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