# Discover the Beauty and Logic of Math with Sneaky Math: A Graphic Primer with Projects that Teaches Algebra, Geometry, Trigonometry, and Calculus

## Sneaky Math: A Graphic Primer with Projects: Ace the Basics of Algebra, Geometry, Trigonometry, and Calculus

## Introduction

Do you think math is boring, hard, or scary? Do you want to learn math in a fun, easy, and creative way? If you answered yes to any of these questions, then this book is for you!

## Sneaky Math: A Graphic Primer with Projects: Ace the Basics of Algebra, Geometry, Trigonometry, and

### What is sneaky math?

Sneaky math is a way of learning math that uses graphics, stories, puzzles, games, and projects to make math more interesting and enjoyable. Sneaky math helps you discover the hidden beauty and logic of math, and shows you how math can be applied to real-life situations. Sneaky math also helps you develop your critical thinking, problem-solving, and creative skills.

### Why is sneaky math useful?

Sneaky math is useful because it can help you improve your math skills and confidence. Math is not only a subject that you have to learn in school, but also a tool that you can use in everyday life. Math can help you understand the world around you better, and make better decisions. Math can also open up many opportunities for your future career and education.

### How to use this book?

This book covers four main topics in math: algebra, geometry, trigonometry, and calculus. Each topic has four sections: an introduction that explains what the topic is about, a lesson that teaches you the key concepts and skills, a project that lets you apply what you learned in a creative way, and a review that summarizes the main points. You can read this book from start to finish, or skip around to the topics that interest you most. You can also use this book as a supplement to your school curriculum, or as a self-study guide.

## Algebra

### What is algebra?

Algebra is a branch of math that deals with symbols and rules. Symbols are letters or other signs that represent numbers or unknown values. Rules are operations or equations that tell you how to manipulate symbols. For example, x + 2 = 5 is an equation that uses symbols (x) and rules (+, =) to express a relationship between numbers (2, 5).

### How to solve equations and inequalities?

An equation is a statement that two expressions are equal. For example, 2x + 3 = 11 is an equation that says that 2 times x plus 3 is equal to 11. To solve an equation means to find the value of the unknown variable (x) that makes the equation true. To solve an equation, you can use different methods, such as adding, subtracting, multiplying, dividing, or factoring both sides of the equation. For example, to solve 2x + 3 = 11, you can subtract 3 from both sides, then divide by 2, and get x = 4.

An inequality is a statement that two expressions are not equal. For example, x - 2

### How to use functions and graphs?

A function is a rule that assigns an output value to each input value. For example, f(x) = x^2 + 1 is a function that takes any number (x) as input, and gives the square of that number plus one (x^2 + 1) as output. A function can be represented by a table, a formula, or a graph. A graph is a visual way of showing the relationship between the input and output values of a function. A graph consists of a horizontal axis (x-axis) and a vertical axis (y-axis), and a curve or line that shows the points that satisfy the function. For example, the graph of f(x) = x^2 + 1 is a parabola that opens upward and crosses the y-axis at (0, 1).

### Project: Create your own secret code

A secret code is a way of transforming a message into a different form that only the intended receiver can understand. One way to create a secret code is to use algebraic functions. For example, suppose you want to send the message "HELLO" to your friend. You can assign each letter a number according to its position in the alphabet: A = 1, B = 2, ..., Z = 26. Then you can choose a function, such as f(x) = x + 5, and apply it to each number in your message. This will give you a new set of numbers: H = 8 -> f(8) = 13, E = 5 -> f(5) = 10, L = 12 -> f(12) = 17, O = 15 -> f(15) = 20. You can then send these numbers to your friend, who can use the inverse function, such as g(x) = x - 5, to decode them back into letters: g(13) = H, g(10) = E, g(17) = L, g(20) = O.

Try creating your own secret code using different functions and messages. You can also use graphs to visualize your code and check if it works.

## Geometry

### What is geometry?

Geometry is a branch of math that deals with shapes and space. Shapes are figures that have boundaries and properties, such as length, area, perimeter, angle, etc. Space is the set of all possible points and directions in which shapes can exist. For example, a circle is a shape that has a boundary made of points that are equidistant from a center point. A circle has properties such as radius, diameter, circumference, and area. A circle can exist in space in different positions and orientations.

### How to measure angles and shapes?

An angle is a measure of how much two rays or lines diverge from a common point. Angles are measured in degrees or radians. A degree is one-360th part of a full circle. A radian is the angle subtended by an arc whose length is equal to the radius of the circle. There are different types of angles based on their size: acute (less than 90 degrees), right (90 degrees), obtuse (more than 90 degrees), straight (180 degrees), reflex (more than 180 degrees), and full (360 degrees).

A shape is measured by its dimensions and attributes. Dimensions are quantities that describe how big or small a shape is, such as length, width, height, diameter, etc. Attributes are qualities that describe how a shape looks or behaves, such as color, texture, symmetry, congruence, similarity, etc.

### How to use congruence and similarity?

Congruence and similarity are two concepts that compare shapes based on their size and shape. Congruence means that two shapes are exactly the same in size and shape. They can be superimposed on each other without any gaps or overlaps. Similarity means that two shapes have the same shape but not necessarily the same size. They can be scaled up or down by a constant factor without changing their shape. Congruence and similarity can be used to prove properties and relationships of shapes, such as angles, sides, areas, volumes, etc.

### Project: Make your own origami models

Origami is the art of folding paper into various shapes and forms. Origami uses geometry to create patterns and structures from a flat sheet of paper. Origami can also teach you about congruence and similarity, as well as other concepts such as symmetry, fractions, ratios, etc.

To make your own origami models, you will need a square piece of paper, scissors, and a ruler. You can follow these steps to make a simple origami crane:

Fold the paper in half diagonally, then unfold. Repeat with the other diagonal.

Fold the paper in half horizontally, then unfold. Repeat with the vertical fold.

Bring the four corners of the paper to the center point. You should have a smaller square.

Fold the top layer of the right corner to the center line. Repeat with the left corner.

Flip the paper over and repeat step 4 with the other side.

Fold the top layer of the bottom corner up to meet the top edge.

Flip the paper over and repeat step 6 with the other side.

Open up the right flap and squash it flat along the crease. Repeat with the left flap.

Flip the paper over and repeat step 8 with the other side.

Fold the top layer of the right flap to the center line. Repeat with the left flap.

Flip the paper over and repeat step 10 with the other side.

Fold down the top point to make a head for the crane.

Pull out the inner layers of the bottom point to make a tail for the crane.

Gently pull apart the wings of the crane and shape them as you like.

You have made an origami crane! You can try making other origami models using different shapes and folds. You can also decorate your origami models with colors, stickers, or drawings.

## Trigonometry

### What is trigonometry?

Trigonometry is a branch of math that deals with triangles and circles. Triangles are shapes that have three sides and three angles. Circles are shapes that have a boundary made of points that are equidistant from a center point. Trigonometry uses ratios and functions to relate angles and sides of triangles, and arcs and angles of circles. Trigonometry can also be used to model periodic phenomena, such as waves, cycles, oscillations, etc.

### How to use trigonometric ratios and identities?

the side opposite to A, adjacent is the side next to A, and hypotenuse is the longest side of the right triangle. Trigonometric ratios can be used to find missing sides or angles of right triangles, or to solve real-world problems involving right triangles, such as heights, distances, slopes, etc.

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables. For example, sin^2(x) + cos^2(x) = 1 is a trigonometric identity that relates sine and cosine functions for any angle x. Trigonometric identities can be used to simplify expressions, solve equations, prove statements, or derive other identities.

### How to solve triangles and circles?

To solve a triangle means to find all the missing sides and angles of the triangle. To solve a right triangle, you can use trigonometric ratios and the Pythagorean theorem (a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse of the right triangle). To solve a non-right triangle, you can use other methods, such as the law of sines (a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the sides and A, B, and C are the angles of the triangle), or the law of cosines (c^2 = a^2 + b^2 - 2ab cos(C), where c is the side opposite to angle C).

To solve a circle means to find all the missing parts of the circle, such as radius, diameter, circumference, area, arc length, sector area, central angle, etc. To solve a circle, you can use formulas that relate these parts. For example, C = 2Ï€r is a formula that relates the circumference (C) and the radius (r) of a circle. A = Ï€r^2 is a formula that relates the area (A) and the radius (r) of a circle. S = rÎ¸ is a formula that relates the arc length (S) and the central angle (Î¸) of a circle.

### Project: Build your own sundial

A sundial is a device that uses the position of the sun to tell the time. A sundial consists of a flat plate (dial) and a stick (gnomon) that casts a shadow on the dial. The dial has markings that indicate the hours of the day. As the sun moves across the sky, the shadow of the gnomon moves along the dial and points to the current hour.

To build your own sundial, you will need a cardboard plate, a pencil or straw, a ruler, a protractor, a compass, and some glue. You can follow these steps to make a simple sundial:

Find out your latitude (the angle between your location and the equator) and your local time zone. You can use online tools or maps to do this.

Cut out a circle from the cardboard plate. This will be your dial.

Draw a line across the center of the dial. This will be your noon line.

Glue the pencil or straw vertically at the center of the dial. This will be your gnomon.

the noon line on both sides of the gnomon. These will be your 6 AM and 6 PM lines.

Use the compass to draw a semicircle that connects the 6 AM and 6 PM lines.

Divide the semicircle into 12 equal parts. These will be your hour markings. Label them from 6 AM to 6 PM.

Place your sundial on a flat surface that faces south (if you are in the northern hemisphere) or north (if you are in the southern hemisphere). Make sure the noon line is aligned with the north-south direction.

Adjust your sundial for your local time zone. To do this, you need to know the difference between your local time and the solar time (the time based on the sun's position). For example, if your local time is one hour ahead of the solar time, you need to rotate your sundial clockwise by one hour. If your local time is one hour behind the solar time, you need to rotate your sundial counterclockwise by one hour.

You have made a sundial! You can use it to tell the time by looking at the shadow of the gnomon on the dial. You can also experiment with different shapes and sizes of dials and gnomons, and see how they affect the accuracy of your sundial.

## Calculus

### What is calculus?

Calculus is a branch of math that deals with change and motion. Calculus uses two main tools: limits and derivatives. Limits are values that a function or a sequence approaches as the input or the index gets closer to a certain point. Derivatives are rates of change of a function with respect to its input. Calculus can be used to study phenomena that involve change and motion, such as speed, acceleration, growth, decay, optimization, etc.

### How to use limits and derivatives?

A limit is a way of finding the value that a function approaches as its input gets closer to a certain point. For example, lim(x->0) sin(x)/x = 1 is a limit that says that the value of sin(x)/x approaches 1 as x gets closer to 0. To find a limit, you can use different methods, such as plugging in values, simplifying expressions, applying rules, or using graphs. For example, to find lim(x->0) sin(x)/x, you can plug in values of x that are very close to 0, such as 0.01 or -0.01, and see that sin(x)/x is very close to 1.

A derivative is a way of finding the rate of change of a function with respect to its input. For example, f'(x) = 2x is a derivative that says that the rate of change of f(x) = x^2 is 2 times x. To find a derivative, you can use different methods, such as applying formulas, using rules, or using graphs. For example, to find f'(x) = 2x, you can use the formula for the derivative of a power function: f'(x) = nx^(n-1), where n is the exponent of x. In this case, n = 2, so f'(x) = 2x^(2-1) = 2x.

### How to use integrals and applications?

int(a->b) f(x) dx = F(b) - F(a) is an integral that says that the area under the function f(x) from x = a to x = b is equal to the difference between the values of an antiderivative F(x) at x = b and x = a. An antiderivative is a function whose derivative is the original function. For example, F(x) = x^3/3 is an antiderivative of f(x) = x^2, because F'(x) = x^2. To find an integral, you can use different methods, such as applying formulas, using rules, or using graphs. For example, to find int(0->2) x^2 dx, you can use the formula for the integral of a power function: int(a->b) x^n dx = (b^(n+1) - a^(n+1))/(n+1), where n is the exponent of x. In this case, n = 2, so int(0->2) x^2 dx = (2^3 - 0^3)/3 - (0^3 - 0^3)/3 = 8/3 - 0/3 = 8/3.

Integrals can be used to model various applications that involve finding the total amount of change or area under a function. For example, integrals can be used to find the distance traveled by a moving object, the work done by a force, the volume of a solid, the center of mass of a system, etc.

### Project: Estimate the area under a curve

A curve is a smooth line that can be described by a function or an equation. The area under a curve is the region between the curve and the x-axis. The exact area under a curve can be found by using integrals, but sometimes integrals are too difficult or impossible to calculate. In such cases, we can use approximation methods to estimate the area under a curve.

To estimate the area under a curve, you will need a graph paper, a ruler, a pencil, and a calculator. You can follow these steps to estimate the area under y = x^2 from x = 0 to x = 4:

Draw the curve y = x^2 on the graph paper. Label the axes and mark the points (0, 0) and (4, 16).

Divide the interval [0, 4] into n equal subintervals. For example, you can choose n = 4 and divide [0, 4] into [0, 1], [1, 2], [2, 3], and [3, 4].

Draw rectangles on each subinterval such that the height of each rectangle is equal to the value of the function at either the left endpoint or the right endpoint of the subinterval. For example, you can choose to use left endpoints and draw rectangles with heights y = 0, y = 1, y = 4, and y = 9.

Calculate the area of each rectangle by multiplying its base and height. For example, if you use left endpoints and n = 4, then the areas are A1 = (1 - 0)(0) = 0, A2 = (2 - 1)(1) = 1, A3 = (3 - 2)(4) = 4, A4 = (4 - 3)(9) = 9.

the area under the curve. For example, if you use left endpoints and n = 4, then the approximation is A = A1 + A2 + A3 + A4 = 0 + 1 + 4 + 9 = 14.

Repeat steps 2 to 5 with different values of n and different choices of endpoints. Compare your results and see how they get closer to the exact area as n increases. For example, if you use right endpoints and n = 4, then the approximation is A = 1 + 4 + 9 + 16 = 30. If you use left endpoints and n = 8, then the approximation is A = 0.5 + 2 + 4.5 + 8 + 12.5 + 18 + 24.5 + 32 = 102.

You have estimated the area under a curve! You can try estimating the area under other curves using different methods, such as trapezoids or parabolas. You can also compare your results with the exact area found by using integrals.

## Conclusion

### Summary of the main points

In this book, you have learned about sneaky math, a way of learning math that uses graphics, stories, puzzles, games, and projects to make math more interesting and enjoyable. You have also learned about four main topics in math: algebra, geometry, trigonometry, and calculus. You have learned how to use symbols and rules to solve equations and inequalities, how to use functions and graphs to model relationships and patterns, how to use shapes and space to measure angles and areas, how to use ratios and functions to relate triangles and circles, how to use limits and derivatives to study change and motion, and how to use integrals and applications to find areas and volumes. You have also applied what you learned in creative projects, such as creating your own secret code, making your own origami models, building your own sundial, and estimating the area under a curve.

### Tips for further learning

If you enjoyed this book and want to learn more about math, here are some tips for further learning:

Read other books or websites that explain math concepts in fun and engaging ways. For example, you can check out Math with Bad Drawings by Ben Orlin, The Joy of x by Steven Strogatz, or

__Mathigon.org__.

Practice your math skills by doing exercises or problems that challenge you. For example, you can use online tools or apps that provide interactive quizzes or games. You can also join math clubs or competitions that suit your level and interest.

Explore your math curiosity by asking questions or doing experiments that relate math to real-life situations. For example, you can use math to analyze data or statistics that interest you, such as sports scores or weather patterns. You can also use math to create art or music that express your creativity.

, you can join o