MTE 280 Miltenberger

Bases can only have one symbol so for anything more than 9 we use 2 symbols.Example:89TEWIf we don't write the bases we assume it is in base 10

Units- 1Longs- 10Flats- 100The long names the base, however many units fit in a long tells us how big the base is.The numbers in each place value must be smaller than whatever base it is in.

When changing from base 10 to another base, rather than counting 1 2 3 4 5 6 7 all the way to say for example 56 instead be can go7 14 21 28 35 42 49 56 and know that we have 6 "longs" because to fill a long we need 7 units which we included by skip counting.

Noticing that there is a pattern

Using prior knowledge to recognize something

Know and count the numbers1 to 1 correspondenceCardinality: counting something they do not need to count10 frames often used, look like longs

Converting from base 10 we can draw it out by skip counting and using tallies, but if the numbers get too big then we can use downward division.

When you are leaving base 10 and going to a different base Ex.243 to base five5 L243__| 5 L48_ | 3 5 L9_ | 3 1 | 4The number would be 1433 base five

When converting from any base to base 10 we can draw it out or solve it with an algorithm.Ex. 321 base 6 to base 10FLAT FLAT FLAT LONG LONG UNIT3 flats 2 longs 1 unitOR3 (6^2) + 2(6^1) + 1(6^0)starting at the number farthest right, we use the exponent 0 and as we go up each place we add 1 to the exponentfor the number 236581 base 9 to base 10 we would do2(9^5) + 3(9^4) + 6(9^3) + 5(9^2) + 8(9^1) + 1(9^0)and find the sum

Zero pair:+_Start with any number you want and how ever many you needPositives on the topNegatives on the bottomGroup of zero pairs = zero bankIf the signs are the same the answer is positiveIf the signs are different then the answer is negative

-5 using 9 tiles+ +_ _ _ _ _ _ _8 using 10 tiles+ + + + + + + + +_4 + (-5)+ + + +_ _ _ _ _Keep Change Change12 + (-20)+ _ _ circle the positive and negative and subtract 12 from 20 and put a negative sign

First number is the # of groupsSecond number is how many is inside of the groupsCircles represent groups2(3) -- 2 groups of 3+ + + ++ +-2(3) + + + + + +_ _ _ _ _ _0 take away 2 groups of 3*circle 6 positive and you are left with 6 negatives = -6

Prime: 2 factorsComposite: More than 2 factorsThe # 1 is neitherWe use a factor tree for prime factorizationWe use PF to be able to find commonalities between numbers

Another way to find the prime factorization2 |_32__ 2 |_16__ 2 |_8__ 2 |_4__ 2 |_2_ 2^52 |_28__ 2 |_14__ 2 |_7__ 2^3 x 7

Greatest Common Factor: small #Least Common Multiple: big #18 2 |_20__3 6 5 |_10__2 3 22 x 3^2 2^2 x 5GCF = 2LCM = 2^2 x 3^2 x 5

3 <------ how much you have_____ 7 <------ size of the piece Denominators are counter intuitive

7/13 > 11/23 emphasizes 1/2

6/11 > 6/13

13/14 < 16/17

Draw same size boxes for each fraction1 box vertical and 1 box horizontal (lines)Transpose and make them the same size by adding the lines vertical and horizontal to each(addition, make 3rd box and add all pieces)(subtraction, take away however many boxes by writing and x)then draw answer and writeDrawing gives us a common denominator

The most important thing when teaching someone and something is making connections that can apply to other aspects of life, problems, classes etc.

Start with a manipulativeDraw representation (show) Move to algorithms (algebra)

A method to use when adding by trying to find numbers that add to 10 or result in so many longs and no units left over. Numbers that are compatible and equal to a tens place.Ex:32 + 16 + 48 + 11 + 34 + 44 + 4530 + 20 + 50 + 10 + 40 + 50

Expanding the numbers to be able to see place value.Ex. 428 400 + 20 + 8+ 56 50 + 6----------- ---------------- 484 400 +70 + 14 484

Taking expanded to the next level and writing out place value but doing it from left to right skipping a line every time you go to a new place value.Ex: 572+ 324---------- 800 90 6---------- 896

Putting numbers in a vertical line and borrowing from one of the numbers to make 0 in the ones place and add 1 to the tens place.Ex. 48 +2 50 +27 -2 + 25---------- ----------- 75 75

A model we use to show place value and allow a student to draw out every flat, long, and unit that makes up a number when multiplied together.

Based off of the base 10 blocks model but is quicker and more efficient not making us have to break down every single unit and or long to get an answer.Ex.13 x 15 10 + 3 __________________ | | 10 | | | 100 | 30 100 + |_________|________ 30 | | 50 5 | 50 | 15 15 |_________|________ -------- 195

-To show students the fundamentals of adding or multiplying then can teach them short cuts.-Some algorithms might be easier or quicker for certain students.-Some may not be as efficient-In some situations, using one algorithm versus another can be more useful or helpful in a certain situation.

Efficiency and Place Value

(3+4) + 7 = 3 + (4+7)

3 + 4 + 7 = 3 + 7 + 4

3 (x + 4)

4 15____ x _____ 9 222 x 2 3 x 5 2 x 2 x 3 x 5_____ X ______ = ____________3 x 3 2 x 11 2 x 3 x 3 x 11 You draw a long 1 across through numbers that would equal 1. So a 2 in the numerator and a 2 in the denominator would be 1 so you you'd draw a long funky through those numbers, along with the one 3 in the numerator and one 3 in the denominator.

Emphasizes place value requires student to be able to estimate though.243 / 624 -24 = 0 243 3_______ = 40 _____ 6 6

BAD worst way to dividenot even worth putting info about

G roupsE xponentsM ultiply/ D ivide ----> A ddition/S ubtraction ------>M/D and A/S move left to rightIn any problem find the + and - Draw lines going down which break up the problem and allows you to do G, E, M/D. Then add left to right

Line up the whole numbers, add 0's so it will still always work (subtraction).Always shown a rectangle with 10 longs in it.

2 x 3 = 62.14 x 3.6 = 214x 36_______7704 we know that the whole numbers multiplied together equals 6 so the most realistic answer we could do is 7.704

2 - Last # is even3 - Sum of digits can be divided by 34 - Last 2 numbers can be divided by 45 - Last # is 0/56 - If 2 and 3 work then six works8 - Last 3 numbers can be divided by 89 - Sum of digits can be divided by 910 - Last number is 0Groups(2, 5, 10)(3, 9)(2, 3, 6)(2, 4, 8)

80% of 4010% = 4x 8 x 8 __________80% = 32 100%-70% discount off of $80_____30% of $8010% = 8x 3 x 3____________30% = $24