MTE 280 Elementary Foundations

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MTE 280 Elementary Foundations

Week 12

Order of Operations

Dividing Fractions

Dividing Fractions

when dividing fractions, always KCF (keep, change, flip)

• also remember to use the funky 1's as seen in multiplying fractions to reduce to limit mistakes

example: 3and3/5 divided by 1and2/10

• first convert numbers to a mixed numbers to get 18/5 and 12/10
• then find out the multiples
• 18 = 3x6
• 10 = 2x5
• 12 = 6x2
• the two 6's and two 5's cancel out so you're left with 6/2 which = 3

remember: when converting to mixed number, use "backwards c"

Week 11

Multiplying Fractions

Multiplying Fractions:

No different than normal multiplication

Example: (3/4)(2/3)=6/12=1/2

Makes the problem much more simple and less prone to simple math mistakes if you try to reduce/ simplify the fractions before multiplying

Example: (24/35)(21/40)

can be simplified by doing the funky "1s" example from class

24=6x4 and 40=4x10 so the 4's can be crossed out

35=5x7 and 21=7x3 so the 7's can be crossed out

then you're left with (6/5)(3/10)=18/50=9/25

Prime factorization and factor trees

Prime Factorization: finding which prime numbers multiple together to make the original number

Example:

90

9 10

3 3 5 2

so... the prime factors= 2x3x3x5

Divisibility Rules, LCM and GCF

Divisibility Rules

2: even #s

3: sum of digits divide by 3

4: last 2 digits divide by 4

5: any # ending in 5 or 0

6: even and sum of digits divide by 3

8: last 3 digits divide by 8

9: sum of digits divide by 9

10: any # ending in 0

Least common multiple

• bigger number and shared multiples
• find prime factorization then multiply the prime factors together

Greatest common factor

• what the two prime number groups share then multiply
• smaller number

Example: GCF and LCM of 60 and 45

• prime factorization for 60=2^2, 3, 5
• prime factorization for 45=3^2, 5

so...

• LCM=3x5=15
• GCF=2^2x3^2x5=180

Week 10

Adding Fractions

Adding Fractions

Tip: if you're given 5+3/8, all you have to do is add it, not convert 5 into a fraction

• 5and3/7 + 4and1/7 goes to be 5+4 then 3/7+1/7 because they have the same denominator

25-4and3/16 is more complicated

• 25-4=21 but then you have to take 1 out and make it a fraction
• so then you have 20and16/16-3/16
• then you get 20and13/16

What fractions mean:

3(5) = 3 groups of 5

3(2/7) = 3 groups of 2/7

(1/3)12 = one shirt of a group of 12

Week 9

Fractions Continued

More Fractions

Important tip:

• the fractions have to be equal to add them correctly, but not to multiple or divide

Ways to show fractions example: 4/6

set model: xxxx~~

area model: [] [] [] [] [] []

linear model: -+-+-+-

-+-+-+-+-+

Intro to Fractions

Intro to Fractions

which fraction is larger?

4/7 or 5/7

• whichever fraction in closer to "1" is larger
• think of it like pieces of a pie

5/7 pieces is more than 4/7 because it is closer to 1

5/8 or 5/9

• 5 out of 8 pieces of pie is more then 5 out of 9 pieces of pie

Easy tip: when doing a problem like 32+8/11

it equals 32and8/11 so don't make it more complicated

numerator= # of pieces

denominator= size of pieces

Week 8

Subtracting Integers

Subtracting Integers

KEEP, CHANGE, CHANGE

Show:

-5-(-2) = - - - - - take away 2 = -3

-4-2 = - - - - - -

take away 2 ++ = -6

Making zeros: a - and a + go together to create a "zero"

Solve:

-35-(-15)

K C C = -35 + 15 = -20

(use Mr. Kilt's student's algorithm shown in adding integers)

5-9

+ -- so it's going to be a negative number = -4

Adding Integers

Adding Integers

Tip: when talking to students about whether the positive or negative number is bigger...

• not "which is bigger" should be "which one would have a bigger tile pile"

Example: 3+(-5)

• 3 is the "bigger" number because it is positive, but for our purposes -5 is actually bigger because it would have the bigger pile of red tiles then 3

SHOW: 3+(-5) Mr. Milt's student's algorithm

+++

_ _ _ _ _ = -2

(there is a bigger pile of negatives, so the answer must be negative)

SLOLVE: 24+(-35)

24 + (-35) = -9

+ - - so...it will end up being a negative number

Week 7

Alt. Algorithms for Subtraction

Alt. Algorithms for Subtraction

Subtraction = the distance between two numbers

show using longs and units

24-12 will go to be two longs and 4 units...then you'll take away one long and 2 units to find the answer

24-18

34-28 =6 the same distance between 2 #

33-27

show using tiles

show 5 using 9 tiles

+++++++

_ _

show 5+(-4)

+++++

_ _ _ _ = 1

show -3(-2)

_ _ _ _ _ = -5

Alt. Algorithms for Division

Alt. Algorithms for Division

Tips:

• don't say "divided by", say "goes into"
• spend the first day of lessons showing students simply how to set up problems
• doing multiples that students already know and are comfortable with is the best way to start

Repeated Subtraction

Example: 146/8

• see notes for how to set up
• start by slowly subtracting multiples of 8 and that will go off to the side
• it allows students to practice their multiples
• has a margin for error, but is a good way to introduce division

eventually you'll take 8 away a certain amount of times until you find the answer

Alt. Algorithms for Multiplication

Alt. Algoritms for Multiplication

Multiplication = Area

Area/base 10 block expanded form

example: 27(36)

20+7

30+6

------- then you multiple each number by its vertical and diagonal counterpart

= 600+210

=120+42 then add them all together

Array Multiplication

Example: 3(4)

o o o o

o o o o

o o o o =12

Lattice Multiplication

Example: 25(15)

• see notes for set up
• create box with lattice lines then multiple each box
• after than add the lattice lines (diagonally) then you'll get your answer

= 375

Week 6

Exam

81/82 on exam 1

Exam Review

see notes

Week 5

Multiplication

Multiplication

1st number: # of groups

2nd number: what is inside the groups

What order to teach times tables in:

teach first:

1s

2s

10s

5s

Teach second:

3s

4s

9s

Teach 3rd:

everything else

MULTIPLICATION = AREA OF A RECTANGLE

showing using base 10 blocks:

• draw a square for 10x10
• add sections and lines for each additional unit

example: 14(12)

have a 10x10 flat then have 4 units added on one side and 2 on the other than add everything to get the answer = 168

Multiplying with Alt. Algorithms (show/ solve)

Introduction to Alt. Algorithms for Multiplication

Area Model:

• somewhat like expanded form addition

Example:

24(28)

20 + 8

20

+

4

• multiple each number by its boxed counterpart (20x20 and 20x4, 20x8 and 4x8)
• then add each row to get the answer

this helps reinforce place value and multiplying with numbers ending in 0

Week 4

Writing Problems

How to Write Problems

Show:

convert from base ten to other bases

• 34 to base 6

convert from other bases to base 10

• 207eight to base ten

Solve:

convert from base ten to other bases

• 33 to base four

convert from other bases to base ten

• 3,425six to base ten

Alt. Algorithms for Addition

Alt. Algorithms for Addition

Friendly numbers:

• adding/subtracting to make both numbers end in zero

example: 28+62 --- 30+60 = 90

Trade off:

• adding/subtracting to make one number end in zero

example: 46+25 --- 50+21 = 71

Left to right:

• adding from left to right instead of right to left
• reinforces place value

example: 37+42 --- add do 30+40 then add the 7 and 2 = 79

Expanded form:

• expanding each number so they're multiples of 10
• reinforces place value

example: 748+165 --- 700+40+8 +100+60+5 = 913

Scratch:

• scratch when numbers make up the base
• add remainders then carry to the next column
• helps students add multiple numbers without having to keep track of everything in their heads

example: 34+12+15

you would scratch when the ones place adds up to 10 then carry over

Week 3

Adding and Subtracting with Different Bases

Week 2

Counting

Counting

Counting and how to works in different bases

• practice the bizz/buzz game to practice how to count
• helps with counting in different bases

Basic counting tools

• 10 frames
• flats
• longs
• units

One to one correspondence

• difficult for younger students to start to comprehend
• it is easy for children to count from 1-10, but they don't know what that means and how it is represented
• practice one-to-one counting to get this idea in their head

Converting Bases

Converting Bases

Diagrams

• using longs and units to represent numbers
• one long equals whatever base you're in
• if there are the number of long that equal what base you're in, it turns into a flat
• flats = 100

Examples:

24six --- 2 long and 4 units

each long = 6 and each unit = 1

so...6+6+4=16

13 to base eight --- 1 long and 3 units

each long = 10 and each unit = 3

so...13 will make 1 long and 5 units = 15

Week 1

Syllabus Video

Class expectations and grading

(see syllabus in canvas)

Intro to Class

Juggling Mr. Milt