MTE 280 Elementary Mathematics
Weeks 11-15
Mentally Calculating %
Find 10% First
- When trying to find a percent of a number, we must first ask ourselves what 10% of that number would be, for example, 10% of 40 would be 4.
- If we were trying to find 50% of 40, we would take the 10% = 4, and multiply both numbers by 5 to get 50% and to get the answer of 20 because 4 x 5 = 20.
- When dealing with a percent like 55% of 40, we would again refer back to what we know about 10% of 40.
- If we know that 10% = 4, then that means 5% would = 2, since half of 10 is 5, and half of 4 is 2.
- We would then add the 5% to the 50% we already found to create 55%, and add 2 to 20 to get 22. Therefore, 55% of 40 =22.
+, -, x Decimals
Showing Decimals
- Similar to adding, subtracting, and multiplying fractions like we have learned previously, we will create models to show how we add, subtract, and multiply decimals.
- If the decimals have the same decimal placement, (i.e. 0.4 + 0.5) you would only need to cut the rectangle into 10 boxes and add or subtract the number required in order to get your answer.
- If decimal places are not the same, numbers are in the hundredths place, or you're multiplying, you will need to create a rectangle cut into 10 boxes horizontally and vertically in order to then count and solve.
Estimate Before Solving
- When multiplying decimals, for example, (3.04 x 2.75) we can estimate what the answer should be by multiplying 3 x 2 to get a whole number of 6.
- With this information we know that the answer should be some where around 6 and where the decimal point of our answer should go so that it is also very close to our 6 estimate.
- The only time estimating will not work is when we have numbers like (0.08 x 0.65), because 0 x 0= 0.
- To alleviate this issue, we can also count how many places it takes to get to each number's decimal, add these together, then move the decimal to the left that many times over.
LINE UP WHOLE NUMBERS!!!!
- When adding or subtracting decimals, it is important to remember to line up whole numbers so numbers are being added and subtracted in the correct place value.
- Be sure to carry the decimal directly down into the answer after solving.
- See attached video for more information on how to add and subtract decimals properly.
Order of Operations
Not P.E.M.D.A.S., but G.E.M.D.A.S.
GEMDAS:
- Grouping
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
- When solving a problem that deals with order of operations, it is important to group the problem first by drawing lines under every addition and subtraction sign, solving within these groups, then solving the problem together as whole to then get our final answer.
Divisibility Rules
2,3,6...3 & 9...2,4,8...5 & 10
Divisibility Rules:
- 2 - Last digit has to be even.
- 3 - if digits add up to a number divided by 3.
- 4 - last two digits divisible by 4.
- 5 - ends in 5 or zero.
- 6 - if divisible by 2 and 3.
- 8 - last 3 digits divisible by 8.
- 9 - if digits add up to a multiple of 9.
- 10 - ends in zero.
- grouping the numbers that have similar rules will make it easier to remember, hence the groups, 2, 3, 6,...3 & 9,...2,4,8,...5 & 10.
Algorithms for +, -, x, / Fractions
Backwards "C" and "KCF"
- When multiplying mixed fractions, we must do a "Backwards C" on the mixed fraction in order to make it into an improper fraction.
- We do this by multiplying the whole number with the denominator, then add the numerator number to get our answer for the new numerator of the improper fraction.
- Remember to use the same denominator the fraction had before when writing the new improper fraction.
- As mentioned before, simplify both fractions before multiplying across to receive your answer.
- When dividing, we must remember KCF, otherwise known as stay, change, flip.
- The first fraction remains the same, we change the sign to multiplication, and flip the second fraction.
- From there we simplify as we normally would in multiplication, then solve.
- When dividing mixed fractions we must do our backwards c to both, then keep the first improper fraction as is, change the sign, and flip the second improper fraction to then simplify and solve as usual.
Simplify First
- When multiplying fractions, it is important to always simplify first.
- You can simplify fractions diagonally, for example if a numerator and denominator of a separate fraction is 20 and 4, you can break the 20 down to 4 x 5, and cancel out the 4s.
- Similarly, you can simplify within a fraction itself (up and down).
- Think of these cross-out lines as "funky ones".
- When both fractions have been simplified as much as possible, multiply across to receive your final answer.
Remember What We Did with Factor Trees?
- When adding or subtracting fractions, if if the fraction is mixed, for example (8 1/3 + 2 2/5) you would simply add the whole numbers first (8+2) and save this answer to put in front of the final answer when the problem has been completely worked out. Do not forget about it!
- When adding or subtracting fractions with different denominators, think back to when we learned how to make factor trees. When we look at the two different denominator numbers, we must think of a number(s) they have in common to make a common denominator for both. For example, if we had the denominators of 12 and 10, we know can break 12 down to 6 x 2 and 10 down to 5 x 2. We can see that they both share a common factor of 2. Then, in order to make the denominators the same we would times 5 to the entire fraction (num. and denom.) that has 12 as a denominator, and times 6 to the entire fraction that has 10 as a denominator.
- Now that denominators are the same, we can now add or subtract the new fractions we have created because they now have common denominators.
- As mentioned before, do not forget the previous whole numbers we added or subtracted in the beginning, as this is a part of our answer.
Weeks 6-10
Showing how to Compare/Solve Fractions with Diagrams
Rectangles are Better
- When showing how to add, subtract, and multiply fractions, it is best to draw rectangle diagrams instead of the typical circle or "pie" diagram because it could get messy dividing the pie up with odd numbers.
- When showing how to add fractions you begin by drawing two rectangles for each fraction, cut them up and then shade them to represent the fraction.
- It is important to note that when cutting them up, when denominators are different, one rectangle must be cut horizontally and the other must be cut vertically to help us solve later on.
- Once they rectangles have been cut, we will add the respective horizontal and vertical lines to the rectangles that have been cut oppositely in order for each rectangle to have the same number of boxes in both.
- After that, we will be able to see how many boxes are shaded in each rectangle and add these totals together to create a new third rectangle that is shaded with how much they have in total, thus, giving us our answer.
- With subtraction, we follow the same steps, but this time instead of adding up how many shaded boxes they both have, we will be taking away the least amount of boxes that they have. For example, if one rectangle has 3 boxes shaded and the other has 4, we will take away 3 of the 4 shaded boxes in the rectangle, and this will be our numerator over the amount of boxes in total that the rectangles have.
- Lastly, with multiplication, we make one rectangle of the fraction that is inside the ( )s, making a diagram like how we have been doing.
- With the fraction outside the ( )s, it is telling us how many groups we have of the fraction inside the ( )s, so we will cut it into how many pieces the denominator has.
- Next, we will make slashes in how ever many number rows the numerator tells us to. For example if it's 2/3, we would add slashes over two rows in the rectangle to differentiate that from the shaded boxes.
- Finally, we look at the boxes that are only shaded with slash marks to find our numerator of the new fraction, while counting how many boxes in total the rectangle has to give us the denominator, and this is our answer.
Pac-Man Wants the Fraction Missing the Smaller Piece
- When comparing fractions, it is important to know that we always want the piece that is missing less than a fraction missing a larger piece.
- For example, if we had the fractions 7/8 and 12/13, by looking at the numerators and denominators we can see that pieces cut into eighths would be larger pieces than pieces cut into thirteenths. Therefore, we know that the 7/8s piece has a bigger chunk missing than the 12/13s piece would.
- Pac-Man wants to eat the fraction that has a smaller portion missing, so 7/8 < 12/13.
- If numerators are the same we always go with the denominator that will give Pac-Man a larger piece to eat. We can also use an anchor fraction to help figure it out by seeing which piece has more than half and which piece has less than half.
- If denominators are the same, we always go with the numerator that has the largest number because it has more pieces and Pac-Man wants to eat the most pieces.
LCM & GCF
Big Number vs. Small Number
- With the least common multiple (LCM) it is important to remember that you're writing out all the listed numbers that are in both groups, but making sure to add the larger exponent if there is two of the same number listed. For example if we have 5, 5 to the fourth power, and 5 to the seventh power, we would only include the 5 to the seventh power in our list since it is the bigger exponent.
- With greatest common factor (GCF) it is important to remember that we are only writing the numbers that the two sets of number groups have in common together. It is also important to remember with GCF that we will be taking the smaller exponent of the number they have in common. For example, if both sets have a 2, but one has 2 to the second, and the other has 2 to the fourth, we would only write down 2 to the second since it is the smaller exponent out of the two 2s listed.
Prime Factorization
Factor Trees and Upside Down Division
- When using a factor tree for prime factorization, we are breaking down a number into other numbers that we would multiply to in order to get that number.
- For example, if we needed to make a factor tree to find the prime factorization of 20, we would need to think of multiples of 20 in order to break it down. For this example, I will choose 5x4.
- We will then make two separate branches for 4 and 5. Looking at 5, we know that 5 is a prime number and cannot be broken down further than into 5 times 1, so we will circle it since it is done. However, with 4, we know that we can continue to break 4 down to 2 times 2.
- Making two separate branches for the 2s under 4, at this point we know that the 2s are prime numbers and cannot be broken down further than into 2 times 1, so we would circle them both.
- Since our tree is now completed, we must now right out what we have in order to complete our answer. So, from biggest to smallest numbers, we would write 2 x 2 x 5. To simplify this further would then write 2 to the second power x 5, and this concludes the prime factorization for 20 using the tree method.
- The upside down division (aka "ladder method") example is shown in the attached video provided.
Showing how to Solve +, -, x, and / Integer Problems with Diagrams
Grouping and Zero Banks
- When multiplying integers, we would read the problem as groups of a number. For example, we would read 5(-2) as 5 groups of 2 negatives.
- When showing this we would draw 5 circles, then add two negative signs in each circle.
- When counting up all the negatives we have, we would get -10. Therefore, 5(-2)=-10.
- With a problem with a beginning negative or both negative, we would instead draw out plenty of positive and negative signs in order to take away how ever many groups it asks for. For example, with -2(4) we would need to take away 2 groups of 4 positives. Drawing out 10 positive and 10 negative signs, we would take away 2 groups of 4 positive signs from our diagram, the zero bank cancels out the 2 positive and negative signs that are aligned with one another, leaving us with 8 negatives in our diagram.
- Therefore, -2(4)= -8.
- In division problems with integers, all we must remember is that when signs are the same the answer is positive, and when signs are different, the answer is negative.
- Say for instance, we have -6/-3. By making a diagram, we see that we have a -/-, therefore the answer is positive 2 because signs are the same.
- However, if we had -6/3, making a diagram again we would see that we have -/+, therefore the answer would be -2 because the signs are different.
"Hector's Method"
- Hector's Method was created for when we need to add or subtract large positive and negative numbers so we are not drawing out a bunch of positive and negative signed diagrams.
- We begin by looking to see which number in the problem is "bigger" excluding the sign in which it has. For example, in -20+16, excluding the negative sign the 20 has, we know that 20 is more than 16.
- Next, once we've figured out what the "bigger" number is, we will draw an extra + or - sign under the "bigger" number, where as the smaller number will only have one + or - sign under it. For example, looking back at our previous problem, since we indicated that 20 is the larger number, we would draw two negative signs since it is a -20 (- -), whereas under 16 we would draw one positive sign (+).
- Continuing, we will then circle one of each sign from the numbers we are adding or subtracting, leaving the left over sign that is outside of the circle to know if our answer will be positive or negative. If the signs in the circle are different we subtract the two numbers as is they were positive, and if the signs in the circle are the same we would add the numbers as if they were positive.
- Finally, using our example problem, since in the circle we would have a + and - sign, we would subtract 20 from 16 to get 4, then we would attach the left over sign that is outside of the circle (-), so that therefore the answer to -20+16 is -4.
White/Yellow Tiles Means Positive and Red Tiles Means Negative
- When using tiles to show the answer of addition or subtraction problems, it is important to remember that white/yellow means positive, and should always be the top row.
- Red tiles means negative numbers and should always be the row underneath the positive tiles.
- To understand an example of what this would look like, let's take for instance 3+(-5).
- You would begin by putting 3 white/yellow tiles on top, then 5 red on the bottom. The white/yellow and red pairs cancel each other out (zero bank) leaving us with 2 negative tiles left over.
- Therefore, the answer would be -2.
Weeks 4-5
Using Alternative Algorithms for Multiplication
Multiplication Made Easy
Base Ten Blocks, Area Model, Expanded Form, and Lattice.
- When multiplying using the Base Ten Blocks Model, similar to how we use our blue base ten blocks in order to create rectangles to find their area, we instead draw out a diagram of how a multiplication problem would look like if we used the blocks to figure out the area.
- To do this, we will use 13(12) as an example. We would begin drawing out a rectangle, and on the length and height of the rectangle we would put the numbers 13 and 12 in long and unit form. So we would draw one long labeled "10" and 3 units with dots on them to represent 13, for example. Finally, we draw the lines we have made for the longs and units into the rectangle to figure out how many units, longs, and flats are in our rectangle to thus know the area inside the rectangle.
- When using the Area Model, similar to how we draw a rectangle like how we do using the base ten blocks model, we will also be putting the numbers being multiplied in expanded form on the length and height of the rectangle, using the pluses from the expanded form numbers for where we draw our lines in order to create boxes inside the rectangle.
- In each box created, we multiply the numbers that match up with that space in order to find the answers for each box. If we were multiplying 70+5 and 50+6, we would have four answers in their four designated boxes.
- Lastly, we add up the top boxes and the bottom boxes answers together in order to receive our final answer.
- Using Expanded Form, similar to the left to right method in addition, we will first be putting the numbers being multiplied in expanded form, then multiplying the top left and bottom left together, then bottom left and top right, bottom right and top left, and finally the top right and bottom right together. Once we add all these numbers together, we receive our answer.
- Lastly, when using Lattice in multiplication, similar to how we add using lattice, we are again using boxes and diagonals to find our answer, however, what is different is that one number will be on the top of the square, and the other number will be written vertically on the right side of the square.
- Similar to the area model, we multiply the numbers that match up to that particular box, writing the numbers how we are used to, from left to right, with the tens diagonal slightly above the ones diagonal. Once finished completing the boxes, we add the diagonals in order to receive our final answer.
Using Alternative Algorithms for Addition
Timely, but Worth It
Expanded Form, Left to Right, Scratch Method, and Lattice.
- When using Expanded Form to add numbers, we simply break the numbers down to their hundreds, tens, and ones place, then add each column. From there, we simply add these columns together in order to receive our answer, while yet again not having to worry about carrying like in the traditional algorithm of addition.
- For example, let's take for instance 35+42. In Expanded Form, we would begin by writing 30+5 and 40+2. Then, we add each column: 30+40=70 and 5+2=7. Now adding from left to right, 70+7= 77.
- With Left to Right, it is a similar concept to expanded form, just without the break down of the hundreds, tens, and ones place like it is required in expanded form.
- For example, we could leave the problem 55+32 lined up how it would be in a traditional addition problem, but under the line drawn we would write what the answer would be from Left to Right as if it were seeing it in expanded form. So, we would write 80 (50+30), then 7 (5+2). When adding these numbers up, we would get 87.
- Scratch Method involves keeping count when adding each column of how many times we reach the number 10 (or base number) by scratching out the number that is exactly the base number or if it has gone over the number. If it has gone over, we minus the number it has gone over by the base number and put that answer next to the scratch.
- Before adding the next column we must count the number of scratches in the previous column and use that number to add over to the next column. This process will repeat until there is no more columns left in which the last number of scratches will be the beginning of the answer that we need to write.
- Finally, Lattice allows us to write our answers how we normally would (left to right) in boxes cut by diagonals, then adding up the diagonals in the end to get our final answer. An example of what this looks like will be provided in this topic's video.
Numbers are our Friends
Compatible/Friendly Numbers and Trade Off.
- When you have compatible/friendly numbers, this means when adding a large list of numbers, by looking at the ends of each number you can see if it is possible to give to a number in the list in order for it to move up to the nearest tens, so it will make the numbers easier to add.
- For example, if we have a list of numbers such as 34+12+26+43+18+27+11, we can see that by giving 4 from the number 34 to the number 26, the 26 will now become 30, and the 34 has also became 30 by getting rid of 4. Likewise, if we gave 2 from the number 12 to 18, 18 would now become 20, and the 12 would now be 10.
- Trade off has a similar concept, except that one of the numbers is trading from the amount it has in order to make the number it's being added to move up to the nearest tens.
- For example, if we were to add 36+28, 36 could trade 2 from its current amount (now becoming 34) in order to make 28 become 30.
- Both methods eliminate having to carry numbers over to the next column when going over 10 like we would see in the traditional algorithm of addition.
Downwards Division
LLLLLLeaving Base Ten
- Downwards Division is typically used for when we are leaving base ten and converting to a different base.
- There is emphasis on "LLLLLeaving" because unlike the traditional way in which we draw the divisor when getting ready to solve long division, in downwards division we create an "L" divisor putting the number given to us on the inside of the "L" and putting the base number we are converting to on the outside of the "L".
- For example, if we have the number 20 and we want to convert to base seven, 20 would go on the inside of the "L", and the number 7 would go on the outside.
- To solve, we would solve similar to regular long division, however, the remainder left over from subtracting goes to the right of the "L" in it's own column because the remaining numbers help us form our answer.
- You know you get the right answer when the number on the inside of the "L" is less than the base number.
- In order to form the answer, we write the number starting from left and working our way up. For example, if you have 3 left over and a remainder of 5 and then a remainder of 4 above that, we would read the number as 354.
Weeks 1-3
Changing Bases
Converting from Base Ten
- When converting from base ten, there is yet again a simpler way in which we do not to need to draw out, for example, 136 "treats" in order to find the number of "treats" there are in the other desired base.
- Likewise to how we previously learned how to convert to base ten, we can use skip counting as a way to convert from base ten. The following example will demonstrate how to skip count using the number 136, converting to base eight.
- Using what we know about the rules of "FLU", we know that in base eight, one long consists of eight units. With this knowledge, we can begin to draw a diagram, counting every long drawn equal to eight. So, for example, 8, 16, 24 and so on. Continue this process. If need be, write the number under the long that you are on.
- Next, when we have reached 8 rows of 8 longs, it is important to remember to then draw that group of longs into a flat, since we know that in base eight, eight rows of eight is equal to one flat.
- Continue counting longs (and remembering to group 8 rows of longs into a flat) until you have reached the number 136.
- By using this method, we should have two flats, one long, and zero units. Therefore, the answer would be 210 base eight.
- To check our math, we can do 64+64+8, and this would equal 136, which was our original base ten number before we began converting.
Converting to Base Ten
- When converting to base ten, especially when working with large numbers (i.e. 271 base nine), instead of drawing out 271 "treats" or circles, we can instead draw a diagram of the number using "FLU" in order to then do simple multiplication and addition to get to our answer. The following will be an example of how to use this method using 271 base nine as an example.
- Using our previous knowledge about "FLU", we know to draw two flats, seven longs, and one unit to represent 271.
- Because it is 271 base nine we know that one flat is equal to nine rows of nine, which equals eighty one. Because there are two flats, we would begin to write our equation as 9 squared plus 9 squared, or simply, 2(81)+__.
- Next, because we know that one long is equal to nine units in base nine, and there are seven longs, we can add to our equation: 2(81)+7(9)+__.
- Continuing, because we know that one unit is equal to one, and there is only one unit in the number 271, we can finish writing our equation by adding in: 2(81)+7(9)+1=__
- Finally, when first multiplying 2(81) and 7(9) then adding all the numbers together, our answer is 226. Because it is assumed when there is no base number written that it is automatically in base ten, our answer is simply 226.
Introduction to Working with Bases
The Smallest Base that Numbers Can Be In
- To figure out what the smallest base certain numbers can be in, we must first figure out what the largest number is.
- For example, take the number 15. The largest number in 15 would be 5.
- Next, add one to the largest number in order to determine the smallest base it can be in.
- For example, looking back on our previous number 15, since we have determined that the number 5 is the largest number, when adding 5 plus 1, we get 6. Therefore, the smallest base the number 15 can be in, is base six.
- More examples: Smallest base of 214 = base five. Smallest base of 25 = base six. Smallest base of 31 = base four.
Reading Numbers with the "FLU"
When reading a number, we must follow the "FLU" (starting from left to right) rule to determine how many flats, longs, and units there are in order to make a diagram representing this, or to help skip count to convert bases.
- For example, take for instance the number 123. Reading this number left to right and following FLU, we can determine there is one flat, since "1" is under the "F" place value for flats. There are two longs, because the number "2" is under the "L" place value for longs. Finally, there are three units, because the number "3" falls under the "U" place value.
- When making a diagram of this, we would draw a box to represent a flat, two lines to represent two longs, and three dots to represent three units.
Standards of Math Practice
Problem Solving Strategies
- Look for a Pattern.
- Do a Similar/Easier Problem:
- Identify What Makes the Problem Hard, and Get Rid of the Hard Part.
- Draw a Diagram (not Picture).
- Guess and Check!
- Write an Equation.
UnDev CarLO
There is a four step process to problem solving:
- Un-Understand.
- Dev-Develop a plan.
- Car- Carry out plan.
- LO- look back.
