The Other Side of Zero

1Bela’s Building of Fun

Floors can sit above and below a ground mark named 0. Going up is +; going down is −. The building is the picture; the floor-number is the integer.

Calling basement floors “broken counts” misses the place below zero.

How it works

1. Name the ground markFloor 0.
2. Name up as positive, down as negativeTwo sides of 0.
3. Read a floor as an integerThe place.

2The Token Model

A plus-token and a minus-token of the same size cancel: a zero pair. A pile of five plus and two minus is three plus after cancels.

Tokens that never pair still count. Colour without pairing is not the model.

Figure. A token is a walk. One plus token and one minus token make a zero pair and cancel. The leftover tokens are the integer you keep.

How it works

1. Pair one + with one −A zero pair.
2. Remove those pairsThey make 0.
3. Read what remainsThe integer.

3Integers in Other Places

Temperature below 0, a lift below ground, a gain-and-loss score — same integers, new pictures. The number line is the shared home.

A new picture is not a new kind of number.

Figure. Integers live in places, not only on a homework line: below zero on a thermometer, a basement floor, sea level as the zero mark. Same number, three landmarks.

How it works

1. Name the pictureTemp, lift, score.
2. Place it on a number line with 0The integer.
3. Use the same + and − rulesThe picture is costume.

4Explorations with Integers

You may add a movement to a start: start + movement = landing. 3 + (−5) = −2. The exploration is trying several movements and recording landings.

Adding the digits 3 and 5 as 8 ignores the sign of the movement.

How it works

1. Write startAn integer.
2. Write movement with its sign+ or − a hop.
3. Land on the number lineStart plus movement.

5A Pinch of History

People needed a name for “the other side of zero” for debts, temperatures, and coordinates. The history explains why the house exists; it does not replace the number-line read.

A tale without a landing is history class, not the integer.

How it works

1. Hear why a below-zero name was neededA reason.
2. Then read this integer on the lineThe maths.
3. Keep tale and read apartTale explains; line names.

6Starting floor plus movement is the target floor

Target = start + movement. Movement up is positive; down is negative. This is the one sentence for the building.

Target is not “start minus the floor number you like”.

How it works

1. Write startCurrent floor.
2. Write movementSigned hop.
3. Add on the lineThe target.

7A token pair that cancels is a zero pair

+1 with −1 is 0. You may add or remove zero pairs without changing the value. That is why 5 + (−2) can be thought of as three leftover plus-tokens.

Removing an unpaired token changes the value.

How it works

1. Find a + and a − of one unitA pair.
2. Cancel themA zero pair.
3. Read the leftover tokensThe same integer.

8Below zero is a place, not a broken count

−3 is three hops left of 0 (on a standard line). It is as honest a place as +3. Empty is 0, not “broken”.

Refusing to write the minus sign makes +3, a different place.

Figure. Below zero is a place on the same line, not a broken count. Floor −1 is as real as floor 1; the minus names the side of zero.

How it works

1. Find 0The origin.
2. Hop left three equal hops−3.
3. Keep the minusThe place-name.