Alphabetical order for organizing
A,B,C…
Imagine if the 26 letters of the alphabet had no regular order…
The same holds true for math.
The order in which we operate on an expression has
a significant impact on the result!
Which one should be correct?
4 X 2 + 6 4 X 2 + 6
= 8 + 6 = 4 X 8
= 14 = 32
OR
A pre-defined “Order of Operations” will allow everyone to always agree
on the correct answer….
Operating on a model is better because:
1. It is easier
2. It is cheaper
3. It is faster
4. It is safer
RememberA variable represents something
f: length
h: another length
f2: could be an area
fh: could also be an area
To simplify an expression, combine only like terms
20A + 26C = 20A + 26C
The money can not be combined because they are of different currencies (they are not like terms…)
• An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). (LS=RS)
is exactly the same as….
is exactly the same as….
• An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent).
is exactly the same as….
2 + 2 4
=2 + 2 4
x + 4 = 10
x = 6
- 4 - 4
3m = 153 3
1
1
m = 5
To isolate x, eliminate the +7
The opposite of add 7, is subtract 7
You must therefore impose this operation to both sides of the equation
To isolate m, eliminate the 3 The opposite of
times by 3, is divide by 3You must
therefore impose this operation to both sides of the equation
zero
1. Solve for x (isolate x)
2x + 6 = 16 -6 -6
2x = 102 2
1
1
x = 5
Most equations reduce to this form. This is our favoured form.
Once here, do the subtraction step first. Leave the division until the end.
It is possible to have a variable term on both sides of the equal sign.
We solve this equation using the same techniques we have used so far…
Multi-Variable Equations
1. Solve the Following:
7m - 8 = 5m + 4 For convention, eliminate the smaller variable term first…(5m is smaller than 7m)-5m -5m
2m -8 = +4 +8 = +8
2m = +12 2 2
m = 6
Now we are back to a two step
Solve the Following:
7p + 6 = 16 + 9p-7p -7p +6 = 16 +2p
-16 = -16 -10 = 2p
2 2 -5 = p
For convention, eliminate the smaller variable term first…(7p is smaller than 9m)
Now we are back to a two step
Remember, ratios must be in the same units
Write as a ratio -
1 minutes to 30 seconds
This is not 1 to 30
Convert 1 minutes into 60 seconds
Rates compare numbers of different units:
• 100 km / h (401)
• $50.00 / week (allowance)
• $12.00 / person (movies)
Ratios can be used to analyze information
• 75 hits in 225 bats
Consider 2 baseball players:
• 180 hits in 270 bats• Who is the better hitter ?
ProportionsYou build the bike jump below.
The ramp needs to be 5 times longer than the height. If the ramp is 4m long, what should the height be?
Length (4m)
height
length
height=
5
1
4
h=