Answer Key and Notes

Intermediate —Part I

Aquaculture, Welding, Welding Fabricator

Lesson 2 — Equation Solving I 5

Lesson 3 — Applications I 7

Lesson 4 — Evaluate Polynomials. 9

Lesson 5 — Ratio and Percent 11

Lesson 6 — Geometry – Terms. 13

Lesson 7 —Triangles. 15

Lesson 8 — Perimeter, Area and Volume. 17

Lesson 9 — Transforming Formulas. 19

Lesson 10 — Review & Test 21

Course Organization

Each lesson covers a different topic. The topic is introduced followed by some basic examples of the operations in a table like the following where room is left for notes. You should make notes here as you go through the material.

	Answers and Notes

The picture of the book is next to the text reference. Our texts have plenty of practice, however, if you need extra practice please ask.

Primary Text: MAP 360 — Stein’s Algebra in Easy Steps by Edwin Stein, Allyn and Bacon

Key points are provided here: hints, potential problems and special things to memorize.

We like to provide some examples of what we expect you to be able to do by the time you finish all of the practice. Remember, this is a mastery course and we expect you to achieve at least 80% on tests before moving on to the next section.

Alternate texts and internet links are provided here. These extra resources are helpful but they are not required.

Secondary Texts: MAP 903 — Mathpower Eight, McGraw-Hill Ryerson; MAP 904 — Mathpower Nine, McGraw-Hill Ryerson

A test follows lesson 9 the review lesson. We expect you to achieve at least 80% on the test.

Lesson 1 —Signed Numbers

In this lesson, you will review operations with signed numbers. The problems below are representative of basic problems that you might see.

	Answers and Notes
(+5) + (+12) =	17
(+12) + (–15) =	-3
(–5) + (+10) =	5
(–3) + (–22) =	-25
(+5) – (+30) =	-25
(+5) – (–42) =	5+42=47
(–30) – (+60) =	-30+-60=-90
(–5) – (–75) =	-5+75=70
(3)(5) = ( this is multiplication)	15
(–7)(8) =	-59
(10)(–5) =	-50
(–6)(–4) =	24
	4
	-4
(30)/(–10) = ( / means divide)	-3
	3

In MAP 360 read sections 2-10 to 2-14 (sec 2-1 to 2-9 are background) and do: page 68 # 1-7; page 77 # 14-17, 25;

page 86 # 11-13; page 90 #11 & 12.

Try to construct a real-world view of these operations. For example (–10) – (20) can be viewed as: the temperature today is 10 below zero and overnight it goes down by 20 degrees. What is the temperature in the morning? The more you are able to do this the easier it is to keep the operations straight.

After completing the material you should be able to perform operations involving signed numbers with relative ease. If you are not comfortable with these problems then ask for further practice.

–12 – (–15) = 9 + 3(–5) – 7 = (–5)(–7)(-2)4 =

3 -13 -280

http://amby.com/educate/math/integer.html — some good explanations with online practice. (Uses a “math party” to explain multiplication)

http://www.math.com/school/subject1/lessons/S1U1L10GL.html — makes heavy use of the number line.

Lesson 2 — Equation Solving I

This lesson will focus on the principles of solving equations. Below are a number of problems that represent some of the equation solving techniques you will need to master.

		Answers and Notes
x + 5 = 12	subt 5 from both sides	x=7
x – 4 = 7	add 4	x=11
5x = -20	div by 5	x=-4
	mult by 3	x=-6
	mult both sides by 3/2	y=27
2x + 3x + 4 = –6	5x=-10	x=-2
	3/4 x = 21	x=28
7x = 63 – 2x	9x=63	x=7
5t – 12 = 8t + 2t	5t - 12 = 10t -5t = 12	t=-12/5 or -2.5
–2x + 7 – 8x – 8= –x + 6 + 2x – 7	-10x - 1 = x - 1	x=0

In MAP 360, read sections 6-4 to 6-8 (sec 6-1 to 6-3 are background) and do: page 212 # 1-4; page 215 # 1-4;

page 219 # 1-6; page 222 #1-6; page 227 # 1-27 (a)(c)(e) then do # 28 to 31 as a self test.

Learn the correct order for performing the operations: collect like terms; add or subtract to place all the variable terms (the ones with the letter) on one side and the numeric terms on the other; divide to solve.

After completing the material you should be comfortable solving equations like:

3m – 5m – 12 = 7m – 88 – 3

m = 8.78

http://www.sosmath.com/algebra/solve/solve0/solve0.html the first five of these

http://www.redcomet.org/Preview/Mpa5bPrv.html very detailed explanation. Includes exercises with answers.

http://www2.shastacollege.edu/cberisso/a101/a101ch02.htm colourful and includes percent and ratio.

Lesson 3 — Applications I

In this lesson you will look at the fundamentals of applying your equation solving skills by solving problems. You can probably solve many of these problems intuitively; however, we want you to learn to structure your problem-solving the same way mathematicians do:

step 1: name the variable;

step 2: write an equation;

step 3: solve the equation;

step 4: draw a proper conclusion.

EXAMPLE: The sum of three consecutive even numbers is 60. Find the numbers

SOLUTION

Step 1: Let the first number be x. The second x+2 and the third x+4

Step 2: the sum of the numbers is 60

x + (x+2) + (x+4) = 60

Step 3: 3x + 6 = 60 (collect like terms)

3x = 54 (subtract 6 from both sides)

x = 18 (divide by 3)

Step 4: \the numbers are 18, 20 and 22

In MAP 903, read the examples on page 190 and 191 then do questions # 1 to 7.

The key to applications is to carefully translate the problem into mathematical symbols. Remember to follow the four-step problem solving process.

After completing the material you should be comfortable solving problems like:

· When you finish making 12,000 mL of wine, you plan to put it in 75 mL bottles. How many bottles will you need?

12000 ¸75 = 160

· A farmer purchases 5 sacks of pig feed at $8.00 a sack, one sack of chicken scratch at $7.95, and ten bags of laying pellets. The total bill was $132.95. How much did one bag of laying pellets cost?

let the cost of pellets = p

5(8) + 7.95 + 10p = 132.95

p = $8.50

Additional questions are available from the MathPower “blackline” masters. — ask an instructor

http://www.purplemath.com/modules/translat.htm – Purplemath (excellent teaching and examples)

http://algebrahelp.com/lessons/wordproblems/basics/ – Good explanations

http://www2.hawaii.edu/suremath/lessonPlans/lesson1.html an interesting approach

Lesson 4 — Evaluate Polynomials

In this lesson, you master evaluating polynomials for given values of the variable. You must first master exponents and radicals (square roots).

	Answers and Notes
as an exponent	2⁵
3⁴ = 3x3x3x3	81
(exact answer)	7
(answer to 2 decimals)	7.28
For x=7 and y=–3 evaluate:	7² - 6(7) + (-3)² = 49 - 42 + 9 = 16
For x=2 and y=–1 evaluate:	2²(-1)² - 6(2)(-1)² + 2⁴ = -4 - 12 + 16 = 0

In MAP 360, read section 1–4 and do questions 1–5 on page 21. Work through chapter 3 doing the diagnostic tests and, where necessary, completing the related practice examples. Make sure you can do the exercise at the top of page 110.

Remember that when you evaluate x² for x= –3 you get +9 for an answer. If you get a different answer, then you don’t know the difference between –3² and (-3)². You might have to ask for help on this

Note the difference:

After completing the material you should be able to evaluate the following:

= 1 =2 =3 =4 =5 =6

=7 = 8 =9 =10 =11 =12

the above should be memorized

Evaluate: for x=-3 and y=-1

= (-3)³ - 4(-3)²(-1) + 3(-1)⁵ = -27 + 36 - 3 = 6

MAP 903 Page 4 & 5 (exponents)

MAP 903 Page 29 to 30

http://www.purplemath.com/modules/evaluate.htm good examples (some harder ones too) `

Lesson 5 — Ratio and Percent

A number of applications in trades involve ratios or percents. This section reviews and deepens concepts mastered in the Foundation part of this course. The emphasis here is to solve problems using ratio and percent.

	Answers and Notes
What is 37% of 70?	25.9
45 is what percent of 82?	45/82 x 100 = 54.9%
25 is 20% of what number?	25/N = 20/100 N=125
10 is to 8 as 25 is to what number?	10/8 = 25/x x=20
Solve	7N = 84 N=7
Solve	15N = 7(45) N=21

In MAP 360, do page 261 # 2 to 8(read section 7-4 for examples); read through the examples on page 590 to 592 and do page 593 # 3-7, 12, 14, 20 (a o n), 21, 22 .

For ratio problems cross-multiply. For example if you have to solve then multiply “across” by multiplying 52 x 7. When you get your answer then divide by “the odd one out” which is 13.

Working with ratios is central to most trades. If you have troubles here get some help.

One trick for percent problems is “is over of equals percent over 100”. . For example “25 is 20% of what number?” Using the is-of-% method 25=is; 20=%0 and x=of.

. To solve this cross-multiply and simplify.

After completing the material, you should be able to solve real problems like:

· In a 12 cm by 18 cm photograph a man is 5 cm high. How high will he be if the photo is enlarged to be 20 cm wide? 8.3 cm

· The blue-prints call for a measure of 82.237 inches. Convert 0.237 inches to 64^th of an inch. 15/64

· 14% of a load of salmon have flukes. The load has 2,327 fish. How many have flukes? 326 (to nearest fish)

· A swimming area sample has 32 ppm (parts per million) e-coli bacteria. How many gallons of e-coli will be in a tank of 3000 gallons? 0.096 gal

· A model of the building you will work in is 18 cm wide and 10 cm high. If the finished building will be 7 m high, how wide will it be? 12.6 m

http://www.math.umd.edu/~jnd/Difficult_Word_Problems.html This teaches solving ratio problems using a unique method (similar to the one I learned in grade 7). It works. Highly recommended.

http://www.purplemath.com/modules/ratio.htm another good explanation.

Lesson 6 — Geometry – Terms

Student notes depend on what the student needs to remember.

	Notes
Line
Angle
Triangle
Polygon	many sides
Quadrilateral	any four sided figure
Circle
Radius	half the diameter
Arc
Diameter	double the radius
Parallel lines
Intersecting lines
Straight angle	180°
Right angle	90°
Acute angle	less than 90°
Obtuse angle	more than 90°
Supplementary angles	two angles that add to 180°
Complementary angle	two angles that add to 90°
Vertically opposite angle	these are always equal

In MAP 912, go through section 1 doing the practice exercises. You may omit the section titled “7. Corresponding Angles on parallel lines”.

After reading the material you should be familiar with all of the terms listed above and be able to: measure angles—with a protractor; name angles using the proper naming rules;

This is ÐABC or ÐCBA. Note that the B must be in themiddle.

identify obtuse, right, reflex and acute angles; identify vertically opposite angles, supplementary angles and complementary angles.

http://www.mathleague.com/help/geometry/basicterms.htm — basic terms

http://www.mathleague.com/help/geometry/angles.htm — angle and angle terms

http://ccins.camosun.bc.ca/~jbritton/jbescher.htm to learn about the geometry behind this:

Lesson 7 —Triangles

You will put your own notes here.

	Notes
Acute	all angles are less than 90°
Right	one angle is 90° (the other two add up to 90°)
Obtuse	one angle is greater than 90° (can two be greater?)
Equilateral	all sides are equal (this means that all angles are equal and 60°)
Isosceles	two sides are equal (all equilateral triangles are also isosceles)
Scalene	no two sides are equal

Read MAP 912, section 2 and do the practice in parts 1, 2, and 3 .

It is important to know the relationships involving angles: sum in a triangle is 180°; angles around a point add to 360°.

After completing the material, you should be comfortable solving problems like:

Find the measure of x in each of the following. 113° and 106° (wasn't drawn to scale)

Text Box:

90°

180°-90°-23°=67°

Note: the exterior angle theorem (EAT) can be used on the left triangle that says an exterior angle is the sum of the two "remote" interior angles.

x = 90 + 23 = 113°

In MAP 903, read page 274 and do page 275 # 1-6; read page 280 and do page 281 # 1-12, 14

http://www.aaamath.com/B/geo612x5.htm looks at a key concept with questions.

Lesson 8 — Perimeter, Area and Volume

The emphasis here is on solving problems involving two and three-dimensional objects. You need to be familiar with the concepts of radius and diameter of a circle as well as the altitude of a triangle. Some formulae need to be memorized and some will be provided on any test.

	Notes
Perimeter (memorize)
Rectangle	2l + 2w or 2(l + w)
Square	4s
Circle	C = pd or 2pr
Triangle	P = a + b + c
Area (memorize)
Rectangle	A = l x w
Square	A = s²
Circle	A = pr2
Triangle	a = 1/2 bh

Volume (do NOT memorize – formulas will be given)
Rectangular prism (box)	V = l x w x h
Triangular prism (wedge)	V = (base area) x height
Cylinder	V = pr²h
Cone	V = 1/3 pr²h
Square pyramid	V = 1/3 b²h
Sphere	V = 4/3 pr³

In MAP 360, read section 7-6 on pages 280 to 284. Do questions page 284 # 1 (a) to (e). If the concepts are totally new, then do MAP 912 section 4

After completing the material you should be able to solve problems like those in MAP 360.

http://www.aaamath.com/geo.html — good explanations with interactive component

http://www.cut-the-knot.com/triangle/altitudes.shtml click on the pop window to explore altitudes

http://www.mathleague.com/help/geometry/area.htm — area and perimeter

MAP 912 Section 4 covers this in quite a bit of detail.

Lesson 9 — Transforming Formulas

Sometimes in practice you may know a formula in one form but need it in a completely different way. You may know that the circumference of a circle is but you need to find the radius when given the circumference. You then need to transform the formula to

Transform:		Notes
V = bh, to find h	divide both sides by b	h = V/b
, to find E	multiply by R	E = RI
, to find R	divide by I²	R = W/I²
A = L + T, solve for L	subtract T	L = A - T

Example: Given the formula P = 2L + 2W, re-arrange to find L

P = 2L + 2W subtract 2W from both sides

P – 2W = 2L + 2W – 2W

P – 2W = 2L divide both sides by 2

cancel the 2s and reverse the formula

In MAP 360, read section 16-2 on pages 595 and 596.

Do questions page 597 # 1 to 11

After completing the material you should be able to solve: Given the formula A = p + prt, re-arrange the formula to find t.

A = p + prt
-p -p	subtract p from both sides
A-p = prt pr pr	divide both sides by pr
(A-p) / pr = t	cancel the pr on the right side

Can you solve for p?

MAP 355 has more examples and an exercise.

http://regentsprep.org/Regents/math/formulas/litless.htm - has a lesson and some practice.

http://ca.geocities.com/magriesti_rc/october/notes3_6/notes3_6.html - some good examples (one easy and one hard).

http://www.quia.com/tq/245711.html - an online quiz.

http://www.purplemath.com/modules/solvelit.htm - Purplemath has good examples.