At a glance. One lesson of about 60 minutes. Students meet integers as a distance and a direction from a chosen zero, compare and order them, and find opposites.
Calculator: no calculator. Nothing here needs one, and reading a number line is the skill being built.
What students will learn
Know
- An integer is a whole number, its negative, or zero.
- The opposite of a number is the same distance from zero on the other side.
- < means is less than; > means is greater than.
Understand
- A sign says which way from a chosen zero, and the zero is a human choice.
- Moving zero changes every number but not the order or the gaps.
- Further left (or lower) means smaller, even for numbers far from zero.
Do
- Place integers on horizontal and vertical number lines.
- Compare and order integers and write the result with < or >.
- Write the opposite of any integer.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: choosing and applying the ordering idea in familiar situations (questions 1 to 6).
- Criterion B. Investigating patterns: the zero-moving figure. Ask what changes and what stays the same; listen for a student who says the gaps never change, and ask them to say why.
- Criterion C. Communicating: writing comparisons with < and > correctly and justifying them with a number line (question 9).
- Criterion D. Applying mathematics in real-life contexts: heights and temperatures, and judging whether a sign makes sense in the situation (questions 3 and 8).
The one thing to do with the figure
Ask for a prediction before anyone moves the slider. With zero at sea level the diver is at −7. “If zero moves up to the harbour wall, what is the diver's number?” Take the answers −3 and −11 from the room, and let the figure settle it.
Then ask what did NOT change. The places did not move, the order did not change, and the lamp stayed 33 m above the seabed whatever the zero. That last fact is the seed of subtraction and of distance later in the unit, so name it on the board.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: Dead Sea and Kilimanjaro heights. The harbour figure on the board, with predictions first. |
| 10 min | Integers and opposites; the second figure. Students set a and b on their own devices and say the comparison aloud. |
| 10 min | Worked example on ordering; students finish the faded one in pairs. |
| 5 min | Question 1, the check. Hands up for each letter; if many choose A or D, reteach with a thermometer before going on. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 35, 53, 305, 350; −1 °C; 25. |
|---|---|
| Worked example | −12 < −8 < −1 < 0 < 3. |
| Faded example | −155 < −86 < −28 < −3 < 5199. |
| 1. Greatest of four negatives | B, −4. Picking −20 or −15 means the digits are being read as size; go back to the thermometer. |
| 2. Opposites | 9. |
| 3. Warmer of two | B, Oslo. −6 < −2. |
| 4. Counting integers | 6: −3, −2, −1, 0, 1, 2. |
| 5. Moving zero | −11. 4 m down to sea level, 7 m more to the diver. |
| 6. Ordering a list | B. C orders by distance from zero; D is reversed. |
| 7. Integer riddle | −6. Candidates −6, −5, −4; even: −6, −4; further from zero than −4: −6. |
| 8. Making opposites | 3.5 m above sea level. Gap 12 + 5 = 17 m, half is 8.5 m; −5 + 8.5 = 3.5. Then the bench is +8.5, the pool −8.5. |
| 9. The digits trap | B. A or D: the digits are being read as size. C: unsure what the sign means; return to the first figure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or D in question 1; −10 > −3 | Reads the digits as the size and ignores the side of zero. | A vertical number line or thermometer: which is lower, which is colder? |
| −3 or 11 in question 5 | −3: moved the diver with the zero. 11: right distance, lost the side. | Ask “is she above or below the new zero?” before any counting. |
| 7 or 8 in question 4 | Counted one or both ends. | Write the list out; circle the two ends and cross them out. |
| 5 in question 4 | Left out zero, often because zero “is not a number”. | Ask whether 0 °C is a temperature. |
| 8.5 in question 8 | Found the distance to the new zero, not where zero goes. | Ask them to point to zero on the figure, then say its height from sea level. |
Questions worth asking
- Is zero positive or negative? Neither: it is the reference point. Listen for “it is where we measure from”.
- Why do we need a sign at all? Because a distance alone does not say which way. 430 m could be up or down.
- Who chose sea level as zero? Could we choose differently? Yes: the ground floor, the freezing point of water, the start of a race. What would go wrong if every map chose its own zero?
- Is there a largest negative integer? A smallest? −1 is the largest; there is no smallest, because you can always go one further left.
Differentiation
A way in for anyone stuck
- Start with a vertical number line (a thermometer): up and down feels more natural than left and right for “greater”.
- Give everyone a printed number line from −20 to 20 for every question.
- Pair the warm-up thermometer question with question 2 before any abstract comparison.
Stretch
- Question 8, then the general case: where must zero go so that places at heights p and q become opposites? (Halfway, at (p + q) ÷ 2.)
- How many integers lie between −50 and 50, not counting the ends? Ask for the answer without listing them, and an explanation that would work for −n and n. (99: 49 negatives, 49 positives and zero; in general 2n − 1.)
Students learning in English as an additional language
Several of these words have everyday meanings that are close to, but not the same as, the mathematical ones. Say each word while pointing to the number line.
| Word | Plain meaning | Example |
|---|---|---|
| integer | a whole number, positive, negative or zero | −4, 0, 17 |
| positive | more than zero: above it, or to its right | +5 m, above sea level |
| negative | less than zero: below it, or to its left | −5 m, below sea level |
| opposite | the same distance from zero, on the other side | −6 and 6 |
| greater than, > | further right, or higher | 3 > −2 |
| less than, < | further left, or lower | −9 < −4 |
| sea level | the zero that heights are measured from | 0 m |
Sentence starters for explaining aloud or in writing:
- −8 is less than −1 because on the number line −8 is …
- The opposite of … is … because both are … from zero.
- When zero moves to …, the number for … becomes …
Two things not to say
“The bigger the negative number, the smaller it is.” True, but it hides the picture and confuses the word “bigger”. Say “further left” or “lower”, which works for every number.
“Minus means take away.” In this lesson the minus sign is a direction. The two meanings are kept apart on purpose until Lesson 3.
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.
- OppositesWrite the opposite of 14, of −3 and of 0.
−14, 3 and 0. Zero is its own opposite. - OrderingOrder from smallest: 2, −11, −2, 7, −7.
−11, −7, −2, 2, 7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- LiftsA lift panel shows floors −3 to 12, and 0 is the ground floor. How many floors are below the ground floor, and how many buttons are there altogether?
3 below (−3, −2, −1), and 16 buttons: 3 + 1 + 12. - TemperaturesAt 6 am it was −5 °C in one town and −9 °C in another. Which was colder, and by how many degrees? (Count on a thermometer.)
The town at −9 °C, by 4 degrees.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningIs there a smallest negative integer? Explain in one sentence.
No: whatever negative integer you name, one less than it is smaller still. - Choosing zeroWith zero at sea level a cave is at −40 m and a peak at +60 m. Where must zero go for the peak to be +100?
At the cave, 40 m below sea level: the peak is then 100 m above zero.
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. Under each one, “Show me the steps” gives the same states in words, for a screen reader or for anyone who prefers to read them.
No device? Print a number line from −20 to 20 and run the harbour figure as a drawing on the board.