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.