MTE 280 Investigating Quatity

Strategy: A method or trick to help students comprehend mathAlgorithm: A step-by-step solutionStrategies:Decomposition: Separating numbers into their components (To divide a number into smaller partsComprehension: Understanding concepts, operations, and relationsOpen Number line: Visual representation for recording and sharing students' thinking strategies during the process of mental computationBase 10 Blocks/Pictures: Strategy used for visual representation while working through math problems such as: addition, multiplication, subtraction, division, etc. This strategy greatly helps children who cannot understand how to work through a problem fully as the blocks or pictures allows them to visually see the quantity of a numberAlgorithmsPartial Sums: The sum of part of a sequence (a set of numbers that is in order)Expanded Notation: Writing a number to show the value of each digit Standard Notation: Number is completely written out using numerical digits

Hindu Arabic- Known as the U.S. version of the base system*There are 10 digits in this system*System goes up to number 9 --> (0,1,2,3,4,5,6,7,8,9)

*Known as the binary system*Contains the numbers 1 & 2*"0..1.."

* Contains 5 digits*"0..1..2..3..4"

*System primarily used by African tribes*"1..2..3..4..5..6..7..8..9..x..3..10"*3(Backwards 3): El*x: Dec*10 is ACTUALLY 12 in this sytem

Digits: Set of singular numbers --> (1,2,3,4,5,6,7,8,9)* Digits are the foundation of ALL numbers

Integer Concepts:Chip Method: When modeling integers, we can use colored chips to represent integers. One color can represent a positive number and another color can represent a negative numberNumber Line: A number line can be used to represent positive and negative quantities, and the number line model can illustrate properties of signed arithmetic.Absolute Value: The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero.

Number line Model: used to represent positive and negative quantities, and the number line model can illustrate properties of signed arithmetic.Pattern Model: The first digit of the sequence stays consistent while the digit being added changes each time the pattern is repeated, until reaching the opposite of the sumCharged Field Model: Positive and negative charges are used just like a chip model, and the field has 0 charged if it has the same number of positive and negative charges. Chip Model: Positive integers are represented with black chips and the negative charges are represented with red chips. A red chip can neutralize a black chip

Number line Model: used to represent positive and negative quantities, and the number line model can illustrate properties of signed arithmetic.Pattern Model: The first digit of the sequence stays consistent while the digit being added changes each time the pattern is repeated, until reaching the opposite of the sumCharged Field Model: Positive and negative charges are used just like a chip model, and the field has 0 charged if it has the same number of positive and negative charges. Chip Model: Positive integers are represented with black chips and the negative charges are represented with red chips. A red chip can neutralize a black chip

Number line Model: used to represent positive and negative quantities, and the number line model can illustrate properties of signed arithmetic.Pattern Model: The first digit of the sequence stays consistent while the digit being added changes each time the pattern is repeated, until reaching the opposite of the sumCharged Field Model: Positive and negative charges are used just like a chip model, and the field has 0 charged if it has the same number of positive and negative charges. Chip Model: Positive integers are represented with black chips and the negative charges are represented with red chips. A red chip can neutralize a black chip

Number line Model: used to represent positive and negative quantities, and the number line model can illustrate properties of signed arithmetic.Pattern Model: The first digit of the sequence stays consistent while the digit being added changes each time the pattern is repeated, until reaching the opposite of the sumCharged Field Model: Positive and negative charges are used just like a chip model, and the field has 0 charged if it has the same number of positive and negative charges. Chip Model: Positive integers are represented with black chips and the negative charges are represented with red chips. A red chip can neutralize a black chip

Use base 10 blocksUnit 1/100long 1/10flat 1cube 10ComparingExpanded formWord formStandard FormBase 10Terminating: Comes to an endNon-Terminating: Repeats

Addition: Line up the decimalsSubtraction: Line up the decimals Multiplication: Whole X DecimalDecimal X WholeDecimal X DecimalDivision:Partition (number of groups)Measurement (How many in each group?)

*Given any whole numbers, a and b with be not equaling zero, there exist unique whole numbers q (quotient) and r (remainder) such that a = bq + r with 0 < r < b.*Partial Quotients*Column Division

*Children use manipulatives, which are physical items that they can interact with to create their own algorithms. *Left-to-Right: Adding from left to right. Adding the larger pieces then the smaller ones.*Lattice: Add single digit numbers by place value on top to the single digit numbers on bottom then add the sums from the diagonals.Scratch: Adding complicated additions by adding only two single digitsExpanded Notation: Separating a larger number into smaller components that still equal the same number when addedCompensation: Adding a number that does not existPartial Sums: Sum of part of the sequence

*Using base 10 blocks to create a concrete model for subtraction.*Equal-Additions: The difference between two numbers does not change if the same amount is added to both numbers*Trade First *Counting Up*Partial Differences

* Partial Products*Lattice

*Set Model: The combining of two sets of discrete objects (individually different and distinct objects)*Linear/Number line Model: Combining two continuous quantities (measured quantities like time, distance, quantity, etc). Shown on a number line to show the change.

*Take Away: Starting with an initial quantity and removing a specified amount*Missing Addend: The need to figure out what quantity must be added to a specified quantity to reach a target amount*Comparison Problem: Comparison of the relative sizes of 2 quantities to determine how much smaller or larger one is than the other*Linear: On a number line using arrows to show a change

*Repeated-Addition: Putting equal-sized groups together to reach a quotient*Rectangular array and Area Model: Objects are arranged with the same number of objects in each row*Cartesian: Creating a tree diagram to show numerous outcomes of the product

*Partition: Diving a group of numbers into smaller equal groups*Missing Factor: Using a related multiplication fact to find the answer*Repeated Subtraction: Subtracting the number that we want to divide by its dividend the same number of times as the quantity of the dividend to reach the final answer

*Property for multiplication AND addition*(a+b)=(b+a) and (a x b)=(b x a)*Addition: Changing the order of the addends will result in the same sum*Multiplication: Changing the order of the factors will result in the same sum. Addend: Numbers in an addition problemFactors: Numbers in a multiplication problem

*Property for Addition AND Multiplication* a+(b+c) = (a+b)+c OR a x (b x c) = (a x b) x cAddition: When adding three or more numbers, the grouping of the numbers will not change the sumMultiplication: When multiplying three or more numbers, the grouping of the numbers will not change the quotient

*Works for Addition AND Multiplication*Addition: If you add any two whole numbers, the sum will be a whole number*Multiplication: If you multiply any two whole numbers, the quotient will be a whole number

*Property for Addition AND Multiplication*Addition: a+0=a *Multiplication: a x 1 = a = 1 x aAddition: When adding a zero to any number, the sums stays the sameMultiplication: When multiplying 1 to any number, the quotient stays the same

*Distributive Property of multiplication over Addition/Subtraction: ex. 5(3+4) = 5 x 3+5 x 4*Distributive Property of Multiplication over Subtraction for Whole Numbers: a(b - c) = ab - ac*Distributive Property of Multiplication over Addition for Whole Numbers: a(b+c) = ab + ac*Distributive Property DOES NOT work for division

*Zero Property of Multiplication: Anything multiplied by zero is zero*a x 0 = 0

Divisibility TestA number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.A number is divisible by 3 if the sum of the digits is divisible by 3.A number is divisible by 4 if the number formed by the last two digits is divisible by 4.A number is divisible by 5 if the last digit is either 0 or 5.A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3.A number is divisible by 8 if the number formed by the last three digits is divisible by 8.A number is divisible by 9 if the sum of the digits is divisible by 9.A number is divisible by 10 if the last digit is 0.Factor RainbowA factor rainbow is a rainbow-shaped diagram that factors of a number in pairs

Addition:Use the are, set, or linear modelSubtraction: Use the are, set, or linear model (cross off pieces to show that you are taking them away)Multiplication: Usually use the area model (although you can use set and linear model but you probably wont have a fun?easy time with that)*1/2 x 3/4 (1/2 of a group of 3/4)Division: Usually use the linear model (although you can use set and linear model but you probably wont have a fun?easy time with that)*For addition and subtraction you need to make the denominators the same number so you create equivalent fractions