the expression it shows, |x + 5|. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 0; x = 5; 11.7 cm. |
|---|---|
| Hook | Neither is accepted: |494 − 500| = 6 and |505.5 − 500| = 5.5 are both more than 5; a 5% error in a side is 10.25% in the area. |
| Worked examples | 3 < x < 7; x = −4 and x = 1; about 10%. |
| Faded example | x = 3 and x = −7/3. |
| 1. An interval from a distance | C. A forgets the lower end; B centres on +7; D forgets the shift. |
| 2. Counting solutions of an inequality | 13. −4 to 8 inclusive. |
| 3. Solving an absolute value equation | 6. The other solution is −3. |
| 4. Two absolute values equal | 5. x = 4 and x = 1. |
| 5. Percentage error in an area | 8.2. 204 ÷ 2 500 = 8.16%. |
| 6. Possible values of |a + b| | B. A needs the same sign; C allows other sizes; D multiplies. |
| 7. Combining error bounds | 0.8. 0.5 + 0.3. |
| 8. Error through a volume | 9. 1.03³ = 1.0927…. |
| 9. A statement true for all numbers | D. A needs the same sign; B reverses the triangle inequality; C can be negative. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Forgot a side, centred on +7, or forgot the shift. | Write |x + 7| as |x − (−7)|, mark −7 on a number line and ask what is 2 to the left and 2 to the right. |
| 12 or 11 in question 2 | Did not count both ends, or used r = 5. | List the integers at each end and count them. |
| 3 or 12 in question 3 | Sign slip when moving 3, or did not halve. | Substitute the answer back into |2x − 3|. |
| 4 or 1 in question 4 | Found only one of the two cases. | Ask: can the insides be opposite as well as equal? |
| 4 or 8 in question 5 | Gave the side's error, or doubled it. | Work out both areas as numbers and then the percentage. |
| A, C or D in question 6 | Took one sign case only, allowed other sizes, or multiplied. | List the four possible pairs (a, b). |
| 0.2 or 0.5 in question 7 | Subtracted the errors or used only one. | Find the largest and smallest values of x − y. |
| 3 or 27 in question 8 | Used the side's error, or cubed the percentage. | Cube the factor 1.03 rather than the number 3. |
| A, B or C in question 9 | Believed a statement that fails when the signs differ. | Test each statement with a = 3 and b = −2, then with a = 2 and b = 5. |
Questions worth asking
- Why does |x − a| = r usually have two solutions? Listen for “there is one point r to the right of a and one r to the left”.
- When is |a + b| = |a| + |b|? When a and b have the same sign, or one of them is zero.
- Can an absolute value inequality have no solutions? Yes: |x − 3| < −1 has none, because a distance cannot be negative.
- Why is an error of 2% in a radius about 4% in a circle's area? The area is proportional to r², so the factor 1.02 is squared.
Differentiation
A way in for anyone stuck
- Always draw the number line first and mark a, then a − r and a + r.
- For |A| = |B|, write the two cases in two columns before solving.
- Use numbers for the error questions: work out both areas and then the percentage.
Stretch
- Solve |x − 1| + |x + 2| = 5.
- Show that if |a − b| ≤ 0.1 and |b − c| ≤ 0.1, then |a − c| ≤ 0.2.
Students learning in English as an additional language
“Absolute value” here means “distance from zero”. “Within” means “not further than”. “Error” is how far a measured or calculated value is from the true one. “Tolerance” is the error that is allowed.
| Word | Plain meaning | Example |
|---|---|---|
| absolute value | the distance of a number from zero, never negative | |−9| = 9 |
| interval | all the numbers between two ends | from 10 to 12 |
| tolerance | the greatest error that is allowed | ±5 mL on a bottle |
| percentage error | the error as a percentage of the true value | 2 out of 50 is 4% |
| triangle inequality | |a + b| is never more than |a| + |b| | |3 + (−5)| ≤ 3 + 5 |
| boundary | the end of an interval, included or not by the sign | the 12 in x ≤ 12 |
Sentence starters for explaining aloud or in writing:
- The centre is … and the distance is …, so the interval is …
- There are two cases: … or …
- The measured value is … times the true value, so the area is …
Two things not to say
“Absolute value makes it positive.” Say “it is the distance from zero”; |x − 3| is the distance from 3, not a sign change.
“The error is 5%, so the area error is 5%.” The factor 1.05 is squared: the area error is about 10%.
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.
- SolvingSolve |x| = 7.
x = 7 or x = −7. - IntervalWrite |x − 1| < 4 without absolute value.
−3 < x < 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA recipe in Lagos needs 250 g of flour with a tolerance of ±8 g. Write this as |m − 250| ≤ 8 and give the least and greatest mass.
242 g and 258 g. - Earlier work (Lesson 2)Write (3 × 10⁴)(2 × 10⁻⁷) in standard form.
6 × 10⁻³.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve |2x + 1| > 7.
x < −4 or x > 3. - ErrorA circle's radius is measured 3% too large. About how much too large is its area?
1.03² = 1.0609, so about 6%.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.