Sec. 1.3 – Adding and Subtracting in Egyptian; Base Ten Blocks and Combining like Terms
Chapter 1, Section 3
Math Topics – Adding and Subtracting in the Egyptian Numeration System; combining like terms in algebra
Elementary Education – Base Ten Blocks; Addition and Subtraction as “Trading;” Partial Sums Addition
Using Base Ten Blocks to Create Numbers, and to Add and Subtract
Base ten blocks can be used to represent the powers of ten.
Base Ten Blocks
Cubes
Flats (squares)
Rods
Units
Ten flats make one cube. In the picture below, imagine stacking tens flats on top of each other to make a cube.
Ten rods make one flat. In the picture below, imagine stacking ten rods up to make a flat (a square).
Ten units make one rod. In the picture below, you can see ten little units stacked up inside the rod.
The smallest sized block
Thousands
Hundreds
Tens
Ones
1,000
100
10
1
103
102
101
100
Children Using Base Ten Blocks
Base 10 blocks can be a way for students to learn about place value as early as first grade. In later grades they can use the blocks to learn about addition and subtraction before learning paper and pencil methods. They can then move to doing paper and pencil arithmetic alongside the blocks, until they can eventually work without the blocks.
Example 1 Make 213 using base ten blocks.
213 has two hundreds, so we use two flats; it has one ten, so we use one rod; it has three ones so we use three units. 213 = 100 + 100 + 10 + 1 + 1 + 1
Example 2 Add 56 + 79 using base ten blocks.
56 is 5 rods and 6 units, 79 is 7 rods and 9 units. When we add them, we get 12 rods and 15 units.
Now we see that we have have 9 + 6 =15 units, so we can trade 10 units for one rod. We can also trade 10 rods for 1 flat (100).
Now we have 1 flat, 3 rods, 5 units = 135.
When we add the “normal” way that we are used to:
1 1
56
+ 79
135
the two trades that we just did are represented by “carrying the 1,” the little ones you see above the 56.
It is definitely faster to show the addition this way, but it has the disadvantage that, when a child is first learning how to add, “carry the 1” makes no sense. I remember when I was in third grade, I asked my teachers, “How do you know which one to carry and which one to put on the bottom?” They told me that I should just memorize it. It is a strong memory, etched in great anxiety for me, because I knew I would not be able to remember what to do.
Instead of jumping right to “carry the 1,” if students spend time trading blocks and seeing patterns, when they are ready, they will be happy to use the “faster” way because they will understand why it works.
Question: Now you try!
Example 3 Subtract 31 – 8 using base 10 blocks.
In order to take 8 units away, we need to trade one of the rods for ten unit blocks:
First, trade a rod for ten units.
Now we have 2 rods and 11 units and can take 8 units away. We end up with 2 rods and three units, which is 23.
Subtraction in base 10 is similar to addition because we are still trading in base 10. But now instead of gathering together 10 blocks and trading for 1 larger block, we trade 1 large block for 10 smaller ones.
Partial Sums Addition
After working with base ten blocks for awhile, children can start to use paper and pencil techniques alongside of the blocks. Partial sums addition is a great method to use with base ten blocks, because it shows each place value being added, just as we do with base ten blocks. Partial sums addition does not involve “carrying the 1,” but that method can be introduced afterwards as a quicker version of partial sums addition. Partial sums addition is a stepping stone toward learning how to carry.
Example 4 Add 56 + 79
56 =
+ 79 =
50 + 6
+ 70 + 9
120 + 15 = 135
We can add the tens: 50 + 70 = 120, then the ones: ones: 6 + 9 = 15; or add the ones and then the tens. Unlike with the “carry the one” method, the order does not matter.
Question: Now you try!
Example 5 Add 795 + 638
795 =
+ 638 =
700 + 90 + 5
+ 600 + 30 + 8
1,300 + 120 + 13 = 1433
With three digit numbers, you can add hundreds, then tens, then ones, or the other way around.
795 = 700 + 90 + 5
638 = 600 + 30 + 8
Adding the hundreds we get 700 + 600 = 1,300, adding the tens we get 90 + 30 = 120, and adding the ones, we get 5 + 8 = 13.
Then we add all our results together to get 1,433.
Question: Now you try!
The Algebra Connection
The way we add separate place values in partial sums addition is similar to the way
we add algebraic expressions. For example, to add (3x2 + 9x + 7) + (5x2 + 2x + 8), I would combine the “like terms” by adding 3x2 + 5x2 = 8x2, then combine 9x + 2x = 11x, and 7 + 8 = 15.
If we write this horizontally, we can see the similarity to adding 397 + 528:
3x2 + 9x + 7
397 = 300 + 90 + 7
+ 5x2 + 2x + 8
+528 = 500 + 20 + 8
8x2 + 11x + 15
800 + 110 + 15 = 925
This similarity exists because the algebraic expressions using increasing powers of x, while the numbers use increasing powers of 10.
If we let x = 10 in each algebraic expression, we get
3x2 + 9x + 7
= 3(10)2 + 9(10) + 7 = 300 + 90 + 7 = 397
and 5x2 + 2x + 8
= 5(10)2 + 2(10) + 8 = 500 + 20 + 8 = 528
The similarities stop at the final addition — with (3x2 + 9x + 7) + (5x2 + 2x + 8), we stop once we get 8x2 + 11x + 15. But with 397 + 528, we continue on to get 800 + 110 + 15 = 925.
Egyptian Addition and Subtraction
We can add and subtract using Egyptian symbols, using the same method as with base 10 blocks. This could be fun for kids to do as an extension exercise. For adults, using Egyptian symbols can make the trading easier, because it’s different enough that you can focus on it with new eyes. Also, with Egyptian numbers, we can use values higher than 1,000. With base ten blocks, we have to stop at 1,000 because cubes are the largest dimension we can show physically.
General principles of Egyptian addition and subtraction:
When we add, we circle and trade ten of a symbol for one of the next higher symbol.
We know we have to trade in addition when our symbols add up to ten or more.
When we subtract, we trade one symbol for ten of the next lower symbol.
We know we have to trade in subtraction when the number we are subtracting from is smaller than the number we are subtracting.
Example 6 Add the two Egyptian numbers below, without translating them into Hindu Arabic (our system.)
Trade tens lines for one horse shoe.
Trade ten coils for a flower, and trade ten flowers for a raised finger.
The answer is now gotten by finding the totals of each symbol – there are three total raised fingers, two flowers, eight horse shoes and four lines. Notice that there are no coils in our answer because we traded them all for a flower.
We can translate the above addition problem into Hindu Arabic if we like, to check our work, but it is important to try the problem first without translating, to really understand the base ten trading.
15,536
+ 16,548
32,084
Example 7 Subtract the two Egyptian numbers below.
To start, we subtract the lines. No need to trade since we have enough lines on top.
We don’t have enough horse shoes on top to subtract, so we trade one coil for ten horse shoes.
Now we have 13 horse shoes and can subtract the six on the bottom.
We can subtract the coils without needing to trade: 5 coils take away 4 coils leaves one coil. But to subtract the flowers, we need to trade one raised finger for ten flowers.
Now we finish the subtraction by subtracting four flowers from 11 flowers, to get seven flowers. Notice that there are no pointing fingers in our answer because we traded them all for ten lotus flowers.
Again, we can translate the above subtraction problem into Hindu Arabic if we like, to check our work, but it is important to try the problem first without translating, to really understand the base ten trading.
Remember, if you are using the paper homework, use the file below to check your answers! But never just copy the answers in here. These answers are for you to check your own work against. Remember to show all your work!
Note: Some people like to check when they are all done, but it is usually better to check as you go, so that you don’t end up practicing an incorrect method.
