At a glance. One lesson of about 60 minutes. Absolute value as distance from zero; the distance between two integers by adding or subtracting their distances from zero.
Calculator: no calculator. The numbers are small on purpose: the work is deciding whether zero lies between two points.
What students will learn
Know
- |n| is the distance of n from zero.
- An absolute value is never negative.
- Every positive number is the absolute value of two integers.
Understand
- Distance has size but no direction.
- Whether zero lies between two points decides whether their distances from zero add or subtract.
Do
- Find the absolute value of any integer.
- Find the distance between two integers on a number line.
- Solve problems with heights, depths, floors and temperatures.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding absolute values and distances in familiar contexts (questions 2 to 6).
- Criterion B. Investigating patterns: with the figure, ask students to predict the gap before each move and to state when the distances add and when they subtract. Look for a rule stated in words.
- Criterion C. Communicating: correct use of the |n| notation, and the exit ticket's justification.
- Criterion D. Applying mathematics in real-life contexts: the lift and submarine problems, and checking that a distance is never negative.
The one thing to do with the figure
Put a at −6 and b at 9, then ask for the gap before showing it. Many will say 3. Move b slowly towards zero and past it to the left, and ask them to watch the band below the line shrink.
Then ask: when do the distances add, and when do they subtract? The answer they find (zero between them, or not) is exactly what Lesson 3 turns into subtraction of integers: the gap between a and b is |a − b|. Do not give them that formula today.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 3 (8 steps, not −8). |
|---|---|
| 10 min | Hook: the lift. Take wrong answers (4) seriously and ask what they assumed. |
| 15 min | Absolute value; the figure with predictions; the rule in their own words. |
| 10 min | Worked examples; the submarine in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 13; −4 °C; 8 steps. |
|---|---|
| Worked examples | Lift: 10 floors. Yakutsk: 13 degrees. |
| Faded example | 650 m apart. |
| 1. Absolute value | B, 12. A means the bars are being read as brackets. |
| 2. Absolute value | 17. |
| 3. Distance across zero | 15 = 6 + 9. |
| 4. Distance, same side | 12 = 20 − 8. |
| 5. Lift | 15 floors. |
| 6. Two values | B. D confuses the distance between the pair with each one's distance from zero. |
| 7. Working backwards | 12: they are 6 and −6. |
| 8. Finding a point | −13. |
| 9. Comparing values | A. B means |−9| is being read as −9; C means the digits are being read as size. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; −17 in question 2 | Reads |−12| as (−12). | Ask “how many steps?” with a finger on the number line. |
| 3 in question 3 | Subtracted the digits although zero is between the points. | Figure: set those two points and look at the band. |
| 28 in question 4 | Added although both points are on the same side. | Ask whether the path from one to the other passes zero. |
| 16 in question 5 | Counted floors (fence posts) rather than gaps. | Draw three floors and ask how many journeys between them. |
| 7 or 13 in question 8 | Went the wrong way, or lost the side of zero. | Ask which way “left” is before counting. |
Questions worth asking
- Can an absolute value ever be negative? No. Ask for a sentence using the word “distance”.
- Is |a| always equal to a? Only when a is 0 or positive. Ask for a counterexample.
- Why does the lift travel 10 floors and not 4? Listen for “it has to go through the ground floor”.
Differentiation
A way in for anyone stuck
- Use a vertical number line: floors and depths make “through zero” visible.
- Let students count steps on the figure before using the add-or-subtract rule.
- Start Level 1 with the warm-up question 3 alongside.
Stretch
- Find every integer x with |x| < 4 (seven of them: −3 to 3).
- Is |a + b| always equal to |a| + |b|? Test with a = 5, b = −2 (3 against 7): the answer is no.
Students learning in English as an additional language
Pair each word with a gesture: two flat hands for “distance”, a finger moving through zero for “between”.
| Word | Plain meaning | Example |
|---|---|---|
| absolute value | how far a number is from zero | |−5| = 5 |
| distance | how far apart, with no direction | the lift travels 10 floors |
| between | in the space separating two things | 0 is between −3 and 7 |
| same side | both above zero, or both below it | −20 and −8 |
| level / floor | a storey of a building; below ground is negative | level −3 |
Sentence starters for explaining aloud or in writing:
- The absolute value of … is … because it is … steps from zero.
- Zero is / is not between … and …, so I add / subtract.
- The distance cannot be negative because …
Two things not to say
“Absolute value makes a number positive.” It points students at a sign-flipping trick and they then write |−3 − 2| = 3 + 2. Say “how far from zero”.
“Just subtract the smaller from the bigger.” That is the error the lesson is built to catch: it gives 3 for −6 and 9.
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.
- Absolute valueFind |−23|, |23| and |0|.
23, 23 and 0. - DistanceHow far apart are −4 and 11?
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TemperatureOvernight the temperature fell from 6 °C to −9 °C. By how many degrees did it fall?
15 degrees. - Air and seaA gull flies 35 m above the sea and a seal swims 12 m below the surface, directly beneath it. How far apart are they?
47 m.
3Challenge
Reasoning, generalising and working backwards.
- Working backwards|x| = 7 and x is less than 2. What is x?
−7. - From Lesson 1Order from smallest: |−5|, −5, 0, |3|, −8.
−8, −5, 0, |3|, |−5|, which is −8, −5, 0, 3, 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: two coloured counters on a printed number line do the same job as the figure.