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Base 10 Representations
One cube is equal to 10
One flat is equal to 1
One long is equal to .1
One unit is equal .01
For example, 6.84 is six flats, eight longs and four units.
For example, 2.06 is two flats, zero longs and six units.
Names of the four place values representing base 10 blocks:
The cubs is the tens place, the flat is the ones place, the longs is the tenths place and the units is the hundredths place.
Comparing Decimals
3.4 or 3.21? Which is larger?
Word form: Three and four tenths or three and twenty one hundredths?
Three and twenty one hundredths is larger because once you draw the base 10 blocks out, you will see that 3.4 has two more longs than 3.21. 3.4>3.21
If you still have trouble seeing that you can add a zero on 3.4 to make it 3.40, so that it is three and forty hundredths, and then you can more easily compare it to three and twenty one hundredths to tell which is larger.
Different Forms of Decimals
Example: 21.34
1.Standard Form- 21.34
2. Word Form- twenty one and thirty-four hundredths
3. Block Form- 21 flats, 3 longs and four units
4. Expanded Form- 2(10)1+1(10)0+3(10)-1+4(10)-2
(note:) the number in the smaller font that tells you the place value should be raised up to the top right hand corner, but there is no feature in these notes that lets me do that.
Another example of expanded form: 345.07
3(10)2+4(10)1+5(10)0+0(10)-1+7(10)-2
Fractions to change to decimals
1/4=.25 .25
4/1.00
- 8.0
-----
20
When dividing decimals the place values should be aligned.
Above was an example of a terminating decimal which means the decimal stops. A non-terminating decimal goes on forever and repeats. An popular example of that is pi, or 3.14 repeated.
Fraction Addition
You may use any of the three models to add and subtract fractions.
For example, let's add 5/9+1/3 using the set model.
Draw nine circles or any other type of unit. Then, color in five to show five ninths. We must add 1/3 in now, so in order to do that we need to break up the nine units in three groups. We find that there is three in each group which represents one third. Use a different color or marking, and mark the three units that shows 1/3 after the five ninths. If we add up all the marked/colored units, you will see eights colored units out of nine totals parts to get a answer of 8/9.
Fraction Subtraction
For example, let's solve 3/4-1/8 using the linear model. First draw 3/4 and mark in on the number line between 0 and 1, then split the number line in eighths and mark it it starting with 1/8 and then 2/8 and so on. When you subtract 1/8 from 3/4 you get 5/8, because it is 1/8 away from 3/4, so the answer is 5/8.
Fraction Multiplication
For multiplication, an efficient way to represent fractions uses the area model of multiplication which is slightly different than subtraction and addition.
For example, lets solve 2/3 times 4/5.
This literally means there are 2/3 rows of 4/5.
First draw a unit square and cut it into five pieces and color in four. Label 4/5 at the top of the square. Then split the square in thirds to make three rows. Since we need two-thirds of the rows, color in or mark the two of the three rows we made. Then count how many in the colored in pieces there are, and you should get 8, and count how many pieces there are overall in the square and you should get 15, so the answer is 8/15.
Fraction Division
An easy way to show fraction division is using rulers, which is similar to the measurement model because we are finding how many groups there are by using rulers, which are like number lines.
For example, lets solve 2 1/2 divided by 3/8.
This literally means how many 3/8 are there in 2 1/2.
To solve, first mark 2 1/2 on the ruler. Then start at zero and find three eights on the ruler and jump to it. Label this as 1 jump. Then, keep going by moving three eighths (if you need help count the tick marks), until you reach or go past 2 1/2. In this problem, you will end up going past 2 1/2.
Count the number of jumps which is 6 jumps, and count the number of pieces in the jump, and it turns out that there is 3 pieces. 2 and 1/12 lies on the second piece out of three pieces. So the answer to 2 1/2 divided by three eighths is 6 and 2/3.
Link
(The link will take you to the home page, but you will see the web pages for addition, subtraction, multiplying and dividing which you can click on. They call the linear model, a line model, and the area model and circle model, and unfortunately there is no set model, but it is a good link for seeing the area and line model. )
Fraction Concepts
A proper fraction represents a number between zero and one. An improper fraction is always greater than one because the numerator is bigger than the denominator.
Counting Fractional Parts: Iteration
In the fraction 1/4, the top number is called the numerator and it tells us how many, and the button number is called the denominator and it tells us the number of pieces. So 1/4, is a count of one part called fourths.
Fraction Misconceptions
ex.) a fraction times a fraction is smaller
The link attached introduces all of the three models, area, set, and linear/length model. Note that in the link, when the examples/manipulatives you can use to do the set model is discussed, there are more examples that is listed on the website. For the set model, you may use any object, even stars or triangles.
Teacher should be familiar with how to represent any fraction using any of the three models.
Equivalent Fractions
You can use any of the 3 models to represent equivalent fractions.
For example, show 2/6=1/3. For the area model, you can draw a rectangle, and first split it into six even parts and color in two parts to represent 2/6. Then split the rectangle in thirds. You will see that 2/6 is equivalent to 1/3 in the drawing.
For the linear model, you may draw a number line and mark 2/6, and split the number line in sixths. Then think how many sixths are in one third by drawing thirds on the number line. After marking the numbers you should find that two sixths is the same as 1/3 on the number line, just as four sixths is the same as 2/3.
For the set model, one way you can show equivalent fractions is to draw six units, such as circles, and color in two of them to show two sixths. Then, split the units in thirds so that there is two units in each of the three groups. You can use a different color or circle them and you will visually see that one third is also two sixths.
**For mixed numbers, you do not have to change them to an improper fraction to represent them using the three models. Just be sure to show that the model is showing either the mixed number or the improper fraction, because for example, visually representing 1 2/3 is sometimes different than showing 5/3
Estimating and Comparing Fractions
To estimate fractions in terms of how much bigger or smaller there are compared to other fractions, and to find out what number the fraction is close to we use benchmarks.
Bench marks are 0,1/2,1
For example, where is four ninths close too?
One way to figure this out is to draw out nine units in the set model and color four in. Where does this drawing look close too? You can see that it looks close to 1/2, so four ninths is going to be slightly less that 1/2.
Another example to compare fractions using benchmarks:
3/8 or 4/7? Which is bigger?
4/7 is greater because it is bigger than 1/2, and 3/8 is less than one half.
Addition
Integers can be added using:
The Chip Model- positive integers are represented by yellow chips and negative integers are represented by red chips ( in our class). One red chip neutralizes the yellow chip, also called a zero pair. Every integer can be represented in different ways, the number -4, could be shown as four red chips, or seven red chips and three yellow chips. To use the chip model for addition, lets say we are adding -4 + 6, join four red chips (four negatives) with six yellow chips (six positives). Because one yellow chip and one red chip together sum to zero, four pairs of yellow chips and red chips zero out. What remains are two black chips (two positives, or +2).
The Charged-Field Model for Addition
A field has a charge of zero if it has the same number of positive and negative charges. An integer can also be shown many different ways. To use this model for addition lets say we are adding -2+5. We would draw two negative charges and then add five positive charges. Now we end up with two zero pairs, because two of the positive charges we drew can be matched up with two of the negative charges. What remains is three positive chargers, so the answer is positive three.
The Number line Model for Addition
A number line can be used for addition of integers. One idea to show how integers are added on a number line is using a drawing or concept of a person walking on a number line. If we are adding a positive integer, the man would move forward on the number line when facing the positive direction. If we are adding negative integers the man would move backward on the number line, in the negative direction.
Subtraction
Chip Model
This model can also be used for integer subtraction. For example, 5-7, can be solved by drawing five yellow chips first. Then, adding seven red chips. Now, we have five zero pairs that cancel each other. What remains is two red chips (negative) so the answer is negative two.
Charged Field Model
The charge field model for subtraction is similar to the way it works in addition. For example, lets say we are solving for 5-(-3). We first draw five positive charges and three negative charges. Remember, a negative and positive charge cancels out. In the problem, we need to take away three negative charges but we do not have three negative charges. So, we draw three plus-minus pairs that cancel out. Now we have a drawing a eight positive charges and three negative charges. When you take the three negative charges away, what remains is eight positive charges, which is the answer.
Number line Model
The number line model is similar for subtraction as it is for addition of integers with the idea of the walking man, but subtraction is modeled by turning around. So for 7-(-4), you would start at zero and move forward seven units for seven. Then the subtraction sign tell you to turn around and you would move backward four units for -4, stopping at eleven.
**When subtraction integers with more than two integers, you may use the rules of order of operations to find the right answer.
Multiplication
Chip Model
For the chip model, you can show multiplication. For example, 4(-3) is four groups of 3 red chips. When you draw this, you can see that you get twelve red chips or negative twelve as the answer. Another example is (-6)(-4). This means that the negative six times negative four means to remove six groups of four red chips. We first draw six groups of four red chips. Then we need to draw twenty four yellow chips to remove the six groups of four red chips. We are left with a positive twenty four.
Charged Field Model
For this model, you can represent multiplication as well. For example, 4(-3), would be four groups of three negative charges to get negative twelve. Another example is (-6)(-4). We first start with the value of zero, and remove six groups of four negative charges, we are left with twenty four remaining positive charges.
Number line Model
To multiply on a number line, we move to the left if we are traveling in a negative direction and we move to the right if we are traveling in a positive direction. For example, (-4)(-3), we begin at zero and then jump four groups of -3, and get the sum of negative twelve. But in this problem, we have -4 (not +4), so the opposite of four groups of- 3 is positive 12, which is the answer.
Pattern Method
You can find out the rules for multiplying integers by using a pattern as evidence. For example, why is a negative times a negative a positive?
-3(-4)=12
3(4)=12
2(4)=8
1(4)=4
0(4)=0
-1(4)=-4
-2(4)=-8
-3(4)=-12
Why is a negative times a positive a negative?
-3(5)=-15
3(5)=15
2(5)=10
1(5)=0
0(5)=0
-1 (5)=-5
-2 (5)=-10
-3(5)=-15
Division
Partition Model- In this model, you are given the quantity and number of groups but you have to solve to find out the number in each group
For example, lets solve -15/3. First draw three groups and then pass the units out (partition them) until you reach 15. You will find that you will get five units in each group. So -15/3 is negative five.
Repeated Subtraction Measurement Model- In this model you are given a quantity and the number in each group, but you have to find out the number of groups formed.
For example, lets solve -12/-4. Start by drawing or passing out -4 in each group. Then determine the number of groups. You should have got three groups.
Inverse Operations/ Pattern Method
If an integer division problem is structured in a way that neither the measurement model or the partition model will work to solve, integer operations can be used to solve it. For example, in 30/-5, you cannot use the measurement model or partition model because you cannot have negative five groups.
So this is how to use inverse operations and the pattern method for 30/-5:
30/-5=X
X(-5)=30
3(-5)=-15
2(-5)=-10
1(-5)=-5
0(-5)=0
-1(-5)=5
-2(-5)=10
-3(-5)=15
-4(-5)=20
-5(-5)=25
-6(-5)=30
So X is -6 and that is the answer.
(The link to these notes show integer addition and subtraction on the number line, the pattern method, and something called counters which is another word for the charged field model. It briefly shows inverse operations for integer division and the concept of difference for subtraction of integers.)
Extra Notes not mentioned in link:
An integer is the opposite of counting numbers, counting numbers and zero. Using the number line approach, an integer and it's opposite is the same distance from zero.
The absolute value of a number is its magnitude, or the distance of the given number to zero on the number line. Numbers to the right of zero are positive, while numbers to the left of zero are negative. Using the chip method, absolute value is the quantity of chips present. It does not matter what color the chips are, negative or positive, but the number of chips is what the absolute value is.
The chip method can also be used to compare the relative size of integers. For example, if they are all positive chips, the more chips you have the greater the integer. If the chips are all negative, the fewer chips you have the greater the integer. Also, of course the positive chips (yellow color) are greater than the negative (red color).
Divisibility Rules
2-If the last digit is even, the number is divisible by 2.
3-If the sum of the digits is divisible by 3, the number is also.
4-If the last two digits form a number divisible by 4
5-If the last digit is a 5 or a 0, the number is divisible by 5.
6-If the number is even and sum of digits is divisible by three.
8- If the last three digits form a number divisible by 8,
then so is the whole number.
9- If the sum of the digits is divisible by 9, the number is also.
10 -If the number ends in 0, it is divisible by 10.
11-The sum of the even digits is subtracted from the sum of the odd digits. The result is either 0 or divisible by 11.
Divisibility rules help us determine the factors and help us save time dividing larger numbers.
Factors
The factors of a number are any whole numbers that can be multiplied by another whole number to get that number.
A factor rainbow is one way you can practice finding the factors of a whole number. It is a rainbow-shaped diagram that shows the factors of a number in pairs through a series of arcs.
For a prime number (for example, 3 or 7), there will be only one pair of factors, one and the number itself.
A prime number can be divided evenly only by 1 or itself, but it has to be two different factors. (But, 1 is not a prime number because it does not have two different factors).
Composite numbers are positive integers that are not prime and they have more than two factors.
An algorithm is different from a strategy because an algorithm is going to be the same steps and roles every time in the math problem, but a strategy is process that changes depending on the numbers and how you approach the problem.
Traditional algorithms is those that are taught in U.S schools but alternate algorithms are done in different countries or situations, and students may make up their own alternate algorithms.
Addition and subtraction strategies
1. Compensating-
48+33
(+2) 50+33=83-2=81
48-33
(-8) 40-33=7+8=15
2. Decomposing- *difficult to show because you cannot draw on notes
48 + 33
^ ^
40 8 30 3
70 + 11=81
17 - 6
^
10 7-6
1
10+1=11
3. Base 10 blocks-
symbols- (I=one long .= one unit )
48+33
IIII........
+III...
=IIIIIIII.
48-33
IIII........
-III...
=I.....
4. Number line-
A number line can be used to illustrate and solve addition and subtraction problems. I cannot draw a number line in the notes for an example, but here is a link to show how a number line is used to solve for addition and subtraction problems.
https://nzmaths.co.nz/content/addition-and-subtraction-number-line
1. Traditional Algorithm
48 48
+33 -33
81 15
Alternate Algorithms
2. Partial Sum-
425+176
425
+176
500 (sum of hundreds)
90 (sum of tens)
11 (sum of ones
601
786-325
786
-325
400 (subtract the hundreds)
60 (subtract the tens)
1 (subtract the ones
461 (add 400+60+1)
*Please note that in the link attached to this note what they call "expanded algorithm" is what we call "partial sums" algorithm.
3. Expanded Form
425+176
100 10
400+ 20+5
+100+70+6
600+ 0+ 1=601
786-325
700+80+6
-300+20+5
400+60+1=461
Multiplication and Division
[This link attached will teach you about multiplication algorithms such as lattice multiplication and the Russian Peasant model, which I never heard of before and is actually easy to do. You will also learn the concrete model and scaffold algorithms for division in this link.]
Multiplication Strategies
6. Open Area Model (You create this model by first decomposing the factors you are multiplying usually by place value so 5x16, would be 5, 10, and 6, then set up the numbers above the boxes in the rectangle and to the side of the boxes, multiply all the numbers in the rows and columns, and put the product in the box. Finally, add all of the products found in each of the boxes to get the total.
Multiplication Algorithms
1. Standard/traditional
2. Partial Product Algorithm (Multiply the ones, then tens, then hundreds if there is a hundreds place, and list all the partial products, then add them together to get the total product).
Division Stratagies
Division Algorithms
1.Traditional/long division
2. Short Division (In long division, each step of the solution is written down, whereas in short division, the steps are performed mentally and are not written down, you are still dividing, multiplying and subtracting. In short division, the remainder is written on the top left side of the next digit of the dividend).
3. Repeated Subtraction or partial quotients (This involves a series of repeated subtraction until you get a remainder. At each step, you find a partial quotient, then you find the product of the partial quotient and divisor and subtract it from the dividend. Finally, you add all the partial quotients to find the final quotient).
Fact Families are three related numbers. Fact families require that students understand the communicative property of multiplication and addition and inverse relationships in order to do them.
Example of a fact family:
4+9+13
9+4=13
13-4=9
13-9=4
3x4=12
4x3=12
12/4=3
12/3=4
Addition
Set Model-characterized by the combining of two sets of discrete objects
Linear Number line- Characterized by the combining of two continuous quantities (time, distance,quantity)
Subtraction-
Take Away- Starting with some initial quantity and taking away a certain amountfrom the initial quantity
Missing Addend-determine what quantity needs to be added toreach the target, this model relates subtraction and addition because one uses related addition facts to find the missing addend in a subtraction equation.
Comparison- comparing two quantities to find out how much larger or smaller one is
Linear- used on a number line to show change, wheresubtraction of whole numbers are modeled on a number line using directed arrows
Multiplication
Repeated Addition (Set)- repeatedly adding a quantity of objects aspecified number of times (discrete)
Repeated Addition (Linear)- repeatedly adding a quantity of continuousquantities a specified number of times.
Area Model- product of two numbers representing the sides of a rectangular region,such that the product represents the numbers of unit size squares within that region which equals the area
Division
Partition (Sharing): distributing a given quantity among a specified number of groups to find out how many are in each group
Measurement: using a given quantity to create groups of a specified size to determine the number of groups formed
Hyperlink attached to some properties of multiplication and addition. All of them listed below.
Listing of Properties-
Addition
1. Closure Property of Addition
2. Commutative Property of Addition
3. Associative Property of Addition
4. Identity Property of Addition
Subtraction
1.None of the properties apply to subtraction
Multiplication
1. Closure Property of Multiplication
2. Associative Property of Multiplication
3. Identity Property of Multiplication
4. Distributive Property of Multiplication over Addition for whole numbers
5. Distributive Property of Multiplication over Subtraction for whole Numbers
6. Zero Property of Multiplication
7.Commutative Property of Addition
Division - none of the properties are true can be shown with examples when division problems to each property
Bases
Base 10- Has ten digits, 0,1,2,3,4,5,6,78,9. The Hindu Arabic system is in this base.
Base 5: has five digits: 0,1,2,3,4. Base five is based on hands and fingers, so one counts 0,1,2,3,4 and 10 for one hand. Then, 11,12,13,14 and 20 for two hands. So 41 would be represented as four hands and one finger, or 41 subscript five.
Base 2: is a binary base with the digits 0 and 1. One counts like 0,1 then 10, 11, 100, 101, 110, 111, 1000, where 1000 would be eight in a base ten system.
Base 12: Has twelve digits with two added placeholder symbols to represent 10 and 11. The symbols chosen are X and E. One counts like 0,1,2,3,4,5,6,7,8,9, X,E, 10. For example, after 39, comes 3X, 3E, 40.
A numeration system consists of properties and symbols that represent numbers. A few examples of numeration systems are below:
Tally- uses tally's to represent each single thing counted. Tallies can be grouped together by using a diagonal line against four straight tally marks to represent five.
Egyptian- uses tally marks but then uses symbols to group these tally marks. For example, ten tally marks is the symbol of a heel bone and one million tally marks is the astonished man symbol.
Mayan- a system where the symbol for zero looks like a clam shell, one is represented as a dot, and groupings of five is written as a horizontal line. The Mayan system is based on twenty with vertical groupings.
Hindu Arabic system has 10 digits, 0,1,2,3,4,5,6,7,8,9 and 10, and place value is on powers of ten, used in the United States.
Roman Numerals - a european system based on using subtractive and additive properties. For example, IV equals 4 (5-1), and CM equals 900 (1000-100). Also, a bar placed over a numeral means to multiply it by 1000.
The link attached explains the types of numeration systems such as positional.
