# Actually Applicable Application Problems and Brainteasers

## Purpose

editThe founding author, Avrila Frazier, is a math teacher and tutor with a standing complaint against the Suzy's Freaking Pocket Change kind of problem, as described in a Facebook comments thread:

When the alleged application problems are made up bullcrap that no one would ever actually apply, like Suzy has $1.57 and 12 coins and three of them are nickels so how many dimes, which you would never really use simultaneous equations for because

- who gives a flying flip?
- you would have counted each coin type to find out the total
the hidden curriculum is This Doesn't Matter.

She was irrationally pleased with herself for cleaning it up to bullcrap and flying flip.

The intention of this wikibook is to serve as a repository of interesting problems, with emphasis on actual application problems, but also inclusive of interesting brainteasers for students who enjoy them, and clearly labeling the two rather than presenting brainteasers as application problems. (Since interesting is a value judgment, we will in general err on the side of inclusiveness, while being honest about the fact that it is a brainteaser so if someone does not find it interesting that's OK too.) It is to be organized mainly by topic rather than grade or age because expectations vary and overlap, though the approximate categories early elementary, later elementary, junior high, high school, and adult will be used to try to include problems applicable to a range of learners.

## How to Use this Book

edit### For Students

editIf you're wondering how you'll use something you're learning in math class, this book should give you some ideas about that. Please be aware that it is a work in progress -- at a very early stage at this point, as you can see -- so don't conclude that something is not useful just because its uses are not listed here yet.

If you know of something math is useful for that isn't listed here yet, please add it! Don't be shy -- the rest of us can help with explaining and formatting it if needed. Just put the ideas out there.

#### A note for students on learning math in general

editRemember that some of these ideas may be based on math that you haven't studied yet. That doesn't mean you can't ever do it -- it just means that, the way I've explained it, you might not be able to do it right now. There's an idea called the Zone of Proximal Development, often abbreviated ZPD; it's the range of things that are not too easy or too hard for the person. **Every** person, in **every** subject, has a ZPD.

A ZPD can move forward within the subject at different speeds depending on if the person is studying within their ZPD, and how much they practice, and if they're getting explanations that work well for them (see below in the "if it's tricky" tips), and probably other things too. A ZPD can also move backward if you take time off from studying a subject -- that's why most classes start with a bunch of review lessons after summer break, because most students have slipped.

It gets complicated for teachers because they are trying to plan lessons for many people's ZPDs, which might not overlap, and which are always changing so even if the teacher knew everyone's exact ZPD last week they don't this week. However, when it's just you, you can do a few things that are mostly simple to use your ZPD to make yourself as good a mathematician as you need or want to be.

- If it's too easy, skip it.
- Or, if it's easy but fun, do it during times when you'd like to unwind a bit.
- If it's for a grade, though, do it anyway to get it over with, and then look for something in your ZPD.

- If it's tricky, try to get a handle on it. Here are some things you can try.
- Sketch things.
- Round the numbers to make the calculations easier (unless you're specifically trying to learn a calculation strategy).
- Look online (or in a different book, or ask someone) for an explanation that makes more sense to you. People have different styles of understanding math and communicating, so just because one teacher is great for lots of people doesn't mean they're the best teacher for you. Another teacher's style might be more compatible for you and it's OK to go looking for that.

- If it's too hard, shelve it for later.
- However, don't be too quick to conclude that things are too hard. If your classes in a subject come easily to you for a while (because the teacher was giving lessons below your ZPD because other people needed them), and then gets to a point where it's tricky (because the lessons have finally caught up to your ZPD), that can feel like "too hard" at first because you're not used to it. Please don't give up because of this!
- If you're really not getting anywhere, try asking someone who's ahead of you in that subject (I'm thinking about math but I think it's true in all subjects) if there's something else you should study first.

### For Teachers

editIf you want to assign better application problems, look for the topic you're currently teaching in our list below. Similar to what we ask of students, please be aware of the stage of the project and bring additional ideas to our attention. We also want to include printable worksheet and problem-organizer pages to make these problems as easy to include in your classroom as possible -- please contribute ideas for that as well!

## Table of Contents

edit### Types of Calculations

edit- Whole Number Operations
- Addition
- Subtraction
- Multiplication
- Division
- Tip Estimation 1
- Modular Arithmetic
- Time-Speed-Distance
- Fuel Burn

- Decimal and Percent Operations
- Addition
- Purchase Price 1 (several items, each listed separately)
- Tip Estimation 2

- Subtraction
- Multiplication
- Simple Interest
- Compound Interest
- Rain Accumulation
- Sales Tax
- Purchase Price 2 (repeated items)
- Purchase Price 3 (bulk goods)
- Tipping

- Division

- Addition
- Fraction Operations
- Addition
- Subtraction
- Multiplication
- Division

- Signed Number Operations
- Absolute Value
- Addition
- Subtraction
- Multiplication
- Division

### Branches of Mathematics

edit- Algebra
- Elementary Algebra
- Evaluating Expressions
- Solving Equations and Inequalities
- Single Variable Equations
- Multivariable Equations
- Simultaneous Equations

- Abstract Algebra
- Group Theory
- The Algebra of Matrices

- Ring Theory
- Distributive Property of Multiplication Across Addition

- Field Theory

- Group Theory

- Elementary Algebra
- Calculus
- Limits
- Derivatives
- Integrals

- Discrete Mathematics
- Graph Theory

- Functional Analysis
- Basics
- Function Types
- Polynomial Functions
- Degree 1 (Linear)
- Degree 2 (Quadratic)
- Higher Degree

- Rational Functions
- Exponential, Radical, and Logarithmic Functions
- Trigonometric Functions
- Other Functions

- Polynomial Functions

- Geometry
- Shapes and Formulas
- Euclidean
- Triangles
- Quadrilaterals
- Rain Accumulation
- Area of Rectangles

- Other Polygons
- Circles
- Other Shapes

- Non-Euclidean

- Euclidean
- Coordinate Geometry
- Trigonometry

- Shapes and Formulas
- Measurement
- Number Theory
- Prime Numbers
- Encryption

- Divisibility
- Common Multiples
- Common Factors

- Prime Numbers
- Probability
- Sets
- Statistics
- Descriptive Statistics
- Displays
- Numerical Summaries of Data Sets

- Inferential Statistics
- Point Estimates
- Interval Estimates
- Hypothesis Testing

- Descriptive Statistics