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Directed Numbers

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**Problems involving directed numbers can sometimes look quite confusing**

Problems involving directed numbers can sometimes look quite confusing. This is because we use + and – signs to show two different things. This is how we see sums written when we first encounter addition and subtraction at an early age. This is how we see sums involving directed numbers written (usually). 3 + 5 = 8 = 8 This + sign means “start at 3 and count on 5”. This + sign still means the same thing: “start at 3 and count on 5”. These signs tell you that you are dealing with positive amounts. It is only really useful to include the direction signs when dealing with negative amounts – this is why you don’t write them all the time.

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**Adding heat increases the height.**

Hot air balloons have burners inside the basket. The burner fills the balloon with hot air, which makes the balloon rise. The balloon starts at 0. As heat is added, the balloon rises. = +3 5 4 3 2 1 This + sign tells us about the operation (add). This + sign tells us about the direction of the number (it’s a positive amount, or heat in our story). + + + Adding heat increases the height.

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**Adding weight decreases the height.**

Balloons also carry sandbags, which weigh them down. If we added some sandbags to our balloon, the balloon would drop because of the extra weight. The balloon is at 3. As weight is added, the balloon drops. = 1 5 4 3 2 1 This + sign tells us about the operation (add). This - sign tells us about the direction of the number (it’s a negative amount, or weight in our story). + + + Adding weight decreases the height. - -

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**Subtracting heat decreases the height.**

If the pilot then turns the burner in the balloon off, the amount of heat will drop, causing the balloon to fall further. The balloon is at 1. As heat is subtracted, the balloon drops. = 1 5 4 3 2 1 This - sign tells us about the operation (subtract). This + sign tells us about the direction of the number (it’s a positive amount, or heat in our story). + Subtracting heat decreases the height.

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**Subtracting weight increases the height.**

If the pilot throws out more sandbags, the balloon will rise up again. The balloon is at 0. As weight is subtracted, the balloon rises. = 2 5 4 3 2 1 This - sign tells us about the operation (subtract). This + sign tells us about the direction of the number (it’s a negative amount, or weight in our story). Subtracting weight increases the height.

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Adding heat Adding weight 3 + 5 = 8 3 - 5 = -2 Subtracting weight Subtracting heat 3 + 5 = 8 3 - 5 = -2

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