At a glance. One lesson of about 60 minutes. It assumes absolute value from Year 8, percentages from Year 7 and significant figures from Lesson 2. Students evaluate absolute values, use their properties, solve |x − c| = d, and find absolute, relative and percentage errors.
Calculator: GDC optional. The lesson works by hand. The optional section shows GDC users the abs( function and how lists give a whole class's percentage errors at once, which suits the measuring activity in Practicalities.
What students will learn
Know
- |x| is the distance of x from 0, and |a − b| is the distance between a and b.
- |ab| = |a||b| and |a + b| ≤ |a| + |b|.
- Absolute error = |measured − true|; relative error = absolute error ÷ true value; percentage error = relative error × 100%.
Understand
- An absolute value is a distance, so it is never negative and it ignores direction.
- |a + b| is less than |a| + |b| when a and b have opposite signs, because the second step turns back.
- The same absolute error matters more on a smaller true value, which is why relative error exists.
Do
- Evaluate expressions with absolute values and find distances between numbers.
- Solve equations of the form |x − c| = d.
- Calculate and compare absolute, relative and percentage errors, and work back from a percentage error.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: absolute value, its properties and the three kinds of error (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Students find every pair a, b with |a + b| = |a| + |b| and state the rule (a and b not of opposite signs).
- Criterion C. Communicating: writing |measured − true| before any division, and stating which value is divided by (worked example, faded example).
- Criterion D. Applying mathematics in real-life contexts: bread, a taxi meter, a running watch, a swimming pool and a market scale; the hook asks which error matters more and why.
The one thing to do with the figure
Ask for a prediction before every slide. In the first figure set a = 3 and ask, before moving b to −5: “Where will the walk end? Is that further from 0 than 8?”
In the second figure, keep the measured value 2 above the true value and move the true value from 60 down to 20. Ask what stays the same and what grows. This settles the hook.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: two loaves, both 20 g short. Collect predictions and reasons. |
| 10 min | First figure; the first half of the key idea; the warning card's first mistake. |
| 10 min | Second figure; the errors; settle the hook (2.5% and 5%); both worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to a = 3, b = −5 in the first figure. |
| 20 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 | 9; 3.5%; 0.042. |
|---|---|
| Hook | Both are 20 g short, but 20 ÷ 800 = 2.5% and 20 ÷ 400 = 5%: the roll's shortfall is twice as large relative to its size. |
| Worked examples | 10 g, 0.04, 4%; x = 3 or x = −5. |
| Faded example | 2.5%. |
| 1. Properties of absolute value | C. A fails for opposite signs; B can give a negative; D fails for negative a. |
| 2. Evaluating absolute values | 4. 8 − 4. |
| 3. Absolute error | 0.2. |4.8 − 5.0|. |
| 4. Percentage error | 7.5. 1.2 ÷ 16 = 0.075. |
| 5. Distance between two numbers | 8. 3.5 + 4.5. |
| 6. Solving |x − c| = d | B. A and C give one direction only; D uses the wrong centre, −2. |
| 7. Working back from a percentage error | 408. 400 + 0.02 × 400. |
| 8. Comparing two measurements | 0.6. 0.3 ÷ 50 × 100. |
| 9. Percentage error | A. B divides by the measured value; C forgets to multiply by 100; D gives the absolute error. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Thinks absolute value can be split over a sum or difference, or that |a| = a. | Test each statement with a negative number before deciding. |
| −12 or 12 in question 2 | Kept a sign inside the bars, or dropped a pair of bars. | Work out the inside of each pair of bars on its own line. |
| −0.2 or 4 in question 3 | Did not take the absolute value, or gave the percentage error. | Ask: is an error a distance or a direction? |
| 7, 0.075 or 1.2 in question 4 | Divided by the measured value, gave the relative error, or gave the absolute error. | Write the three errors as three separate lines, labelled. |
| 1 or −8 in question 5 | Added instead of subtracting, or left the sign. | Mark both numbers on a number line and count the gap. |
| A, C or D in question 6 | Found one solution only, or used the wrong centre. | Mark 2 on a number line and step 5 in each direction. |
| 392 or 402 in question 7 | Went the wrong way, or added 2 m instead of 2%. | Ask: 2% of what? Then which direction does the watch err? |
| 0.8 or 0.3 in question 8 | Chose the larger error, or gave an absolute error. | Find both percentage errors in a table before choosing. |
| B, C or D in question 9 | Divided by the measured value, forgot × 100, or gave the absolute error. | Ask which value is the true one, and what the % sign means. |
Questions worth asking
- Why is |a + b| never bigger than |a| + |b|? Listen for “walking a then b, the furthest you can get is when you never turn back”.
- Why divide by the true value and not the measured value? The error is measured against what is correct; dividing by a wrong measurement would make the error depend on itself.
- Which is worse: 1 cm wrong on a 10 cm pencil or 1 cm wrong on a 2 m door? The pencil: 10% against 0.5%. The same absolute error, a very different relative error.
- Can you ever know a length exactly? No measurement is exact; we can only say how large the error could be.
Differentiation
A way in for anyone stuck
- Draw a number line for every absolute value question and mark both numbers before calculating.
- Give an error card with the three formulas and a worked example underneath each.
- Start with whole-number measurements, then move to decimals.
Stretch
- For which values of b is |8 − b| greater than |b − 2|? Explain with a number line.
- A length measured as 12.0 cm has a percentage error of at most 0.5%. Find the smallest and largest possible true lengths, to 4 significant figures (12.0 ÷ 1.005 = 11.94 cm and 12.0 ÷ 0.995 = 12.06 cm: the percentage error is measured against the true length).
Students learning in English as an additional language
“Absolute” here means “without a sign”; “relative” means “compared with something”. “True value” and “measured value” should be written as labels next to the numbers every time.
| Word | Plain meaning | Example |
|---|---|---|
| absolute value | the distance of a number from 0, written |x| | |−7| = 7 |
| distance between | how far apart two numbers are, |a − b| | |−3 − 4| = 7 |
| absolute error | the distance between a measured value and the true value | |9.7 − 10| = 0.3 s |
| relative error | the absolute error divided by the true value | 0.3 ÷ 10 = 0.03 |
| percentage error | the relative error written as a percentage | 0.03 × 100 = 3% |
| true value | the correct value of a quantity | the pool is 50 m |
| measured value | the value a person or instrument records | 49.6 m |
Sentence starters for explaining aloud or in writing:
- The absolute error is … because the distance between … and … is …
- I divide by … because it is the true value.
- |x − c| = d means x is … away from …
Two things not to say
“Absolute value makes it positive.” Say “it is the distance from zero”; the slogan leads to |−3 + 1| = 4.
“Divide by the measurement.” Always the true value; the slogan gives the wrong answer in every question that compares them.
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.
- EvaluateWork out |−6| × |2 − 5|.
6 × 3 = 18. - ErrorA 30 cm ruler is read as 29.4 cm. Find the absolute and percentage errors.
0.6 cm; 2%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA census estimates a town's population as 48 000. The true figure is 50 000. Find the percentage error.
4%. - Earlier work (Lesson 3)Convert 2.5 m² to cm².
25 000 cm².
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind every integer x with |x − 1| < 3.
−1, 0, 1, 2 and 3. - Working backwardsA measurement of 81 cm has a percentage error of 10% and is too small. Find the true value.
90 cm, since 81 = 0.9 × 90.
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.
Measuring activity: pairs measure a sheet of A4 paper, a desk and a pencil to the nearest centimetre and the nearest millimetre, then compare their percentage errors. The With a GDC section shows how lists do every calculation at once.