Elementary School Mathematics

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Elementary School Mathematics

Week 9: Fractions

The reason we need common denominators when adding and subtracting and not when multiplying or dividing is because in the adding or subtracting case, we need equal sizes of the pieces. When multiplying or dividing isn't combining. It is more a group times what is inside the group.

Using boxes to add or subtract fractions

When adding, you use 3 boxes, when subtracting you use 2 boxes, and multiplying fractions requires 1 box when using simple numbers. Using complex numbers requires a simplifying method and then common denominators.

Linear Model

Comparing it to length and it talks about straight line corresponding to length

Compare sizes of shapes so you need the same shape

Set Model

Shape does not matter and they have common characteristics

Week 8: Multiplication

Multiplication is easy to learn if the two rules are remembered. The way problems are read is also important when solving these types of problems.

Rules

If signs are different, the answer will be negative.

6:40

If signs are the same, the answer will be positive.

Zero Banks: Only used if the first number has a (-).

With diagrams you have to match what the question is asking. Ex. 3(-6) would be 3 groups of negative 6. It is important to remember that -9(3) is the same as 0-9(3) so take away 9 groups of 3.

Week 6: Integers

Integers help with the idea of negatives. The idea of zero banks then is more apparent because in some situations you do not have enough of a given positive or negative. Also, when adding you get introduced to the idea of KCC (Keep Change Change).

Addition

For addition you draw a diagram. Writing the value of the numbers on top, (-- +, ++ -). If - +, subtract the numbers, if - - or + +, add the numbers and use the sign that is left.

Week 5: Alternative algorithms for adding and subtracting & alternative algorithms for multiplication

Similar to add/sub, these algorithms are useful in visually seeing what is being multiplied. Remembering place value is important to make sure the correct value, such as hundreds or ones, is being multiplied correctly. Place value is important!

Multiplication

Array

Once again, this helps visually see what is being multiplied ex. 4(3) would look like: . . . . . . . . . . . .

Groups

Grouping helps to visually see what is being done ex. 4(3) = 4 groups of 3, usually in a circle

Lattice works the same as in add/sub. Set up is similar, just multiply

Area Model

Area model creates a box, almost like expanded form making it easier to multiply. After multiplying, simply add across then down.

Expanded form

Like addition and subtraction, only this time you will be multiplying the values

Base 10 blocks

Using this helps students visually see what is being multiplied. It also makes it easy to see longs, flats, and cubs, be made.

Alternative Algorithms for adding/subtracting

Lattice

This helps to keep number in order and it makes it easier to have students write down numbers in the way they normally would. Draw the box under the addition or subtraction problem then add lines between the gaps of the numbers to create squares. Then connect diagonal lines from coroner to corner.

Equal add ins

This allows for the number to be put into simpler form making it easier to add/subtract 47-16 : add 3 to both 50-19 =31

Week 4: Alternative algorithms for add/sub

The alternative algorithms work similar for both adding and subtracting bases. It is important to note an important part:

When adding/subtracting in different bases, it is important to look for when units, longs, flats, etc... turn into the next bigger thing. Such as if you are adding 2 base 3 and 1 base 3, the answer will be 1o base 3 because the units turn into a long and zero units. Likewise if you are subtracting.

Scratch

Having the numbers aligned makes it easier when adding multiple numbers together. Depending on the base, counting the columns right to left, you will count until the base you are in and scratching when you get there. The number of scratches then transfers to the next column and so on. *The base is 10 12 04 11 +03 = 20

Left-Right

Once again, this makes it easier as you can keep the values of the numbers in place and makes more sense than "carrying a one" 123 +321 (100+300) +(20+20) +(3+1)

Expanded Form

This method teaching students to expand their numbers and learn place value once again. 234 + 432 = (200+30+4 ) + (400+30+2).

Friendly Numbers/ Trading Off

This helps to get numbers to end in zeros which then allow to make adding or subtracting easier.

Week 3: Adding & Subtracting (in bases) & converting using algorithms

When converting bases using algorithms, it is important to know the value of the numbers you are using.

For example, going left to right, the values of the numbers 162085 would go as follows:

5 4 3 2 1 0

1 6 2 0 8 5

These values become important when using algorithms to convert others bases into base 10.

When converting other bases into base 10, backwards division is used (LLLLLLLeave). For example, turning 37 base 10 into base 8 would look like

8|37

|__

*It is important to remember that the numbers you get from dividing should never be bigger than the base you are converting to. So in the previous example, the biggest number can only be 7. Also, your remainder will go on the outside of the L.

8|37

|__

4|5₈

So the answer will be 45 base 8.

Converting using algorithms *To and from base 10

To base 10: You will be solving! Almost like expanded form, you will be separating the number involved and using the desired base to convert to a different base. The numbers place value is important in knowing what number is used to squared to the base being converted. for example, 423, the 3 will be squared to 0, the 2 to 1, and the 4 to 2. So the problem "423 converted to base 5" will look like: 4(5^2)+2(5^1)_3(5^0).

From base 10: Remember LLLLLLLLeave base 10=Backwards division. It is important to remember that numbers cannot be bigger than the base that you wish to convert to!! The answer will be going from your last number divided and going right up after that.

Add/Sub in bases (Using base 10 blocks)

Line up, to show place value, your units, longs, flats, ect.. in such a way that it makes it easier for you to add/sub. It is important to note if adding results in a higher units, such as in base 3 you get 3 units, you must turn it into a long to avoid mistakes.

Week 2: Converting Bases *Using base 10 blocks

The normal base used in the United States is base 10. When converting to base ten, it is important to note that there are units, that then turn to longs, that then turn into a flat, then to a cube. After this cycle, it becomes a bigger unit, bigger long, etc...

*Numbers can NEVER be = to the base you are in Ex. 45 base 5 does not work so 44 base 5 would be a better option*

Converting from Base 10 (using base 10 blocks)

The number that was once 10 turns into the base you are wanting to convert to. Such as before, one long was 10, now, depending on the new base, it can vary from any number besides 10.

Converting to Base 10 (using base 10 blocks)

Count the number of units (using the pervious base as a reference) to turn into base 10.