MATH 112 Section 3.2: Understanding Subtraction

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1 MATH 112 Section 3.2: Understanding Subtraction Prof. Jonathan Duncan Walla Walla College Fall Quarter, 2006

2 Outline 1 Interpretations of Subtraction 2 Subtraction Properties and Algorithms Subtraction Algorithms 3 Mental Subtraction Methods 4 Conclusion

3 Set Model of Subtraction Just as with addition, subtraction can be modeled in several different ways. Again, sets provide one of the most basic models. Jane had 7 oranges. She gave 3 of them to Billy. How many oranges does she have left? The take away model is easy to see and works well when the problem is phrased in a particular way. Jane has 7 oranges. Billy has 3 oranges. How many more oranges does Jane have than Billy?

4 Number Line Model of Subtraction Another model of subtraction which works well for certain applications is the number line model. Sam walked 6 miles home from school and passed a candy store on the way. After raiding his piggy bank, Sam walked one mile back to the candy store. How far is the candy store from Sam s school? In this problem, it is perhaps easier to see that we can define subtraction in terms of addition. Subtraction undoes addition. Subtraction and Addition For whole numbers a, b, and c the following are equivalent. c b = a a + b = c

5 Bringing the Models Together So if subtraction and addition are tied together, their models should be tied together as well. Modeling Addition In general, a + b is modeled by combining two parts to make a whole as shown below. Modeling Subtraction In general, a + b is modeled by combining two parts to make a whole as shown below.

6 Properties of Subtraction In the last section, we used algebra to express several properties of addition for whole numbers. What about subtraction? Properties of Subtraction For whole numbers a, b, and c which of the following properties which were true for subtraction are also true for addition? Identity Property: 0 a = 0 a = a Commutative Property: a b = b a Associative Property: (a b) c = a (b c) Closure: a b is another whole number Counting: a 1 is the previous whole number Solutions None but the last of the properties above are true of subtraction.

7 Modeling Base 10 Subtraction As you have seen in your lab work, we can use manipulatives to model numerals (think of flats, longs, and units) and operations such as subtraction. Use base 10 blocks to model each subtraction problem

8 Subtraction Algorithms The Standard Algorithm As with addition, there are several different algorithms for subtraction. Most of you have probably learned the standard algorithm in school. Standard Subtraction Algorithm In the standard algorithm, subtraction is done column-by-column starting with the right-most column. If the bottom digit is larger than the top digit, then we borrow one from the digit at the top of the next column to the left and add 10 (or whatever the base) to the top digit in the current column. Use the standard addition algorithm to find the following difference

9 Subtraction Algorithms Pay-Back Subtraction Another popular method for subtracting is the pay-back method. Instead of borrowing from the top digit in the next column, we pay back the bottom digit in the next column. Pay-Back Subtraction Algorithm In the pay-back algorithm for subtraction, we start with the right-most column and subtract the bottom digit from the top. If the bottom digit is larger than the top, we add 10 (or our base) to the top and continue the subtraction. Then, we pay-back our use of 10 by adding one to the bottom digit in the column to the left. Use the pay-back subtraction algorithm to find the following difference

10 Subtraction Algorithms Indian Subtraction Algorithm The Indian algorithm uses borrowing just as does the standard algorithm. However, there are two major differences we subtract from left-to-right and borrow from the answer. Indian Subtraction Algorithm In the Indian subtraction algorithm subtraction is performed from right to left. If the bottom digit of a column is larger than the top digit, then the one is is borrowed from the solution to the previous column and 10 (or whatever the base) is added to the top digit in the current column. Use the Indian subtraction algorithm to find the following difference

11 Subtraction Algorithms The Multicultural Subtraction Method The multicultural subtraction method has been used in a wide variety of places and times. According to your text this method was used in 15th century Italy and is currently used in Japan. Multicultural Subtraction Algorithm In the multicultural subtraction algorithm subtraction is performed in columns from right-to-left. If the bottom digit is smaller than the top, then subtraction is performed as normal. If the bottom digit is larger than the top digit, then the bottom digit is subtracted from 10 (or whatever the base) and that difference is added to the top digit. Finally, the top digit of the next column over is reduced by one. Use the multicultural algorithm to find the following

12 Algorithms for Mental Addition As with addition, many of the algorithms we use to perform subtraction with pencil and paper do not work as well when subtracting without these tools. In the next few slides we will look at several ways to subtract mentally. Find each of the differences in your head as quickly and accurately as possible

13 Adding Up Method Since addition and subtraction are closely related, it makes sense that we might be able to use addition to make a subtraction problem easier. To subtract using the adding up method, follow these steps: We want to know what do we add to 28 to get 65? Adding yields 68 which is close to 65. Since 8 5 is 3, we need to add 40 3 = 37 to 28 to get 65. Try this method with the other differences: and

14 Add to Nearest 10 Since our system of numeration is base 10, multiples of 10 are particularly easy to work with in addition and subtraction problems. To subtract using the add to the nearest 10 method, follow these steps: First note that 28 is 2 less than 30 (the nearest 10). Next, yields 65 which is the number we are subtracting from. Finally, = 37 giving the amount we must add to 28 to get 65. Use the nearest 10 method to find the other differences: and

15 Piecemeal Subtraction As with the break and bridge addition method, the piecemeal subtraction method emphasizes breaking numbers up to make the process easier. To subtract using the piecemeal method, follow these steps: First break 28 into 20 and 8. Next subtract 20 from 65 to get 45. Finally, subtract the remaining 8 from 45 yielding 37. Use piecemeal subtraction method to find the other differences: and

16 Using Negative Numbers So far we have only been working with whole numbers. However, negative numbers can allow us to more quickly carry out subtraction problems. To subtract using negative numbers, follow these steps: Subtract the first digits, 6 2 = 4. Next subtract the second digits, 5 8 = 3. Finally, put 40 and -3 together to get 37. Use negative numbers to find the other differences: and

17 Estimating Addition and Subtraction In many instances we do not necessarily need the exact answer to an addition or subtraction process an estimate is close enough. Use strategies seen in the mental addition/subtraction sections we have covered to estimate each sum or difference , , ,

18 Important Concepts Things to Remember from Section Ways to model subtraction 2 Properties of subtraction 3 Alternative subtraction algorithms 4 Mental subtraction algorithms and estimations