Hills and Dales

Put one hill and one dale on the base. Explain how, if there is one hill and one dale, a bulldozer could be used to push the hill into the dale and leave a flat (zero) surface. Ask  the students what other combinations of hills and dales would result in flat earth that 50 would give a zero surface: +2 and –2, +3 and –3, etc. Record these combinations as equations, for example: +3 +  –3 = 0.

Show how calculations like these can be done on a calculator. For example, press 3 + 3 ± =. Note that the ± (+/–) key changes the sign of the number shown in the calculator window from positive to negative (or vice versa). Now place three hills and two dales on the base. Ask the students how they could describe this “landscape” using numbers (+3 + –2). Ask them how they could use a bulldozer to fill in the two dales. What would remain? (One hill). Show them how to complete the equation +3 + –2 = +1

Get the students to perform the operation on a calculator as a check: 3 + 2 ± =. Give them a range of addition problems to solve, using hills and dales. The students can use calculators to confirm the answers.

Good problems might be:

–1 + +4 = +3      +3 + +2 = +5      -2 + –3 = –5     +5 + –3 = +2      +1 + –4 = –3

–3 + +7 = +4      –3 + 0 = –3        –2 + +2 = 0       0 + –3 = –3        –1 + –4 = –5

Now give your students some subtraction problems that involve only hills, or only dales, and get them to describe these problems using equations. For example: +3 – +2 = +1 and –5 – –2 = –3.

Care needs to be taken when explaining how to do subtractions that involve both hills and dales. It can be useful to think of a subtraction as a removal. Suggest to your students that they imagine they are trucking contractors who earn their living from earthmoving. Ask them what similarity there is (in the amount of earth to be trucked in or away) in these pairs of situations:

• Removing a hill; creating a dale (by excavating it)

• Removing a dale (by filling it in); creating a hill.

Consider these two scenarios:

 From the trucker’s point of view, carting away the earth from the demolition of two hills or from the excavation of two dales is the same thing. (The same amount of earth has to be shifted.) Likewise, from a mathematics point of view, +3 – +2 has the same answer as +3 + –2. Both equal +1. Subtracting +2 has the same effect as adding –2. 

Similarly, from the trucker’s point of view, removing two dales (by filling them in) requires exactly the same amount of earth as creating two hills. (The same amount of earth has to be shifted.) So –3 – –2 has the same answer as –3 + +2. Both equal –1. Subtracting –2 has the same effect as adding +2.

 Give your students a range of problems involving removing hills and dales, and ask them to rewrite each one (using the trucker’s principle) so that the problem is entirely about hills, or entirely about dales, and then solve. Here are some to start with:

+4 –-1 [= +4 + +1 = +5]             -6 – +4 [=-6 + -4 = -10]             

+5 – -2 [=+5 + +2 = +7]             -6 – +3 [= -6 + -3 = -9]            

+4 – -2 [=+4 + +2 = +6]             -7 – +3 [= -7 + -3 = -10]

+8 – -5 [=+8 + +5 = +13]           -1 – +6 [= -1 + -6 = -7]        

+12 – -7 [= +12 + +7 = +19]      -4 – +6 [= -4 + -6 = -10]                        

+2 – -2 [=+2 + +2 = +4]             -9 – +9 [= -9 + -9 = -18]

Using Imaging

Provide equation sets so students can use both imaging of hills and dales and recursive relationships within the equations to establish results. For example:

+3 + +2 = +5                  –3 + +2 = –1                  +3 – +2 = +1                 

–3 – +2 = –5                  +3 + +1 = +4                  –3 + +1 = –2                 

+3 – +1 = +2                  –3 – +1 = –4                 +3 + 0 = +3                   

–3 + 0 = –3                    +3 – 0 = +3                    –3 – 0 = –3

+3 + –1 = ?                –3 + –1 = ?                +3 – –1 = ?                –3 – –1 = ?

+3 + –2 = ?                –3 + –2 = ?                +3 – –2 = ?                –3 – –2 = ?

Summarise the results of the calculations using an integer number line. Draw specific problems on number lines. For example, 0 + –3 = –3

 Develop a number line model that shows the directional effect of adding and subtracting positive and negative integers:

 Give the students the opportunity to apply this model, folding back to materials and imaging of the hills and dales, if necessary, to confirm results.

Using Number Properties

 The students can demonstrate their understanding of the number properties by solving integer addition and subtraction problems mentally. Suitable problems are:

+3 + –4 = –1                  –1 + +5 = +4                 +3 – –3 = +6     –4 – +2 = –6

+2 + +2 = +4                  –3 + –1 = –4                 +3 – +2 = +1     –3 – +2 = –5

+6 – +3 = +3                  –3 – –2 = –1                  –1 – +5 = –6      –3 + +2 = –1

+39 + –26 = +13           –64 + +58 = –6          +72 – –28 =     –47 – +45 = –92

Independent Activities

Material Master 4–40 is the game of Integer Invaders. Through this game, the students learn to add integers and plot co-ordinates on an integer number plane. Material Master 4–41 is the game of Integer Cover-up, which is a game to consolidate addition and subtraction of integers.