MTE 280 Investigating Quantity

Numeration Systems

Bases

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Base 10 has 10 digitsBase 5 has 5 digitsBase 2 has 2 digitsConcrete --> with numbers (blocks)Pictorial --> with pictures (drawings)Abstract --> numbers & symbolsStandard form, draw model, word form, expanded form, and underline digits are all different ways to show different bases.

Different Systems

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Base 10 is a part of the Hindu-Arabic system of math.

Integer Concepts

Understanding Integers

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The Opposite of an Integer:The opposite of a number is better known as the additive inverse; that is, the opposite of the number a is the number which must be added to a to produce the additive identity 0.Number Line Approach: A number line is one method of visualizing the integers. To investigate the opposite of an integer graph both the given integer and its opposite on the number line provided.Chip Method: Another method for visualizing integers is to use colored chips. Typically the chips are red on one side and another color (yellow or white) on the other side which permits representing positive numbers with white or yellow chips and negative numbers with red chips. The idea of opposite seems rather natural when using these manipulatives since there are only two colors. To find the opposite of a number all that is necessary is to turn each chip to the opposite side.Absolute Value:Most students when asked what the absolute value of a number is reply with what they perceive as the definition of absolute value: The absolute value of a positive number is the number itself and the absolute value of a negative number is the numbers’ opposite. In reality, the absolute value of a number is its magnitude. It is the case of real numbers the method mentioned does in fact produce the magnitude of the number but masks what is meant by finding the absolute value of a number. We look at two ways of investigating absolute value.Number Line Approach: A number line is one method of visualizing the integers. In looking at magnitude in this context, absolute value is the distance of the given number to zero.  Chip Method: In looking at magnitude in this context, absolute value is the quantity of chips present. For our purposes the shaded/colored chips are negative and the white chips are positive.Ordering Integers:By convention, the number line is structured so that numbers increase from left to right.  

Operations

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Addition:There are three appealing ways to understand how to add integers. We can use movement, temperature and money. Traditionally, zero is placed in the center. Positive numbers extend to the right of zero and negative numbers extend to the left of zero. In order to add positive and negative integers, we will imagine that we are moving along a number line. The temperature model for adding integers is exactly the same as the movement model because most thermometers are really number lines that stand upright. The numbers can be thought of as temperature changes. Positive numbers make the temperature indicator rise. Negative numbers make the temperature indicator fall. It can be helpful to think of money when doing integer addition. The positive numbers represent income while the negative numbers represent debt.Rules for AdditionPositive + Positive = PositivePositive + Negative = DependsNegative + Positive = DependsNegative + Negative = NegativeSubtraction:The technique for changing subtraction problems into addition problems is extremely mechanical. There are two steps:Change the subtraction sign into an addition sign.Take the opposite of the number that immediately follows the newly placed addition sign.Multiplication:Now we have to understand the rules. The first rule is the easiest to remember because we learned it so long ago. Working with positive numbers under multiplication always yeilds positive answers. However, the last three rules are a bit more challenging to understand.Rules for MultiplicationPositive x Positive = PositivePositive x Negative = NegativeNegative x Positive = NegativeNegative x Negative = PositiveDivision:Integer division is division in which the fractional part (remainder) is discarded is called integer division and is sometimes denoted . Integer division can be defined as , where "/" denotes normal division and is the floor function.

Decimals

Concepts

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Place Value- Long = 1/10 = .1- Flat = 1 = 1.00- Cube = 10 = 10.00 flats- Unit = 1/100 = .01Expanded- 34.26 = 3(10)1st power + 4(10)0 Power + 2(10)-1 power + 6(10)-2 power --> 30+4+2(1/10)+6(1/100) --> 30+4+.2+.06 = 34.26Base 10 Blocks- 21.34 = 2 cubes, 1 flat, 3 longs, 4 units- 3.4 = 3 flats, 4 longs- 6.84 = 6 flats, 8 longs, 4 unitsDecimals as Fractions- .2 = 2/10 = 1/5- .125 = 1/8 = 125/1000 divided by 125/125 = 1/8- .625 = 625/1000 divided by 25/25 = 25/40 = 5/8Fractions as Decimals-Find a number you can multiply by the bottom of the fraction to make it 10, or 100, or 1000, or any 1 followed by 0s.-Multiply both top and bottom by that number.-Then write down just the top number, putting the decimal point in the correct spot (one space from the right hand side for every zero in the bottom number)Terminating and Non-Terminating Decimals- Terminating: Have an end (ex: 3/4 = .75)- Non-terminating: Repeating fraction (ex: 4/3 = 1.333 repeating)Writing Decimals:- Five hundred, fifty-five and three tenths = 555.3- Twenty-four and ninety-seven hundredths = 24.97- Two and forty-two thousandths = 2.042- Seventy-five ten thousandths = .0075

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Operations

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AdditionWrite down the numbers, one under the other, with the decimal points lined up.Put in zeros so the numbers have the same length (see below for why that is OK)Then add using column addition, remembering to put the decimal point in the answer.SubtractionWrite down the two numbers, one under the other, with the decimal points lined up. 2. Add zeros so the numbers have the same length. 3. Then subtract normally, remembering to put the decimal point in the answer.MultiplicationLine up the numbers on the right - do not align the decimal points.Starting on the right, multiply each digit in the top number by each digit in the bottom number, just as with whole numbers.Add the products.DivisionIf the divisor is not a whole number, move decimal point to right to make it a whole number and move decimal point in dividend the same number of places.Divide as usual. ... Put decimal point directly above decimal point in the dividend.Check your answer.

Whole Numbers

Properties

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Addition Properties:- Closure Property of Addition: If you add any 2 whole numbers, the sum will be a whole number- Commutative Property of Addition: When adding, changing the order of the addends will result in the same sum- Associative Property of Addition: When adding 3 or more numbers, the grouping of the numbers will not change the sum- Identity Property of Addition: When adding zero to any number, the sum will be the same numberSubtraction Properties:- Closure Property of Subtraction: If you subtract any two whole numbers, does the different exist and is it a whole number? NO, you would end up with a negative number- Commutative Property of Subtraction: Does this property apply to subtraction of whole numbers? NO, subtracting would result in different differences- Associative Property of Subtraction: Does this property apply to subtraction of whole numbers? NO, grouping different quantities gives different solutions- Identity Property of Subtraction: What do you think a property called the Identity Property of Subtraction would do? It can't work because you should be able to commute the numbers.Multiplication Properties:- Closure Property of Multiplication: If you multiply any two whole numbers, is the product a whole number? Yes!- Commutative Property of Multiplication: Does this property apply to multiplication of whole numbers?- Associative Property of Multiplication: Does this property apply to multiplication of whole numbers? Yes!- Identity Property of Multiplication: Does this property apply to multiplication? Yes!- Zero Property of Multiplication: When multiplying any number by zero or zero by any number, the product is zero. Is this true? Yes!

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Problem Types/Models

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Addition Problem Types:- Set Model: combination of two sets of discrete objects- Linear/Number Line Model: combination of two continuous quantities, shown on a number line using arrows.Subtraction Problem Types:- Take away: characterized by starting with some initial quantity and removing a specific amount.- Missing Addend: characterized by the need to determine what quantity must be added to a specific quantity to reach a target amount.- Comparison: characterized by a comparison of the relative sizes of two quantities to determine how much larger or smaller one of the quantities is than the other quantity. - Linear: characterized on a number lie using arrows to show change.Multiplication Problem Types:- Repeated Addition (set): repeatedly adding a quantity of objects a specified number of times- Repeated Addition (linear): repeatedly adding a quantity of continuous quantities a specified number of times.- Area Model: a product of two numbers representing the sides of a rectangular region such the the product represents the number of unit-sized squares within that region.

Fact Families

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Addend: Any number being addedInverse Operations: An opposite operation that undoes another operationAddition/Subtraction:6+5=115+6=1111-5=611-6=5Elementary school children need to understand inverse operations or "opposites" when it comes to the relationship between addition and subtraction.

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Algorithms/Strategies

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Addition:Partial SumsUnderstanding place value, finding partial sums, and adding partial sums to get the final answer.Column AdditionRecording numbers in place-values columns, adding in place-value columns, making and moving groups of 1s, 10s, 100s, and so on.Expanded notation --> alternate algorithmMultiplication:Partial ProductsUsing the distributive property, thinking about expanded notation, using extended facts to calculate partial products, adding partial products to find the final answer.LatticeUsing basic facts knowledge, organizing a multiplication problem using a grid based on place value, using the distributive property, and following several well-defined steps to find the product.Subtraction:Trade FirstMaking all place-value trades first, don't have to switch back and forth between subtracting and trading.Partial DifferencesThinking about numbers in expanded notation, using place value to determine partial differences, and adding partial differences.Counting UpSubtracting by finding the distance between two numbers, using benchmark numbers, and adding smaller distances to get the final answer.DecomposingDivision:Partial QuotientsSimpler way to do long division. It's easier to understand than some other methods. Involves finding multiples of the divisor, finding partial quotients, and finding the sum of the partial quotients.Column Division

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Number Theory

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Divisibility Rules:One whole number is divisible by another if, after dividing, the remainder is zero.If one whole number is divisible by another number, then the second number is a factor of the first number.A divisibility test is a rule for determining whether one whole number is divisible by another. It is a quick way to find factors of large numbers.Factor Rainbows:A factor rainbow is a way of writing factors for numbers using a series of arcs.It helps students ensure they have all the factors by listing them in consecutive order.Ex: A number's factors are the pairs of smaller numbers that can be multiplied to make the larger number. When factoring 28, the problem solver begins with one and 28 then moves up to two and 14 and four and seven.Prime vs. CompositeA prime number is a whole number that only has two factors which are itself and one. A composite number has factors in addition to one and itself. The numbers 0 and 1 are neither prime nor composite. All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

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Fractions

Concepts

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3 models:Area/region modelLinear/length modelSet ModelUsing fraction language:Counting fractional parts: IterationThe top number counts (numerator)The bottom number tells what is being counted (denominator)Use your faction circles to represent 3/4. Is is a count of three parts called fourths.Fraction notation is a convention and giving explicit attention to the meaning of numerator and denominator comes from iteration.Fraction size is relativeEquivalent-sized fractional piecesFraction Misconceptions:Thinking of the numerator and denominator as separate and not as a single valueNot recognizing equal partsThinking that fraction 1/5 is smaller than 1/10 because it has a smaller denominatorUsing the rules from whole numbers to compute

Operations

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To add (or subtract) two fractions      1) Find the least common denominator      2) Write both original fractions as equivalent fractions with the least common denominator.     3) Add (or subtract) the numerators.     4) Write the result with the denominator.To multiply two fractions:     1) Multiply the numerator by the numerator.     2) Multiply the denominator by the denominator.To divide by a fraction, multiply by its reciprocal.Mixed numbers can be written as an improper fraction and an improper fraction can be written as a mixed number.A fraction is in lowest terms when the numerator and denominator have no common factor other than 1. To write a fraction in lowest terms, divide the numerator and denominator by the greatest common factor.

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