MTE 280

Problem solvingProblem solving is not a content, it is a developable skill/component similar to critical thinking.Students must first understand a problem and what it is looking for before they can start problem solving.Understand the problemDevelop a plan and strategiesImplement the planCheck workGeorge Polya created this strategy in a book called “How to Solve it”A Mathematical Tug-of-WarUse the information given to determine who will win the third round in a tug-of-war.Round 1: On one side are four acrobats, each of equal strength.On the other side are five grandmas, each of equal strength.The result is dead even.Round 2: On one side is Ivan, a dog. Ivan is pitted against two of the grandmas and one acrobat. Again, it’s a draw.Round 3: Ivan and three of the grandmas are on one side and the four acrobats are on the other.George Polya created this strategy in a book called “How to Solve it”Four acrobats and five grandmas are equal in strength.This means the total strength of four acrobats is the same as the total strength of five grandmas.4A = 5G   4A = 5G4A = 5GTo find one acrobat’s strength in terms of grandmas, I divided both sides by 4:A = 5/4GThis told me that one acrobat is equal to 5/4 of a grandma’s strength.Ivan, the dog, is placed against two grandmas and one acrobat, and it ends in a tie.This means Ivan’s strength is equal to the strength of two grandmas plus one acrobat:I = 2G+ASo I used substitutionI = 2G+5/4GI converted everything to have a denominator of 4:I = 8/4G + 5/4GI = 13/4GTo express everything with a denominator of 20 I multiplied by 5:I = 65/20GTotal strength of Ivan's teamI + 3G = 65/20G + 60/20G = 125/20GTotal Strength of the 4 Acrobats4A = 4 X 25/20G = 100/20G125/20 > 100/20Therefore Ivan's team is stronger

A Cartesian product is when you take two groups and pair each item from the first group with each item from the second group. This creates all possible combinations. The total number of pairs is found by multiplying the number of items in each group.Example:Dan has 3 shirts and 3 pairs of pants, how many possible combinations of shirts and pants does he have?3 x 3 = 9Problem SolvingMake problem solving problems as physical as possible when teaching. Base 10 blocks and props will help facilitate their understanding.

Numeration systemsPeople use symbols to indicate a quantity (numbers)Our system is a base 10 system, because there is a consistent 1-10 relationship, and is a decimal system that uses a base of 10Positional system, numbers get their value from the place where they sit, in a number, there is the hundred spot, the tens spot, and the ones spot in a ten systemThis 1-10 relationship is always there, no matter how big or small the number is, including into decimals, each place is a multiple of 10 or a division of 10.375= 300+70+5      = (3x100)+(7x10)+(5x1)      = (3x10^2)+(2x10^1)+(5x10^0)Base 5 is a way of counting that only uses the digits 0, 1, 2, 3, and 4. Instead of counting in groups of ten (like we do in base 10), we count in groups of five.Each place in a number represents a power of 5, just like in base 10 each place represents a power of 10.For example, in base 10:The first place is ones (1).The second place is tens (10).The third place is hundreds (100).In base 5:The first place is the ones (1).The second place is the fives (5).The third place is the twenty-fives (25).

Numeration SystemsBase 5 to Base 10Multiply each digit by 5 raised to its place value.Then add them togetherExample: Convert 13 base 5 to base 10( 1 × 5^1 ) + ( 3 × 5^0 )( 1 × 5 ) + ( 3 × 1 ) = 5 + 3 = 8​Converting Base 10 to Other BasesMake a representation of each digit of base 10 which needs to be converted, for this example it is 12.Remove the number from the base is. For example, for base 9, remove 9. It counts as the equivalent of 10 for that base. 9 in base 9 is equivalent to 10 in base 10.Count the remainder, and it becomes the second digit in the number.Example: Convert 12 in base 10, to base 7,8, and 912 base 10 = xxxxxxxxxxxx = 13 base 912 base 10 = xxxxxxxxxxxx = 14 base 812 base 10= xxxxxxxxxxxx = 15 base 7

Numeration SystemsBases 1-10Base 3 (digits 0-2)ones- 3^0threes- 3^1nines- 3^2twenty-sevens- 3^3Base 4 (digits 0-3)ones- 4^0fours- 4^1twelves- 4^2sixty-fours- 3^3Base 5 (digits 0-4)ones- 5^0fives- 5^1twenty-fives - 5^2one-twenty-fives - 5^3Base 6 (digits 0-5)ones- 6^0sixes- 6^1thirty-sixes - 6^2two-hundred-sixteens - 6^3Base 7 (digits 0-6)ones- 7^0sevens- 7^1forty-nines - 7^2three-hundred-forty-three - 7^3Base 8 (digits 0-7)ones- 8^0eights- 8^1sixty-fourths - 8^2five-hundred-twelve - 8^3Base 9 (digits 0-8)ones- 9^0nines- 9^1eighty-ones - 9^2seven-hundred-twenty-nines - 8^3Base 10 (digits 0-9)ones- 10^0tens- 10^1hundreds - 10^2thousands - 10^3

Properties of Multiplication1. Commutative PropertyThe order of the numbers does not change the product.Example: 3 × 4 = 4 × 32. Associative PropertyThe grouping of numbers does not change the product.Example: ( 2 × 3 ) × 4 = 2 × ( 3 × 4 )3. Distributive PropertyA number outside parentheses multiplies each term inside.Example: 5 × ( 3 + 2 ) = ( 5 × 3 ) + ( 5 × 2 )4. Identity PropertyAny number multiplied by 1 stays the same.Example: 7 × 1 = 7

The Standard American algorithm for long division:Divide – Check how many times the divisor fits into the first part of the dividend.Multiply – Multiply the divisor by the quotient (the number you just wrote).Subtract – Subtract the result from the dividend section you used.Bring down – Bring down the next digit of the dividend.Repeat – Continue dividing, multiplying, subtracting, and bringing down until there are no more digits left.

Addition Algorithms1. American Standard AlgorithmAdds from right to left.Standard method everyone is taught.2. Partial Sums MethodAdd each place value separately (ones, tens, hundreds).Combine all partial sums to get the final answer.Example:348 + 275 → (300 + 200) + (40 + 70) + (8 + 5) = 623.3. Lattice MethodDraw a gridAdd digits and place results in the grid,Add the numbers diagonally to get the final answer.4. Expanded Notation MethodBreak each number into place values before adding.Example:346 + 275(300 + 40 + 6) + (200 + 70 + 5)Add each place value: (300 + 200) + (40 + 70) + (6 + 5) = 623.5. Partial Sums with Place ValueSimilar to the Partial Sums method, but it keeps numbers in place value columns.Each place value is added separately in order from left to right.This is the best method to teach children first as it focuses on maintaining place value6. Left-to-Right Addition MethodWork from left-to-rightThis makes sense for children as that is the direction we read in the United States.Essentially the same as the American Standard algorithm, except it is worked the opposite directionSubtraction Algorithms1.  American Standard AlgorithmSubtract from right to left.The standard method commonly taught in schools.2. Lattice Method for SubtractionUses a grid to do subtraction, helps students line up place value, however, does not necessarily help them understand the role of place value when subtracting.3. Expanded Notation MethodBreak numbers into place values before subtracting. Example:543 - 278(500 + 40 + 3) - (200 + 70 + 8)Subtract each place value separately and then subtract.4. Integer Subtraction AlgorithmSubtracting where instead of borrowing, there are negative numbers.It is not a good method to teach elementary students, however, could be fun and effective for middle schoolers learning negative numbers.5. Reverse Indian AlgorithmSubtracts from left to right instead of right to left.Borrowing is done in advance before starting subtraction.

DecimalsDecimals are another way to show parts of a whole, like fractions.Use a 100-block grid:The whole block = 1Rods = tenths (1/10)Small squares = hundredths (1/100)Ex:1/10 of 1/10 = 1/1003/10 = 30/100Place Value:Each place to the right gets 10 times smaller.Each place to the left gets 10 times bigger.Example: 10 → 375.75Money Example:$111.11 = 111 dollars and 11 cents$375.35 = 375 dollars and 35 cents1 penny = 0.01 or 1/100Common Mistakes:Thinking more digits mean a bigger number (e.g., 0.9 is more than 0.51)Not understanding place valueDecimal Point:Separates whole numbers from partsRead as “and” (e.g., 111.11 = "one hundred eleven and eleven hundredths")Zeros:Sometimes zeros can be left out (e.g., 0.50 = 0.5), but they can help with understanding

Adding, Subtracting, Multiplying and Dividing DecimalsLine up the decimal points/place values.Subtraction is like borrowing in fractions . use visuals (ex. break $1 into 10 dimes, then into pennies).Multiplying DecimalsEstimate to place the decimal at the end.Focus on the multiplication first, then figure out where the decimal goes using place value.Ex: multiply as if there are no decimals, then count decimal places in the original numbers.Dividing DecimalsUse long division; bring the decimal straight up in the answer.Use real-world examples like sharing money.If the divisor is a decimal, move the decimal point to the right in both numbers to make the divisor a whole number.The answer stays the same even if you do this.Fractions to DecimalsEasiest when the denominator is 10 or 100.If not, find an equivalent fraction with a power of 10.Example: 2/5 → 4/10 → 0.4Remember: fractions = division problems.Turn the numerator into a decimal by dividing.Add zeros if needed and divide (e.g., 5/6 → 50/60 → divide out).

Percentages“Per cent” literally means “out of 100” (from Latin root cent = 100).A percentage is a way to show a part of a whole using 100 as the total.Example: 30% off means you save $30 for every $100 you spend.Important:Percentages over 100% are only used when something increases beyond the original amount (like "150% of the goal").You can't say things like “110% off”, that doesn’t make sense.In elementary math, always round percentages to 2 decimal places or fewer.Types of Problems:3 common types of problemsa) What percent of _ is _?The answer will be a percentage Formula is n = a ÷ bEx: 8 is what percent of 22? n = 8 ÷ 22 = 0.36 (repeating) 36%b) _ % of _ is what number?”The answer will be a numberFormula is n = a × b. Ex: 8% of 22 is what number? n = 0.08 × 22 = 1.76c) _% of what number is _?The answer will be a number (the original whole)Formula: n = b ÷ aEx: 8% of what number is 22? n = 22 ÷ 0.08 = 275Things to remember:“is” = equals (=)“what” = the unknown (use n)"of” = multiply (×)Always convert percentages to decimals before calculating (Ex: 8% = 0.08)

Percentages ContinuedThink of percentages, decimals, division, and fractions as proportions.Ex:37/45 = 45/45 = 100%45/37 = 100/? → Use cross multiplication to solve.Cancel out common factors when dividing.Focus on reasonable thinking:2% of 7009/25 = 36/100

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Positive and Negative NumbersPositive numbers: greater than 0Negative numbers: less than 0Zero (0): neutral value, separates positives and negativesNumber Line:Typically shown horizontally in schoolVertical number line is more effective:Acts like an elevator (0 = ground level)Easier to relate to place valuePositive = up, Negative = downChip Method:Used for:AdditionSubtractionProbabilityNumber familiesChips represent integers:One side = positiveOther side = negativeOn tests, draw chips using:"+" inside a circle for positives"−" inside a circle for negativesWhen a positive and a negative chip are together, they form a zero pairAlways circle zero pairsTo subtract, add zero pairs if needed and remove as instructedLeaves remaining chips as final answerMultiplying IntegersUse commutative property when multiplying positive and negative integersRemember integer sign rules:Positive × Negative = NegativeNegative × Negative = PositivePositive × Positive = Positive

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Number theory and divisibility rulesDivisibility rules involve numbers where the result is a whole number.All numbers can be divided by each otherA is divisible by b if there is a C that meets the requirement cxb=aEx. 10 is divisible by 5 because there is a number 2 that meets the requirements (2x5=10) therefore, all the numbers are divisors and factorsRules by Number5 is a factor of 102 is a factor of 102 is a divisor of 105 is a divisor of 1010 is a divisor of 210 is a divisor of 5-end of the numberBy 2: 0,2, 4, 6, 8By 5: 0,5By 10, 0-sum of the digits By 3: if the sum of digits is divided by 3Ex. 24, 2+4=6 6 divided by 3= 2By 9: if the sum of digits is divided by 9By 6: if it’s divisible by both 2 and 3-last digitsBy 4: if the last 2 digits are divisible by 4By 8: if the last 3 digits are divisible by 8By 7:   double last digit Subtract from the remaining number RepeatBy 11: the “chop off” methodChop off the last 2 digitsAdd them to the remaining numberRepeatEx: 29,194 divided by 1129,194 to 291+94 = 385385 to 3+85=88Composite and Prime NumbersNumbers with a lot of factors are called composite numbersNumbers with only 2 factors, 1 and itself,  are called prime numbersEx, 13 factors are (1, 13)The numbers 1 and 0 are neither prime numbers nor composite numbers.0 is the additive identity elements1 is the multiplicative identity element

Prime Factorization and Prime factor trees24: 1, 2, 3, 4, 6, 8, 12, 24Prime tree for 24- 4 and 6, the result is 24= 2x2x2x3It doesn't matter which 2 factors you choose for a tree, they will always end with the same base prime numbersexpress the prime factorization as 24= 2x2x2x3, you have to express this through multiplicationGreatest Common Factor, GCF, and Least Common Multiple, LCMMultiples go on forever and are a number multiplied like 2: 2,4,6,8,10, etc.Divide each side of the fraction by a common number (factor) to work on simplifying the fractionEx. 25/100 to simplify divide by 25 which is a common factor to get 1/4If you cant divide, you have to multiply by a common multipleEx. ½+⅙ have the multiple 6 in common, the smallest,  so you can multiply to become 3/6 +1/6 = 4/6Methods for GCF and LCM:List method:Find GCF 24: 1,2,3,4,6,8,12,2436: 1,2,3,4,6,9,12,18,36The GCF is 12, because it is the biggest factor in commonFind LCM24: 24, 48, 72, 96 …36: 36, 72, The LCM is 72 because it is the smallest multiple in commonPrime Factorization Method:(using prime trees)24= 2x2x2x336= 2x2x3x3GCF= 2x2x3=12LCM= GCF x 2x3       = 12x2x3=72

Meanings of FractionsA fraction is a way to express a part of a whole.Quotient ex. 3 divided by 6 is the same as 3/6Ratio ex. 5 girls and 7 boys, 16 total, the ratio of boys/girls is 7/9 or 7:9, the ratio of girls/boys is 9/7 or 9:7When teaching fractionsModels: area (pattern block, shading), length (number line, folding paper), sets or groups (demonstrating with groups)The most important rulesWhen the numerator and the denominator are the same, it equals one. (this is important!!)The more pieces my whole is divided into, the smaller the pieces get.When we talk about fractional parts, we are talking about equivalent parts.

Spring Break

Adding and Subtracting FractionsIf the denominators are the same, add the numerators.When the denominators are different, find equivalent fractions, the LCM is often the best choice.Factor in the value of the whole when comparing fractions.Remember the a/a=1 rule when converting mixed numbers into improper fractions.Multiplying FractionsMultiply straight across, denominator x denominator, numerator x numeratorThere is no need for a common denominatorWhen multiplying fractions, the result is smaller because it is a part of a part.ex. 1/2 of 1/2 equals 1/4Dividing FractionsChange the problem into a multiplication problemKeep, Change, FlipMultiplying a fraction by one "gets rid of" the denominator.Every number has a denominator, it is just generally ignored if it is a whole number.

Notes From Video1st GradeStudents learn the word partition (dividing shapes into parts).Focuses on halves and quarters (fourths).Students learn terms like "half" "fourth" or "quarter"2nd GradeIntroduced to thirdsThey learn that different-looking shapes can have the same area when divided equally.3rd GradeFirst-time students see fractions written with numerators and denominators.Heavy use of visual modelsConnect fractions to whole numbers:Example: 4 is made of 1+1+1+1.Learn to use a number lineLearn counting by fractions Use number lines to show equivalent fractions.Students must learn that the size of the whole number stays the same, only the fractions can change.Students learn that whole numbers can also be written as fractions Understanding common denominators and numerators when determining greater than and less than4th gradeStudents learn whatever is done to the numerator must be done to the denominator. Students should discover this pattern instead of being told.Learn benchmark fractions, like using one half to help determine greater than or less than.Use visual models to help students understand equivalent fractionsMy Response to Video1. I did not realize how early children start learning about fractions. I always thought of it as a 3rd or 4th-grade skill. I do not remember doing partitions or learning fractions in school before 3rd grade. As a teacher, I will follow the timeline laid out in the video as I think something similar would have helped me.2. I like how much the video emphasizes the importance of visual models. When I was in school, I remember using the number line but not other visual models, and I think it would have helped my understanding. I always hated fractions because they were confusing, but that may have helped fix that confusion.3. The strategy of having students fold paper in order to see equivalent fractions is the same activity we did in class before spring break. While I already understood the concept, I found that this method made it much easier to explain fractions and lead the students to their own understanding. This is a method I will definitely use in the future, and I think solidifies students understanding.

Problem Solving with Fractions Using DiagramsJim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of all the bars, and Ken and Len each took ⅓ of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get?Jim got 12 barsKen got 16 barsLen got 16 barsMax got 4 barsJim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of the bars. Then Ken took ⅓ of the remaining bars. Next, Len took ⅓ of the remaining bars, and Max took the remaining 8 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get? How is this problem (with regard to fractions) different from Problem 1?Jim got 6 barsKen got 6 barsLen got 4 barsMax got 8 barsThese questions are different because the first question has each person taking a different fraction of the whole. The second question has each person taking fractions of the remainder of what is left. I used different visual models using a rectangle for both questions. I attempted to attach an image of my model, but that is a paid feature on Mindomo

Class spent working on Mind Map project and reviewed homework for test.

Word Problems with PercentagesA studetn takes a test with 46 questions and gets 37 questions right. What is their percent on the tst.37 is what % if 4537 = n x 45 n=37/45cross multiply and divide 45/37 and 100/xx = (37/100)/45

Multiplying numbers with decimals Use logic and reasoning to place the decimal; otherwise, multiply like normal Don't need to line up the decimals, line up the numbers insteadTeaching why and place value is super super important when teaching children to multiply numbers with decimalsDivision for numbers with decimalsUse the long division algorithmMost teachers teach to bring the decimal point up and continue with the normal algorithm, but it doesn't teach them why, which is that it represents the place value