Number Sense and Algebra Mid-Term Exam Review

Order of Operations Remember, just copy down the text in black

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 = = 4 X 8 = 14 = 32 OR A pre-defined “Order of Operations” will allow everyone to always agree on the correct answer….

B E D M A S rackets xponents ivision ultiplication ddition ubtraction Left to right

1. Simplify the following a)(4)(5) + 8 Multiply first = = 28

b) 7 + (9 - 1) X 5Brackets first = X 5 = = 47

c) 10 + (12 4) = 10 + (3) = 13 d) 15 – 10 X 2 = = -5

e) 11 – X 7 = 11 – = f) (4 + 7) – 25 5 = 11 – 25 5 = = 20 = 6

The Algebra of Construction

The process of modeling is extensively used throughout all construction and engineering…

Playing with this…

Can lead to building this…

Operating on a model is better because: 1. It is easier 2. It is cheaper 3. It is faster 4. It is safer

You need a way to keep track of everything you need (here comes the algebra….) f thingalgebra h

You can design any kind of yard you like…

f f f2f2 Backyard enclosed by all fence

fh f h Using both

The Concept of a Polynomial

Remember A variable represents something f: length h: another length f 2 : 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…)

Simplify: 3f + 7h + 2f = + 7h -2a + 7b + 3a + 9a =10a 5f + 7b 4x + 7y + 3x + 9y 2 = 7x+7y +9y 2

For example: 2(4x + 3y) = 8x+ 6y

For example: 3(5a + 6b) = 15a+18b

3(1a + 4b) + 2(a + 7b) = 3a+ 12b+2a+ 14b = 5a+ 26b

One Step Equations

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). (LS=RS) 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… = 4

x + 4 = 10 x = m = 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 3 You must therefore impose this operation to both sides of the equation zero

C = X X C = y = -5 y =- 3 +2

Two-Step Equations

1. Solve for x (isolate x) 2x + 6 = x = 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.

Solve for x (isolate x) 7x - 1 = x = x = 4

2(x + 3) = x = x = 4 2x + 6 = 10 Check this answer !!!

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 2m -8 = = +8 2m = m = 6 Now we are back to a two step

Solve the Following: 7p + 6 = p -7p +6 = 16 +2p -16 = = 2p 2 -5 = p For convention, eliminate the smaller variable term first… (7p is smaller than 9m) Now we are back to a two step

Ratios

Ratios should always be reduced: 4 : 12Divide both sides by4 1 :3 Demonstrate with people

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 ?

Which is the better deal? 210g of popcorn for $1.00 or 750g of popcorn for $2.70

Proportions You 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 = h =

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