Addition and Subtraction Algorithms What is an algorithm? - a set of ordered steps for solving a problem. A place-vale part is a number, such as 6000, 20, or 5 that has a single leading digit followed by some number of zeros. The number 270 is not a place-value part but is the sum of two placevalue part, 200 + 70. (you know this as expanded form) Write each number as the sum of place-value parts (write in expanded form). 1. 1982 2. 305 3. 39,105 Addition Using Columns – the traditional addition algorithm follows these steps. 413 + 254 + 91 Millions of people trust this algorithm. It gives the correct answer, but not many people ask why it works. In a small group, you will discuss one or more algorithms and understand why each algorithm works. For exercises 1-7, answer these questions using sentences. How and why does the technique or algorithm work? Include the basic rules of arithmetic in your explanation, if possible. Why is the technique or algorithm useful? 1. Add more and then subtract. Find the sum. $17.99 + $19.99 Think about $17.99 as $18 - $0.01, and $19.99 as $20 - $0.01. Then, add $18 and $20 and subtract $0.02. 2. Reorder the terms. When you add many numbers, changing the order of the numbers can make finding the sum easier. How can you make this addition easier? 148 + 3 + 60 + 37 + 152 3. Subtract using columns. The traditional subtraction algorithm uses columns. Instead of carrying, you “borrow” a unit. You borrow when a column shows a subtraction such as 3 – 4, that does not give a positive result. 4. Subtract some more. Suppose you want to subtract 341 from 537. You can break up the number 341 into 337 and 4. Then you can subtract 537 – 337 = 200 and 200 – 4 = 196. 5. Add from left to right. Tyler has a method for adding numbers mentally. He explains his method this way. “Look at the sum 42 + 37 + 55 + 96 Start with the largest place value. These are all 2-digit numbers, so the largest place value is the tens place. First you add just the tens. 40 + 30 + 50 + 90 = 210. Next, take 210 and add the ones. 212, then 219, then 224, then 230. This is much easier!” 6. Subtract by counting up. To find the difference 1017 – 345, start with 345 and add numbers until you reach 1017. Here is one way to do this. 345 + 600 = 945 945 + 60 = 1005 1005 + 10 = 1015 1015 + 2 = 1017 Because 345 + (600 + 60 + 10 + 2) = 1017, (600 + 60 + 10 + 2) is the difference 1017 – 345 = 672. 7. Add consecutive integers. Here is a trick for adding a list of consecutive integers. Add the integers from 1 to 9. Then regroup the numbers so that many sums are the same. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 = 10 + 10 + 10 + 10 + 5 = 45