MTE 280- Elementary Mathematics

Show:Draw out + and - for what you have (the first number)Draw out + and - for what is addedCircle zero pairs (pairs of one + and one -, which add up to zero)Answer is what is left without zero pairsSolve:Draw two + or - under whichever number is larger (absolute value)Draw one + or - under whichever number is smaller (absolute value)Circle one sign under one number and one sign under the other numberIf the signs circled are the same, addIf the signs circled are different, subtractThe sign left out of the circle is the sign the answer will have

Say "take away" instead of "minus" or "subtract"Difference between take away and negative:Take away is an actionNegative is a descriptionShow:Draw out + and - for what you haveCircle what you are taking away and draw arrowAnswer is whatever is left after taking awaySolve: Use "Keep, change, change"Keep the first number's sign, change the subtraction to addition, change the second number's sign

First number is # of groups, second is what's in each group2(3) is 2 groups of 3-2(3) is like 0-2(3)Show: Draw circles to represent groups, fill in groups with + or - to represent what's in each groupFor problems where the first number is negative: Create zero banks firstCircle and arrow what is being taken awaySolve: Multiply numbers like normalIf the signs are the same, answer is positiveIf the signs are different, answer is negativeExample: -2(3)=-6-2(-3)=62(3)=62(-3)=-6

Solve: Divide numbers like normalIf signs are the same, answer is positiveIf signs are different, answer is negative

Value of a number without its signExample: Abs. value of -12 is 12. Abs. value of -53 is 53.

Factors: numbers multiplied together to get an answerPrime: a number with exactly 2 factorsComposite: a number with more than 2 factorsPrime factorization: factoring a number using only prime numbers

Draw any two factors coming down from numberKeep drawing two factors for each next number until left with only prime numbers24^12*2^6*2^3*224 = 2*2*2*3 = 23*3

Use downwards division, but the number on the left must be primeExample:3 L242 L82 L4 224 = 23*3

The largest factor that is common for two different numbersUsing each number's prime factorization, the GCF is all of the factors the two have in common, with the lowest exponent present for eachExample: 20 and 1820 = 2*3218 = 22*5GCF = 2

The smallest multiple that two numbers have in commonList out all factors of each numberAdd the biggest exponent on each numberExample: 20 and 1820 = 2*3218 = 22*5LCM = 22*32*5

What is a fraction?Parts of a wholeNumerator (top number): how many pieces you haveDenominator (bottom number): what size the pieces are

Show:Draw two same-sized boxesDivide first box vertically based on the first fraction's denominatorDivide second box horizontally based on the second fraction's denominatorShade in each box based on its respective numeratorCopy the dividing lines from one box onto the other and vice versaDraw a third box with the vertical and horizontal dividing linesShade in the third box by counting the number of small sections in the first and second boxesSolve: Two fractions must have common denominators to be added or subtractedIf denominators are the same, add or subtract the numerators and keep the denominatorIf denominators are different, find a common denominator:Factor each denominator using one common factorWhatever is missing from each denominator, multiply by it as a fraction equal to one (x/x=1)Denominators are the same, add or subtract numerators and keep the denominator

Show: When # of groups = fraction:Draw a full group and fill in with the number inside the groupDraw a box around how much of the group you wantExample: 1/2(4)Draw a group with 4 dots insideDraw a box around 1/2 of the groupAnswer = 2 because 2 dots are boxedWhen number inside the group = fraction:Draw number of groupsInside each group, draw a box to represent the fractionAdd the fractions togetherWhen both numbers are fractions:Draw a boxDivide box vertically by first denominator and divide box horizontally by second denominatorShade in vertically based on first numerator and shade in horizontally by second numeratorAnswer: numerator = number of boxes that are double shaded, denominator = numbers of boxes totalSolve: Multiply numerator by numerator and multiply denominator by denominatorExample: (2/3)(1/4) = (2/12)

Determine if fraction is >, <, or = to another fraction> greater than< less than= equal thanMethods: Anchor fractions: simple fractions to compare each fraction to (1/2, 1, etc.)Whole numbers: whichever number has the larger whole number is greaterSame denominator: whichever fraction has the same numerator is greaterSame numerator: whichever number has the smaller denominator is greater (because they have the same number of pieces, but a smaller denominator means bigger pieces)

Write division problem as a fractionDivide denominator into each digit left-to-rightExample:474=59 2/8847/88*5=4047-40=774/88*9=7274-72=2

Students don't need to know many multiplication facts Takes longerExample:52/33|52 -30| 10 22 -15| 5 7 -6| 1=17 1/3

A number is divisible by x if...2: Ones place is even3: Sum of all digits is divisible by 34: Last two digits is divisible by 45: Ones place is 5 or 06: Divisible by 2 and 38: Last 3 digits is divisible by 89: Sum of all digits is divisible by 910: Ones place is 0

GEMDASMake groupsUse EMDAS to simplify groups

Order to teach multiplication table: Ones, twosTens, fivesThrees, NinesDoubles

Show:Flat = unitLongc= tenthUnit = hundredthDraw out like fractionsAddition: draw out box divided into ten, fill in decimalsSubtraction: draw out box divided into ten, fill in first decimal, take away (cross out) second decimalSolve: Line up the whole numbers and add normally

Show:Draw out box, divide into tenDraw first numberDivide box into ten the opposite wayDraw second numberCount how many boxes are shaded twiceSolve: Estimate answer by multiplying the whole numbersWrite out multiplication problem without the decimalsMultiply like normalCompare answer to estimate and add decimal where the answer is closest to the estimate

Find 10%Multiply to get desired percentageAdd 5% if neededSubtract from 100 first for discount problemsExample:40% of 7010% = 7x4 x440% = 28

Understand the problemDevelop a plan to solve the problemCarry out the planLook back- does the answer make sense?

Make sense of the problemUse abstract/quantitative reasoningUse logical argumentsUse toolsUnderstand structureLook for patterns of reasoningModelUse precision

Look for the patternDo an easier problemModify the problem (get rid of the hard part)Draw a diagramGuess and checkWrite an equation

Inductive ReasoningFiguring something out based on a patternDeductive ReasoningFiguring something out based on or using prior knowledge

Standard system of countingCreated because humans have 10 fingersAssume base ten if no base is specifiedCounting in Baseten: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Base is specified as subscriptCounting in basefour: 1, 2, 3, 10Counting in basethirteen: 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E, W, 10

Base TenOnes, tens, hundredsOnes=100Tens=101Hundreds=102Other BasesUnits, long, flatsUnits=x0Longs=x1Flats=x2x=base number

Diagram Method (Show)Draw out the given number using units, longs, and flatsWrite down the value of each shapeUnits=1, longs=x, Flats=(x)2Where x=number of baseAdd up the value of each shapeExample: Convert 23five to baseten2 longs and 3 unitsEach long=5, each unit=12(5)+3(1)=13Algorithm Method (Solve)Multiply number of flats by x2Multiply number of longs by x1Multiply number of units by x0Sum the numbersExample: Convert 23five to baseten2(51)+3(50)=13Example: Convert 4,120,367nine to baseten4(96)+1(95)+2(94)+0(93)+3(92)+6(91)+7(90)=2,198,239

Diagram Method (Show)Skip count to draw the given number using units, flats, and longsCount up the number of each: units, longs, and flatsWrite the number of each shape in their respective place valuesExample: Convert 31 to base eightSkip count: 8, 16, 24 (draw 3 longs)Draw one unit for each remaining number (7 units)Write the answer as 37eightAlgorithm Method (Solve)Use downwards division, dividing the base number into the given numberExample: Convert 31 to base eight8 goes into 31: 3 times with 7 leftover31=37eightExample: Convert 47 to base five5 goes into 47: 9 times with 2 leftover5 goes into 9: 1 time with 4 leftover47=142five

Used for adding many numbersWrite numbers verticallyStart in ones column and add going downScratch a number off once you get to 10 (or whatever the base is) and write the remainder next to itContinue adding going down and place the last remainder of the ones column under the addition barCount up the number of scratches and write that number on top of the tens columnAdd up the tens column using the same process of scratches and remaindersExample:231827 62 5162+ 23 2----------202

Use for adding large numbers with in hundreds or moreWrite out each number in its expanded formAdd the expanded numbersExample:4615+3548 4000+600+10+5+3000+500+40+8________________ 7000+1100+50+13=8163

Write numbers verticallyDraw box underneath each place valueDraw diagonal line through each box, extending from top right through bottom left cornerStarting with the ones column, add numbersWrite the sum's tens digit in the top section of the box and the ones digit in the bottom sectionMove onto the next column and add the same wayAdd down the diagonals, placing sum just outside the boxAnswer is read left to rightExample:

Instead of adding starting at the ones place and working left, you start from the left and work right. Example: 348+ 891---------1100130+9-----------1239

Take away some from one number and give that amount to the otherGoal of getting numbers to multiples of tenExamples:37+63=(37+3)+(37-3)=40+60=10028+54+36+22=(28+2)+(54-4)+(36+4)+(22-2)=30+50+40+20=140

Same process as friendly numbersSubtract an amount from one number and add that amount to anotherWill not get all numbers into multiples of tenGoal: to turn some or all numbers into compatible numbers (numbers that are easy to use mentally)Examples:24+39+12=(24+1)+(39+1)+(12-2)=25+40+10=7543+62+21=(43+2)+(62-2)+21=45+60+21=126

Write numbers out in their expanded formMultiply each separate numberAdd the productsExample:275+182=(200)100=20000(200)80=16000(200)2=400(70)100=7000(70)80=5600(70)2=140(5)100=500(5)80=400(5)2=1020000+16000+400+7000+5600+140+500+400+10=50,050

Draw a rectangleAlong two adjacent edges of the rectangle, separate units, longs, and flats (or ones, tens, and hundreds)Continue the lines through the rectangle to separate the interior into flats, longs, and unitsAdd up each sectionAdd the section totals togetherExample:

Similar to base ten blocks, but slightly more advancedDraw a rectangleOn adjacent sides of the rectangle, write each number in its expanded formExtend lines through the rectangle wherever there are plus signsMultiply the numbers on the edges to find the area of each individual squareAdd the areas together to get the total area of the rectangleExample:

Draw the lattice structureWrite each digit of each number along the top and right edges of the rectangleMultiply each edge, write the first digit in the top section and the second digit in the bottom sectionAdd down the diagonalsAnswer is read from top left corner to bottom right cornerExample:

Goal: change some or all numbers to be multiples of tenAdd to one number, subtract that from another numberSubtract new numbersExample:57-21=(57+1)-(21-1)=58-20=38

Write each number in its expanded formSubtract numbers from the same place value normallyAdd the expanded number togetherExample:525-422=(500+20+5)-(400+20+2)=(500-400)+(20-20)+(5-2)=100+0+3=103

Draw out the base ten blocks representing the number being dividedStarting with the largest place value, separate into groups of the what the number is being divided byConvert the remainder of that place value into the one below itContinue separating into groups and regrouping until there are units left that cannot make a groupWrite out the answer based on how many groups of each shape there areExample:726/6*Brackets represent groupings7 flats = [6 flats] + 1 flat1 flat + 2 longs = 12 longs = [6 longs] + [6 longs]6 units = [6 units]6 flats + 2(6 longs) + 6 units =121

Works if there is a non-zero remainderRepeatedly subtract from the number being dividedAnswer is the number of subtractions completedExample:18/6=31 subtraction: 18-6=122 subtractions: 12-6=63 subtractions: 6-6=0

Write out the division problemDivide like the traditional algorithm, but with emphasis on place valueExample:860/2400+30=430

Grouping (parenthesis placement) does not make a difference in the answerWorks only for addition/multiplicationExample:(1+4)+6=111+(4+6)=11

Order does not make a difference in the answerWorks only for addition/multiplicationExample:2+9+3+1=159+1+1+3=15