Elementary Mathematics: By Nathan McCormick
Week 6:
Positive and negative integers
Integers- these are whole numbers that can booth positive and negative. ( …. -4, -3, -2, -1, 0, 1, 2, 3, 4,…..)
-4 is opposite of 4
Absolute value is simply the distance from zero
20 = 20
-5 = 5
0 = 0
- when teaching negative numbers try to have the student develop the concept that a -3 is just three spaces away from zero
- Have students use a number line and have them plays zero at the center of that number line 1st go over the positive number
- then show them that numbers exist to the right of the zero and these are the negative numbers
- Learning negative numbers is important because they often come in real life situation such as discounts on a receipt or building things
Week 7
Types of Numbers: Exploring rational numbers
Real: any number on the positive and negative number line
Natural: so Counting numbers (one, two, three, four,…..)
Rational: Any number that can be written as a fraction of two integers a/b
Irrational: A number that cannot be written as a fraction - Pie, e, f
Imaginary:
Integers: A whole number both positive and negative
- rational numbers can be made by dividing two integers- you can identify whether or not a number is irrational or rational by converting the number into its decimal form if the number in decimal form is terminating or non-terminating with a repeating pattern then the number is rational if it is anything other than less than a number is irrational
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Add and subtracting Real Numbers
Add and subtract on a number line; use the first number as your starting point, move left if middle sign is/are negative, move right if middle sign is/are positive, answer is at the end point.
positive x positive = positive
positive x negative = negative
negative x negative = positive
charge field model
- use dots that represent positive and negative value
- group any dots that may cancel out and circle said dots
- then any remaining dots should be the answer to the equation
- Circle everything together
Commutative property- This means that you can switch numbers around and still get the same result
Associative property Dash when the expression has three terms they can be grouped in any way to Xpress the same result
Multiplicative identity property - Any number multiplied by one will remain the same
Distributive property – when a number is distributed to others within parentheses
distributive property – when a number is distributed to others within parentheses
multiplying by zero results in zero
Dividing real numbers:
Inverse property of multiplication –
- Every real number besides zero has a reciprocal
- The product of any number and its reciprocal is one
- Dividing by number is the same as multiplying by its reciprocal
- dividing by zero is impossible this will result in no solution
Week 8
Multiplication of Decimals
Blocks simplify multiplication with decimals by giving a physical manipulative for a physical representation.
blocks can be used to group together both numbers in the equation and then be reused by adding them together to get the answer for the multiplication equation.
AND - represents the decimal when put in written form.
expanded form with decimals will look like this-
21.34
20+1+.30+.04
Decimals
Teaching Decimals:
- 12.61843
- twelve AND sixty-one-thousand…..
Identifying place Value-
. Tens, Hundreds, thousands…
- AND means the decimal point in written form
- When you go to teach decimals start by explaining place value with whole numbers
- describe decimals as the numbers that are in between other numbers
- next go over the place values and use a 10 x 10 grid to explain how these numbers work and then relayed them to something that they previously know such as fractions
- show students how to convert one to another by using another concepts that they are familiar with such as multiplication and division
Week 9
Decimals, fractions and percentages
The relationship between fractions decimals and percentages- A fraction can be written as a decimal and a decimal can be converted to a percentage because they percentage is just a ratio the same as what a fraction represents.
- Decimals fractions in percentages are all just variations of the same concept.
Using 10 x 10 grids can be a good tool for learning the relationship between decimals fractions in percentages.
- Percentages are based off 100 so using a 10 x 10 grid allows you to see the exact decimal and therefore percentage of a number
- Fractions can be converted to decimals by dividing the numerator by the denominator a decimal can then be converted into a fraction by multiplying the numerator by 100 and then dividing by the denominator.
- Repeating decimals can occur in this will be marked with a line above the place values that are repeating
- When working with exponents remember that a positive exponent move the decimal place to the right however many times the exponent is represented by and if the exponent is negative then it will be moved to the left creating a smaller number.
Week 10
Decimals and percentages
To convert a decimal to a percentage divide the number by 100
-to convert the decimal to an actual percentage just move the decimal 2 spaces to the right.
- Rounding decimals is even more simple- when asked to round a decimal keep in mind what place value you are rounding to, for a percentage you will always be rounding to the hundredth (the second place value to the left of the decimal) then will make a binary choice as to whether or not a number is at or above 5 (example: .485); if it is at or above 5 then you will round up making the new number .49. on the other hand if the number reads .484 then you will round down making the new number .48
Week 12: Fractions with pictorials
To be able to understand and identify fractions with visual aids and manipulative's. Understanding how to identify the correct fraction portion when comparing it to a whole number can be aided or eased with the use of manipulative such as rods. Understand the difference between an improper fraction (5/2), proper (1/2) or a mixed number (1 1/2).
Comparing Fractions both concrete and pictorial-
- When comparing fractions you can either ensure they have the same denominator or you can convert them to a percentage.
- number actors must be multiplied by the same number as the denominator
- you can use the lowest common multiple or you can just settle for a common denominator
- The above notes represents the abstract form
Showing fractions pictorially can help change the denominator by giving a visual representation of what the fraction actually looks like
Week 13
Unit Fractions
- Associative property - relating our grouping two values together. It does not matter witch values are associated with each other, you can group any two that you like.
- Identity property of multiplication - if you multiplied a value by 1 you will get the same number
Adding fractions with fraction bars- to find the common detonator place one line on top of one of the others.
Do not accept improper fractions as the final answer- convert these types of answers to mixed numbers.
We use the fraction bars to find and explain the common denominator.
- All fractions on a ruler are terminating decimals
- When using a ruler for addition or subtraction use the largest fraction as your marker and then count the dash marks for the smaller fraction. Then add or subtraction the dash marks to the largest.
Fraction Bars-
- fraction bars should have a reference as to what is the whole or 1.
- all numerators should be 1
- numbers align when they are either possessing an even denominator or when they are an a multiple
Week 14
Multiplication and Division with fractions
Multiplication with fractions:
- Fraction bars can be used to represent addition with fractions: for addition the answer should be represented in the denominator
- Fraction bar: for subtraction the answer should be present in the numerator
Using Pattern boxes- use shapes to represent fractions- say a hexagon is equal to 1 use the other shapes to represent fractions of that one whole.
Another way of saying multiplication problems with fractions is to say the problem in reverse: 1/2 x 3/4 = 3/4 of 1/2
By using the blocks you can make a constant shape representing the whole, then you can form the first fraction representing a potion of that whole- from here take you second fraction and take that portion from the second hope formed- now relate that new amount to the whole that is the constant.
Dividing Fractions
- Diving is to find out how many of something can go into something else
- Division with fractions can be thought of in the same light as multiplication- 3/ 1/2= 6 not 1.5
- multiply the number by the denominator when dealing with a whole number and a fraction
- When dealing with 2 fractions flip and multiple the second fraction
- fractions are another way to write division this is why finding the reciprocal leads to flipping and multiplying
- Don’t use cross multiplication as a teaching method, it causes confusion.
Week 5: Exploring divisibility
Divisibility rules used to make solving division problems much easier and also provide a deeper understanding into the relationships between numbers.
Rules:
Divisor-
2- A number must be even, it must end in a 0,2,4,6,8
3- The sum of the digits is dividable by 3
4- the number formed by the last two digits must be divisible by 4
5- the last digit must be 5 or 0
6- the numbers must be divisible by 2 and 3
9- the sum of its digits must be divisible by 9
11- The sum of the digits in odd numbered places will be equal to the sum of the digits and even number places or will the fire by a multiple of 11. 1 1 1 the numbers go odd-even odd.
Greatest Common Facotr and Least Common Multiple
Finding the Greatest common Factor is all about finding a number that is the largest number within the original numbers that can be multiplied to make both the original numbers. Using Prime Factorization is the easiest way to do this. To use prime factorization start by making factor trees for each number; then identify all prime numbers and take what they have in common to make you GCF
To Find the Least common multiple use prime factorization as well for the quickest method. Start with your number trees and then from there identify all prime numbers- here you will take the largest sums of each factor tree from what they have in common- then you will either have your LCM or you will need to place the number into a equation and find it that way.
See picture for example problems
LCM Video
GFC Video
Week 4: Subtraction in other bases.
Multiplication- Bulid. Up strategy is most useful when teaching others multiplication. Buildup strategy is take
5x7=
5x5= 25
5x2= 10
10=25=35
Use visual aids such as dots for full understanding.
- Separation into two different problems using visual aids and manipulative’s.
- Using blocks is a good manipulative
- Multiplication using arrays- this means to understand that multiplication is repeated addition . 4+4+4= 12. This can and should be represented along side with visual aids.
Place value is key when working with subtraction in other base systems. As you move from right to left in a problem its important to understand that you will hit the next place value in base 5 much quicker then you traditionally would in base 10. See video tot he right for a example of subtraction in base 6. Watch the video to the right for exact instruction on subtraction of different base systems.
Week 3: Order of operations is quite simply the order in witch you do your math. Going against the order of operations will always result in wrong answers
Addition in Base systems other then 10.
Adding different base systems is all about remembering the place value and keeping in mind the base you’re working in. In base 5 4+4 = 13; this represents 1 five and 3 units.
Problem Solving strategies-
1. Guess and check method
2. Solve an easier Problem
3. Look for a pattern
4. Try a different way.
5. Reflect on the problem and method of use
PEMDAS- This stands for parentheses, exponents, multiplication, division, addition, subtraction. This is one of the more basic orders of operation however it is applicable to all equations.
WEEK 2: Binary Code and Mayan Math
We've become so comfortable working with base 10 simply because 10 is a number we see frequently. However these other base systems work the same way. Once you get to the peak of the base you then restart and move onto the next place value and this is true for all base systems.
Mayan numbers are all based of symbols that represent base 20. They more then likely used this base system because of 10 toes and 10 fingers.
Understanding base systems is all about understanding place values. For example the number 101 in binary code is not one hundred and one- this humber is actually 5.
This is because 1-0-1 are palce holders for 2^2, 2^1 and 2^0. you have one 4 zero 2's and one 1.
1. We began to learn about both binary code and Mayan math. Both of these were obvious segways from working with common bases lower than 10.
Mayan Number system Video
Week 1: Base Systems
1. Base systems 1-9 vary greatly in comparison to our standard base 10. It’s important to remember that with base systems the “number” you’re seeing is not a real number but rather place holders that represent the true number.
Base 10 consists of numbers 0-9, once you hit 9 you get an additional place value in the tens spot, 10. Base 7 works the same way; Base 7 consists of 0-6 and once you hit 7 it is now represented by 10- this is saying that you one 7 and zero units… this is NOT the number 10!!!
2. The numbers that correlate with the base will be multiplied by the power that corresponds with the base and the place holder; for example if you have the number 11 in Base 9 it would actually be (1X9^1 +1X9^0) = 10 in base
Numbers and Limitations of each base systems 1-10
3. When thinking about base systems its easier to understand with manipulatives such as blocks. Use units, longs, flats and cubes as visual aids to understand each individual basis System. The image below represents the limitations of each base system which can be used to understand what a unit, long, and flat will look like inside of each base system.
