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§8.2 Adding and Subtracting Integers
How do we use models to add and subtract integers?How do we add or subtract integers without models?
Today We’ll Discuss
One of the main reasons for having negative numbers is that they give us additive inverses.Specifically, if a is an integer,a + (-a) = 0
Additive Inverse Property
The integer counter model extends the set model of addition and subtractionBUT! Students must be aware that we now have two types of counters: Positive Integer CountersNegative Integer CountersThe point of this model is to emphasize additive inverses by calling each pair: a zero pair
To add with integer counters:Model each integer using the appropriate amount of positive or negative counters.Place them all in the same pile and pair up with until you no longer can.Remove all zero pairs from the pile.Identify the integer represented by the remaining counters.
Integer Counters (Addition)
When adding integers, if they have… Same Sign: Add their absolute values. Use the common sign.Opposite Signs: Subtract the lesser of the two absolute values from the greater. Use the sign of the number with greater absolute value.
Find each sum.
a) 5 + (-10)
b) -3 + 7
c) -12 + 12
d) -8 + (-14)
The number line model is useful for helping students to get a “number sense” for working with negative numbers.It allows us to visualize how these numbers “make us go up or down.”
Number Line Model
To add with integers on a number line:Place an arrow of appropriate length with its tail starting at 0.The arrow points right if the number is positiveThe arrow points left if the number is negativePlace another arrow of appropriate length and direction with its tail starting at the head of the last arrow.The head of the last arrow is the sum.
Number Line Model
To subtract integers on a number line:Follow the general process as in addition of always starting at 0 and then placing the tail of a new arrow on the head of an old arrow.But, be sure you know the direction of the arrow… Recall that when we “subtract an arrow,” we place it in the opposite direction.
Number Line Model
Adding an Arrow:Positive = RightNegative = LeftSubtracting an Arrow:Positive = LeftNegative = Right
Number Line Model Rules of Addition and Subtraction
Similar to the number lines, we could make rules for adding and subtracting positive or negative integers…. But because of a simple rule, it is often easier to change the problem rather than memorizing even more rules.
Let a and b be integers. To subtract b from a, add the opposite of b.a − b = a + (-b)
Find each difference.
a) 3 − 12
b) -8 − (-13)
c) 5 − (-4)
d) -20 − 7
The highest elevation in North America is 6194 meters, while the lowest is -86 meters. Meanwhile, the highest in Africa is 5895 meters, while the lowest is -156 meters.
Which continent has a greater range in elevations?
The bottom of the shallow end of a swimming pool is represented by -3 feet. The bottom of the deep end is 9 feet deeper. Write an expression that represents the bottom of the pool at the deep end. What is the depth at the deep end?
To subtract using integer counters, lay out the counters for the first number, and then prepare to remove the second number from the pile.However, if you cannot remove the counters it asks for, you must put them in the pile without changing the problem.We do this by first adding as many zero counters as necessary.