Math Fundamentals for Statistics I (Math 52) · PDF fileThe Math 52 curriculum is designed to force you out of the box you consider to be mathematics. ... 1.5: Geometric ... Molly

Math 52 – Unit 1 – Page 1

Math Fundamentals for

Statistics I (Math 52)

Unit 1: Problem Solving

and Patterns

By Scott Fallstrom and Brent Pickett

“The ‘How’ and ‘Whys’ Guys”

This work is licensed under a Creative Commons Attribution-

NonCommercial-ShareAlike 4.0 International License

3rd Edition (Summer 2016)


1.0: Something About Learning

A) What is mathematics all about (in your experience)?

B) What feelings do you associate with “math?” Do you believe you are good at math?

C) What is something you are good at? (cooking, biking, etc.)

D) How long have you been working that skill/talent?

E) What do you think separates people who are good at math from people who are not as good at

math?

F) Why is homework assigned – and what are the best/worst parts about homework?

G) Do you ever have the situation where you seem to be understanding while you are in class but

then when you get home, the same questions look like a foreign language? What classes do

you have this happen in?


One of the main reasons that we forget something is that we don’t have the time to put it from our

short term memory into our long term memory. Our minds are subjected to a massive amount of

information every minute, hour, and day. Maybe you can remember every car you saw on the way to

school today, but probably not. Our short term memory releases most things quickly so that we aren’t

overwhelmed. That may also mean that we lose some of what we want to learn and have to train our

brain a new way to deal with information we want to remember.

This was studied by Hermann Ebbinghaus in the late 1800’s, and showed that this is actually normal

– people forget what they have learned unless they actively work to retain it.

The stronger a memory is, the longer a person is able to retain information. For this math course, we

have designed this text to work differently. Concepts are presented and then examples are done – both

working with the instructor and on your own or in small groups. This creates many opportunities to

practice the concepts in the classroom.

Math 52 and Math 95 are each 4 units, but have 5 hours in the classroom because part of the time is

structured as a “lab” – where we are actively working to learn the ideas, as opposed to “lecture” –

where a teacher talks to you and you listen. Let’s face it, the “lecture” method is not effective for most

students.

Here’s how to make the learning last longer and be better retained:

1. Be an active participant in class. When working, if you do not understand something, raise

your hand so the instructor or an imbedded tutor can come and help clarify.

2. Start homework as soon after class as possible. If you get stuck, bring in questions to ask – it’s

amazing how many other students may have the same question.

3. Get help outside of class. There are tutors in TASC and the Math Learning Center who are

able to help you get back on track.

4. Be prepared for class – we forget quicker when we are sleepy or distracted, try to put yourself

in the best position to succeed by getting enough sleep and removing any distractions you can.

a. What distractions could keep you from being prepared for class?

5. Do some math every day. Spaced repetition is extremely beneficial – doing some each day

instead of cramming will help you retain more of the information.


Table of Contents

1.1: A First Puzzle to Play With ........................................................................................................... 5

The Math 52 curriculum is designed to force you out of the box you consider to be mathematics.

The first problems may look nothing like mathematics at all, but we start here to build

connections as our knowledge grows.

1.2: Problem Solving Strategies ......................................................................................................... 11

One of the themes of the course is understanding that there is not only one way to do

mathematics. We cover multiple ways to solve problems to give you the tools to attack

problems in the future.

1.3: Number Patterns .......................................................................................................................... 17

Exploration with patterns requires experience and we hope to give you plenty of experience

with these patterns. However, we also hope to bring you to understand the idea that there may

be more than one pattern that matches a particular group of numbers.

1.4: Arithmetic Sequences .................................................................................................................. 19

With some knowledge of patterns, we now begin to group similar ideas together. Some patterns

require addition (or subtraction) and we study these patterns here.

1.5: Geometric Sequences .................................................................................................................. 23

Continuing to group similar ideas, we see that some patterns require multiplication (or division)

and we study those patterns in this section.

1.6: Connections and Other Patterns ................................................................................................ 27

Not every pattern is arithmetic or geometric; it's good to know that and to see that other patterns

both exist and are useful.

1.7: Inputs and Outputs...................................................................................................................... 29

Seeing the patterns in the previous sections allows us to begin viewing objects and patterns with

relationships; perhaps the most important and the one used most often is the input-output

relationship. This is also where we discover the concept of ordered pairs.

1.8: Graphing Patterns ....................................................................................................................... 32

While there is nothing wrong with input-output as groups of numbers, it is often good to be able

to visualize these numbers and graphing allows us to do just that! We have numbers, tables, and

now graphs.

1.9: Functions ..................................................................................................................................... 40

The input-output relationship can be formalized using new notation. This notation simplifies

description and helps us to communicate more quickly and more clearly about the information

we know.

INDEX (in alphabetical order): ......................................................................................................... 47


1.1: A First Puzzle to Play With

EXPLORE! Molly and the Bully

Molly is trying to avoid a bully and still get to school on time. She lives five blocks south of the

school and two blocks west (see diagram). To avoid the bully she must take a different route to

school every day, but she can only walk 7 total blocks without being late.

Determine if it’s possible to take a different route (always 7 total blocks) to school every day for 4

weeks (20 school days).

School

Molly


M

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The initial question asked about Molly moving north 5 blocks and 2 blocks east for a total of 7

blocks. That’s a pretty challenging question to find how many different ways she can make that trek.

Even though our goal was to find how many ways for the initial problem, there’s nothing stopping us

from solving a simpler problem like this:

Interactive Example 1: What if Molly lived two blocks west of the school; how many different

(smallest) routes are there from Molly’s house to school now?

Interactive Example 2: What if Molly lived two blocks west and one block south of the school; how

many different (smallest) routes are there from Molly’s house to school now? How does this relate to

the previous example?

Interactive Example 3: What if Molly lived two blocks west and two blocks south of the school; how



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Interactive Example 4: What if Molly lived two blocks west and three blocks south of the school;

how many different (smallest) routes are there from Molly’s house to school now? How does this

relate to the previous example?

Interactive Example 5: What if Molly lived two blocks west and four blocks south of the school; how



Interactive Example 6: What if Molly lived two blocks west and five blocks south of the school; how



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What we hope you see from this example, that while we are often asked to solve problems that can be

hard, we are in charge of the problem. We have the power to decide to solve simpler problems and

build up to the solution. Being able to take a challenging problem and break it down is a big part of

our course and we hope you’ll enjoy this tool in your math-tool-belt!

EXPLORE! Find the number of 1-by-1 squares in each big square, and answer the questions below.

A) ** Draw the 4th level and find the number of 1-by-1 squares in the 4th level.

B) Find the number of 1-by-1 squares in the 5th level.

C) Fill in the table for the pattern above:

Number of squares

in the top row 1 2 3 4 5 6 7 8 … n

Total number of

1-by-1 squares 1 4 9 49 …

1

1


This could also be listed in a different way by writing both table values side-by-side. The first column

would be written as (1, 1). The next column would be (2, 4). Fill in the next pieces using this format:

(1, 1), (2, 4), (3, 9), (4,______ ), (5, ______ ), (6, ______ ), (7, 49), (8, ______ ), …, (n, ______ )

D) ** If the number of squares in the top row of the table was 15, find the total number of 1-

by-1 squares. Use your calculator if necessary when the symbol shows up.

E) (L) If the number of squares in the top row of the table was 89, find the total number of 1-

by-1 squares.

F) (R) If the number of squares in the top row of the table was 250, find the total number of

1-by-1 squares.

G) If the total number of 1-by-1 squares is 441, find the number of squares in the top row.

H) If the number of squares in the top row of the table was M, find the total number of 1-by-1

squares.


1.2: Problem Solving Strategies

EXPLORE! The Parking Lot

20 children come to school, each bringing either a wagon (4 wheels) or a bicycle (2 wheels). Once

everyone arrives and the parking lot is full, there is a total of 52 wheels in the parking lot. How many

children rode their bike to school?

Be prepared to explain how you came up with your answer and how you checked your answer.


There are many strategies to solving problems. We will solve this previous problem using a few

different strategies/methods:

Method #1: Guess and check [Working on the initial problem with 20 children and 52 wheels.]

In this method you make an educated guess (or sometimes just any old guess will do) and then check

to see if that guess is right. Sometimes that guess can help you make a better guess to get closer to

your answer. We could start by guessing that half are bicycles and half are wagons.

Guess: 10 wagons and 10 bicycles

10 wagons have 40410 wheels

10 bicycles have 20210 wheels

Check: Together these add up to 60 wheels. The problem states that we only have 52 wheels, so we

need to reduce the number of wheels. To do that, we’ll need more bicycles and fewer wagons.

EXPLORE! Write down 2 or more guesses and add up the total number of wheels. Modify your

guesses as needed. See if you can get to the correct answer, and then check your answer to be sure it

is correct.

What would you change about your process if you still had 20 children, but now had 66 total wheels?

NOTE: Guess and check is not a great strategy, especially when the numbers are not whole

numbers. However, we feel it is a strategy most of you have seen before and it is always better to do

something than nothing when confronted by a problem. We hope that you’ll look at the next few

strategies and see that they are much more efficient than random guessing!


Method #2: Draw a picture. [Working on the initial problem with 20 children and 52 wheels.]

Here’s our parking lot.

A) ** If everyone brought a bike it would look like in terms of the wheels? Draw the

corresponding wheels in the parking lot above.

B) How many wheels are there in the picture?

C) How many more do you need to complete the problem?

D) Can you solve the problem with this picture? Explain how.

E) Could you solve this modified problem using this technique: there are still 20 children, but

now there are 70 wheels?


Method #3: Logic and computation [Working on the initial problem with 20 children and 52

wheels.]

If everyone had a bike we would have 40 wheels. We need 52 wheels so we need 12 more.

A) ** How many more wheels does a wagon have?

B) How many more wagons would we need to get 12 more wheels? Use this to come up with the

solution.

C) ** Let’s go the other direction and assume everyone had a wagon. That would be 80 wheels,

so we need only 52 which is 28 less. How many fewer wheels does a bicycle have?

D) How many wagons must be traded out for bicycles to get 28 fewer wheels? Use this to come

up with the solution.

EXPLORE! How about 20 children with 46 wheels? Could you solve this modified problem using

this technique?


Method #4: Make a comprehensive list of all options. [Initial problem with 20 kids and 52 wheels.]

Since there are only 20 parking spaces, we can list the number of bikes and wagons there will be in the

parking lot. After that, we can count up the number of wheels from the bikes and the wagons, and then

find the total number of wheels. Some values have been filled in, so fill in any missing values.

Number of

bikes

Number of

bike wheels

Number of

wagons

Number of

wagon wheels

Total number

of wheels

20 40 0 0 40

19 38 1 4 42

18 36 2 8 44

17 34 3

16 32 4

15 30 5

14 6

13 7

12 8

11 9

10 10

9 11

8 12 48

7 13 52

6 12 14 56 68

5 10 15 60 70

4 8 16 64 72

3 6 17 68 74

2 4 18 72 76

1 2 19 76 78

0 0 20 80 80


Method #5: Write an equation. [Initial problem with 20 kids and 52 wheels.]

If x represents the number of bikes and the total number of vehicles is 20, then x20 represents the

number of wagons. Look at the table on the previous page to see if this matches up with the table.

Since each bike has two wheels, the number of bike wheels can be written as x2 . Similarly, the

number of wagon wheels can be written as x204 .

So, if we count the bike wheels, and count the wagon wheels they need to add up to 52 which can be

written as the equation: 522042 xx

Example: Solve the equation: 522042 xx .

14

2

28

2

2

282

805280280

52280

524802

522042

x

x

x

x

x

xx

xx

Since x = 14, that means 14 is the number of bicycles. With 20 total vehicles, there must be 6 wagons.

You may have seen this algebraic notation in another math class, but it is not expected that you can

solve this equation on your own at this point. In fact, we won’t be solving equations until Unit 7. So

take heart: you will not be given test questions using these skills until they have been taught to you in

full.

For Love of the Math: Many problems in math and science classes, as well as real life problems,

can be solved using the techniques shown here. There are also other methods of solving problems as

well as these five methods: look for a pattern, find a similar problem, work backwards, solve a

simpler problem, and make a diagram/graph. One goal of this class is to help students feel more

comfortable starting a problem they haven’t seen before. Remember these techniques as we continue

through the course!


1.3: Number Patterns

There are lots of patterns involving numbers, and some are more important than others. First, we will

work on identifying patterns, finding terms in patterns, and then we will look more closely at some

special patterns.

EXPLORE! Look at the start of the list of numbers. Try to find a pattern and write out the next few

terms that would continue your pattern.

A) ** 2, 4, 6,

B) ** 26, 27, 28,

C) (L) 3, 6, 12,

D) (R) 64, 32, 16, 8,

E) (L) 3, 2, 1,

F) (R) 1, 1, 2, 3, 5, 8,

Interactive Example:

A) Did you use a problem solving technique to finish these patterns? Which one(s)?

B) Is there more than one way to continue the patterns? Which ones?


When there is a list of numbers, we often naturally want to continue a pattern. But just because we see a

way to continue the list doesn’t mean that it is the only way to continue the pattern. And it also doesn’t

mean that the way we see to continue the pattern is the way that everyone else will see to continue the

pattern.


A) Think about the list of numbers that starts with 1, 2,… What number do you think comes next?

B) What math operation were you using to get from the first number to the second? Was it

addition? Multiplication? Something else?

C) If we gave you the clue that the next number was 4, does that change your thinking about how the

pattern was made? If so, what do you think the next term is now?

D) (L) If we told you that to get the next number, the rule “ multiply the number by itself and then

add 1.” With the pattern starting 1, 2, … what are the next three numbers?

E) (R) If we told you that to get the next number, the rule is “ multiply the number by three and

then subtract 1.” With the pattern starting 1, 2, … what are the next three numbers?

F) Now think back to the pattern that starts with 1, 2, … How many different “correct” patterns

could you create?


1.4: Arithmetic Sequences

EXPLORE! Fill in the missing boxes in this pattern.

1 2 3 4 5 6 7

3 7 11 15

As mentioned earlier, some patterns show up in the world more often than others. Because they tend to

show up often, we’ve given these pattern types special names. This is a very common trend in

mathematics – if something happens often, we’ll name it so we can quickly refer to it. The pattern in the

box above is the first of two pattern types we’re going to focus on.

A sequence of numbers is called arithmetic if the way to move from one term to the next is by adding

the same amount over and over again. Notice that in the box above, we moved from 3 to 7 by adding 4.

Then we moved from 7 to 11 by adding 4. If we kept adding 4, we’ll be able to fill in the next terms.

When we talk about an arithmetic sequence in general, we use variables to hold the place of the many

different numbers that could be used. A variable is a symbol or letter that could represent many different

numbers. The first term in the sequence is often referred to by writing a and the amount added to get

from one term to the next is called the common difference and is referred to by writing the variable d.

EXPLORE! Circle whether the sequence is arithmetic. For those that are arithmetic, identify a and d.

Sequence Arithmetic? a d

A) ** 2, 4, 8, 16, 32 Yes No

B) (L) 5, 8, 11, 14, 17, 20 Yes No

C) (R) 55, 52, 49, 46, 43, 40 Yes No

D) (L) 1, 1, 2, 3, 5, 8, 13, 21 Yes No

E) (R) 3, 3, 3, 3, 3, 3, 3 Yes No

F) (L) 0, – 2, 4, – 6, 8, – 10, 12 Yes No

G) (R) – 1, – 3, – 5, – 7, – 9, – 11 Yes No

So in general, we’ll add d to get from one term to the next. Let’s see how this could work with an

example using variables.


Interactive Example: Using a first term of a and the common difference of d, write the terms of an

arithmetic sequence.

Term 1 2 3 4 5 … n

Value a …

Sometimes we do use a mathematical short-hand for this to make it quicker: when we write the first term

of a sequence, we’ll write 1a . The second term would be written 2a , the third term as 3a , and the nth

term as na . If we wanted to find the 37th term, we would be looking for 37a .

This shows there are two ways to write the first term in an arithmetic sequence: a or 1a . Because it is

shorter to write, we mostly use a to represent the first term.

Example: Interpret the given information and then find the missing piece: 14a and 3d . Find 3a .

Solution: This information indicates that the first term in the sequence is 14 and the common difference

is 3. So the sequence would be 14, 17, 20, … This means 203 a .

EXPLORE! Interpret the given information and then find the missing piece.

A) ** 141 a and 5d . Find 2a .

B) 6 11a and 2d . Find 9a .

C) (L) 405 a and 50d . Find 7a .

D) (R) 147 a and 11d . Find 5a .

Example: Create an arithmetic sequence where a = 2, and d = 5; find the first 5 terms and the nth

term.

Solution: Since the first term is 2 and the common difference is 5, we can start writing the terms out

as we go, adding 5 each time to get to the next value. The sequence would be 2, 7, 12, 17, 22, … and

the nth term is 355525121 nnndnaan , which simplifies to 35 nan .


EXPLORE! Give the first 3 terms of the following arithmetic sequence and the nth term, na .

A) ** a = 3 and d = 4 na =

B) (R) a = 14 and d = – 3 na =

C) (L) a = 31 and d = 2 na =

EXPLORE! Which of the patterns from the first page in section 1.3 could be arithmetic? The list is

shown here. Explain your conclusion.

A) ** 2, 4, 6,

B) 26, 27, 28,

C) (L) 3, 6, 12,

D) (L) 64, 32, 16, 8,

E) (R) 3, 2, 1,

F) (R) 1, 1, 2, 3, 5, 8,

Since we know that the nth term is dnaan 1 , we can find specific terms for any arithmetic

sequence. Let’s see how to use this to find a specific term.

Example: Find the 5th term for the sequence with a = 6 and d = 2.

Solution: Using the formula above, 21565 a .

We simplify this by the following steps:

21565 a

2465 a

865 a

145 a So the 5th term in the sequence is 14.


We could also have done this visually by making a chart. Making a chart/list works well when there

are just a few terms to consider. Write out the numbers to find the 5th term, adding 2 each time. What

is the 5th term?

1 2 3 4 5

6

Since we could write out terms or use a formula for these problems, it will be up to you to determine

which approach is the best one to use depending on the information given to you. Let’s practice

deciding which one to use.

EXPLORE! Determine whether you would rather use the formula or write the terms out to find the

requested information. Put a in the box you would rather do. Neither answer is wrong, but often

one method is longer than the other. Try to think which method would be most efficient.

Write out the terms Use the formula

A) ** Find 4a when a = 20 and d = 5.

B) Find 6a when a = 40 and d = – 5.

C) Find 4a when 6a = 40 and d = 5.

D) Find 49a when a = 30 and d = 1.

E) Find 25a when a = 37 and d = 11.

EXPLORE! Find the term that is requested.

A) ** Find 4a when a = 20 and d = 5.

B) (L) Find 6a when a = 40 and d = – 5.

C) (R) Find 4a when 6a = 40 and d = 5.

D) (L) Find 49a when a = 30 and d = 1.

E) (R) Find 25a when a = 37 and d = 11.


1.5: Geometric Sequences

We have dealt with arithmetic sequences, but what about sequences that aren’t arithmetic.

Interactive Example: Answer these questions about 2, 4, 8, 16, 32.

A) Would this sequence be considered arithmetic? Why or why not?

B) If you were describing the sequence to someone, how would you describe moving from one

term to the next?

A sequence of numbers is called geometric if the way to move from one term to the next is by

multiplying by the same amount over and over again. Notice that in the sequence above, we moved

from 2 to 4 by multiplying by 2. Then we moved from 4 to 8 by multiplying by 2. If we keep

multiplying by 2, we’ll be able to fill in the next terms.

When we talk about an geometric sequence in general, we again use variables to hold the place of the

many different numbers that could be used. The first term in the sequence is often referred to by

writing a and the amount multiplied to get from one term to the next is called the common ratio and

is referred to by writing r.

EXPLORE! Circle whether the sequence is geometric. For those that are geometric, identify a and

r.

Sequence Geometric? a r

A) ** 1, 2, 4, 8, 16, 32 Yes No

B) ** 5, 8, 11, 14, 17, 20 Yes No

C) (L) 16, 8, 4, 2, 1, 1 1

,2 4

Yes No

D) (R) 1, 1, 2, 3, 5, 8, 13, 21 Yes No

E) (L) 3, 3, 3, 3, 3, 3, 3 Yes No

F) (R) 3, – 3, 3, – 3, 3, – 3, 3 Yes No


So in general, we’ll multiply by r to get from one term to the next. Let’s see how this could work with

an example using variables.

Interactive Example: Using a first term of a and the common ratio of r, write the terms of an

geometric sequence.

Term 1 2 3 4 5 … n

Value a …

Sometimes we do use a mathematical short-hand for this to make it quicker: when we write the first

term of a sequence, we’ll write 1a . The second term would be written 2a , the third term as 3a , and

the nth term as na . We also write rr as 2r ; in 2r we refer to r as the base and 2 as the exponent.

Example: Interpret the information and then find the missing piece: 41 a and 3r . Find 3a .

This information indicates that the first term in the sequence is 4 and the common ratio is 3. So the

sequence would be 4, 12, 36, … This means 363 a .

EXPLORE! Interpret the given information and then find the missing piece.

A) ** 141 a and 2r . Find 2a .

B) ** 171 a and 11r . Find 3a .

C) (R) 111 a and 2r . Find 2a .

D) (L) 405 a and 50r . Find 7a .

E) 287 a and 2r . Find 5a .


Example: Create a geometric sequence where a = 5, and r = 2; find the first 5 terms and the nth term.

Solution: Since the first term is 5 and the common ratio is 2, we can start writing the terms out as we

go, multiplying by 2 each time to get to the next value. The sequence would be 5, 10, 20, 40, 80, and

the nth term is 11 25 nnn raa .

For geometric sequences, the final result is more challenging to simplify and as a result, we often just

leave it as it is written.

EXPLORE! Give the first 3 terms of the following geometric sequence and the nth term, na .

A) ** a = 3 and r = 4 na =

B) ** a = 3 and r = – 2 na =

C) (R) a = 2 and r = 5 na =

D) (L) a = 2 and r = – 3 na =

Since we know that the nth term is 1 nn raa , we can find specific terms for any geometric

sequence. Let’s see how to use this to find a specific term.

Example: Find the 5th term for the sequence with a = 6 and r = 2.

Solution: Using the formula above, 155 26 a .

We simplify this by the following steps: 15

5 26 a

45 26a

1665 a

965 a The 5th term in the sequence is 96.


We could also have done this visually by making a chart. Making a chart/list works well when there

are just a few terms to consider. Write out the numbers to find the 5th term, multiplying by 2 each

time. What is the 5th term?

1 2 3 4 5

6

EXPLORE! Determine whether you would rather use the formula or write the terms out to find the

requested information. Put a in the box you would rather do. Neither answer is wrong, but often

one method is longer than the other. Try to think which method would be most efficient.

Write out the terms Use the formula

A) ** Find 2a when a = 20 and r = 5.

B) Find 6a when a = 4 and r = 3.

C) Find 7a when a = 11 and r = 2 .

D) Find 49a when a = 30 and r = 4.

EXPLORE! Find the term that is requested.

A) ** Find 2a when a = 20 and r = 5.

B) Find 6a when a = 4 and r = 3.

C) (R) Find 7a when a = 11 and r = 2 .

D) (L) Find 49a when a = 30 and r = 4.


1.6: Connections and Other Patterns

Sometimes we are given a sequence of numbers and need to find the pattern. We’ve seen both

arithmetic and geometric sequences so far, and each type had a formula that we could use to find

specific terms or even a general nth term.

Sometimes, there are other ways to find the nth term of a sequence by spotting patterns as you go.

Example: Find the nth term for the following sequence: 2, 4, 6, 8, 10, 12, …

Solution: If we take a moment to rewrite the terms, we might spot a nice pattern.

The 1st term is: 2(1) = 2

The 2nd term is: 2(2) = 4

The 3rd term is: 2(3) = 6

The 4th term is: 2(4) = 8

This continues as we increase the term, so our nth term is 2(n) = 2n.

It’s important to notice that this sequence started as arithmetic, and we could have written it using the

formula dnaan 1 where a = 2 and d = 2.We are now going to work on identifying some

patterns and writing the nth term.

EXPLORE! Could this pattern be arithmetic (A), geometric (G), or maybe neither (N)? Circle the

answer. If it is (A) or (G), write the nth term for the sequence.

Start of the Sequence Type of Sequence nth Term

A) ** 1, 2, 3, 4 A G N

B) ** 1, 4, 9, 16 A G N

C) (R) 5, 8, 11, 14, 17, 20 A G N

D) (L) 16, 8, 4, 2, 1, 1 1

,2 4

A G N

E) (R) 1, 1, 2, 3, 5, 8, 13, 21 A G N

F) (L) 3, 3, 3, 3, 3, 3, 3 A G N

G) (R) 1, 3, 9, 27 A G N

H) (L) 1, 2 A G N



Could a pattern be both arithmetic and geometric at the same time? Explain your answer.

EXPLORE! Now we continue with examining other patterns. Determine the next 3 possible terms

of these patterns.

A) ** ,3

3 ,

3

2 ,

3

1

B) ** ,5 ,4 ,3 ,2 ,1

C) (L) ,5 , 3 ,1

D) (R) – 2, – 4, – 8,

E) (R) – 1, 1, – 1,

F) (L) 2, 1, 0, – 1, – 2,

G) (R) 1.01, 1.02, 1.03,

H) (L) ,15

9 ,

12

7 ,

9

5 ,

6

3 ,

3

1

eeeee


1.7: Inputs and Outputs

In the past sections, we worked with patterns that came from a pair of numbers. We had the term

number and the value. 236 a would mean that the 6th term is 23. This concept could be applied to

more realistic situations, like buying apples.

Number of apples: 1 2 3 4 5 6 7

Cost: $ 0.50 $ 1.00 $ 1.50 $ 2.00 $2.50 $ 3.00 $ 3.50

Since the number of apples we buy corresponds directly to the cost of the apples, the pattern above

makes sense. However, we could write this information in a slightly different way that gives all the

information without needing to write the entire table out:

(1, 0.50), (2, 1.00), (3, 1.50), (4, 2.00), (5, 2.50), (6, 3.00), (7, 3.50)

These groups are called ordered pairs since they contain two numbers and the order of the numbers

matters. The two values are often called coordinates, since we have two numbers together “co” and

they are in order “ordinates”. Our table could even be modified to include these ordered pairs:

Number of

apples: 1 2 3 4 5 6 7

Cost: $ 0.50 $ 1.00 $ 1.50 $ 2.00 $2.50 $ 3.00 $ 3.50

Ordered pairs: (1, 0.50) (2, 1.00) (3, 1.50) (4, 2.00) (5, 2.50) (6, 3.00) (7, 3.50)

You will notice that you can’t determine the total cost until you know how many apples you’re

buying. In a grocery store you would have to put in the number of apples to get out the cost. We call

the number that naturally comes first the input and the second number the output.

So another way to write this grid would be:

Input: 1 2 3 4 5 6 7

Output: $ 0.50 $ 1.00 $ 1.50 $ 2.00 $2.50 $ 3.00 $ 3.50

Ordered pairs: (1, 0.50) (2, 1.00) (3, 1.50) (4, 2.00) (5, 2.50) (6, 3.00) (7, 3.50)

For Love of the Math: In later courses, like statistics, these are not called input and output.

Instead, since the second number depends on the first, the second is called the dependent variable (or

response variable), and the first number is called the independent variable (or predictor variable).

Relating this to the apple example, think about how we walk through the store and independently

select a number of apples; the cost of the apples depends on the number of apples we chose.


Example: Fill in the blank spaces of the table based on the pattern given being either geometric or

arithmetic.

Input: 2 4 6 8 10 12

Output: 1 2 4 32

Ordered pairs:

Solution: The output values seem to be geometric with ratio of 2, but let’s check that to see if it

matches with the rest of the pattern.

Input: 2 4 6 8 10 12

Output: 1 2 4 8 16 32

Ordered pairs:

It does, since the 32 fits perfectly! Now we can fill in the ordered pairs. [Note: There could be other

numbers that fill in the spaces too. Unless it is specifically stated to be one particular pattern, this

work is nothing more than an educated guess following one of the patterns we know about.]

Input: 2 4 6 8 10 12

Output: 1 2 4 8 16 32

Ordered pairs: (2, 1) (4, 2) (6, 4) (8, 8) (10, 16) (12, 32)

EXPLORE! Fill in the blank spaces of the table based the pattern given being either geometric or

arithmetic.

A) **

Input: 1 2 3 4 5 6

Output: 2 4 6

Ordered pairs:


B)

Input: 1 2 3 4 5 6

Output: 1 3 9 27

Ordered pairs:

C)

Input: 0 5 10 15 20 25

Output: 0.2 0.4 0.6

Ordered pairs:

D)

Input: 1 2 3 4 5 6

Output: 1 – 1 – 1

Ordered pairs:


1.8: Graphing Patterns

Drawing pictures was one of our strategies for solving problems in section 1.2. We could draw a

picture called a graph for single numbers. If there is a single number, we could graph it on a number

line which shows the size and location of a number. We indicate the scale on the graph so we know

where different points will be located.

Example: Graph the number 7 on the number line below:

Solution:

We can count in from the left until we find the mark that would represent 7. Then write the symbol 7

in the space below the correct mark.

As you can see, the 0 and the 10 give us the scale, so we know exactly where to put to the 7.

With the ordered pairs that we were just dealing with, each coordinate requires a number line. So for

a point like (2, 7) we need a number line to represent the first coordinate (2) and a different number

line to represent the second coordinate (7).

When doing this, we write the first number line horizontally (↔) while the second number line is

written vertically (↕). For an ordered pair like (2, 12), we can find the location by moving

horizontally until we get to the number 2 and then moving up or down vertically until we get to the

number 12.

By graphing two number lines in this way, we end up with a grid-like shape. That grid gives a layout

and allows us to quickly find location. In one sense, this might remind you of the game “Battleship”

where a row and column is selected.

An ordered pair will then correspond to exactly one location, which is called a point.

Interactive Example: Describe (3, 11) in words using: coordinate, input, output, point, and ordered

pair.

0 10 7

0 10


Interactive example: Find the appropriate letter that corresponds to the ordered pair and write it in the

box.

Input: 2 4 6 8 10 12

Output: 12 10 8 6 4 2

Ordered pairs: (2, 12) (4, 10) (6, 8) (8, 6) (10, 4) (12, 2)

Letter A

A) Can you see both number lines in the picture above – one for the input and one for the output?

B) Describe (in words) how to find the point A so someone who is new to this could do it.

C) Does your description require a “starting” point? Where did you start to end up at point A using

your description above?

This starting point is often referred to as the origin. What are the coordinates of the origin?

0 0 14

4

7

20

12

Input

Output A

D

E

C

F

B

8


EXPLORE!

Find the letter that corresponds to the given point, if possible.

Ordered pair Letter (Point) Ordered pair Letter (Point)

A) ** (6, 3) B) (2, 5)

C) (9, 7) D) (19, 15)

E) (4, 13) F) (12, 5)

Now backwards: Find the coordinates of the following points:

Letter (Point) Ordered pair Letter (Point) Ordered pair

G) ** H H) K

I) M J) D

Origin or (0, 0)

5 20

15 10

5

10

15 M

A

B

C

D

E

F

G H

K

J

N

L


EXPLORE! Work with a partner – one person turn back a page and one person keep this page open.

All these questions refer to the graph on the previous page:

A) (L) List all the points that have 2 as their second coordinate.

B) (R) List all the points that have 5 as their first coordinate.

C) (L) List all the points (letters) that have their first coordinate greater than 11.

D) (R) List all the points that have their second coordinate less than 4.

E) List all the points that have their first coordinate between 6 and 13. [The numbers 7, 8, 9, 10,

11, and 12 are between 6 and 13.

F) List all the points that have their second coordinate between 6 and 13.

G) List all the points that have their first coordinate between 4 and 14 while also having their

second coordinate between 1 and 8.


We can also use this pattern for graphs that have both positive and negative numbers in them. In these

cases, the number lines extend in both directions. Take a look at the graph below:

Example: Find the ordered pair that represents point D.

In order to find D, just move left 2 and up 1 from the origin. This represents the ordered pair (– 2, 1).

EXPLORE!

Find the letter that corresponds to the given point, if possible.

Ordered pair Letter (Point) Ordered pair Letter (Point)

A) ** (– 5, 0) B) (– 5, – 3)

C) (3, – 1) D) (0, 3)

Now backwards: Find the coordinates of the following points:

Letter (Point) Ordered pair Letter (Point) Ordered pair

E) ** I F) E

G) J H) G

I) A J) Origin

K) Draw in these points on the graph above: M** (– 1, 6), N (– 3, – 4), P (5, – 5), and Q (2, 6)

A

B

C

D

E

F

G

H

I

J

6

– 6 6

– 6


Now it’s time to look at some of the patterns we’ve seen and think about how we would graph them.

EXPLORE!

A) ** Consider the arithmetic pattern given by a = – 5 and d = 1. Graph and label these points;

do your best to smoothly connect the points when done.

A B C D E F G

Input 1 2 3 4 5 6 7

Output – 5 – 4 – 3 0

B) Also, on the same grid graph the arithmetic pattern given by a = 5 and d = – 1. Graph and

label these points; do your best to smoothly connect the points when done.

H I J K L M N

Input 1 2 3 4 5 6 7

Output 5 2 – 1

C) Now explain the type of graph that an arithmetic pattern creates (what does it look like):

6

0

- 6

12

1


EXPLORE! Graph these points; do your best to smoothly connect the points when done.

A) ** Graph the arithmetic pattern given by a = 1 and d = 2, we would have:

Input 1 2 3 4 5 6

Output 1 3 5 9

B) On this same grid, graph the arithmetic pattern with a = 12 and d = – 2.

Input 1 2 3 4 5 6

Output 12 10 4

C) Do these graphs match the arithmetic pattern we saw on the previous page?

12

12

6

6


EXPLORE! Graph these points; do your best to smoothly connect the points when done.

A) ** Graph the geometric pattern given by a = 1 and r = 2, we would have:

Input 1 2 3 4 5 6

Output 1 2 32

B) Graph the geometric pattern given by a = 1 and r = 3, we would have:

Input 1 2 3 4 5 6

Output 1 3

C) What are differences between the graph of an arithmetic pattern and a geometric pattern?

60

0

20

6 0 12

40


1.9: Functions

In a previous section, we saw that the cost of apples depended on how many apples we wanted to buy.

Similarly, the amount of money you are paid at work depends on the number of hours you work. If

you worked 30 hours one week, you’d be paid a certain amount. And if you worked 30 hours the next

week, you’d expect to be paid the same amount.

This type of relationships between certain quantities can be more clearly defined. In mathematics,

a function is a relation between a set of inputs and a set of outputs with the property that each input is

related to exactly one output. We’ve been graphing functions for the last few pages without knowing

it!

EXPLORE! Determine which of the following tables are functions. Explain your answers.

A) ** B) C)

Input Output Input Output Input Output

1 5 3 – 2 6 11

2 8 4 3 8 11

3 12 7 10 10 11

4 2 3 7 5 – 3

EXPLORE! Determine which of the following graphs are functions. Explain your answers.

A) **

B)

C)

In some of our graphs, we let x be the input and y be the output, but the most common notation for a

function is to have x as the input and xf as the output. This is called function notation, and the f

refers to the function’s name.


EXPLORE! Identify the following information from function notation.

Function

Notation Function Name Input

Output

(one way)

Output

(another way)

A) xxf 7 f x 7x f x

B) 24 mmP

C) 357 A

D) 435 ppQ

E) 28xW

Imagine having a function which took every input and doubled it. In function notation we would write

that relationship as: xxf 2 . This notation is useful to find output values when we know input

values. We could look at this function and see that the input of 1 corresponds to an output of 2. Let’s

look at how to use this new notation to find the output value when we know the input value.

Example: Find 1f using the function xxf 2 .

Step 1 Step 2 Step 3 Step 4

Start with the function

notation.

Place ( ) where x was

in the output Insert the input of 1. Simplify.

xxf 2 2f 121 f 2121 f

Example: Find 3f and 4f using the function xxf 2 .

Solutions:

xxf 2 2f 6323 f and xxf 2 2f 8424 f

Notice that when replacing the input x with a number, we always write a set of parenthesis ( )

wherever there is an x; after that, we write the number inside the ( ). Without this step, we could

make mistakes in the future so it is critical that we make this step.

Here’s what would happen if we found the value of 4f without using the parenthesis.

2424 f , which makes it look like the coordinates would be (– 4, – 2) instead of (– 4, – 8).

It’s important to be careful when working with functions – slow down, and take your time!


EXPLORE!

A) Find xf for x = 1**, – 3, and 9, using the function 23 xxf .

B) Find mg for m = – 1**, – 2, and 4, using the function mmg 7 .

Functions are used to describe many real world applications – even ice cream!

EXPLORE! You want to start the Deluxe Cookie Ice Cream Bar Company. Your research shows

that it takes $5 worth of ingredients to make each bar, and $25 per hour for payroll and utilities for

your small factory. We can describe the cost, C, to make x ice cream bars per hour with the function:

255 xxC .

A) ** What is the value of 2C ? Once you find it, interpret what this represents.

B) (L) Find and interpret 20C .

C) (R) Find and interpret 500C .

EXPLORE! Evaluate 4R ** and 900R for xxR .


Formulas for certain quantities, like the area of a circle, can also

be written in function notation. The area of a circle is dependent

on the radius, and we could use the function: 2rrA . Since

this is a fairly complex formula, mathematicians often create

tables to represent the input and output values. Here’s what it

would look like for a circle with different radii the area is rounded

to two decimal places. Note: A circle with radius 1 has a lot less

area than a circle with radius 4, as seen in these pictures.

Area of a Circle:

r Area r Area r Area

0.5 0.79 5 78.54 9.5 283.53

1 3.14 5.5 95.03 10 314.16

1.5 7.07 6 113.10 10.5 346.36

2 12.57 6.5 132.73 11 380.13

2.5 19.63 7 153.94 11.5 415.48

3 28.27 7.5 176.71 12 452.39

3.5 38.48 8 201.06 12.5 490.87

4 50.27 8.5 226.98 13 530.93

4.5 63.62 9 254.47 13.5 572.56

Example: Use the area table to find the area of a circle with the radius 8.5 inches (to two decimal

places).

Find the space in the table corresponding to a radius of 8.5, and locate the area next to it.

r Area

8 201.06

8.5 226.98

9 254.47

The area of a circle with radius 8.5 inches is approximately 226.98 square inches.

EXPLORE! Use the area table to answer the following:

A) ** Find the area of a circle with radius 2.5.

B) Find the area of the circle with radius 4.5.

C) Find the radius for a circle with approximate area of 113.10.

1

4


The area of a circle depends only on the radius, other functions might need more than just one input.

For a rectangle, we need to know the length and the width of the rectangle in order to calculate the

area. We could write this in function notation as: wlwlA , .

A table could be created to quickly find the area without having to repeat the calculations.

Area of a Rectangle:

Wid

th

Length

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45

4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105

8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120

9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135

10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165

12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180

13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195

14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210

15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225

Example: To find the area of a rectangle with width 7 and length 13, we would find where those two

meet in the table. That intersection is shown in the table above by following the shaded portions.

EXPLORE! Use the area table to find the area of a rectangle with:

A) ** length 5, width 3

B) (R) width 5, length 7

C) (R) length 13, width 14

D) (L) length 14, width 13

E) (L) width 14, length 5

Use the area table to find the length and width if the rectangle has:

F) ** Area = 121 G) Area = 180 H) Area = 60


Using tables to look up information is important here, but is extremely important later in statistics.

Statistics tables are complicated functions that have been computed for you. With circle area, we had

one variable. With rectangle area, we had 2 variables. In statistics, the challenge of the functions is

that we have to know 3 pieces of information to get one value out!

In the t-distribution chart below we have to know if the value is:

1) how many degrees of freedom you have

2) in one tail or 2 tails

3) what the area in the tail(s) is

For example, if we have 6 degrees of freedom, in 1 tail, with an area of 0.05, the output from the table

would be 1.943 (where the grey regions meet as shown below).

t Distribution: Critical t values

Area in one tail

0.005 0.01 0.025 0.05 0.10

Degrees of

Freedom

Area in two tail

0.01 0.02 0.05 0.10 0.20

1 63.657 31.821 12.706 6.314 3.078

2 9.925 6.965 4.303 2.920 1.886

3 5.841 4.541 3.182 2.353 1.628

4 4.604 3.747 2.776 2.132 1.533

5 4.032 3.365 2.571 2.015 1.476

6 3.707 3.143 2.447 1.943 1.440

7 3.499 2.998 2.365 1.895 1.415

8 3.355 2.896 2.306 1.860 1.397

9 3.250 2.821 2.262 1.833 1.383

10 3.169 2.764 2.228 1.812 1.372

To help with a table like this, be sure that you do the following:

1) Identify which area group you are working with (one tail or two tails) at the top of the table.

2) Find the column that corresponds to the amount of area.

3) Follow that column down until you intersect with the degrees of freedom you desire.

.


t Distribution: Critical t values

Area in one tail

0.005 0.01 0.025 0.05 0.10

Degrees of

Freedom

Area in two tail

0.01 0.02 0.05 0.10 0.20

1 63.657 31.821 12.706 6.314 3.078

2 9.925 6.965 4.303 2.920 1.886

3 5.841 4.541 3.182 2.353 1.628

4 4.604 3.747 2.776 2.132 1.533

5 4.032 3.365 2.571 2.015 1.476

6 3.707 3.143 2.447 1.943 1.440

7 3.499 2.998 2.365 1.895 1.415

8 3.355 2.896 2.306 1.860 1.397

9 3.250 2.821 2.262 1.833 1.383

10 3.169 2.764 2.228 1.812 1.372

EXPLORE! Find the appropriate critical t-value using the table above.

A) ** Area in two tails at a 0.05 level with 2 degrees of freedom.

B) (L) Area in two tails at a 0.05 level with 9 degrees of freedom.

C) (L) Area in one tail at a 0.05 level with 9 degrees of freedom.

D) (R) Area in one tail at a 0.005 level with 3 degrees of freedom.

E) (R) Area in two tails at a 0.10 level with 7 degree of freedom.

F) How could you get out a critical t-value of 2.998? Is there more than one way?


INDEX (in alphabetical order):

area table ....................................................... 43

arithmetic ...................................................... 19

base ............................................................... 24

between ......................................................... 35

common difference ....................................... 19

common ratio ................................................ 23

coordinates .................................................... 29

dependent variable ........................................ 29

Draw a picture ............................................... 13

exponent ........................................................ 24

function ......................................................... 40

function notation ........................................... 40

geometric ....................................................... 23

Guess and check ............................................ 12

horizontally .................................................... 32

independent variable ...................................... 29

input ............................................................... 29

Logic and computation .................................. 14

Make a comprehensive list ............................ 15

number line .................................................... 32

ordered pairs .................................................. 29

origin ............................................................. 33

output ............................................................. 29

point ............................................................... 32

t-distribution .................................................. 45

variables......................................................... 19

vertically ........................................................ 32

Write an equation .......................................... 16

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Math Fundamentals for Statistics I (Math 52) · PDF fileThe Math 52 curriculum is designed to force you out of the box you consider to be mathematics. ... 1.5: Geometric ... Molly