Fundamentals of Elementary Math
RATIOS
PRACTICE
Molly bought six heads of cabbage for 9.30. If willie buys 22 heads at the same price per head, how much will it cost him?
6 l 3 l 12 l 2 l 10 l 22
____________________________
9.30 l 4.65l 18.60 l 3.10 l 15.50l 34.10
One fruit basket is 15$
How many fruit baskets can you buy with 135$?
1 l 2 l 10 l 5 l 9
__________________
15 l 30 l 150l 75 l 135
The proper way to write ratios is with a colon, or to write it out
2 boys, to 34 girls
2:34
or
2 to 34
DIVIDING FRACTIONS
Dividing with common denominators is very simple, because the denominators cancel out, and you are just left dividing the top numbers.
SUBTRACTING FRACTIONS
<---4&3/4---5----------------7-----7&1/3--->
__ __________ ____
1/4 2 1/3
1/4+2+1/3
2+ 3/12 + 4/12
2 7/12
Finding the distance between two fractions can give you the answer without using dificult subtraction!
7 1/3 - 4 3/4
ADDING FRACTIONS
Decomposing the numbers, to make a whole
(You find numbers you can break out of the second one, to get closest to a whole)
1 3/10
3/5 +(4/10+3/10)
3/5 +7/10
MULTIPLYING FRACTIONS
The correct way to "cancel out"
5 3
_ X _
7 5
3X7
___
7X5
*Use associative property of multiplication to switch the numbers
3X7 3 7
___ --> __ X __ (1)
5X7 5 7
EQUALS
3
_
5
Miah had 4/6 of a pan of browines left, and Sarah wants to buy 3/4 of what's left
______________________
l XXXXXXXXXXXXXXXXXXXl
l XXXXXXXXXXXXXXXXXXXl
l XXXXXXXXXXXXXXXXXXXl
l XXXXXXXXXXXXXXXXXXXl
l l
l_____________________l
X= 4/6 of what is left
X= 3/4, what Sarah bought
Through drawing a picture, we were able to quickly realize that Sarah bought 1/2 of the pan of brownies.
This technique proved to be very effective when multiplying fractions
PERCENTAGES
Using number lines
100% of 72 = 72
50% of 72= 36
25% of 72= 18
75% of 72= 54
10% of 72= 7.2
5% of 72= 3.6
15% of 72= 10.80
20% of 72= 14.40
40% of 72= 28.80
Finding simple percentages is easy, by just making something similar to a ratio table.
EXPANDED FORM
12345.6789
(1X10)^1+(2X10)^0+(4X10)^-1+(7X10)^-2
One is always the balance point!
10^4 10^3 10^2 10^1 10^0 10^-1 10^-2 10^-3 10^-4
1 2 3 4 5 . 6 7 8 9
10000 1000 100 10 BALANCE 1 1 1 1
_____ ___ ___ __ __ ___ ____ _____
1 1 1 1 10 100 1000 10000
EXPANDED FRACTIONS
12.47
10+2+.4+.07
1342
1000+300+40+2
To further expand it you can..
(1X10)^3+(3X10)^2+(4X10)^1+(2X10)^0
In expanded form, every single number is represented seperately
FRACTION SENSE
Tell which fraction is greater
3/7 < 5/8
1/24 < 17/32
3/8 < 4/10
6/7 < 8/9
We solved the more complex ones through placing them on number lines
We found a lot of these answers through benchmarks
Part-part-whole: Knowing 3/4 can be broken into parts; 1/2+1/4, 1/4+1/4+1/4+1/4
We went into further detail of this and related it to decomposing.
How can 7/8 be decomposed?
7/8+3/8
7/8+ (1/8+2/8)
8/8 + 2/8
1 whole and 2/8
7/8+0
6/8+1/8
5/8+2/8
4/8+3/8
Benchmarks: 0, 1/2, 1, 1&1/2. Is it above or below 1/2? How far away is it from 1?
One/Two (Units) more and less: If I have 5/4 what is 1/4 more or less? What is 2/4 more or less?
*As long as we're in the same unit, the bottom stays the same
Spatial Relationships: have a picture of the number, including where it lies on the number line
FRACTIONS
This week we started with different ways of representing things, I.E.; Write/Draw everything you can that represents 3/4.
Many of us drew circles and rectangles, the rectangle proved to be the best choice.
Mixed Numbers
How many wholes can you make?!
47/36
You can find the mixed number by
36/36+ 11/36= 47/36
36/36= 1 Whole
11/36
1 and 11/36 is your mixed number.
Numerator: the NUMBER of pieces
Denominator: the DENOMINATION (size) of pieces
We worked with wooden shapes to practice and further learn about wholes and fractions.
Do NOT say 1 out of 5
DO say 1, one fifths
Fractions are out of one WHOLE thing, not multiple!
What is a "Unit?"
A Unit is a Whole.
A whole can be 1/2 of a banana,
or multiple eggs.
Partitioning
Splitting the whole into equivalent parts
Unit Iteration
Consistently repreating a unit to build a whole
We then did a problem on how to find a whole, when you know a fraction of it. For example, is 6 stars is 3/5, then how many stars are in 1 unit? (5/5) In this problem, 10 stars is a whole.
We talked about a problem about a case of coke, the case of coke may be looked at as a whole in many different ways, this was the case in this problem.
It could be stated that "I took 18 home", in which case the viewer considered each individual can a whole.
It could be stated as "I took 1 and 1/2 home, refferring to the rows of coke as each whole. There are many different ways of viewing a whole, and none of them are incorrect.
