How do we know if a number is a divisor of another number?
Understanding a set of divisibility rules goes a long way.
Factors are numbers that can divide evenly into another number. 4 is a factor of 12. In the multiplication problem 4x3=12, 4 and 3 are called factors and 12 is called a product. Factors can also be called divisors. 4 is a divisor of 12.
Products are defined as the answer to a multiplication problem. Products are also considered multiples of each factor. For instance, 12 is a multiple of 4. 12 is also a multiple of 3. The product of 12 x 3 is 36.
Some numbers have many factors, while other numbers have only a few. Prime numbers cannot be divided evenly by any numbers except for 1 and the number itself. Prime numbers include 2,3,5,7,11,13,17,19 and so on.
Composite numbers on the other hand have more than 2 factors. 6 can be divided by 1,2,3, and 6. This is what makes 6 a composite number.
Some skills we are working with:
Decomposing numbers is taking them apart.
So 14 + 17 can be thought of as 10 + 10 + 7 + 3 + 1
Or 12 x 15 can be thought of as 6x2x15 or 6x30
Composing them is just putting them back together.
Decomposing numbers often makes use of “friendly numbers” that can be any number people have familiarity with. (25, 10, 12 etc.)
Compressibility describes the idea that while we learn complex things it takes up a large portion of our brain and attention, but when we get it down it takes up very little… like riding a bike. This is why we can keep learning new and interesting math concepts and our brains don’t fill up!
Examine Mr. Olmsted's Factor Pair Mountain
The following diagram illustrates the concept of the Factor Pair Mountain for the number 110. Each pair of factors multiply together to get 110. You can make similar mountains for any number you choose.
Every factor has a factor pair. These two numbers multiply together to make the product, which in this case is 110. When searching for factors, you can use this knowledge to ensure that you have found all of the factors for a number.
To find factors, you need to use a system! Start with ONE and work your way to Factor Pair Mountain Summit. You will know you've reached the summit when two consecutive factors form number you are looking for. 10 times 11 is 110, so this is the mid-point, or summit. Now just find the factor pair for each number you found on the way up.
Notice that all of the high factors of 110 (22 and 55) are paired with low factors. This is intuitive, but also instructive. In fact, all factors fall evenly on both sides of the square root of the number you are factoring. This means that if you enter the number into a calculator and press the square root button, all of the factors you need to check are below that number. When you find one factor, you should then have its pair. (Divide one factor into your number to get the pair.)
All factors are whole numbers, which have no fractional (decimal) component.
Fractions to Decimals
All numbers can be written as fractions. All whole numbers would be written in the following way:
Any number over itself is an expression of the number 1
Fractions indicate division. 1/2 is one divided by two. 1/3 is one divided three ways. If we actually do that division, we can turn a fraction into a decimal.
1/3 = 0.333 (repeating)
1/10 = 0.1
You can use long division (or a calculator) to get this result.
Some fractions allow you to take short cuts. We can divide any number (including fractions) by one and get the same number. This is often called "simplifying fractions."
In this case we might decide to divide this fraction by 2/2, which doesn't change the value because 2/2 = 1.
Twenty-hundredths as a fraction is twenty-hundredths as a decimal!
Percents are the decimal representation for a fraction multiplied by 100. Just slide the decimal over to the right two places!
0.47 = 47%
1.2 = 120%
0.012 = 1.2%
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
There is a great deal of vocabulary in geometry - we begin with a few basic terms:
Polygon - from the Greek poly (many) gonia (angle) is a many sided shape made of line segments. The polygon itself refers only to the line segments and not the space inside or outside of the lines. The line segments connect to form a closed shape. Simple polygons (which we study) do not have any intersecting line segments. Where line segments meet an angle or vertex is formed.
The Julian Calendar (named for Julius Caesar) added July (named after him) and August (named after his son) to correct the calendar. These additions displaced the months that come after.
You should begin to master is the multiplication table through 12 times 12.
If you think about it... there aren't that many.
Because 6x4 is the same as 4x6, almost half of the facts are duplicates.
You already know that anything times 1 is itself.
And you probably know your 2s, 5s, 10s, most 11s, and some 3s.
Learn them systematically, notice patterns, practice.