Elementary School Mathematical Concepts

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Elementary School Mathematical Concepts

Week One

Adding Bases

When presented with two bases to add, first use base blocks to construct each number using the proper base amount for each place value. When we add in this manner, I like to think of it as "combining." *Similar meaning, but the connotation helps the imagery I feel* Once you have each number constructed, combine all of the blocks into as many complete six base tens columns as you can. Whatever is remaining will be a part of the ones place. Doing this requires some regrouping, or, combining of ones from each number to create tens. Count each column & record the amount of columns as the tens place value. Do the same with the ones. Since we regrouped by a base of six, we already know that our base remains six.

To add bases, it is most convenient to first make sure that the bases in which are being added are the same. This means that we should at least first begin to add bases that have equal value. Again, using base blocks for this will further strengthen meaning & understanding as it provides a visual & tangible action that is happening as a representation of the numbers & symbols.

Converting Bases

Converting bases to base ten begins with the non-base-ten number. Using base ten blocks is an effective way to understand this conversion/ relationship. Let's say we are working with 23 _ base six. First, build the number using base ten blocks. This should result in two sets of "tens" & three blocks in the ones place. o o o o This equals 23 due to the base being 6. o o o o o o o o o o o However, to convert it to a base ten, we simply count how many blocks exist; there are 15. 23base6 equals 15base10.

The base is relative. Therefore, we can use bases interchangeably as long as we understand how to convert them.

Although it is typical to base our math upon ten (hence, "base ten" blocks), we can also apply the same rules to any base. When we think about what it means for a number to be a base number, we are considering, in a sense, what makes the ones, tens, & hundreds places. For example, if seven is the base at-hand, comparatively to ten, 49=100. Because...

Introduction to Base Ten Blocks

Base ten blocks are tangible manipulatives which represent the quantitative value of number symbols. Typically, they are used in representation of the ones, tens, & hundreds places.

Week Two

Introduction to Multiplication

Furthermore, 4 x 5 is not the same as 5 x 4. Of course the answer is the same, BUT when guiding students to a genuine understanding of what it means to multiply x by x, it is crucial that there is differentiation in the order of the variable. Diagrams effectively support the connection of these groupings.

Diagrams in the teaching & learning of multiplication are essential for a few different reasons. Diagrams can be photos, drawings, or manipulatives.

Alternate Algorithms

Subtraction: Much like addition, there are a few different ways to systematically subtract. Here is one of those strategies:

Addition: Standard addition algorithm: 123+123=246

Expanded form: *adding by place value 100+20+3 100+20+3 _________ 200+40+6 = 246 Left to Right: *add beginning with the first digits, using their appropriate value 123+123 200 40 6 ______ 246 Friendly numbers: *first adding the numbers that make ten 62+28 2+8= 10 60+20=80 10+80=90 Trading off: *transferring a key amount from one side of the plus sign to the other to create a more simplistic addition 75+28 (+2)73+30 = 103 Scratch: * rounding numbers in an equation up or down to create more easeful adding; typically done in factors of five or ten, adding & subtracting the difference accordingly to account for the difference 4+6+9+11=20 5+5+10+10=20 Lattice: *using a box to separate the digits of the equation, only adding single digits, then adding the columns 2 5 +4 8 _______ 0 1 6 3 7 3

Week Four

Fractions

Ways to make fractions compatible

How to add & subtract fractions: If the fraction shares the same denominator... - simply add the numerator, leaving the denominator the same If the fraction does not share the same denominator.. - find a common denominator & repeat the action to the numerator; then, add the numerator only - simplify if needed

Vocabularly Numerator: how many of something is being represented Demoniator: the size of each "something"

Solving Integer Addition/Subtraction

A mini diagram should be used to help a student keep tabs, per se, on the integers in an equation. This strategy can show students, depending on the problem, whether they should add or subtract as well as whether their answer should be positive or negative. The idea is to still represent & keep track of the actions of the integers, using more efficient symbols.

Transitioning from Manipulatives to Diagrams ~ Once students can master the demonstration of adding positive & negative integers using manipulatives (tiles, chips), they can begin to represent these actions systematically using symbols as opposed to tangible items. ~ One way to do this is through mini diagrams.

Week Three

Integers

A Basic Understanding of Adding Integers: ~ What is a positive integer? ~ What is a negative integer? ~ Is 0 a positive or negative integer?

"Cancelling Out" While this is a relatively simple term, it is essential for students to understand what it means & what the action of cancelling out does. When adding opposite integers especially, students must be able to recognize how the value of the final product is effected & changed based upon where positive & negative integers are present.

Adding integers Before "+" & "-" can have any meaning or explicit understanding, students must make it tangible. Using manipulatives such as two-colored tiles or chips is effective in bringing this concept to life. Designating a color to positive-representing integers & the opposite to negative-representing integers aids students in actually seeing & interacting with the relationship between positive & negative integers. Once students are taught that there are numbers less than zero, it can be incredibly challenging to "picture it." Having visuals help to demonstrate the implied number line of negative-positive.

Division

Disadvantages of Long Division ~ Students can be unsure in how to set up the problem properly. It is easy to become unsure of which number to put inside the bracket versus outside of it. This is due to the problem needing to be set up in a way that is not the same as we would speak or say the problem. _____ _____ 5) 25 versus 25) 5 These problems are not the same. If the written problem is "twenty-five divided by five," it seems logical to set the problem up in the order in which we hear the words. However, this is incorrect.

Alternate Algorithms Repeated subtraction: ~ This method might look like long divison, but it is more practical as students get to consider factors in using repeated subtraction. ~ Using the number in which is being divided by, find multiples to subtract from the number being divided (the similarity to loooong division). ~ Keep track of what numbers are being multiplied by the constant number, or the number which is being divided into; these will be necessary to add in solving for the product & remainder.