MTE 280_Spring 2019

Make sense of problem and PERSERVEREReason Abstractly/QuantitativelyConstruct Viable ArgumentsUse Appropriate ToolsLook for & make sense of StructureLook for Repeated ReasoningModelAttend to Precision

Four Step Solving Process:Un- UnderstandDev- Develop planCar- Carry out planLO- Look back

Look for a PatternDo a Similar/Easier ProblemIdentify What Makes the Problem Hard. Remove the Hard part.Draw a Diagram (Not a Pic)Guess and Check!Write an Equation

UnitsThe number of an individual unitLongsA group of units that are determined by the Base(Ex: 10 Base five = five units equals a long)FlatsA group of longs that is also determined by the Base (Ex: 100 Base five = five longs made up of five units per flat)*If there is no written base, it is in base 10*Whatever fits in a long names the base*Any digit in a number cannot equal or exceed the number of the base.

Determine the number of units in a given numberMultiply the number of Longs by the Base, then add the Units Ex: 23 Base five = 2(5) + 3 = 13 Ex: 302 Base four = 3(16) + 0(4) + 2 = 50

Use Downward DivisionDivide the given number by the Base numberEx: 24 to Base five = 24 \ 5 = 4 with a remainder of 4, so answer equals 44 base fiveEx: 64 to Base seven = 64 \ 7 = 9 remainder 1; converting to Base seven, the 7 must be converted to a flat, so the answer equals 121 base seven*Remember any digit in a number cannot equal or exceed the number of the base.

Numbers are represented with a + or - depending if the number is positive or negative.A pair of a positive and a negative is a zero pair. A collection of zero pairs is a zero bank.Can have an unlimited amount of zero pairs as neededNegative is different from Subtracting/taking awaySeparate numbers into groups and units when multiplying integersThe groups contain a number of positives/negativesTake away groups of units for negative numbers

3+(-2)=(+++)+(--)2 positives and 2 negatives create a zero bank One negative is left overAnswer= -1(-1)+(-1)= (-)+(-)Add negative togetherAnswer= -2

(-3)-4= (---) - (++++----)Take away the one positive created by the zero bank(++++) -->Seven negatives are leftAnswer=-7(-3)-(-1)= (---) - (-)Take away one negative (-) --> Answer= -2

2(2)= (++) + (++)Two groups of two positivesAnswer= 4-3(-2)= (++(--) -->) + (++(--) -->) +(++(--) -->)Take away three groups of two negativesOnly positives are left Answer= 6

One is neither prime or compositeComposite NumbersNumbers with more than two factors4, 6, 8, 9, 10Prime NumbersNumbers that only have two factors2, 3, 5, 7, 11

Smallest number that goes into two or more numbersUse smallest exponents for individual numbers18= 2*3220=22*5GCF=2

The greatest number that two or more numbers are divisible by.Use greatest exponents for individual numbers18= 2*3220=22*5LCM=22*32*5

Factors of a specific numberMethods:Factor Tree18= 2*9, 9=3*3factors: 2 and 3Prime factorization: 2*32Upside Down L Division20= 20/2=10/2=5factors: 2 and 5Prime factorization: 22*5GCF= 2LCM= 22*32*5

Numerator: represents what you have ( how many pieces)denominator: represents the size of the pieces of a whole

Fractions must have like denominators so only the numerators are added or subtracted when adding or subtracting fractions.Denominators do not need to be the same when multiplying fractions

Adding- Draw two boxes and divide them up according to the fraction. Shade in the amount the numerator reads. Draw a third box then use it to show how many individual spaces are shading both original models2/3 + 1/2= 1 1/6Subtracting- Draw two boxes. A second set of shading is used to show how much of the first is being subtracted.3/4 - 1/5= 11/20Multiplying- One box is used to. The firsts set of shading is labeled by the second set showing the how many individual boxes that are shaded are marked by the second shaded set.1/3(2/5)=2/15

Numbers can be added vertically or horizontallyEliminate digits as the base value has been reachedAdd with each scratch representing a base value then carry remainders Ex: 3+2+1+4+6+2+3+4+2 Base eight 3+2+1+4=10 SCRATCH 4 with remainder 2 2+6=0 SCRATCH 6 with remainder 0 2+3+4=9 SCRATCH 9 with remainder 1 1+2=3 Carry ALL SCRATCHES over to next place value Answer= 33 Base eight

Separate the units, longs, etc, much like the area model then set numbers vertically like traditional addition algorithmadd numbers separatelyAdd sums together to determine answer Ex: 315+216=(300+10+5)+(220+10+6) 300+220=520 10+10=20 5+6=11 520+20+11=531

Add numbers vertically like traditional algorithmAdd sums diagonallyAnswer is read left to right Ex: 256+138 6+8=14 5+3=8 2+1=3 Add digits diagonally= 394

Keep digits of numbers separateAdd each place value separatelyAdd final numbers. Ex: 35+12=30+5+10+2 40+7=47

Change one number to be a factor of tenSubtract from one number then add to otherAdd Ex: 38+12 = (38+2)+(12-2) 40+10=50

Similar to Friendly NumbersChange one number to be a factor of ten or in base tenAdd numbers vertically Ex: 48+27 (48+2)+(27-2) 50+25=75

Separate the units, longs, etc, much like the area model then set numbers vertically like traditional multiplication algorithmMultiply numbers separatelyAdd products together to determine answer Ex: (34)(68)=(30+4)x(60+8) 30x60=1800 30x8=240 60x4=240 8x4=32 1800+240+240+32=2,312

Designed to multiply each individual digit by the otherSized to fit the numbers being multiplied (box)The multiplicand is placed along the top of the lattice so that each digit is the header for one column of cells; the multiplier is placed along the right side of the lattice so that each digit is a header for one row of cellsNumbers of products are added diagonallyAnswer is read from left to right Ex: (34)(56) 3x7=21 4x7=28 3x8=24 4x8=32 Add diagonally in diagram= 3,354

Calculating the area of a squareunits, longs, etc. are separated to solve multiplication Ex: (34)(68)=(30+4) and (60+8) 30x60=1800 30x8=240 4x60=240 4x8=32 1800+240+240+32=2,312

AssociativeGrouping different numbers together Ex: (3+4)+7=3(4+7)CommutativeReordering the numbers Ex: 3+4+7=3+7+4DistributiveMultiplying a number by every number in the parentheses Ex: 3(6+4)= 3(6)+3(4) 18+12=30

Order of operations - GEM/DA/SGGroupingEExponentsM/DMultiple/DivideA/SAdd/Subtract*Groups can be separated with +/-

2- If the one's place has an even digit or zero then the number is divisible by 23- If the sum of all digits is divisible by three then the number is divisible by 34- If the ten's & one's are divisible by four then the number is divisible by 45- If the one's digit is a five or zero then the number is divisible by 56- If the number is even and the sum of all digits is divisible by two and three then the number is divisible by 68- If the last three digits are divisible by three then the number is divisible by 89- If the sum of all digits is divisible by nine then the number is divisible by 910- If the one's digit is a zeroSimilar Divisibility Groups2, 4, 82, 3, 63, 95, 10

Adding- Draw a box and divide them up according to the decimal. Shade in the amount the tenths or hundredth place reads then do the same for the other decimal being added. Add shaded parts together.3.13 + 6.4= 9.53Subtracting- Draw a box. Shade in the amount the tenths or hundredth place reads. A second set of shading is used to show how much of the first is being subtracted.*Add zero for missing hundredths place2.7 - 1.03= 1.67Multiplying- Draw one box. The first set of shading is labeled by the second set showing the how many individual boxes that are shaded are marked by the second shaded set.* Remove decimal and multiply numbers like you would multiply whole numbers. Multiple the whole numbers ONLY from the numbers being multiplied to know where to place decimal after multiplying2.14 x 3.6= 7.704

Mentally Calculating Percentages Example: What is 40% of 70?10% of 7010%= 710% x 4 = 7 x 440% = 28