Negative numbers in Sec 1: think up and down

Every voyage needs a vessel. This one is a cardboard box in the laundry room. The washing machine's round door is the porthole, a kitchen-roll tube is the periscope and a blue bedsheet is the ocean. The Mastermind, in a paper captain's hat, gives the order: "Dive." The mission has a name, "Project Deep Dive", and a plan: "Five questions a day." The Partner, sonar operator for the day, with toy headphones on and a juice box in hand, is not convinced. "...Five?"

That is Project Deep Dive, the fourth episode of our comedy series, The Mastermind, and our first Secondary episode. Watch it first (it is under a minute), then read on. The treasure at the bottom of this ocean is a Secondary 1 sum.

Project Deep Dive: five questions a day. Deep down, it's just practice.

Every practice question gets a flat "Ping." from sonar. On the fifth ping, a cardboard hatch stamped TOP SECRET creaks open on a depth problem: the sub is at −30 m and rises 12 m. An arrow climbs a vertical number line from −30 to −18: −30 + 12 = −18, which is 18 m below the surface. The Partner sums up the mission: "Deep down, it's just practice."

He has a point. And the picture inside that hatch is one of the most useful ideas in Secondary 1 math.

Why negative numbers feel slippery

Negative numbers and integers are a Secondary 1 topic. Most everyday numbers count or measure something you can see. A number below zero is harder to picture: nobody can hold −5 apples.

A few things make the topic feel harder:

The signs do two jobs. In −7 + 3, the minus sign tells you where to start (7 below zero) and the plus sign tells you which way to move. A child who sees only "a sum with a 7 and a 3" may answer 10, −10 or 4. The answer is −4.
Bigger-looking numbers are smaller. 30 is more than 18, but −30 is less than −18. Comparing by the digits alone gives the wrong answer.
Crossing zero. A move that passes through zero, such as from −5 up to 3, is where most slips happen.

The good news: one picture handles all three.

The up-and-down method

Stand the number line on its end, like the depth gauge on the side of the cardboard sub. Zero is the surface. Positive numbers are above it and negative numbers are below it. Now every addition becomes a move:

Start at the first number.
Move up to add a positive number. Move down to add a negative number.
Read where you land. Above zero, the answer is positive. Below zero, it is negative.

The same picture settles comparisons: higher up is always greater, so −18 is greater than −30. Standing the line up ties each sign to words your child already uses: a lift goes up, the temperature goes down, a diver goes deeper.

Worked example 1: the sub rises 12 m

The sub is at −30 m. It rises 12 m. Where is it now?

Step 1: Start. Put a finger on −30, which is 30 m below the surface.

Step 2: Choose the direction. Rising is moving up, so we add 12: −30 + 12.

Step 3: Move in friendly jumps. Up 10 takes the sub to −20. Up 2 more takes it to −18.

Step 4: Read the answer. The sub stops below zero, so the answer is negative: −30 + 12 = −18. The sub is at −18 m, which is 18 m below the surface.

Step 5: Check with words. The sub was 30 m down and came up 12 m, so it is now 30 − 12 = 18 m down. Same answer.

Worked example 2: going deeper

The sub is at −4 m and dives 6 m deeper. Where is it now?

Diving deeper is moving down, which means adding a negative number: −4 + (−6). Start at −4 and count six steps down: −5, −6, −7, −8, −9, −10. So −4 + (−6) = −10, and the sub is 10 m below the surface.

Adding a negative number made the answer smaller. That surprises many children, and the picture explains it: adding a negative is a move down.

Worked example 3: crossing zero

On a winter holiday, it is −5 °C outside at 7 a.m. By noon it is 8 °C warmer. What is the temperature at noon?

Warmer means up, so we work out −5 + 8. This move crosses zero, so split it at zero:

Jump	From	To
Up 5	−5	0
Up 3 more	0	3

The jumps add up to 8, so −5 + 8 = 3: the temperature at noon is 3 °C.

"Go to zero first" is the safest habit for any move that crosses the surface. It works going down too. One of the practice cards in the episode is 4 − 9. Taking away 9 is moving down 9, so go down 4 to reach 0, then down 5 more: 4 − 9 = −5.

Common slips

Moving the wrong way. −30 + 12 answered as −42. Before any numbers, ask, "Is the sub rising or sinking?"
Comparing by size alone. "−30 is greater than −18, because 30 is bigger." Ask, "Which one is nearer the surface?"
Losing the sign across zero. −5 + 8 worked out as −3. Stop at zero first, then count what is left.
Dropping the minus sign. "−18 m" and "18 m below the surface" say the same thing; "18 m" on its own does not.

Three ways to practise at home

Masking tape, a sticky note, a coin and a die are enough.

A door-frame depth gauge. Stick masking tape up a door frame. Mark 0 at about shoulder height, 1 to 10 above it and −1 to −10 below it. Take turns to call out an addition, such as "start at −3, add −5", and move a sticky note to find the answer. Whoever moves the note writes the number sentence: −3 + (−5) = −8.
Coin-and-die dive. Flip a coin for the direction (heads is up, tails is down) and roll a die for how far. Start at 0, take five turns each, and write every move as an addition. At −2, tails and a 4 gives −2 + (−4) = −6. Five turns each, like five pings.
The lift puzzle. Basement levels are a natural way into negative numbers: B2 is two floors below the ground. But in many Singapore buildings the ground floor is Level 1 and there is no floor 0. Ask, "Going from B2 up to Level 3, how many floors does the lift travel?" On a number line, −2 up to 3 is 5 steps. In the lift, it is only 4 floors: B1, 1, 2 and 3. Explaining the difference (the missing zero) shows the picture has really sunk in.

Keep each session short and stop while your child still wants to play. When your child is stuck, ask "Up or down?" before "How far?", and praise the method rather than the speed.

If your child has just started secondary school, our guide to moving from P6 to Sec 1 covers the bigger picture.

Five calm questions a day

In the episode, the whole top-secret mission is five questions a day, each one logged with a flat "Ping.", while the day counter goes Day 1, Day 7, Day 30. It only ever goes up. Negative numbers become comfortable the same way: a few moves, met often and checked with a picture your child trusts.

That is the idea we built Singapore Math Drills around:

Daily Five drills: five hand-picked drill cards are waiting each day, so your child never has to decide what to practise.
Streaks and mastery stars: steady progress markers that reward turning up and getting things right, never speed.
Calm by design: no guilt-tripping over a missed streak and no dark patterns. Your child's progress is only ever measured against their own.

Integers and Negative Numbers is one of our live Secondary 1 topics, alongside Linear Equations and Angles, Triangles and Polygons, the other two topics on the episode's mission chart.

The Mastermind keeps his periscope up until the very end. Then, quietly: "Periscope down." Deep down, it's just practice.

Try it free

Try a free interactive Secondary 1 worksheet and see how calm practice feels, with no signup or credit card required.

Try the free Secondary 1 sample worksheet