1. Chapter 3 Whole Numbers Section 3.5 Algorithms for Whole-Number Addition and Subtraction

2. In this section we look at some algorithms for adding and subtracting whole numbers. An algorithm is a method to do something, in this case addition and subtraction. We will look at the standard way you learned to add and subtract and some alternatives. What is common to all these arithmetic algorithms is that they involve breaking numbers up into parts (digits), doing something with each part (digit), then putting it together to get the overall answer. Addition Algorithms +0123456789 00123456789 112345678910 223456789 11 33456789101112 445678910111213 5567891011121314 66789101112131415 778910111213141516 8891011121314151617 99101112131415161718 The addition problems shown to the right are the basic addition facts. Those are the addition problems that show how all the single digit numbers are added to all other single digit numbers. These facts are eventually memorized through frequent use and from some techniques such as flash cards.

3. The addition algorithm you remember from school is a breaking apart type of method. It is known as the partial sums algorithm and is based on knowing the basic addition facts and the concept of carrying. 45=4  10+5Write in expanded form ( break apart ) + 38=3  10+8 =7  10+13Use the basic addition facts =7  10+ 1  10+ 3Write 13 in expanded form =(7+1)10+3 Distributive property of mult over addition =8  10+3Basic addition Fact 83 This can also be represented in terms of Dienes blocks by making a series of trades. carrying  = 4 longs 5 units 3 longs 8 units 7 longs 13 units 4538 = 8 longs 3 units 83 Trade 10 units for 1 long

4. Error Patterns in Addition As teachers you would be expected to analyze a series of problems that students have done and be able to identify the mistake they are making. Look at the addition problems below and see if you can spot the mistake that is being made. Show what the child would get if they did the next problem and continue to make the same mistake. What mistake is made here? 127 49 453 + 54+ 78+247 12754 4978453247 They are writing the numbers together one after the other. This is called an amalgamation error. What mistake is made here? 127 49 453 + 54+ 78+247 171111176910 They are not carrying the tens digit up to the next place value.

5. Subtraction Algorithms The standard subtraction algorithm that you learned is known as the partial differences algorithm. It also breaks apart the number to do smaller number computations. 56=5  10+6write in expanded form - 32=3  10+2 (5-3)  10+ 6 – 2subtract corresponding digits 2  10+4subtraction facts 24 This can also be represented using Dienes blocks. 56 subtract 3 long and 2 units leaves 24 The difficulty for subtraction is when you want to take a larger digit away from a smaller digit. (i.e. the digit on the bottom is larger than the digit on the top) This is when we need to introduce the concept of borrowing.

6. Subtraction With Borrowing We look at when we want to do the following type of subtraction problem. 56=5  10+6write in expanded form - 27=2  10+7 = (4 + 1)  10+6basic addition fact =2  10+7 = 4  10 + 1  10+6distributive property =2  10+7 =4  10 +16expanded form in reverse =2  10+7 = (4 – 2)  10+ (16 – 7) =2  10+9subtraction fact 29 56 borrow subtract 2 longs and 7 units trade 1 long for 10 units leaves 2 longs and 9 units Using Dienes blocks we get:

7. Equal Addends Subtraction Method The equal addends subtraction method is an alternate method to the standard subtraction that eliminates the need for borrowing. It adds the same amount to both numbers so that the need to borrow is eliminated. The idea is when you have a digit in the number you are subtracting that is larger than the number you are subtracting from add the correct number to both to get the digit in the number being subtracted to be 0. This is slightly different than the way that is demonstrated in the book, but I think it is easier. Example: 246 - 158 + 2 = 248 - 160 + 40 = 288 - 200 88

8. Lets do the following problems with the equal addends subtraction method. 5164 - 327 + 3 = 5167 - 330 + 700 = 5867 - 1030 4837 2629 - 845 + 60 = 2689 - 905 + 100 = 2789 - 1005 1784