We are now going to multiply two numbers $mcan and $mlier. The algorithm can be stated simply as:

1) Begin with the rightmost digits of each number

2) Multiply digits

3) Add carry if there is one

4) Go to the next digit

5) When there are no more digits, add the last carry

<==== CLICK CANVAS TO SEE NEXT STEP

So lets begin. The rightmost digit of $mcan is $digit. The multiplier is $mlier.

<==== CLICK CANVAS TO SEE NEXT STEP

When we multiply these two digits, we get $prod. There is no carry to add.

<==== CLICK CANVAS TO SEE NEXT STEP

The rightmost digit $digit becomes a part of our product.

<==== CLICK CANVAS TO SEE NEXT STEP

The leftmost digit $digit becomes a carry to be added to the next product.

<==== CLICK CANVAS TO SEE NEXT STEP

Now we need to choose the next digit in $prod.

<==== CLICK CANVAS TO SEE NEXT STEP

The next digit in $prod is $digit.

<==== CLICK CANVAS TO SEE NEXT STEP

When we multiply these two digits, we get $prod. This time there is a carry to add.

<==== CLICK CANVAS TO SEE NEXT STEP

Next we add the carry $carry to the product $prod.

<==== CLICK CANVAS TO SEE NEXT STEP

Again the rightmost digit $digit in $prod becomes a part of our product.

<==== CLICK CANVAS TO SEE NEXT STEP

And the leftmost digit $digit in $prod becomes the new carry.

<==== CLICK CANVAS TO SEE NEXT STEP

Since there are no more digits to multiply, we simply bring down the newest carry

<==== CLICK CANVAS TO SEE NEXT STEP

The newest carry is $digit. We add to get our final result $prod.

<==== CLICK CANVAS TO SEE NEXT STEP

NO MORE STEPS

<==== CLICK CANVAS TO RESET ALGORITHM

We are going to round the number $number. Our first step is to determine our range, or what would we round up or down to.

<==== CLICK CANVAS TO SEE NEXT STEP

To find the lower bound, we take our number and change the one's digit to zero. So our original number is $number. When we change the one's digit to zero, we get $lower.

<==== CLICK CANVAS TO SEE NEXT STEP

To find the upper bound, we take our lower bound and increase the ten's digit by one. So our lower bound is $lower. When we increase the ten's digit by one, we get $upper.

<==== CLICK CANVAS TO SEE NEXT STEP

So we can say, we will either:

Round down to $lower.

or

Round up to $upper.

<==== CLICK CANVAS TO SEE NEXT STEP

Examine the one's digit xdigit of our number $number to see what to do.

Since xdigit is less than 5, we round down.
Since xdigit is greater than or equal to 5, we round up.

<==== CLICK CANVAS TO RESET ALGORITHM

We are going to round the number $number. Our first step is to determine our range, or what would we round up or down to.

<==== CLICK CANVAS TO SEE NEXT STEP

To find the lower bound, we take our number and change the one's and ten's digit to zero. So our original number is $number. When we change the one's and ten's digit to zero, we get $lower.

<==== CLICK CANVAS TO SEE NEXT STEP

To find the upper bound, we take our lower bound and increase the hundred's digit by one. So our lower bound is $lower. When we increase the hundred's digit by one, we get $upper.

<==== CLICK CANVAS TO SEE NEXT STEP

So we can say, we will either:

Round down to $lower.

or

Round up to $upper.

<==== CLICK CANVAS TO SEE NEXT STEP

Examine the ten's digit xdigit of our number $number to see what to do.

Since xdigit is less than 5, we round down.
Since xdigit is greater than or equal to 5, we round up.

<==== CLICK CANVAS TO RESET ALGORITHM

There are $numgps groups. Lets draw them first.

<==== CLICK CANVAS TO SEE NEXT STEP

There are $itemsPer items per group. Lets place the items in each group.

<==== CLICK CANVAS TO SEE NEXT STEP

When we count the total number of items, we get $itemTotal.

<==== CLICK CANVAS TO SEE NEXT STEP

We can therefore conclude that
$total = $prod1 X $prod2.

<==== CLICK CANVAS TO RESET ALGORITHM

Joseph uses tally marks to record information about the farm and his family. He may record information such as the number of cattle, goats, chickens. He may also record personal information such as the age of family members.

<==== CLICK CANVAS TO SEE NEXT STEP

Tally sticks, made of wood or bone, have been used since ancient times as a “data recording” device or memory aid to record numbers, quantities, or even messages. The most famous of such artefacts is possibly the Ishango bone. The artefact was discovered during excavations conducted in 1950 in Ishango in the north-east of the Democratic Republic of Congo.

<==== CLICK CANVAS TO SEE NEXT STEP

One of the factors that makes this a base 10 system is each coin (or bill) is 10 times the value of the next smallest coin (or bill). So a dime = ten pennies, a dollar = ten dimes, etc. We can extend this with a ten dollar bill, huindred dollar bill, etc.

<==== CLICK CANVAS TO SEE NEXT STEP

When Joseph goes to the market, this is a time when recording numerical information or performing calculations is especially important. Also depending on the number of transactions, it can be time consuming.

<==== CLICK CANVAS TO SEE NEXT STEP

Another factor, that makes this a base 10 system, is there are ten symbols or digits. This is also a positional system because the value of a digit depends on its position.

<==== CLICK CANVAS TO SEE NEXT STEP

Lets contrast the two systems for a simple example. First lets examine using tally marks to record the value of some money.

<==== CLICK CANVAS TO SEE NEXT STEP

We will convert as many pennies to dimes as possible. Then we will convert as many dimes as possible to dollars. This will give us a more compact representation.

<==== CLICK CANVAS TO SEE NEXT STEP

First lets convert each group of 10 pennies to dimes.

<==== CLICK CANVAS TO SEE NEXT STEP

Next lets convert each group of 10 dimes to dollars.

<==== CLICK CANVAS TO SEE NEXT STEP

When we contrast our base 10 positional system with the tally mark system, we see a more compact system for representing quantities. As we will see later, it also allows more efficient calculation.

<==== CLICK CANVAS TO SEE NEXT STEP

The Mayans used a base 20 positional system. There were 20 "digits" and the powers of 20 ( 1, 20, 20*20 = 400, 20*20*20 = 8000, ...). Unlike our base 10 positional system where a digit's horizontal position determines its value, the Mayans used a stacked system where the vertical position determines its value.

<==== CLICK CANVAS TO RESET PRESENTATION