This book would not exist if not for “Discrete and Combinatorial Mathematics” by Richard Grassl and Tabitha Mingus. Some of the best exposition and exercises here were graciously donated from this source.

Thanks to Alees Seehausen who co-taught the Discrete Mathematics course with me in 2015 and helped develop many of the *Investigate!* activities and other problems currently used in the text. She also offered many suggestions for improvement of the expository text, for which I am quite grateful. Thanks also to Katie Morrison and Nate Eldredge for their suggestions after using parts of this text in their class.

While odds are that there are still errors and typos in the current book, there are many fewer thanks to the work of Michelle Morgan over the summer of 2016.

The book is now available in an interactive online format, and this is entirely thanks to the work of Rob Beezer and David Farmer along with the rest of the participants of the mathbook-xml-support group. Thanks for

Finally, a thank you to the numerous students who have pointed out typos and made suggestions over the years and a thanks in advance to those who will do so in the future.

## Acknowledgements

Techniques of Calculus 1 is licensed under a CC BY-NC-SA, except where otherwise noted.

This text is provided to you as an Open Educational Resource (OER) which you access online. It is designed to give you a comprehensive introduction to calculus with emphasis on applications in economics and the social sciences. It contains both written and graphic text material, intra-text links to other internal material which may aid in understanding topics and concepts, intra-text links to the appendices and glossary for tables and definitions of words, and extra-text links to videos and web material that clarifies and augments topics and concepts.

Chapters and sections were adapted from the the following OER textbooks. Without these foundational texts, a lot more work would have been required to complete this project. Thank you to those authors who shared their work before us.

Business Calculus, Copyright © 2013 Shana Calaway, Dale Hoffman, David Lippman. This text is licensed under a Creative Commons Attribution 3.0 United States License.

- 1.2 Operations of Functions
- 1.3 Linear Functions
- 1.5 Quadratics
- 1.6 Polynomials and Rational Functions
- 1.7 Exponential Functions
- 1.8 Logarithmic Functions
- 2.1 Limits and Continuity
- 2.2 The Derivative
- 2.3 The Power and Sum Rules for Derivatives
- 2.4 Product and Quotient Rules
- 2.5 Chain Rule
- 2.6 Second Derivative and Concavity
- 2.7 Optimization
- 2.8 Curve Sketching
- 2.9 Applied Optimization
- 2.11 Implicit Differentiation and Related Rates
- 3.7 Applications to Business

Calculus Volume 1, Copyright © 2020 Edwin Herman, Gilbert Strang. This text is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International (CC BY-NC-SA)

## 1 Finding intersections

The method for finding line intersections that I will discuss here relies on knowing a general equation for a shape and substituting it’s coordinates with a parameterised line.

Starting with what a parameterised line is. It is defined solely by an origin and a direction that scales with a parameter (a number that can be anywhere between negative and positive infinity). The points on this line are given by the equation . Here is the origin, the direction and the parameter. Our parameter thus denotes the distance from our line’s origin along the direction and can be anywhere between negative and positive infinity.

For those unfamiliar with the notation, the arrows denote vectors with an x, y and z component, like .

Now to find the intersections between this line and a shape we need to know the shape’s equation, substitute it’s x, y and z component with , and solve for . For example, the equation of a sphere would be , with r the radius. To find the intersections we would substitute , and . This would result in an equation that has only one variable, our parameter . Solving this equation for would then give us the distance between the intersection and the line’s origin (if there is an intersection).

So to summarise, the method consists of the following 3 steps:

- Know/find the shape equation
- Substitute coordinates with the parameterised line equation
- Solve the equation for parameter t, the distance to the intersection

Note that this method will also return negative distances, i.e. when the intersection is behind the line’s origin. In most use cases you would either discard these negative distances or use it to say something about your location, e.g. if the shape is convex and there is one intersection behind you and one in front you are within the shape’s volume.

### 1.1 Space transformation

Most of our shapes will point in some direction, like a plane that has its normal pointing upward along the y-axis. For simplicity’s sake we will define each shape equation in its own local space where the shape is centred on the origin and y is always the up axis, z is the forward axis and x is the right axis.

In order to translate and rotate our shapes to any place or direction we will be transforming our input line (that is likely defined in world space) to our shape’s local space, somewhat similar to transforming from world space to object space except we don’t adjust the scale. In order to do this we will provide two orthonormal vectors that form a basis of our shape space, the forward and up direction, in addition to the origin of the shape. These vectors must be defined within the same space as our input line. With this transformation we can easily define our shapes in their local x, y and z coordinates and use a transformation matrix to rotate and the origin to translate the shape to any location/rotation we want.

In short, our equations will be defined in their own space, where it is centred on the origin and always pointing upward. However, our input is defined in world space (or any other space), which consists of the line’s origin, the line’s direction, the shape’s origin and the shape’s direction (up and forward vectors). Finally, we use the shape’s origin and direction to transform our line to shape space.

I know this might be a bit much and the next sections will show it implemented in code, but let me know if you want a more in-depth tutorial on matrix transformations. In the meanwhile you can also check out this tutorial by Catlike Coding on matrix transformations.

### 1.2 HLSL include file

The actual HLSL implemenation of our line intersections will be in the form of a HLSL include file that we can add to any shader. Create a new text file and change its name (including format) to “**LineIntersections.hlsl**” and add the following lines.

The **constructTransitionMatrix** function uses the shape’s forward and up direction to construct a rotation matrix to transform from our input space to the shape’s space. Notice the definition lines starting with a #. These are used to ensure that we don’t accidently include our functions twice in a shader. Basically, it checks if **LINE_INTERSECTIONS_INCLUDED** is defined, if not, define it and add our functions.

### 1.3 Plane intersection

Now onto the actual stuff, finding intersections. We’ll start with the simplest shape, as it always has exactly one intersection, the flat plane. We can define a plane with the following equation [1]:

Here the shape parameters determine the normal of the plane and an offset along the normal. Now in order to find the intersection with the line we need to substitute our coordinates with the line parameterisation . Because this is a simple linear equation we can solve it for to find the distance between the line’s origin and the intersection. This process is written out below.

Because we’re transforming our line to the shape’s local space, which already defines the shape’s direction and origin/offset, we don’t have to provide the shape parameters as they overlap. In our shape space the plane always points up (the y-direction), this allows us to simplify the equation to .

The code below shows how this is implemented in HLSL, where we transform the input line to shape space and calculate . As input we provide the line origin, line direction, shape origin, and shape up direction. Add the following lines to our HLSL file below the **constructTransitionMatrix** function but before #endif.

In the previous section I noted we must always provide two orthonormal vectors, but because our plane is infinite and thus symmetric within the plane we can get away with only providing one vector, the normal or up direction.

### 1.4 Visualise shader

Having this **LineIntersections.hlsl** is great and all, but we can’t see anything yet as we do not have a shader or material. Let’s change that by making a new shader called **VisualiseIntersection.shader** and fill it with the code block below.

This tutorial is primarily focussed on the mathematics and implementation behind finding line intersections, I will thus only shortly gloss over what the shader does.

We use 3 properties to influence our shapes, the shape parameters which usually say something about the scale, the distance scaling which is used to colour the shape and the cap heights which will be used in section 3 for the cap positions.

We include our **LineIntersections.hlsl** file at line 21. The #include requires the path to our HLSL file, if this is in the same folder we can simply put the name there. Otherwise it would look something like the following.

We start our line at the camera’s world space location, with its direction being towards the fragment world position of the gameObject to which we apply the material. The shape’s origin is set to the gameObject’s origin and the shape’s forward and up direction are set to the gameObject’s local forward and up direction.

Finally we get the intersection information to determine the output colour. If there is an intersection and it is in front of the camera we colour it depending on its distance with respect to the shape’s origin, otherwise it is black to show the bounds of our gameObject.

I’ve applied the shader to a standard Unity sphere with all scales set to 50. The image below shows a visualisation of the plane intersection. The red and blue line correspond with the world x-axis and z-axis respectively.

Plane visualisation on a 50 scale sphere object with distance scale = 50, the red and blue line correspond to the x and z axis of the scene

## Math Tags

The following tags are used for the intermediate math representation:

represents a math token. It may contain text for presentation. Additional attributes are:

the name that represents the *meaning* of the token this overrides the content for identifying the token.

the OpenMath content dictionary that the name belongs to.

the font to be used for presenting the content.

whether scripts should be stacked above/below the item, instead of the usual script position.

represents the generalized application of some function or operator to arguments. The first child element is the operator, the remainig elements are the arguments. Additional attributes:

the name that represents the meaning of the construct as a whole.

combines representations of the content (the first child) and presentation (the second child), useful when the two structures are not easily related.

represents spacing or other apparent purely presentation material.

names the effect that the hint was intended to achieve.

serves to assert the expected type or role of a subexpression that may otherwise be difficult to interpret — the parser is more forgiving about these.

serves to wrap individual arguments or subexpressions, created by structured markup, such as frac . These subexpressions can be parsed individually.

the grammar rule that this subexpression should match.

refers to another subexpression,. This is used to avoid duplicating arguments when constructing an XMDual to represent a function application, for example. The arguments will be placed in the content branch (wrapped in an XMArg) while XMRef’s will be placed in the presentation branch.

## MAT 112 Ancient and Contemporary Mathematics

In mathematics symbols are used to obtain a clearer and shorter presentation. The first of these symbols is the ((ldots)). When we use this symbol in mathematics, it means “continuing in this manner.” When a pattern is evident, we can use the ellipses ((ldots)) to indicate that the pattern continues. We use this to define the integers.

The integer (0) is not considered to be positive or negative.

In the video in Figure 1.1.1 we give an introduction to the integers and statements.

Figure 1.1.2 On the Number line (a) shows the integers, which extend both to the left and to the right. Figure 1.1.2 (b) shows the natural numbers (also called positive integers), which extend only to the right. Figure 1.1.2 (c) shows the negative integers, which extend only to the left.

### Subsection 1.1.1 Comparing Integers

The symbols (= ext<,>) ( e ext<,>) (lt ext<,>) (le ext<,>) (> ext<,>) and (ge) are used to compare integers.

symbol | read as |

(=) | “is equal to” |

( e) | “is not equal to” |

(>) | “is greater than” |

(ge) | “is greater than or equal to” |

(lt) | “is less than” |

(le) | “is less than or equal to” |

The first symbol is the equality symbol, (= ext<.>) Two integers are equal if they are the same integer. To indicate that two integers are not equal we use the symbol, ( e ext<.>)

The other symbols compare the positions of two integers on the number line. An integer is greater than another integer if the first integer is to the right of the second integer on the number line. An integer is less than another integer if the first integer is to the left of the second integer on the number line.

###### Example 1.1.3 . Reading (= ext) ( e ext) (gt ext) (ge ext) (lt ext) and (le).

We give examples of comparisons and how to read them.

(2=2) is read “2 is equal to 2.”

(2 e 3) is read “(2) is not equal to 3.”

(3> 2) is read “3 is greater than 2.”

(3ge 2) is read “3 is greater than or equal to 2.”

(2lt 3) is read “2 is less than 3.”

(2le 3) is read “2 is less than or equal to 3.”

In the Checkpoint 1.1.4 select the correct comparison operator.

###### Checkpoint 1.1.4 . Comparison operators.

### Subsection 1.1.2 Operations

Addition, negation, subtraction, and multiplication are the basic operations of integers. We write “(+)” for plus, “(-)” for minus, and “(cdot)” for times.

###### Example 1.1.5 . Statements involving integer operations.

We give some examples of statements that involve integer operations. As we do not say “is false” we mean that all of these equality statements are true.

(2+3=5) is read “2 plus 3 is equal to 5”

(2+0=2) is read “2 plus 0 is equal to 2”

(2+(-2)=0) is read “2 plus negative 2 is equal to 0”

(2-2=0) is read “2 minus 2 is equal to 0”

(2cdot 5=10) is read “2 times 5 is equal to 10”

(2cdot(-5)=-10) is read “2 times negative 5 is equal to negative 10”

((-2)cdot(-5)=10) is read “negative 2 times negative 5 is equal to 10”

Multiplication of a natural number with an integer can be viewed as repeated addition.

###### Example 1.1.6 .

We give examples of multiplication viewed as repeated addition.

Again, we can use ellipses ((ldots)) to represent a continuing pattern:

Defining the multiplication of two negative integers is more difficult, and we appeal to your previously acquired knowledge about integers for that. Recall that the product of two negative integers is positive.

###### Example 1.1.7 .

We give examples of multiplication of integers and negative integers:

### Subsection 1.1.3 Order of Operations

We use parentheses to indicate the order in which expressions should be executed. We evaluate the expressions in the innermost parentheses first and then work our way outwards.

###### Example 1.1.8 . Order of operations.

We give examples for order of operations. The numbers and the operations are the same only the grouping of the expressions given by the parentheses differs.

###### Example 1.1.9 . Order of operations.

We give examples for order of operations. The numbers and the operations are the same only the grouping of the expressions given by the parentheses differs.

(5cdot left(2+(3cdot 4) ight)=5cdot(2+12)=5cdot 14=70)

By the associative property of addition that the order of operations does not matter for addition. Likewise the associative property of multiplication tells us that the order of operations does not matter for repeated multiplication. We recall these properties in the next section (Example 1.3.17 and Example 1.3.19).

###### Example 1.1.11 .

We illustrate that the order of operations does not matter for repeated addition by computing the same sums in the order indicated by the parentheses.

Usually we write (1+2+3+4=10 ext<.>)

In most cases we will use parentheses to indicate the order of operations. There are other conventions for implicit order of operations (see Figure 1.1.10). One of these conventions is that multiplication is performed before addition and subtraction. We will use this convention when we feel that the additional parentheses will make it hard to read the expressions under consideration.

In the video in Figure 1.1.12 we recap the operations for the integers and give a motivation for the following section.

## Submit Paper

**Please read the guidelines below then visit the Journal’s submission site http://mc.manuscriptcentral.com/mams to upload your manuscript. Please note that manuscripts not conforming to these guidelines may be returned.**

Only manuscripts of sufficient quality that meet the aims and scope of *Mathematics and Mechanics of Solids* will be reviewed.

There are no fees payable to submit or publish in this journal.

As part of the submission process you will be required to warrant that you are submitting your original work, that you have the rights in the work, that you are submitting the work for first publication in the Journal and that it is not being considered for publication elsewhere and has not already been published elsewhere, and that you have obtained and can supply all necessary permissions for the reproduction of any copyright works not owned by you.

**1. What do we publish?**

**1.1 Aims & Scope**

Before submitting your manuscript to *Mathematics and Mechanics of Solids*, please ensure you have read the Aims & Scope

**1.2 Article Types**

*Mathematics and Mechanics of Solids* publishes original, well-written and self-contained research that elucidates the mechanical behaviour of solids with particular emphasis on mathematical principles.

1. Original Research Article

2. Review Paper

3. Letter to the Editor

**1.3 Writing your paper**

The SAGE Author Gateway has some general advice and on how to get published, plus links to further resources.

**1.3.1 Make your article discoverable**

When writing up your paper, think about how you can make it discoverable. The title, keywords and abstract are key to ensuring readers find your article through search engines such as Google. For information and guidance on how best to title your article, write your abstract and select your keywords, have a look at this page on the Gateway: How to Help Readers Find Your Article Online .

**2. Editorial policies**

**2.1 Peer review policy**

*Mathematics and Mechanics of Solids* operates a conventional single-blind reviewing policy in which the reviewer’s name is always concealed from the submitting author.

Authors wishing to submit a paper to the journal must comply with the requirements stated below. Failure to do so will delay acceptance and publication of the paper.

**2.2 Authorship**

Papers should only be submitted for consideration once consent is given by all contributing authors. Those submitting papers should carefully check that all those whose work contributed to the paper are acknowledged as contributing authors.

The list of authors should include all those who can legitimately claim authorship. This is all those who:

- Made a substantial contribution to the concept or design of the work or acquisition, analysis or interpretation of data,
- Drafted the article or revised it critically for important intellectual content,
- Approved the version to be published,
- Each author should have participated sufficiently in the work to take public responsibility for appropriate portions of the content.

Authors should meet the conditions of all of the points above. When a large, multicentre group has conducted the work, the group should identify the individuals who accept direct responsibility for the manuscript. These individuals should fully meet the criteria for authorship.

Acquisition of funding, collection of data, or general supervision of the research group alone does not constitute authorship, although all contributors who do not meet the criteria for authorship should be listed in the Acknowledgments section. Please refer to the International Committee of Medical Journal Editors (ICMJE) authorship guidelines for more information on authorship.

**2.3 Acknowledgements**

All contributors who do not meet the criteria for authorship should be listed in an Acknowledgements section. Examples of those who might be acknowledged include a person who provided purely technical help, or a department chair who provided only general support.

**2.3.1 Third party submissions**

Where an individual who is not listed as an author submits a manuscript on behalf of the author(s), a statement must be included in the Acknowledgements section of the manuscript and in the accompanying cover letter. The statements must:

- Disclose this type of editorial assistance – including the individual’s name, company and level of input
- Identify any entities that paid for this assistance
- Confirm that the listed authors have authorized the submission of their manuscript via third party and approved any statements or declarations, e.g. conflicting interests, funding, etc.

Where appropriate, SAGE reserves the right to deny consideration to manuscripts submitted by a third party rather than by the authors themselves*.*

**2.3.2 Writing assistance**

Individuals who provided writing assistance, e.g. from a specialist communications company, do not qualify as authors and so should be included in the Acknowledgements section. Authors must disclose any writing assistance – including the individual’s name, company and level of input – and identify the entity that paid for this assistance”).

It is not necessary to disclose use of language polishing services.

Any acknowledgements should appear first at the end of your article prior to your Declaration of Conflicting Interests (if applicable), any notes and your References.

*Mathematics and Mechanics of Solids* requires all authors to acknowledge their funding in a consistent fashion under a separate heading. Please visit the Funding Acknowledgements page on the SAGE Journal Author Gateway to confirm the format of the acknowledgment text in the event of funding, or state that: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

**2.5 Declaration of conflicting interests**

It is the policy of *Mathematics and Mechanics of Solids* to require a declaration of conflicting interests from all authors enabling a statement to be carried within the paginated pages of all published articles.

Please ensure that a ‘Declaration of Conflicting Interests’ statement is included at the end of your manuscript, after any acknowledgements and prior to the references. If no conflict exists, please state that ‘The Author(s) declare(s) that there is no conflict of interest’. For guidance on conflict of interest statements, please see the ICMJE recommendations here .

**2.6 Research Data**

The journal is committed to facilitating openness, transparency and reproducibility of research, and has the following research data sharing policy. For more information, including FAQs please visit the SAGE Research Data policy pages .

Subject to appropriate ethical and legal considerations, authors are encouraged to:

- share your research data in a relevant public data repository
- include a data availability statement linking to your data. If it is not possible to share your data, we encourage you to consider using the statement to explain why it cannot be shared.
- cite this data in your research

**3. Publishing Policies**

**3.1 Publication ethics**

SAGE is committed to upholding the integrity of the academic record. We encourage authors to refer to the Committee on Publication Ethics’ International Standards for Authors and view the Publication Ethics page on the SAGE Author Gateway .

**3.1.1 Plagiarism**

*Mathematics and Mechanics of Solids* and SAGE take issues of copyright infringement, plagiarism or other breaches of best practice in publication very seriously. We seek to protect the rights of our authors and we always investigate claims of plagiarism or misuse of published articles. Equally, we seek to protect the reputation of the journal against malpractice. Submitted articles may be checked with duplication-checking software. Where an article, for example, is found to have plagiarised other work or included third-party copyright material without permission or with insufficient acknowledgement, or where the authorship of the article is contested, we reserve the right to take action including, but not limited to: publishing an erratum or corrigendum (correction) retracting the article taking up the matter with the head of department or dean of the author's institution and/or relevant academic bodies or societies or taking appropriate legal action.

**3.1.2 Prior publication**

If material has been previously published it is not generally acceptable for publication in a SAGE journal. However, there are certain circumstances where previously published material can be considered for publication. Please refer to the guidance on the SAGE Author Gateway or if in doubt, contact the Editor at the address given below.

**3.2 Contributor's publishing agreement**

Before publication, SAGE requires the author as the rights holder to sign a Journal Contributor’s Publishing Agreement. SAGE’s Journal Contributor’s Publishing Agreement is an exclusive licence agreement which means that the author retains copyright in the work but grants SAGE the sole and exclusive right and licence to publish for the full legal term of copyright. Exceptions may exist where an assignment of copyright is required or preferred by a proprietor other than SAGE. In this case copyright in the work will be assigned from the author to the society. For more information please visit the SAGE Author Gateway .

**3.3 Open access and author archiving**

*Mathematics and Mechanics of Solids* offers optional open access publishing via the SAGE Choice programme. For more information on Open Access publishing options at SAGE please visit SAGE Open Access. For information on funding body compliance, and depositing your article in repositories, please visit SAGE’s Author Archiving and Re-Use Guidelines and Publishing Policies.

**4. Preparing your manuscript for submission**

**4.1 Formatting**

The preferred format for your manuscript is Word. LaTeX files are also accepted. Word and (La)Tex templates are available on the Manuscript Submission Guidelines page of our Author Gateway.

**4.2 Artwork, figures and other graphics**

For guidance on the preparation of illustrations, pictures and graphs in electronic format, please visit SAGE’s Manuscript Submission Guidelines .

Figures supplied in colour will appear in colour online regardless of whether or not these illustrations are reproduced in colour in the printed version. For specifically requested colour reproduction in print, you will receive information regarding the costs from SAGE after receipt of your accepted article.

**4.3 Supplementary material**

*Mathematics and Mechanics of Solids* does not currently accept supplemental files.

**4.4 Reference style**

*Mathematics and Mechanics of Solids* adheres to the SAGE Vancouver reference style. View the SAGE Vancouver guidelines to ensure your manuscript conforms to this reference style.

If you use EndNote to manage references, you can download the SAGE Vancouver EndNote output file .

**4.5 English language editing services**

Authors seeking assistance with English language editing, translation, or figure and manuscript formatting to fit the journal’s specifications should consider using SAGE Language Services. Visit SAGE Language Services on our Journal Author Gateway for further information.

**5. Submitting your manuscript**

*Mathematics and Mechanics of Solids* is hosted on SAGE Track, a web based online submission and peer review system powered by ScholarOne™ Manuscripts. Visit http://mc.manuscriptcentral.com/mams to login and submit your article online.

IMPORTANT: Please check whether you already have an account in the system before trying to create a new one. If you have reviewed or authored for the journal in the past year it is likely that you will have had an account created. For further guidance on submitting your manuscript online please visit ScholarOne Online Help.

As part of our commitment to ensuring an ethical, transparent and fair peer review process SAGE is a supporting member of ORCID, the Open Researcher and Contributor ID. ORCID provides a unique and persistent digital identifier that distinguishes researchers from every other researcher, even those who share the same name, and, through integration in key research workflows such as manuscript and grant submission, supports automated linkages between researchers and their professional activities, ensuring that their work is recognized.

The collection of ORCID iDs from corresponding authors is now part of the submission process of this journal. If you already have an ORCID iD you will be asked to associate that to your submission during the online submission process. We also strongly encourage all co-authors to link their ORCID ID to their accounts in our online peer review platforms. It takes seconds to do: click the link when prompted, sign into your ORCID account and our systems are automatically updated. Your ORCID iD will become part of your accepted publication’s metadata, making your work attributable to you and only you. Your ORCID iD is published with your article so that fellow researchers reading your work can link to your ORCID profile and from there link to your other publications.

If you do not already have an ORCID iD please follow this link to create one or visit our ORCID homepage to learn more.

**5.2 Information required for completing your submission**

You will be asked to provide contact details and academic affiliations for all co-authors via the submission system and identify who is to be the corresponding author. These details must match what appears on your manuscript. At this stage please ensure you have included all the required statements and declarations and uploaded any additional supplementary files (including reporting guidelines where relevant).

**5.3 Permissions**

*Please also ensure that you have obtained any necessary permission* from copyright holders for reproducing any illustrations, tables, figures or lengthy quotations previously published elsewhere. For further information including guidance on fair dealing for criticism and review, please see the Copyright and Permissions page on the SAGE Author Gateway .

**6. On acceptance and publication**

**6.1 SAGE Production**

Your SAGE Production Editor will keep you informed as to your article’s progress throughout the production process. Proofs will be sent by PDF to the corresponding author and should be returned promptly. Authors are reminded to check their proofs carefully to confirm that all author information, including names, affiliations, sequence and contact details are correct, and that Funding and Conflict of Interest statements, if any, are accurate. Please note that if there are any changes to the author list at this stage all authors will be required to complete and sign a form authorising the change.

**6.2 Online First publication**

Online First allows final articles (completed and approved articles awaiting assignment to a future issue) to be published online prior to their inclusion in a journal issue, which significantly reduces the lead time between submission and publication. Visit the SAGE Journals help page for more details, including how to cite Online First articles.

**6.3 Access to your published article**

SAGE provides authors with online access to their final article.

**6.4 Promoting your article**

Publication is not the end of the process! You can help disseminate your paper and ensure it is as widely read and cited as possible. The SAGE Author Gateway has numerous resources to help you promote your work. Visit the Promote Your Article page on the Gateway for tips and advice.

**7. Further information**

Any correspondence, queries or additional requests for information on the manuscript submission process should be sent to the *Mathematics and Mechanics of Solids* editorial office as follows:

## Math 321 Class Notes

A or is a sentence which is either true or false, but not both.

###### Example 1.1.2 .

Which of the following are logical propositions?

- This is a course in discrete mathematics
- Chocolate cupcakes are the best
- (displaystyle 1 - 3 = 4)
- Wichita is the capitol of Kansas
- What are you doing?

###### Definition 1.1.3 .

Let (p) be a logical proposition. The of (p ext<,>) denoted by ( eg p) has the opposite truth value of (p ext<.>)

###### Example 1.1.4 .

What are the logical negations of each of the following?

- This is a course in discrete mathematics
- (displaystyle 1- 3 = 4)
- Wichita is the capitol of Kansas

###### Definition 1.1.5 .

Let (p) and (q) be propositions. The of (p) and (q ext<,>) denoted (p wedge q ext<,>) is the proposition “(p) and (q)”.

The of (p) and (q ext<,>) denoted (p vee q ext<,>) is the proposition “(p) or (q) (or both)”.

The logical disjunction is an “inclusive or”. On the other hand, we define the “exclusive or” of (p) and (q) to be the proposition “(p) or (q) but not both”. We won't be using it in Discrete 1, so we won't give it a special symbol.

###### Definition 1.1.6 .

Let (p) and (q) be propositions. The is the compound proposition “if (p) then (q)”. The conditional is denoted by (p o q ext<.>)

We call (p) the or antecedent or premise, and (q) is the or consequence.

###### Example 1.1.7 .

Write the following as a simple English expression, letting (p) be the statement “it rains” and (q) be the statement “I complain about the weather”.

- (displaystyle p o q)
- (displaystyle p vee q)
- (displaystyle q o p)
- (displaystyle eg q o eg p )

What is the logical negation of (p o q) in simple English?

###### Note 1.1.8 .

There are many ways to phrase the conditional statement (p o q ext<.>) Here are just a few common ones:

- If (p ext<,>) then (q ext<.>)
- (p) implies (q ext<.>)
- (p) only if (q ext<.>)
- (p) if sufficient for (q ext<.>)
- (q) is necessary for (p ext<.>)
- (q) if (p ext<.>)
- (q) whenever (p ext<.>)
- (q) unless ( eg p ext<.>)

###### Definition 1.1.9 .

Let (p) and (q) be propositions. For the conditional (p o q ext<,>) we define:

###### Definition 1.1.10 .

Let (p) and (q) be propositions. The of (p) and (q ext<,>) is the statement “(p) if and only if (q)”, denoted (p leftrightarrow q ext<.>)

Other ways to phrase an “if and only if” statement:

- (p) iff (q ext<.>)
- (p) is necessary and sufficient for (q ext<.>)
- If (p) then (q) and conversely.

Just as with arithmetic operations ((+, -, imes, div)) on numbers, we need to define an order of operations so that compound propositions can be understood without grouping symbols.

Operator | Precedence |

( eg) | highest |

(wedge, vee ) | next, from left to right |

( o, leftrightarrow ) | lowest, left to right |

### Subsection 1.1.2 Truth Tables for Logical Connectives

allow us to uniquely determine the truth value of a compound proposition, based on the truth values of the simple statements from which it is made. Below are the truth tables for conjunction (wedge ext<,>) disjunction (lor ext<,>) conditional ( o ext<,>) biconditional (leftrightarrow ext<,>) exclusive or (oplus ext<,>) and negation ( eg ext<.>)

An adjacency matrix is a square matrix used to represent a finite graph. The elements of the adjacency matrix L indicate whether pairs of vertices in the graph are adjacent or not. For a simple graph with a set of vertices *V*, the adjacency matrix is a square |L| × |L| matrix such that its element *L*ᵢⱼ is 1 when there is one edge from vertex *i* to vertex *j*, 2 when there are two, and zero when there are no edges from vertex i to vertex j. The diagonal elements of the matrix are all zero, since edges from a vertex *i* to itself (loops) are not allowed in simple graphs. For all step walks of length 1 along the edge set *E*, this gives us the following adjacency matrix for the graph G:

**Solution 1.1.**Edge elements from vertices i to j and adjacency matrix of graph G, showing the number of edges between vertices i and j

The second task in problem 1 asks to find the matrix which encodes all possible walks of length 3 (Knill, 2003). That is, to find the number of different sequences of edges which join every distinct sequence of vertices.

An *n +* 1 step walk from *i* to *j* consists of an *n* step walk from *i* to *k* and then a 1 step walk from *k* to *j*. That is, the *ij* entry of *L*ⁿ⁺¹ is given by the sum:

Which in English for this problem states that “the number of walks of length 3 from vertex i to j" is equal to the sum of “the number of walks of length 2 from vertex i to *k”* multiplied by “the number of walks of length 1 from vertex *k* to j” for *k* = 1,2. By matrix multiplication, for all step walks of length 3 from i to j this gives the following matrix:

The third task in problem 1 asks for the generating function from vertex *i* to *j*. To answer this question, Horváth et al (2010) consider an analytic generating function defined by a power series

Where the coefficient *zⁿ* denotes the number of *n* step walks from *i* to *j*. From task 1.3, we found that ω_n(i → j) is the *ij* entry of the matrix *Lⁿ*. The problem asks for the generating function that gives all the entries simultaneously, and so it makes sense to consider a matrix *L* given by the familiar power series (Horváth et al, 2010):

Where *Lⁿ* is the matrix containing the number of step walks from each vertex *i* to *j* (the general case of the solution to problem 1.2). The sum can be calculated using the familiar identity for geometric power series, that is:

To calculate the inverse of (*I* − *z* × *L)* we can use Cramer’s rule. According to Horváth et al (2010) for a matrix *M* let *Mᵢⱼ* denote the matrix obtained from *M* by removing the *i*th column and the *j*th row. If we do so, we obtain a matrix N whose *ij* entry is

By Cramer’s rule, if M is invertible (there exists some n×n matrix N such that *M*×*N* = *N*×*M* = *I_n*) then

That is, the *ij* entry of of the inverse matrix M is:

Applied to compute the inverse of M = (*I* − *z* × *L*), we obtain:

As Horváth et al (2010) notes, this is Will’s solution in the movie, except his solution omits the term (−1)^(i+j) (likely due to notation), and he denotes the identity matrix with 1 instead of the more common *I*.

To solve task 1.4, we simply apply the general formula for walks from i to j (from task 1.3) to the case of walks from 1 → 3:

Whose determinants are trivial to find:

Giving the following expressions, obtain by using the definition of a determinant:

To obtain the coefficients of this power series, one computes the Taylor series of the function:

For our expression f(z), we can use the quotient rule where g(z) = 2z² and h(z) = 4z³− 6z² −z +1. In the movie, Will provides the values for the first six derivatives of the f(z) expansion, which are:

## Acknowledgements

This book was created for the Ryerson course POH103, Data Management by Ian Young. It has been adapted from the following three OER texts as follows and organized to reflect the content taught in this course:

Wang, M. (2018) *Key Concepts of Intermediate Level Math.* Victoria, B.C.: BCcampus.

Adapted content from Units 2, 4-7, 9, and 11.

This text is licensed under a Creative Commons Attribution license.

Sekhon, R. (2011). *Applied Finite Mathematics.* Houston, TX: OpenStax

Adapted content from Chapters 1, 11, 13, 15, and 17 (section 1-3).

This text is licensed under a Creative Commons Attribution license.

Lippman, D. (2016). *Business Precalculus*.

Adapted content from Chapter 1 (pp. 1-22, 24-26), 4 (pp. 153-163), and 5.

This text is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.

Content was adapted for Pressbooks by CareerBoost Digital Publication Assistant, Angelica Chimal, Ryerson BSc student, with the support of the Ryerson University Library Digital Publication Team.

## What does anti-racism in mathematics look like?

This question is on the front of my mind and is followed by how is anti-racism in mathematics practiced? The differences in how members of underrepresented groups, especially those who identify as Black and African American, are treated in the mathematical community, and our society as a whole is glaring. Protests condemning the murders by the hand of the police of George Floyd, Tony McDade, Ahmaud Arbery, and Breonna Taylor has led mathematicians to ask professional organizations and institutions to take a stand. In particular, through concrete action and by building better support structures to address the many ways systemic racism plays a role in our community.

First and foremost, one must acknowledge that mathematics is part of a societal system that is inherently racist. In this post, I want to share some of the resources that have helped me reflect on how to grow as a better ally, to understand how organizations and institutions promote racism, and what actions could/should we be taking to dismantle racism as a community. There are several resources out there that I encourage you to share and engage with, these are just a few.

Back in January, Dr. Tian An Wong asked ‘*can mathematics be anti-racist?*‘ in the AMS inclusion/exclusion blog, he concludes,

“Nonetheless, one thing is clear: if mathematics is political (and also racial and gendered), then we must be on the side of justice, whatever that may look like. In other words, if mathematics can be antiracist, then it ought to be.[…] I don’t pretend to have the answers to the questions I am asking. This small sampling suggests a handful of possibilities for mathematics as, say, an intersectional, anti-racist, and class-consciously feminist enterprise. In any case, if we can agree that mathematics can operate as whiteness, then we have a moral duty to ask how mathematics might be otherwise. There is much work left to do. With the strength of our combined mathematical creativity, what might we come up with if we dared to imagine?”

*What does anti-racist mathematics look like*? And, *how is anti-racist mathematics practiced?* It is our responsibility to make sure that these questions do not become a passing trend but the foundation in which we build our community. In The Aperiodical, Samuel Hansen shares *R**esources for Anti-Racism and Social Justice in the Mathematical Sciences* *,* a definition of anti-racist from Ibram X Kendi, author of *How to be Anti-Racist and This is what anti-racist America would look like. How do we get there?.*

“There is no such thing as a “not-racist” policy, idea, or person. Just an old-fashioned racist in a newfound denial. All policies, ideas, and people are either being racist or antiracist. Racist policies yield racial inequity antiracist policies yield racial equity. Racist ideas suggest racial hierarchy, antiracist ideas suggest racial equality. A racist is supporting racist policy or expressing a racist idea. An antiracist is supporting antiracist policy or expressing an antiracist idea. A racist or antiracist is not who we are, but what we are doing at the moment.” – This is what an antiracist America would look like. How do we get there? by Ibram X Kendi.

In their post, they lists many of the resources that have been shared in social media including the statements of support to the Black Lives Matters movement by organizations, readings, list of anti-racist mutual aid projects you can donate to, organizations and projects focused primarily on the mathematical sciences you can become a member of, or otherwise support and sponsor, and actions you can take, scaffolded anti-racist resources , among others. For example, you can support the National Association of Mathematicians (NAM), as mentioned in the statement of support of the Black Lives Matter movement, their organization has made a priority promoting the excellence and mathematical development of all underrepresented minorities.

“NAM was founded in 1969, one year after the assassination of Dr. Martin Luther King, Jr. sparked widespread protests throughout the nation, similar to the ones we are seeing today. Indeed, NAM’s founding was a direct result of the marginalization of black people within the professional mathematics community, which then and now serves as a microcosm of the society in which we live. Over 50 years since NAM’s founding, despite the lessons of the civil rights movement, we still see systemic racial inequities in education, economic prosperity, criminal justice, and public health. Today, it should be clear to us all that the consequence of ignoring these racial inequities is dire.” – NAM’s Statement on the Death of George Floyd

On June 10th, there was a call join the Strike for Black Lives . In the post, #ShutDownMath in the inclusion/exclusion blog makes the great point that in these we must avoid ally theater and focus on the actions that will tackle systemic racism in mathematics.

“So yes, go to Black Lives Matter protests, donate to bail funds for protestors, use hashtags to express your outrage at police brutality, but be prepared to commit for the long haul. Donate to NAM (or better yet, get your department to become a departmental member!), donate to Mathematically Gifted and Black , donate to Data 4 Black Lives . Get your department to read anti-racism books . Design your classroom around rehumanizing principles that center your Black students. Change your hiring practices. Think about how you may be complicit in gate-keeping by accepting the status quo. And given that the current national focus is on the police state and how it’s implicated in the murder of Black people, demand that your colleagues stop contributing to the development of algorithms of oppression . Demand that we stop rewarding work that supports policing, inequality, and surveillance . Be intentional and mindful about mentoring graduate students. Read this letter in its entirety. And then do something.”

We can hold conferences, panels, read, and discuss as we acknowledge this conversation is long overdue. Our community is in dire need of action at all levels. For example, a group of mathematicians has urged the community (and professional organizations) to stop using predictive-policing algorithms and other models. As discussed in the Nature article, *Mathematicians urge colleagues to boycott police work in wake of killings * , this is due to the widely documented disparities on “how the US law-enforcement agencies treat people of different races and ethnicities”. Predictive policing, a tool aimed at stopping crime before it occurs, is only one of many ways mathematics can promote racism through algorithmic oppression. As mentioned by one of the coauthors of the letter, Dr. Jayadev Athreya,

“In recent years, mathematicians, statisticians, and computer scientists have been developing algorithms that crunch large amounts of data and claim to help police reduce crime — for instance, by suggesting where crime is most likely to occur and focusing more resources in those areas. Software-based on such algorithms is in use in police departments across the United States, although how many is unclear. Its effectiveness is contested by many.

But “given the structural racism and brutality in US policing, we do not believe that mathematicians should be collaborating with police departments in this manner”, the mathematicians write in the letter. “It is simply too easy to create a ‘scientific’ veneer for racism.”

While exploring resources on Twitter, I discovered an initiative aimed at department chairs to brainstorm and share ideas on how departments can become anti-racist places for the community. You can participate and look at the resources provided at Academics for Black Survival and Wellness (June 19 – June 25) which was organized by a group of Black counseling psychologists and their colleagues who practice Black allyship. Also, you can sign-up to join Math Chairs for Racial Justice by June 23, and find a brief description below.

“Over the next two months, we will be gathering in small groups to read Ibram X. Kendi’s How to Be an Anti-Racist. Weekly discussions (starting as soon as possible) will give you space to brainstorm how you might work to make your department an anti-racist place – a community that is not just open to all people, but one that actively supports and empowers students, faculty, and staff from groups historically underserved by the mathematics community. All discussions will be facilitated by mathematicians with experience tackling issues of racial justice in mathematics.”

In the field of math education, which has a long history with tackling and understanding racism in the classroom, a recent article by principal Pirette McKamey. In *What Anti-racist Teachers Do Differently* , McKamey emphasizes that,

“Anti-racist teachers take black students seriously. They create a curriculum with black students in mind, and they carefully read students’ work to understand what they are expressing.[…] To fight against systemic racism means to buck norms. Educators at every level must be willing to be uncomfortable in their struggle for black students, recognizing students’ power and feeding it by honoring their many contributions to our schools. Teachers need to insist on using their own power to consistently reveal and examine their practice, and seek input from black stakeholders they must invite black parents to the table, listen to their concerns and ideas, and act on them.”

In a lot of ways, this thinking should be adopted beyond K-12 and into higher educations institutions as well. A lot of the resources I shared start or end with an acknowledgment that we must learn, we must do better, we must grow. This is a process that has been happening in subsets of our community but it must become part of the bigger narrative of who the mathematics community is and strives to be. I wanted to end this post with a quote from the book ‘So You Want to Talk about Race’ by Ijeoma Oluo. Join the conversations, follow and listen to diverse voices of Black mathematicians, join the fight to make mathematics an anti-racist place for all, and when you do remember: it is the system of racism that we must fight.

“Ask yourself: Am I trying to be right, or am I trying to do better? Conversations about racism should never be about winning. This battle is too important to be so simplified. You are in this to share, and to learn. You are in this to do better and be better. You are not trying to score points, and victory will rarely look like your opponent conceding defeat and vowing to never argue with you again. Because your opponent isn’t a person, it’s the system of racism that often shows up in the words and actions of other people.”

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter ( @MissVRiveraQ ).

## Table of Contents

PREFACE AND ACKNOWLEDGEMENTS

Chapter 1: Introduction and Review of the SIOP MODEL

Chapter 2: The Academic Language of Mathematics

Chapter 3: Activities and Techniques for Planning SIOP Mathematics Lessons

Chapter 4: Lesson and Unit Design for SIOP Mathematics Lessons

Chapter 5: Pulling It All Together

Appendix A: SIOP Model Components and Features

Chapter 1 INTRODUCTION AND REVIEW OF THE SIOP MODEL

Key Components of the SIOP Model

Why Is the SIOP Needed Now?

Organization and Purpose of This Book

CHAPTER 2 THE ACADEMIC LANGUAGE OF MATHEMATICS

What is Academic Language?

How Does Academic Language Fit Into the SIOP Model?

How Is Academic Language Manifested in Classroom Discourse?

Why Do English Learners Have Difficulty with Academic Language?

How Can We Effectively Teach Academic Language In Mathematics?

The Role of Discussion and Conversation in Developing Academic Language

What is the Academic Language of Mathematics?

Appendix B Academic Math Vocabulary Based on NCTM Content and Process Standards

CHAPTER 3 ACTIVITIES AND TECHNIQUES FOR PLANNING SIOP MATHEMATICS

LESSONS By Araceli Avila and Melissa Castillo

Math Techniques and Activities

SIOP Math Techniques and Activities: Lesson Preparation

Number 1-3 for Self Assessment of Objectives

BLM 3.1 What Do You Know About Geometric Shapes?

SIOP Math Techniques and Activities: Building Background

4 Corners Vocabulary Chart

SIOP Math Techniques and Activities: Comprehensible Input

Math Representations Graphic Organizer

BLM 3.2 Math Representations Graphic Organizer

SIOP Math Techniques and Activities: Strategies

SIOP Math Techniques and Activities: Interaction

Group Responses with a White Board

SIOP Math Techniques and Activities: Practice & Application

SIOP Math Techniques and Activities: Review & Assessment

CHAPTER 4 LESSON AND UNIT DESIGN FOR SIOP MATHEMATICS LESSONS

By Araceli Avila and Melissa Castillo

BLM 1.1 Vocabulary Activity Sheet

BLM 2.1 Let&rsquos Measure the Length of&hellip

BLM 4.1 Measuring Length and Distance

BLM 1.1 4-Corners Vocabulary Activity Sheet

BLM 1.2 Shape Characteristics

BLM 1.1 Integer Dollar Cards

BLM 2.2 Instruction for Who is Colder? Card Game

BLM 3.1 Integers Venn Diagram

BLM 3.2 Weather News Transparency

BLM 3.3 Adding Integers Lab Sheet

BLM 3.4 Simultaneous Round Table Activity Sheet

BLM 4.1 Subtracting Integers Lab Sheet

BLM 4.2 Fun With Integers Instructions

BLM 4.3 Fun With Integers Recording Sheet

BLM 5.1 Where is The Submarine?

BLM 5.2 Applying Integers Lab Sheet

BLM 1.1 Math Representations Graph Organizer

BLM 3.1 Tiling Squared Pools

BLM 4.1 Translating Parent Functions Lab Sheet

BLM 4.2 Ordered Pairs for Quadratic Parent Function

BLM 4.3 Ordered Pairs for Linear Parent Function

BLM 4.4 Ordered Pairs for Exponential Parent Function

BLM 5.1 Go To Your Corner Cards

BLM 5.2 Multiplying x by -1 < a < 0

BLM 5.3 Multiplying x by 0 < a < 1

BLM 5.4 Multiplying a by > 1

BLM 5.5 Multiplying x by a < -1

BLM 5.6 Multiplying x by -1

BLM 5.7 Combining Transformations