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 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….

BEDMAS

rackets

xponentsivisionultiplication

ddition

ubtraction

Left to right

Left to right

1. Simplify the following

a) (4)(5) + 8 Multiply first

= 20 + 8

= 28

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

= 7 + 8 X 5

= 7 + 40

= 47

c) 10 + (12 4)

= 10 + (3)

= 13

d) 15 – 10 X 2

= 15 - 20

= -5

e) 11 – 5 + 2 X 7

= 11 – 5 + 14

= 6 + 14

f) (4 + 7) – 25 5

= 11 – 25 5

= 11 - 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

thing algebra

h

You can design any kind of yard you like…

f

f

f2 Backyard enclosed by all fence

fhf

hUsing both

The Concept of a Polynomial

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…)

Simplify:

3f + 7h + 2f = + 7h

-2a + 7b + 3a + 9a = 10a

5f

+ 7b

4x + 7y + 3x + 9y2 = 7x +7y+9y2

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….

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

C = -26

6 X X 61

1

C = - 12

-2 + y = -5

y = - 3

+2 + 2

Two-Step Equations

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.

Solve for x (isolate x)

7x - 1 = 27 +1 +1

7x = 287 7

1

1

x = 4

2(x + 3) = 10

2 2

1

1x = 2

-6 -6

2x = 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 -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

Ratios

Ratios should always be reduced:

4 : 12 Divide both sides by 4

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 or750g of popcorn for $2.70

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=

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