作者：Elizabeth Hernandez 12 年以前
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a number that can be written as a ratio of two integers
a/b where both a and b are integers
Not all fractions are rational
interpreting the fraction as a division problem 2 2 divided by 3 equal 0.6666... 3
the whole is partitioned into b equal parts of which a of those parts are selected
Ex. 2 Total of 2 of those 3 parts are used 3 Whole is divided into three equal parts
Measurement (Repeated Subtraction Model): Characterized by using a specified quantity to create groups (or partitions) of a specified size (amount) and determining the number of partitions (groups) that can be formed Ex. Sally has 8 eggs and a recepie for brownies requires 2 eggs. How many batches fo brownies can Sally make? KNOW: Size of group FIND: Number of partitions/groups
Partition (sharing): Characterized by distributing a specified number of partisions( groups) and determining the size ( amount) in each partition (group) Ex. Billy comes to school with ten pencils. He Decides to share the pencils with 5 friends. How many pencils does each friend get? KNOW: 5 groups FIND: amount in each group 2
Cartesian Product: Characterized by finding all the possible pairings between two or more sets of objects Ex. Sara has 4 Jackets and 3 scarves. How many jacet and scarve outfits can she wear? J1-s1, s2, s3, (J1,s1) J2- s1, s2, s3 (J2, s2) J3- s1, s2, s3 J4- s1, s2, s3
Area (Arran) Model: Characterized as a product of two numbers representing the sides of a rectangular region such that the product produces unit size squares Ex. Tom is tiling his bathroom that measures 10ft by 15ft. To purschase the tile he needs to find area of bathroom floor. What is the are of the bathroom floor? 10*15
Communative Property: If a is an element of a whole number, b is an element of a whole number, then a*b=b*a a= 5, b=7 a*b is 5*7 and b*a is 7*5 not the same but produce the same value
Closure Property: if a is an element of a whole number and b is an element of a whole number then (a*b) is an element of a whole number Ex. a=5 and b= 7, 5*7 is 35, 35 is an element of a whole number
Repeated Addition- Continuous: Characterized by repeatedly adding continuous quantities a specified number of times Ex. During the week, Monday thru Friday Sandra practice the piano 30 minutes a day. How long did she practice this week 30+30+30+30+30... 30*5
Repeated Addition- Discrete: Characterized by repeatedly adding a quantity of discrete objects a specified number of times Ex. Sammy's brother and sister both gave him 2 cars for his birthday. How many cars did he get? 2+2... 2*2
0 -9 is used when 10 is reached a new palcement value is formed
In base 60, 0-59 units are grouped and when 60 is reached a new placement value is created
In base 20, 0-19 units are grouped and when 20 is reached a new placement value is created
Sequence: an ordered list of objects, events, or numbers
Defines a sequence in which the current term is dependent on previous term(s)
Difference: multiply by 2 then add 1
13 multiplied by 2 equals 26 then add 1
Product of 1st term 13
ex. 6, 13, 27, 55..
Sequence of numbers with a common ratio
common ratio -1/2
ex. 8, -4, 2,-1,..
Sequences or numbers with a common difference
Adding 2 to each time will create the next number
common difference of 2, 4, 6, 8 is 2
ex. 2, 4, 6, 8, 10
a is a factor of b b is a multiple of a a is a divisor of b b is divisible by a
3|15
3 is a factor of 15, because 15= 3* 5
15 is a multiple of 3, because 3*5= 15
3 is a divisor of 15 because 15/3=5
15 is divisible by 3 because 15 divided by 3 is 5
a number with more than two factors
a number with exactly two distinct factors (1 and itself)
is a number that can be wrtten as (2 * n) + 1
is a number that can be written as 2 * n
Four Fact Families
relates the sum of 3 &4 3+4=7 4+3=7 7-4=3 7-3=4
European Algorithm
Adding method in order to subtract from numbers
Identity Property of Subtraction
If a is an element of W can we say a-0 = a and 0-a=a? NO
Ex. If a=4 then a-0= 4 but 0-a= -4
Associative Property of Subtraction
If a is an element of W, b is an element of W and c is an element of W can we say (a-b) -c = a- (b-c)? NO
Ex. If a=5 and b=4 and c=3 (a-b)-c=-2 and a-(b-c)=4 -2 is not equal to 4
Communative Property of Subtraction
If a is an element of W and c is an element of W can we say a-b= b-a? NO
Ex. If a=5 and b=7 a-b=-2 and b-a=2 -2 is not equal to 2
Closure Property of Subtraction
If a is an element of W and b is an element of W can we say (a-b) an element of W? NO
Ex. If a=7 and b=5 a-b= -2 -2 is not a whole number
Missing-addend: the need to determine what quantitiy must be added to a specified amount to reach some target quantity
Ex. Kelsie has 6 blocks. She wants 10 blocks. How many blocks does Kelsie need?
Comparison: comparing relative sizes of two quantities to determine how much smaller one of the quantities is compared to the other quantity
Ex. Emily read 5 books. Jim read 3 books. How many more books did Emily read than Jim?
Take Away: starting with an initial quantity and removing (take away) a specific amount
Ex. Vince came to class with 5 pieces of candy. He gave 3 pieces away. How many pieces of candy does Vince have?
Scratch
Addition done as regular but the "carry over number" is added to the following place value number
Any Column First
Addition is done using place values. Zeros are inserted after the number as "place holders"
Left to Right
Start with the begining place value and work backwards. Make sure to place zeros as "place holders"
Low Stress
Pyramid style. The sum of the first two numbers will begin the pyramid. The following number is added to the ones place but not to the tenths.
Communative Property of Addition
If a is an element of W then a +b = a+b
2 sets, one contains a objects and other contains b objects the two sets will contain the same a+b number of objects
Identity Property of Addition
If a is an element of W then a+ 0 = a = 0+ a
Two sets , one contains a of jects and the other contains zero objects then when combined the new set will contain a objects
Associative Property of Addition
If a is an element of W, b is an element of W and c is an element of W then (a +b) +c = a+ (b+c)
The order of the addition will not matter because both sets will bring the same result
Closure Property of Addition
If a is an element of W and b is an element of W is (a +b) an element of W
Sum of two whole numbers is another whole number.
Addition- Continuous
Characterized by the combining two continuous properties Ex. Paul goes to the gym spends 10 min jogging and 20 min walking. How long are his workouts?
Addition- Discrete
Characterized by combining two sets of discrete objects ex. Bob has 3 apples and 2 oranges. How many pieces of fruit does he have?
Definied as a collection of objects
symbol: { }
ex. {1, 2, 3} ~ {a, b, c}
Both sets contain 3 elements
ex. {1, 2, 3} = {2, 3, 1}
Although different order both sets contain same elemets
Is there an easier way to solve it?
Does it need to be revised?
Does the answer make sense?
Subtopic
Revise the plan if necessary
If the solution is not visible, rethink the plan
Be persistant with your current plan
Use a model
Create charts, lists, and use objects
Work backwards
Solve a simpler problem
Use an easier problem based on your current problem
Guess and check
Think of solutions and plug into the problem to check answer
Understand the problem
Quote by George Polya pulled from Mathematical Discovery and re quoted in A Problem Solving Approach to Mathematics for Elementary School Teachers 11th edition by Rick Billstein.
