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IMP 1 Honor

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Problem Set #1

Geometry

Set Number Name Page

#s

Due

Date

1 I. Basic Geometry Terminology 2 - 3

2 II Angles 4 - 5

3 III. Quadrilaterals 6 - 7

4 IV. & V. Isosceles Triangles and Similar Polygons 8 - 9

5 VI. Congruent Polygons 10 - 11

6 VII. Parallel Lines 12 - 14

Do NOT wait until the last minute to start this problem set.

Get extra help ahead of time – not the day before it’s due.

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I. Basic Geometry Terminology Points, Lines and Planes Point: a precise location, usually represented by a dot. No size. Point A A B

A

Line: extends in 2 directions without ending. No thickness. Line AB A B

Line Segment: part of a line with two endpoints. AB A B

Ray: part of a line that starts at one point and extends in one direction without ending. AB

Plane: like a wall or floor, extends without ending. No thickness. Plane M

Collinear: if all points are in one line.

Ex1 3 collinear points: A, B, and C

3 other points that look collinear: D, B, and E

A ray opposite to BC is: BA or AB

A ray starting at A through B but not c is: not possible

CB + AB = AC (this is known as Segment Addition)

** ANY TWO points are collinear!

Planes

Ex2 A plane parallel (||) to plane EFG: plane CBAD

A plane that intersects EFG at FG : plane BFGA

A point coplanar (all points in one plane) with C, D and F: point G

Two points not coplanar with plane ABC and not coplanar with plane ABE : points C and G

**ANY THREE points are coplanar!!

Ex3 To the right a plane contains points A, B, P & E.

Line CD which intersects the plane at point P

True or False: A, B, C, and E and are coplanar : False

A, B, C and D are coplanar: True

B, C, and E are coplanar : True

Midpoint, Bisectors and Perpendicular

M is the midpoint of AB if:

1. AM = MB

2. AM = ½ AB and MB = ½ AB

A line, ray or segment bisects AB if:

11. AX = XB

2. X is the midpoint of AB

Lines are perpendicular )(⊥ if they intersect to from right angles (90° angles).

If you want to show lines are ⊥ , you only have to show that one angle is a right angle.

M

3

Problems: Answers and work on a separate sheet of paper (you will need to draw the diagrams)

1. You are given that GH bisects DF at E. First you should draw a picture!

(a) If 35=DE and )5(5 xEF −−= , find x .

(b) If 18=DF and )2(3 xDE −−= find EF

2. You are given that 6+= zGH , 7=GE and 42 −= zEH

The diagram may or may not be drawn to scale!

(a) Is E the midpoint of GH (hint: what must be true about the three line segments? Make an equation)?

(b) If E has coordinate -10 on a number line, what are the coordinates of G and H?

3. The top and bottom of the prism to the right are perfect parallel

hexagons. Its sides are rectangles.

Name:

(a) A point coplanar with plane GHB _____

(b) A plane that appears parallel to GHB _____

(c) A plane that appears perpendicular to GHB _____

(d) A point coplanar with G, L and D. _____

(e) The line where AGH intersects ABD _____

4

O

B

A C

II. Angles

An angle is formed by 2 rays that have the same endpoint. The 2 rays are called the sides of the angle, and

their common endpoint is the vertex of the angle. Use three letter to name an angle: BOA∠

Angle bisector is a ray that divides the angle into two equal (congruent) adjacent angles.

Ray OC bisects BOA∠ , so AOCBOC ∠=∠

Two angles are complementary if their measures add up to 90°

Two angles are supplementary if their measures add up to 180°

Adjacent angles are 2 angles that have a common vertex and side, but no common interior points. BOC∠ and AOC∠

Vertical Angles are equal, and look like, so AODBOC ∠=∠ and CODAOB ∠=∠

A O B

D C

Types of Angles Acute: Less than 90° Obtuse: More than 90° Right: 90° Straight: 180°

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1. If )30(6 xEADm −=∠ and )9(12 xBADm −−=∠ , ( EADm∠ means the “measure of” angle EAD)

(a) Find x

(b) Find EADm∠

2. You are now given that AC bisects BAD∠ in the same diagram above.

If 509 −=∠ xBACm and 3412 −=∠ xBADm , find BADm∠

4. Diagrams below contains angles that are either vertical, complementary, or supplementary

For each one write equations to find x, and then y.

(a) NOT TO SCALE! (b) NOT TO SCALE!

5. Use the diagram to the right and proper symbols

(a) Carefully find the value of x.

(b) Is ACBE ⊥ ? Explain why or why not.

(c) Does BD bisect CBE∠ ? Explain why or why not.


3. In the diagram to the right name…

(a) A pair of acute vertical angles.

(b) A pair of obtuse vertical angles.

(c) 2 angles that are complementary and adjacent.

(d) 2 angles that are complementary but not adjacent.

(e) Two angles supplementary to DXE∠

(f) Two supplementary and congruent angles.

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III. Quadrilaterals Parallelogram: A quadrilateral with both pairs of opposite sides parallel.

Properties: 1. Opposite sides are parallel

2. Opposite sides are congruent (equal).

3. Opposite angles are congruent.

4. Diagonals bisect each other

Rectangle: A quadrilateral with 4 right angles. Every rectangle is a parallelogram.

Properties All the properties of a parallelogram, plus:

1. All angles are right angles.

2. Diagonals are congruent.

3. All 4 pieces of the diagonals are congruent.

Rhombus: A quadrilateral with 4 congruent sides. Every rhombus is a parallelogram.

Properties All the properties of a parallelogram, plus:

1. All four sides are congruent.

2. Diagonals bisect angles.

3. Diagonals are perpendicular.

Square: A quadrilateral with 4 right angles and 4 congruent sides. Every square is a rhombus, rectangle

and a parallelogram. Properties

All the properties of a parallelogram, rectangle and rhombus.

Trapezoid: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the other sides are legs.

Isosceles Trapezoid: A trapezoid with congruent legs.

Properties

1. Two legs congruent.

2. Both pairs of base angles are congruent.

3. Diagonals are congruent.

4. Adjacent angles are supplementary

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1. Fill in all angles in the diagram given that Parallelogram ABCD is a ….

(a) Rhombus (b) Rectangle (c) Square

2. Now suppose that ABCD is a rectangle.

Consider all you now know about a rectangle to find x, y, z and w.

Big Hint: Find y first!

3.(a) Multiple Choice: The following is a parallelogram. Which ONE of the following equations is

the only one that HAS to be true.

a. 326 −= yy

b. 52510 +=− yy

c. )52()510( ++− yy

d. 32184 −=− yy

(b) Solve the appropriate equation to find the value of y.

(c) Based on your values for y, is ABCD a rectangle? Why or why not?

4. Each shape below is a parallelogram. Decide if each is a rhombus, rectangle, square, or just a parallelogram.

(a) __________________

(b) __________________

(c) ________________

(d) ________________

(e) ________________

(f) ________________

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IV. Isosceles Triangles

An isosceles triangle has at least two congruent sides.

Fact: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

V. Similar Polygons Similar polygons:

• same shape

• One is a magnification of the other

• The magnification is called the scale factor.

Recall Two polygons are similar if:

1. Corresponding sides are proportional

2. Corresponding angles are equal

We have also learned that for triangles knowing just one of the two things above is enough! To put this in

more technical terms:

SSS~ If you have two triangles where all three sides are proportional, the triangles are similar!

AAA~ If you have two triangles where corresponding angles are equal, the triangles are similar!

It can also be proven:

SAS~ If you have two triangles with two pairs of proportional sides, and the angles between them are equal,

the triangles are similar!

Suppose QRST ~ XYZW

Find and reduce the fraction:

Perimeter of ABC

Perimeter of EFG

Do you notice anything about the ratio of the perimeters?

*The ratio of their perimeters is equal to the ratio of their sides.

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PROBLEMS: Answers and work on a separate sheet of paper (you will need to draw the diagrams) 1. Two triangles are similar. One has sides of length 16, 12, and 8.

The other triangle has perimeter 45. Find the length of each side of the other triangle.

2. In each diagram below ABC∆ is similar to another triangle shown. Explain why they’re similar (using the correct

order), what the scale factor is, and then find the value of x.

(a) ______~ABC∆ by ______

Scale factor = ______

(b) ______~ABC∆ by ______


(c) ______~ABC∆ by ______


3. In the diagram shown, DEFABC ∆∆ ~

Find x and y

4. In the diagram below ABCD ~ EFGH. Find the exact values of w, x, y, and z.

5. A right triangle has sides 5, 12 and 13 cm. A similar triangle has perimeter 90cm

(a) Find the scale factor for the triangles. There are two ways to write it.

Here please write it with the smaller number on top! Reduce fractions.

(b) Find each side of the second triangle.

(c) Find the area of both triangles. Remember, hbA ⋅=2

1

(d) What is the ratio of their areas? Write this too with the smaller area on top!

Reduce!

(e) What do you notice about how their scale factor relates to their area ratio?

12 13

5

10

VI. Congruent Polygons

Def Two polygons are congruent if their corners can be matched up so that the corresponding sides and angles are

equal. The order in which corners match is key!

Ex1 Do the polygons below seem like they might be congruent? Note- Unless you’re given more information, or actually measure stuff, you can’t tell for sure, so never just assume they are!

Ex2 Now, in each problem below you’re given that two triangles are congruent, but

the way the sides match up change in each. Very carefully find the unknowns!

(a) DEFABC ∆≅∆ : find x and y

x= 15 y = 80

(b) DEFABC ∆≅∆ find x, y and z.

x= 7 y = 8 z = 3

Saying that two triangles are congruent means six things match up (3 angles and 3 sides). This section is all about

looking for shortcuts. How many matching sides or angles is really necessary to know the triangles are ≅ ? For

example, must two triangles be congruent if they have the following in common…

S

No

SS

No

SAS

Yes

SSA

No

ASA

Yes

AAA

No (only similar)

SSS

Yes

AAS

Yes

SS90°

Yes

Five New Facts (theorems)

1. SAS If you have two triangles with two pairs of congruent sides, and the angles between them are congruent, the

triangles are congruent!

2. ASA If you have two triangles with two pairs of congruent angles, and the sides between them are congruent, the

triangles are congruent!

3. SSS If you have two triangles where all three pairs of sides are congruent, the triangles are congruent!

4. AAS If you have two triangles with two pairs of congruent angles, and one pair of sides adjacent to them is

congruent, the triangles are congruent!

5. HL (or SS90°) If you have two right triangles with congruent hypotenuses and one pair of corresponding

congruent legs, the triangles are congruent! This is the only situation where SSA is valid.

“Corresponding Parts of Congruent Triangles are Congruent”

Once you know two triangles are congruent, then all six corresponding parts of the triangles are congruent.

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PROBLEMS: Answers and work on a separate sheet of paper (you will need to draw the diagrams)

1. Decide if the two triangles in each picture MUST be congruent. If so, give a reason (SSS, SAS …) Some pictures

look very similar but all are different.

(a) (b) (c) (d)

2. Consider the diagram below. For each set of given information, name the triangle ≅ to ABD∆ and the reason.

The diagrams are all identical. Only the given information changes.

(a) BCAB ≅ , CDAD ≅ ∆ ABD ≅ ∆ ______ by _______

(b) ADAB ⊥ , DCBC ⊥ , BCAB ≅ ∆ ABD ≅ ∆ ______ by ______

(a)

(b)

3. For each set of givens below, name two triangles that you can conclude are congruent.

On the second line give the reason. Note: There are 8 triangles in each picture, not just

(a) ACBE ⊥ , CDBC ≅

________ ≅ _______

by _______

(b) AC bisects

BAD∠ and BCD∠

________ ≅ _______

by _______

(c) ADAB ≅ , CDBC ≅

________ ≅ _______

by _______

(d) ACBE ⊥ , 83 ∠≅∠

________ ≅ _______

by _______

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VII. Parallel Lines

Recall from class:

Same Side Interior Angles are Supplementary, ex: ∠ 4 & ∠ 5

Alternate Interior Angles are Congruent, ex: ∠ 3 & ∠ 5

Corresponding Angles are Congruent, ex: ∠ 1 & ∠ 8

*Remember when looking at complex diagrams, it is helpful to highlight the parallel lines.

When stating two angles are equal because of parallel lines, they both have to be along the same transversal.

Ex 1 From the diagram to the right, name (if possible) an angle that is ….

(a) Congruent to BHI∠ ABH∠

(b) Supplementary to BHI∠ GHB∠ or CBH∠

(c) Congruent to DBC∠ ABG∠ or BGH∠

(d) Congruent to GBH∠ none (this is an

angle not made from

parallel lines)

Above we saw that if two lines are parallel, then we know things about special angles.

Now let’s look at this backwards and determine what needs to be true about the angles to know the lines are parallel.

1. Same side interior angles are supplementary

or

Lines 1l and 2l are parallel if 2. Alternate interior angles are congruent

or

3. Corresponding angles are congruent

Ex 2 In the diagram to the right…

(a) Which two lines are definitely parallel: lines a and b

The 50° angles are corresponding angles with transversal of line c

(b) Which two lines are definitely not parallel: lines c and d

The 50° and 40° angles are corresponding angles and not

congruent, so lines are not parallel.

(c) Which two lines do we know nothing about!: lines e and f

transversal

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PROBLEMS: Answers and work on a separate sheet of paper (you will need to draw the diagrams)

1. Find values of x, y, and z.

2. In each problem below some information is given about the diagram.

Use the information to determine which (if any) lines are parallel. If no lines are parallel write none.

(a) 114 ∠=∠ (b) 125 ∠=∠

(c) 116 ∠=∠ (d) 103 ∠=∠

(e) °=∠+∠ 18067 mm (f) 12117 ∠+∠=∠ mmm

3. Each diagram below shows a number of lines, many of which appear parallel.

Use the angles to name all segments or rays that are actually parallel in the diagrams.

(a) (b)

4. Find values of x, y, and z in the diagram to the right.

5. In each problem below some information is given about the diagram.

Use the information to determine which (if any) lines are parallel. If no lines are parallel write none.

(a) 1317 ∠=∠ (b) 617 ∠=∠

(c) 817 ∠=∠ (d) °=∠+∠ 1801316 mm

(e) °=∠+∠ 180617 mm (f) 12117 ∠+∠=∠ mmm

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6. Triangle Puzzle! Find the missing angles in the figure below. HAVE FUN!!!

a = ________

b = ________

c = ________

d = ________

e = ________

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Problem Set #2 Fractions & Rational Expressions Unit

Set Number Name Page #s Due

Date

1 I. Introductory/Review Exercises 16 - 20

2 II., III. & IV.

Adding and Subtracting Fractions,

Multiplying Fractions, & Dividing Fractions 21 - 23

3 V. Order of Operations (PEMDAS) 24

4 VI. & VII.

Adding Simple Rational Expressions &

Subtracting Simple Rational Expressions 25 - 26

5 VIII. & IX.

Multiplying Simple Rational Expressions &

Dividing Simple Rational Expressions 27 – 28

6 X. Putting It All Together 28

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I. Miscellaneous Introductory/Review Topics A. Least Common Multiple

Defining the least common multiple (LCM) of two or more numbers is as easy as repeating its name. Actually finding

the LCM is a little more involved. For example to find the LCM of 36 and 156, you could start listing the multiples of

36, and then list the multiples of 156 until you find a number that appears on both lists. But that’s time consuming and

an unsophisticated method. You can do better than that! Instead, how about thinking about the divisors of 36 and 156

– specifically the prime number divisors? What prime numbers multiplied together make 36?

So to make 36 you need two 2’s and two 3’s. That means that any multiple of 36 is the product of at least two 2’s, two

3’s, and more.

What prime numbers multiple to 156?

So to make 156 you need two 2’s, one 3, and one 13. So any multiple of 156 is the product of at least two 2’s, one 3,

and one 13, and more.

Therefore, in order for a number to be a multiple of BOTH 36 and 156, it needs to be the product of at least two 2’s,

two 3’s, and one 13, and more. But the least common multiple (LCM) will simply be the product 2*2*3*3*13 = 468.

You can find the LCM by any means necessary, but try thinking about it from this perspective. It’ll make you a

stronger math student!

EXERCISES. Find the LCM of the listed numbers. Show work and final answer on a separate sheet of paper.

1. 15, 27

2. 42, 120

3. 32, 66, 98

4. 48, 72, 528

B. Adding and Multiplying Variables

What’s the difference between 2x and x2? A lot! They’re very different, yet students confuse them all the time. The

term 2x means x + x whereas the term x2 means x*x. It’s a completely different operation and therefore value! So

please be careful.

Adding: You can only add like terms. When working with variables, it’s the power on the variable that determines

families of like terms. For instance, anything like 3x, 17x, and -22x are like terms for they’re all multiples of x1. You

should think of the x like an apple. Therefore, 3x means 3 apples, -22x means negative 22 apples, etc. Meanwhile 3x

and 5x2 are NOT like terms and cannot be added together. You should think of x

2 as something different like oranges.

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Therefore 3x + 5x2 means you have 3 apples and 5 oranges. However, any other term of the form ax

2 (where a is a

constant) can be added together by just summing their coefficients. These same rules apply for expressions with

multiple variables. The terms 3x and 4xy are NOT like terms, but 4xy and 7xy are and 4xy + 7xy = 11xy.

EXERCISES. Simplify the following expressions by adding like terms. Show work and final answer on a separate

sheet of paper.

5. 5x + 7x2 – x – 4 + -2x

6. 6xy + 3x – 5y – x + y

7. 18a + 15a + 4ab – 7a + ab

8. -4z5 + 9z

5+ 28 + 2z – 3z

5 + z

3 + 6

Multiplying: There are no restrictions as to which terms can be multiplied. Simply take the product of the coefficients

and sum the exponents. Here’s an example:

I expect that when you do problems like this you’ll jump right to the answer, but I showed the mental steps here to

prove a point. Multiplication is commutative, meaning the order in which you multiply doesn’t matter. Therefore, you

can multiply the numbers together to get the new coefficient and the variables together to find the new exponent.

Here’s one more example for you do some of your own problems:

EXERCISES. Completely simplify the following expressions. Show work and final answer on a separate sheet of

paper.

9.

10.

11.

12.

C. Distribution

The distributive property of multiplication over addition is simply this: it makes no difference whether you (1) add

two or more terms together before multiplying the result by the factor or (2) multiply each term by the factor first and

then add up the results. That is, adding up the terms first, then multiplying the result by the factor = multiplying

each term by the factor first, then adding up the resulting terms:

18

However, often when dealing with variable expressions (which we are going to do in this unit), it’s impossible to add

the terms together first because they are not like terms. Therefore, you have no choice but to multiply each term by

the outside factor. This is multiplication distributing over addition. Here’s an example:

This process gets more involved when you have to multiply two sums. You may have heard this called FOIL, Lobster

Claws, Smiley Face, Diving Your Dolphins, or maybe simply distribution. Here’s an example:

However, most people prefer to look at it the way shown below (distribute the first term in the first set of parentheses

to each term in the second set, then move on to distributing the second term in the first set of parentheses to each term

in the second set, and so on):

Tips:

1. The product of the number of terms in each set of parentheses is the number of times you should be

multiplying two terms. However, it is most likely not the number of terms in the answer, as you may have to

add like terms. For instance, requires multiplying four pairs of terms (4 = 2*2) which are x*x,

x*3, 2*x, and 2*3. Yet the final answer, , has only three terms because 2x and 3x are like terms

and sum to 5x.

2. If you have to multiple 3 or more sums, do only two at a time, simplify, and then multiply the result by the

next set of parentheses. For example, if you had to simplify , first multiply x + 3

by x – 4, then take the result and multiply it by x2 + x – 1.

3. Go slow as this will prevent many silly mistakes.

4. Show your work so that it is easier to spot and correct any errors without having to start the problem over

again.

5. You can evaluate the given expression for x = 4 (or some other number). Then evaluate your simplified

answer for x = 4 as well. You should get the same number in both cases so long as you did all the algebra

AND computation error-free. This is not a guaranteed way to check your work. It is possible to make a

mistake and yet end up with the same number coincidentally.

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EXERCISES. Completely simplify the following expressions so that no parentheses remain and all like terms are

combined. Show work and final answer on a separate sheet of paper.

13.

14.

15.

16.

D. “Canceling” or Simplifying

Many teachers are hesitant to use the term “canceling.” Unfortunately, it’s been used for so long by so many that

we’re forced into using it all the time. “Canceling” is actually the act of simplifying or reducing an expression. When

dealing with addition and subtraction, “canceling” happens between two terms sum to zero so they “disappear”. In the

case of multiplication and division, “canceling” happens because the product or quotient of two terms is one. Here are

some examples of “canceling”:

In the first example, 2x – 2x = 0, so adding 2x to 5 and then subtracting 2x from the 5 is like doing nothing it all. The

two terms “undo” one another. In the second example, the x is being multiplied by 3, but then it’s being divided by 6.

Dividing by 6 is like dividing by 2 as well as 3. So the multiplication by 3 and division by 3 “undo” one another and

we see the equivalent expression is just x divided by 2.

A common mistake students often make is they “cancel” when they shouldn’t because they just know to cross out

something that appears both in the numerator and denominator and don’t understand what “canceling” really means.

Here’s a common mistake:

What’s wrong? Well, let’s look at it again with numbers instead of x’s.

I just proved that 1.625 = 2!

That should actually be proof enough that I made a mistake when simplifying. You can only simply fractions when

the numerator and denominator have a common divisor. In the example above, neither x+10 nor x+5 are divisible by

20

x (because nothing times x equals them).Actually, x+10 and x+5 have no common divisors, so this fraction is as

simplified as it can be.

Here are some examples of fractions that can be simplified:

Notice that in the middle step I’m showing the opposite of distribution. This is called factoring. I’m rewriting the

numerator and denominators as products of its factors. Then you can see if the numerator and denominator have

common factors. Those can be “canceled”. When you’re not sure whether or not you can “cancel”, you could try

replacing the x’s with a number like I did in the earlier example to prove that cannot be simplified by crossing

out the x’s.

EXERCISES. Simplify the following fractions. If they cannot be simplified, simply say so. Show work and final

answer on a separate sheet of paper.

17.

18.

19.

20.

21

II. Adding and Subtracting Fractions

You probably remember how to add and subtract fractions, but you’ll find that practicing with these simple problems

will prepare you to use the same ideas on harder problems in Worksheets VI and VII.

Here’s the most common mistake that happens when adding or subtracting fractions:

What happened? What mistake was made?

The biggest mistake made was adding the denominators. It’s also incorrect to add the numerators when the

denominators are different. As a result adding a quarter to a half equals a third ( ) which we all know is

LESS than a half. That’s obviously wrong. If you add to a half, you should get more than a half in the end.

What should have been done?

1. Find a common denominator for the two fractions. The common denominator should be a common

multiple of 2 and 4, preferably the LCM (least common multiple).

The LCM of 2 and 4 is 4.

2. Rewrite both fractions with the new denominator.

3. Add the numerators and keep the denominator the same.

4. Simplify if possible.

3 and 4 have no common factors so the fraction is completely simplified

EXERSICES. Completely simplify the following expressions. You must show work. Do not use your calculator’s

>Frac function. Show work and final answer on a separate sheet of paper.

1. 3.

2. 4.

22

III. Multiplying Fractions

You probably also remember that when multiplying fractions, you should multiply the numerators as well as multiple

the denominators. For example:

But why? Let’s think of it in terms of length and area. Imagine a square that’s one foot by one foot. Shade a column in

the square with width of of a foot and a row that’s of a foot.

The region shared by the column and the row has area of or square feet because the area of a rectangle is its

length times its width. And you can see in the diagram that the overlapping region is 4 of the 15 pieces making the

square foot.

Here’s another way to think of it. Calculating is the same as calculating “one third of ”. “One third of ” means

what would you need three of in order to get ? Well, . So one third of is .

Hopefully at least one of these explanations confirms what you already know about multiplying fractions.

EXERCISES. Completely simplify the expressions below. Show all work. Do not use your calculator’s >Frac

function. Show work and final answer on a separate sheet of paper.

1. 3.

2. 4.

23

IV. Dividing Fractions

How many times has a teacher said to you, “What’s the same as dividing by a fraction?” And of course you know to

answer, “Multiplying by the reciprocal!”

Example:

Let’s think about what this means:

If we had divided 8 by 2 we would have been solving for how many times 2 can fit into 8. The answer is 4. Four 2’s

make 8, so .

2 2 2 2

8

So what does it mean to calculate ? Well, it means the same thing. “How many times can fit into ? The answer

is more than 1 since is bigger than , and the exact number of times is . See the diagram below for a visual

representation:

As you can see, one and a third ’s fit into the same space as .

EXERCISES. Completely simplify the expressions below. Show all work. Do not use your calculator’s >Frac

function. Show work and final answer on a separate sheet of paper.

1. 3.

2. 4.

24

V. Order of Operations (PEMDAS)

So far you’ve only been asked to do very simple calculations with just one operation. When the expressions get more

complex, you need to abide by the Order of Operations. Ever hear of PEMDAS? It’s an acronym designed to help

students remember the Order of Operations.

P = Parentheses. Any expression inside parentheses needs to be simplified before starting operations outside the ().

Within a set of parentheses follow the order as given below.

E = Exponents. Exponential expressions should be simplified after all parentheses are eliminated. Remember that -

102 does not mean the same thing as (-10)

2.

MD = Multiplication and Division. Multiplication and division are synonymous operations (as you saw when

dividing fractions). Therefore, they should be carried out in order from left to right (hence multiplication does not

necessarily come before division). Examples: 8/2*7 = 4*7 = 28 and 4*3/6 = 12/6 = 2 and 1/4*16 = 0.25*16 = 4.

AS = Addition and Subtraction. Again, these are synonymous operations (subtracting 2 is like adding -2). Do these

operations as they come from left to right.

EXERCISES. Completely simplify the following expressions. Do not use your calculator’s >Frac feature to do the

problems. However, you can use it to check your work. Show work and final answer on a separate sheet of paper.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

25

VI. Adding Simple Rational Expressions

Rational Expressions are simply fractions that may contain variables in the numerator, denominator, or both. They’re

called rational because they can be written as ratios. Here are some examples of rational expressions:

You have already demonstrated in the first five worksheets of this unit that you have the necessary skills to add

rational expressions.

1. Find the LCM of the denominators. In many cases this may be the product of the denominators. If not,

sometimes it’s helpful to factor the denominators first (aka the opposite of distributing) so you can see if the

denominators already have divisors in common (just like in Worksheet I Part A).

2. Rewrite the fractions as their equivalents with the new denominator.

3. Add the numerators by combining like terms.

4. Keep the denominator the same (for these problems you can leave the denominator as a product of terms).

Example 1: Simplify .

The LCM of 4 and x is 4x. So the first fraction needs to be multiplied by and second one needs to be multiplied by

so as to not change the overall value of the fractions (since it’s like multiplying then by 1) while still changing the

denominators to 4x.

Example 2: Simplify .

The LCM of 2x and x is x(2x+1) and you can leave the denominator like this. You do not have to simplify the

denominator.

EXERCISES for VI. Add the following rational expressions. Simplify the numerators and denominators completely

(this means no remaining parentheses and all like terms are combined). Show work and final answer on a separate

sheet of paper.

1. 2. 3.

4. 5.

26

VII. Subtracting Simple Rational Expressions

Subtracting rational expressions is very much like adding rational expressions except you must be careful to subtract

the ENTIRE second fraction from the first. What do I mean by the ENTIRE second fraction? Well, often times when

the numerator of the second fraction is a sum or a difference of multiple terms, students forget to subtract each term

and instead only subtract the first. Here’s an example of this common mistake:

Can you see the mistake? It’s right here!

Here’s the correct way to subtract these fractions:

It may look like a small difference but it’s really a BIG difference in many ways. It’s not just x2 being subtracted from

5. The sum x2 + x is being subtracted from 5, so both the x

2 and the x must be subtracted.

Moral of the Story: Make sure to distribute that subtraction sign throughout the entire numerator (each term) of the

second fraction!

EXERCISES. Subtract the following rational expressions. Simplify the numerators and denominators completely

(this means no remaining parentheses and all like terms are combined). Show work and final answer on a separate

sheet of paper.

1.

2.

3.

4.

5.

27

VIII. Multiplying Simple Rational Expressions

Once again you’ll find that multiplying is easier than adding rational expressions (just as it was easier with fractions).

Reminders:

1. You must completely simplify the numerator as well as the denominator (that means no parentheses and like

terms are combined).

2. Make sure you distribute multiplication over addition carefully and completely. See Worksheet I Part C for a

refresher.

Here’s an example:

EXERCISES. Multiply the following rational expressions. Completely simplify both the numerator and the

denominator. Show work and final answer on a separate sheet of paper.

1.

2.

3.

4.

5.

28

IX. Dividing Simple Rational Expressions

Again, these problems may look hard at first glance, but you know how to handle it. You know how to divide

fractions and how to multiply rational expressions, therefore, you know how to divide rational expressions.

EXERCISES. Divide the following rational expressions. Completely simplify both the numerator and the


1.

2.

3.

4.

5.

X. Putting It All Together

EXERCISES. Simplify the following rational expressions. Completely simplify both the numerator and the


1.

2.

3.

4.

5.

29

Problem Set #3 Functions and Transformations Unit

Set Number Name Page #s Due

Date

1

I.

Relations, Functions, Domain and Range

p. 93 Exploratory #1,4, 5-13 odd, 25,29,33

p. 94 Written #1,3,18,20,21

p. 98 Written #27-33 odd

Read

p. 30 – 31

Problems

in Green

Book

2

II. Function Notation and Composition

p. 98 Written #18, 19, 20, 42, 44, 46, 47

p. 116-117 Written #6-10, 16-18, 20-22, 25

Read

p. 32

Problems

in Book

3 Corrections Corrections for Sections I & II 30 – 32

4 III. Transformations I - Translations 33 – 34

5 IV.

Transformations II – Scale Changes and

Translations 35 – 37

6 V.

Transformations III – Reflections, Scale

Change and Translations 38 – 39

30

I. Relations, Functions, Domain and Range A large part of mathematics is looking at how one set of numbers relates to another set of numbers. A number of

definitions are necessary to understand so that you can delve deeper into more advanced concepts.

Definitions: Relation: any relationship between a set of input numbers(x) and a set of output numbers (y)

Ex: a relation: {(-1,2),(3,5),(4,7),(-1,3)}

Function: a special type of relation in which each input number is paired with exactly one output number.

Ex: a function: {(1,3),(2,5),(3,1),(4,5)}

NOT a function: {(-1,2),(3,5),(4,7),(-1,3)}

notice that the input, -1, is paired with more than one output

Here are a few more examples of the difference between functions and relations. Notice that pairs of numbers can be

represented in charts, coordinates or graphs.

X Y X Y

1 4 3 9

3 -2 4 4

5 6 8 1

7 -2

Domain: The set of all input (x) values of a relation or a function.

Ex: Find the domain of the function {(1,3),(2,5),(3,1),(4,5)}

Solution: {x:1,2,3,4}

• When asked to find the domain of an equation, it is helpful to first look for values that would NOT be allowed

as inputs. The denominator may not = 0 and you may not have negative numbers under a square root.

Ex: Find the domain of the function 3

4

xy

x

−=

+

Solution: Step 1: set the denominator ≠ 0. 4 0x+ ≠

Step 2. Solve for x. 4x ≠ −

Step 3. Write your answer in domain notation: { }: 4x x ≠ −

Ex: Find domain of the function 4 2 6y x= − −

Solution: Step 1. Set the expression under the square root > 0. 2 6 0x − ≥

We do this because we know we are only allowed to have numbers greater than or equal to 0

under a square root in order to get a result that is a real number.

Step 2. Solve for x. 3x ≥

Step 3. Write your answer in domain notation: { }: 3x x ≥

Range: The set of all output (y) values of a relation or function.

Relation but not a function Relation and a function

31

Ex: Find the range of the function {(1,3),(2,5),(3,1),(4,5)}

Solution: {y:3,5,1,}

• If given an equation, graph it and look at which y-values are used if the graph were to extend on forever.

Ex: Find the range of the function y= (x-3)2-4

Solution: Step 1. Graph the function.

Step 2. Imagine smooshing the graph

onto the y-axis and then list the points that

would be covered up. {y:y > -4}

Now that you have those definitions and examples, read p. 90-92 of Mathematics: A Topical Approach by Bumby and

Klutch for more examples and then complete the following problems:

p. 93 Exploratory #1,4, 5-13 odd, 25,29,33 (remember, the ℜ symbol can be used to indicate all real numbers)

p. 94 Written #1,3,18,20,21 p. 98 Written #27-33 odd (remember, f(x) is function notation and means y)

(3,-4)

32

II. Function Notation and Composition NOTE: the problems are listed in 2 separate sections within the lesson. It is recommended that you read and do them

as you go. They are located in boxes so that you won’t miss them

You have already learned about function notation during the Overland trail. As a refresher, we can write an equation

that is a function such as y=2x-3 using function notation f(x)=2x-3. f(x) is a way to label the function and indicate that

x is the input and f(x) is the output, or y-variable.

In the function f(x)=2x-3, a coordinate pair could be represented as f(1)=-1 or (1,-1) showing that when the input is 1

the output is -1.

As you saw in problem 20, you do not only need to input numbers into a function, you can also input variable

expressions.

Composition functions are functions created when you put one function into another.

Ex: f(x)= 2x+9 find f(3x)

Solution: Step 1. Substitute 3x everywhere you see an x in the original function f(x). f(3x)=2(3x)+9

Step 2. Simplify the left side. f(3x)=6x+9

Now, if g(x) = 3x, you could have been asked to find f(g(x)). This is a composition function where you plug in g(x)

everywhere you see an x in the f(x) function.

f(g(x))=2g(x)+9 Step 1. Plug in g(x) for x in the f(x) function

f(g(x))=2(3x)+9 Step 2. Replace the g(x) on the left side with its expression

f(g(x))=6x+9 Step 3. Simplify

x 3x 6x+9

In a slightly simpler example let’s watch what happens to a specific number that is put through this composition

function.

f(x) = 2x+9 and g(x) = 3x. Find f(g(5)).

g(x) f(x)

g(5)=3(5)

f(15) =2(15)+9

5 15 39 so f(g(5)) = 39.

Practice using function notation by completing the following problems:

p. 98 Written #18, 19, 20, 42, 44, 46, 47

Do the following problems on composition functions:

p. 116-117 Written #6-10, 16-18, 20-22, 25… explain why you think #25 comes out the same either way

the functions are composed.

33

III. Translations

In this unit, you are going to learn how to use equations to shift graphs right, left up and down. The sliding movement

is called a translation.

We will be working with a set of graphs to help us learn these. These graphs we will call the parent functions.

f(x) = x3 f(x) = x

f(x) = |x| (absolute value) f(x) = x2

It is important to know the shapes of these graphs and at least one key coordinate on each… Each of these graphs has

a coordinate at (0,0) and (1,1)

34

Do all of this on a separate piece of paper!

You must complete the following step for #1-3:

• Draw the original parent function on the same axes

• State what translation took place FROM the parent function to the function displayed (translated function)

• Fill in the coordinates as the parent function point listed maps to the graph displayed.

1. The 1st example is completed. 2. 3.

Translation: shift 3 units left Translation: Translation:

(0,0) (-3_,_0_) (1,1) (___,___) (-3,3) (___,___)

4. Explain how the equations and graphs of the parent functions relate to equations and graphs of functions that are

translated to the right or the left.

Repeat the process for #1-3 with the functions below.

5. 6. 7.

2 4y x= +

3 7y x= + | | 3y x= −

Translation: Translation: Translation:

(0.5,0.25) (___, ___) (0,0) (___,___) (1,1) (___,____)

8. Explain how the equations and graphs of the parent functions relate to equations and graphs of functions that are

translated up or down.

9a. Describe the translations to the graph parent function y = x3 when it becomes y = (x + 4.2)

3 + 5

9b. Give one coordinate from the parent function and where it would translate in the translated function.

( __ , __ ) ( __ , __ )

10a. Describe the translations to the graph parent function y = |x| when it becomes y = |x - 2.5| - 4.1

10b. Give one coordinate from the parent function and where it would translate in the translated function.

( __ , __ ) ( __ , __ )

11. If the graph of y = x is translated 2 units up and 3 units left, what is the equation of this new graph?

12. If the graph of the parent function y = x2 is translated 5 units down and 1 unit right, what is the equation of its

image?

35

IV. Scale Changes and Translations

In the last assignment you learned how to translate graphs. A translation is a sliding of a graph. In this assignment we

will investigate a different kind of transformation called a scale change. Again, show all work and answers on a

separate sheet of paper.

1a. Complete the following table for the functions f(x)=x2 and g(x)=4x

2

x f(x)=x2 g(x)=4x

2

-2

-1

0

1

2

b. Given the completed table above, explain the change between the (x,y) coordinates for f(x) and g(x).

c. Describe how the graph of f(x) would change to become the graph of g(x).

2a. Complete the following table for the functions f(x)= x and g(x)=1

2x

x f(x)= x g(x)=

1

2x

0

1

4

9

16



For #3, circle the correct word to complete these sentences.

3a. c●f(x), if c > 1 causes the parent graph to vertically stretch / compress

3b. c●f(x), if 0 < c < 1 causes the parent graph to vertically stretch / compress

36

4a. Notice the difference between g(x) from problem 2 g(x)= 1

2x and h(x) in this problem h(x)=

1

2x . Complete

the following table for the functions f(x)= x and g(x)=1

2x

x f(x)= x x g(x)=

1

2x

0 0

4 4

9 9

16 16

25 25



Circle the correct words to complete the following sentences.

6. Describe the transformation that takes place from the parent function f(x)=|x| to g(x)=|3x|.

7. Describe the transformation that takes place from the parent function f(x)=x3 to j(x)=5x

3.

You will often see many transformations taking place at one time in one function. The order of the transformations is

important otherwise the image may end up in the wrong place. The following rules should be followed when listing

multiple transformations:

• If the transformation is a vertical change (translation, scale change or reflection) then you should list the

transformations using the order of operations (PEMDAS)

• If the transformation is a horizontal change (translation, scale change or reflection) then you should list the

transformations using the OPPOSITE order of the order of operations (SAMDEP)

For example: Using the parent function f(x)=x2, find the transformations that takes place with the function g(x)=2(3x-

1)2-5.

X-transformations Y-transformations

1. translate right 1 unit 1. vertically stretch by 2 times

2. horizontally compress by a factor of 3 2. translate down 5 units

5a. f(c●x), if c > 1 causes the parent graph to horizontally stretch / compress

5b. f(c●x), if 0 < c < 1 causes the parent graph to horizontally stretch / compress

37

8. Describe the transformations that take place. Be sure to list the transformations in the correct sequence

( ) 3 1 4f x x= + − then graph the function on a piece of graph paper.

9. Describe the transformations that take place. Be sure to list the transformations in the correct sequence.

( ) 3( 1 4)f x x= + − then graph the function on a piece of graph paper.

10. Use the graph of g(x) to the right to sketch the graphs below.

a. y= ½ g(x) b. y= 2g(x-1)-3

38

V. Reflections and other transformations

1a. Complete the following table for the two graphs presented below.

c. Complete the following sentence by circling the correct words.

Another way of showing this is g(x)= -f(x)

2. If you wanted to reflect a graph over the y-axis as shown below, which coordinates would need to multiply by -1?

g(x)

x f(x) g(x)

-1

0

1

2

b. What do you notice about the

relationship between the values of

f(x) and g(x)?

If all of the y-values of a function are multiplied by -1, then the graph will go

through a reflection over the x / y-axis.

g(x)

This could also be written as g(x)=f(-x)

39

The following questions will be combining a few different ideas from the past few honors projects.

3. Given the graph of function f(x) below, answer the following questions:

a. The function g defined by ( ) ( ) 2g x f x= + will have a minimum point at (x, y) = _______________

b. The function h defined by ( ) ( 1)h x f x= − − will have a minimum point at (x, y) = ________________

c. The function q defined by 1

( ) ( ) 12

q x f x= + has a range of ________________

4. Describe the transformations that take place if the parent function is ( )f x x= and it is transformed

to ( )( ) 2 3 1g x x= − + − . Make sure you list the transformations in the correct order.

5. State the domain of ( ) 3p x x= − ______________

40

Problem Set #4 SEQUENCES and SERIES

Set Number Name Page #s from green

book

Due Date

1 I. Sequences 440 – 443 + SAT II

2 II. & III.

Arithmetic & Geometric

Sequences 444 – 453 + SAT II

Corrections Set 1 440 – 443 + SAT II

3 IV. & V.

Arithmetic & Geometric

Series 454 – 462 + SAT II

Corrections Set 2 444 – 453 + SAT II

4 VI. Infinite Geometric Series 464 - 468 + SAT II

Corrections Set 3 454 - 462 + SAT II

5 Corrections Corrections Set 4 464 - 468 + SAT II

REFERENCE: Mathematics a Topical Approach, Course 3, Bumby and Klutch,

I. SEQUENCES, Section 13.1

A sequence is a function whose domain is a set of consecutive natural numbers.

Read pages 440 – 442.

Do WRITTEN page 442: 6, 14, 17, 18, 32

If tn = 2n

n + 1, find the first five terms of the sequence.

II. ARITHMETIC SEQUENCES, Section 13.2

A sequence is arithmetic if and only if each term after the first is obtained by adding the same number, d, to the

preceding term. The number d is called the common difference.

Read pages 444 - 446.

Do WRITTEN page 447: 4, 8, 17, 29, 43

If a1 = 2 and an = an−1

2, find the first five terms of the sequence.

If a8 = 4 and a12 = -2, find the first three terms of the artithmetic sequence.

III. GEOMETRIC SEQUENCES, Section 13.3

A sequence is a geometric sequence if and only if each term after the first is obtained by multiplying the preceding

term by the same number, r. The number r is called the common ratio.


Do WRITTEN page 453: 7, 12, 17, 21, 22

Find the seventh term of the geometric sequence 1, 2, 4, … , and the sum of the first seven terms.

The first term of a geometric sequence is 64, and the common ratio is ¼. For what value of n it tn = ¼?

41

IV. ARITHMETIC SERIES, Section 13.4

The sum of the indicated terms of an arithmetic sequence is called an arithmetic series. For example, given the

arithmetic sequence 2, 4, 6, 8, 10, 12, the sum, S6, of 2 + 4 + 6 + 8 + 10 + 12 is an arithmetic series of 6 terms.


Do WRITTEN page 457: 2, 7, 10, 12, 22, 25

Find the sum of the first 28 terms of the series, 2, 5, 8,…

In an arithmetic series, if Sn = 3n2 + 2n, find the first three terms.

V. GEOMETRIC SERIES, Section 13.5

The sum of the indicated terms of a geometric sequence is called a geometric series.

Read pages 459 – 460.

Do WRITTEN page 461: 5, 10, 14, 16, 20

A geometric sequence is 1, 2, 4, … , and the sum of the first seven terms.

VI. INFINITE GEOMETRIC SERIES, Section 13.6

If an infinite geometric series has a common ratio, r, which is less than 1, then the series has a sum.


Do EXPLORATORY page 467: 3, 6, 8, 12, 14

Do WRITTEN page 468: 4, 8, 10, 12, 22

Find the value of the infinite geometric series: 4 + 2 + 1 +1

2+

1

4+

1

8+ ...

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