Three concepts of subtracting are take away concept, the comparison concept, and the missing addend concept. The take away concept is taking something away from a whole. ex) 15-5=10 The comparison concept is like in the title comparing the difference and finding the answer. So if we use the same example with 15 socks and under it 5 socks we can visually see that there are 10 missing. Finally Missing addend Concept, this is when we figure out the middle. For example we have 5 stamps and we need 15 letters, we need 10 more. These are all great ways to teach children how to subtract and help them understand.
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Addition Algorithms
There are 2 ways for addition involve two seperate procedures
1. is adding digits
2. regrouping, or “carrying”
I think it is important that people don’t forget how to add with paper and pencil and not to just rely on calculators to help us.
If we look back and really look into what goes on when you add, you don’t realize what your brain actually does when it adds.
Left to Right Addition goes against everything I personally was taught but to some people I think its easier for their brains to understand. The left to right addition adds the numbers then uses the regrouping method. For example:
227+323
we write it out
2 Hundred + 2 tens + 7
3 Hundred + 2 tens +3
5 Hundred + 4 tens+ 10
we regroup it so the answer 550
Associate Property for Addition- For any whole numbers a, b, and c
a +(b+c)= (a+b)+c
In addition no matter which number you add first you are going to come out with the same number. Even if you interchange the numbers then it will always come out to be the same answer.
Mayan Numeration
The Mayas used a modified base-twenty numeration system that included a symbol for 0. The basic symbols were for 0-19. The mayans used place value to show if a number was bigger than 19.. They wrote numerals vertically with one numeral above another. I thought this was interesting because now when we read numbers we read them left to right but the Mayans you read up to down. With that being said the number on the bottom would be units. the number above that would be 20, the number before that would be 20 squared position etc. Lets make up our own symbols are go through some examples
lets us o for one oo for 2 / for 5 // for 10 /// for 15 etc.
Example 1}
ooo This number is in the 20’s position so it would be 3 (20)
o/ This number is in the ones position so it would be 6(1)
There for making the number 66.
Example number 2}
oo This number is the 20 squared position so 2(20 squared)
// This number is in the 20 spot so 10(20)
000// This is the units spot therefore 13
Adding it all together we would have 4213.
Babylonian Numeration
Babylonians developed a base sixty numeration system. The basic symbols for 1 through 59 were additivley formed by repeating symbols. So base 60 means that the place values go 1, 60, 60 squared, 60 cubed etc. versus what we do now which is base 10. Their basic symols had different values depending on the position or location of the symbol. For example if we had 130 it would be 2(60) +10. Because the “tens” spot really represents the 60 squared.
A few more examples using ? representing 1 and > representing for 10.
??>>
This number would be 22
?? ?> >>>
22(60 squared) + 12(60) + 3
This number would be 79,923
(Side note) I just wanted to let you know these are not the real Babylonian symbols. Thank you.
Egyptian
The Ancient Egyptian numeration system used picture symbols called hieroglyphics. The Egyptians used a base 10 system meaning that it is grouped by ten’s which is what we use now. The Egyptian also use the additive system meaning numbers are written by repeating powers of the base the necessary number of times. Each number has a place value and each number tells us how many hundreds or how many one are in that number. Additive numbers are laid out like this 1, 10,10 squared, 10 the the third power etc.
The Egyptians also used really awesome symbols to represent their numbers. Since numbers weren’t invented yet they came up with some unique symbols. When you see the symbol for a number, how ever many of that symbol their are that is what goes in that place value spot. For example if we had the number 21 and a bow tie represented the number 10 and a stick represented the number 1. We would need 2 bow ties and a stick to show the number 21 in Egyptian.
V Roman Numerals V
There are no historical record of the first uses of number, names or symbols. Numerals were developed before number names. Numerals often resemble tallying or slashes. In present times Roman numerals can be found on clocks, buildings, or preface pages in a book, some movies still use roman numerals. I think it is important for kids to learn roman numerals becaause it is dying out and is rarely used anymore. Roman numerals used base 10 additive system, which means the there are number placements. There are symbols for 5,50, and 500. Other symbols are
I – one
V- five
X-ten
L-fifty
C- one hundred
D- five hundred
M- one thousand
This website has some great games for kids to help learn their roman numerals.
Base 10
In many schools one way they teach numeration and place value is with units, longs, flat, and long flats. These pierces help kids grasp how to add, subtract, multiply and division eventually. I remember using base 10 blocks in second grade. Using base 10 includes unit pieces symbolizing ones place or ones, longs are 10 units put together vertically. Flats are 10 longs put together, which concludes a 10×10 Lastly long-flat are 10 flats by 10 flats.
If we take the number 56- we would need 1 flat to show 50 and 6 units to show 6 making the number 56.
I think this is a great visual tool for kids to help them grasp the idea of addition, subtraction, multiplication, and subtraction.
Below I posted a Great video that shows how to use the base ten and how to add using the base 10 blocks. Below that is a picture of the units, longs, and flats.
LCM
The Least Common Multiple or LCM is the smallest multiple of both the numbers. A multiple is a number that can be divided into another number. Just like the GCF you can use the 2 different ways the prime factorization or the listing of multiples. Example multiples of 20 would be 20,40,60,80,100,120 etc. If we wanted to find the LCM of 12 and 20 using the listing it would look like this: 12: 12,24,36,48,60,72,94 and the multiples of 20 are 20,40,60,80,100. As you can see they both contain the number 60. Therefor 60 is the LCM.
Now finding the LCM of 30 and 54 using the prime factorization.
Now, 30 has 2x3x5 then we look at what the other number has and write down what we DON’T already have… so we already have a 2 so we don’t need one we already have a 3 so we don’t need one. BUT we do need two additional threes so we are left with 2x3x3x3x5. If we multiply those all together we get the LCM which is 270.
Greatest Common Factors
To find the Greatest Common Factor or GCF are 2 types of factorization: prime factorization and listing factors. Among the common factors of two numbers there will always be the largest factor which would be the GCF. Lets run through a couple examples and explain as we go.
1. Lets find the GCF of 12 and 36 using the listing method
The factors of 12 are: 1,2,3,4,6,12
The factors of 36 are: 1,2,3,4,6,9,12,18,36
Thus, the GCF of 12 and 36 is 12.
There are many special cases when finding the GCF could be, maybe the ONLY factor could be the greatest factor. Lets use the prime factorization method to find the GCF. The prime factorization method works best when using larger number.
2. Lets find the GCF of 360 and 72 using the prime factorization method
In any case there can and will be more than one way to branch of and prime factor a number. However in the example as seen above you get the same ending result. 2x2x2x3x3x5
Next step is to factor 72 using the tree form
as you can see it is 2x2x2x3x3
Next step is to see what these to numbers have in common……. which is 2x2x2x3x3.
The final step is to multiply them together which is 72. So 72 is the GCF of 360 and 72.
To find the Greatest Common Factor or GCF are 2 types of factorization: prime factorization and listing factors. Among the common factors of two numbers there will always be the largest factor which would be the GCF. Lets run through a couple examples and explain as we go.
1. Lets find the GCF of 12 and 36 using the listing method
The factors of 12 are: 1,2,3,4,6,12
The factors of 36 are: 1,2,3,4,6,9,12,18,36
Thus, the GCF of 12 and 36 is 12.
There are many special cases when finding the GCF could be, maybe the ONLY factor could be the greatest factor. Lets use the prime factorization method to find the GCF. The prime factorization method works best when using larger number.
2. Lets find the GCF of 360 and 72 using the prime factorization method
In any case there can and will be more than one way to branch…
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Divisibility Tricks and Tips
My favorite thing about math is when there are tricks and short cuts to help you get to the answer faster than the original way. Today we learned a few rules that will help you quickly come to the conclusion if it is divisible by that number or not. So here it is:
Divisibility by two: if the last digit is even, the original number will be divisible by two.
Divisibility by three: if the sum of the digits is divisible by 3, then the original number is divisible by three.
Divisibility by four: If last 2 digits form a number that is divisible by four, then the original number is divisible by four.
Divisibility by five: If the last digit is either 0 or 5, then the original number is divisible by 5.
Divisibility by six: If the number is divisible by 2 and 3, then it is divisible by six.
Divisibility by eight: If the last three digits form a number that is divisible by eight, then is divisible by 8.
Divisibility by nine: if the sum of the digits is divisible by 9, then the original number is divisible by 9.
Divisibility by ten: if the last digit is 0, then the original number is divisible by ten.
Divisibility by twelve: if the number is divisible by 3 and 4, then the it is divisible by 12.
Unfortunately, there are no rules for seven or eleven that are in my opinion short cuts. I would now like to go through a few examples:
example number #1: 2,126 is divisible by 2 because six is divisible by 2, 4 because six plus two is eight which is divisible by four so that makes 2126 is divisible by 4. 6 because this number is divisible by two and three.
example number #2: 2000 is divisible by 2, 4, 5,8, and 10.
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