### Algorithm

An algorithm is a process, a procedure. The word was coined by medieval European mathematicians to give a name to the arithmetic and algebra they were learning from the Arabs. They based it on the name of the great Muslim mathematician al Khwarizmi. The word algebra comes from the title of his great book. To visualize an algorithm picture the long division process. We continue to teach our students paper-based algorithms, though they will never need or use them. Why?

### Choice

We have three levels of Labs, beginner, intermediate, and advanced on a wide variety of topics to give you lots of choice. We think learning math should be like riding an elevator and not climbing a staircase. Go to different floors to choose lessons of interest to you. We add new Labs all the time so keep looking. Send us a note on Facebook to tell us what topics you would like new Labs on.

### Challenge

Spreadsheets are a wonderful tool for exploring math, that’s why we think of them as laboratories. You have all the power to explore, to try, to challenge yourself and other students to find new and different ways to solve problems. Spreadsheets are function machines. They let you look at problems and data as tables, graphs, formulas, and even visualizations.

### Creativity

The most important ability that you will need in the 21st century is to think out-of-the-box. Creativity is not something we are born with, it is something we learn by practice. What if Math can give you that practice. Instead of asking What is… like “What is the answer to this question”, we ask “What if…”What if I do this or change that.” So ask yourself and others, “What if…” and have fun!

### Common Core

What if Math is built on the principles espoused in the Standards of Mathematical Practice in the Common Core State Standards in Mathematics. Our content, as you might expect from a curriculum designed for the 21st century and not the 13th century, is not based on the Standards. But we believe that most of our Labs are valuable for helping students learn many of the Standards of Content because as they learn math from a functional perspective they will naturally deal with and come to understand math designed to develop algorithmic proficiency.

### Concrete vs. Abstract

Functions have long been considered too abstract for most students and relegated to college algebra and beyond. The Common Core drops its introduction into 8th grade with a traditional approach. Teachers tell us that their students don’t get functions, that the notation confuses them, that they have a hard time with the concept of variable having treated x for a long time as just an unknown number, and that their introduction focuses on continuous functions while most of of the math they learned in the passed was discrete. We believe that spreadsheets can enable students of all ages to learn function-based math and functional thinking. Spreadsheets are concrete. The notation does not involve symbol manipulation. And most of the functions students will deal with are discrete. Spreadsheets can make functions accessible to all students.

### Curriculum

What if Math Labs are not a complete curriculum in any sense of the word. Rather, these lessons are models to be used as suggestions for what we hope will be thousands of Labs to cover all the math concepts and skills in a wide variety of ways that provide every student with choices of interest to learn and practice functional thinking in every possible situation.

### Free

Yes, we are Free and the Labs we develop ourselves will remain free of both cost and advertising. We seek to reinvent math education and we contribute our time, effort, knowledge, and creativity to this effort. In the future we will expect to seek contributions to support our efforts. And as we develop an online “math store” that accepts outside contributions, some authors might charge for their Labs. For the present we are a small organization dedicated to reinventing STEAM education by providing the best possible math curriculum materials for students.

### Function

Function is an enormously powerful concept. A function transforms one quantity (an input) into another (the output) following a rule. And it does this in a unique way so there is one and only one value of the output for each value of the input. We use x as the symbol of all the possible inputs, and f(x) as the symbol for its outputs. The rule is a formula which we enter beginning with an = sign. We represent a function with its formula, a table of values, a graph.

### Function Families

The difference between a linear function and an exponential function can be profoundly important in our world today. Recognizing and understanding the major function families should be near the top of the math learning chart. We have focused on these differences in What if Math, and ask students at every opportunity to graph their functions and to tell us what that graph looks like and most importantly to answer the “What if…” questions about them. You might want to ask you students, “What function family do the triangular numbers belong to?

### Function Machines

Thinking of a function as a machine helps us make this abstract concept concrete. It has traditionally been viewed as a box with a funnel on top for its input and an upside down funnel on the bottom as its output. The box contains the rule that converts the input to the output. Spreadsheets are function machines. They enable us to build functions with inputs, outputs, and rules and like any good machine they run those functions automatically converting the input to the output. A spreadsheet is like a factory with something coming in and something new going out.

### Functional Thinking

Functional thinking is “What if…” thinking. “What if I change the input to this, what will the output be?” “What if I change the rule to this, what will happen to the output?” “What if I try this, what will happen then?” It is the science experiment, which is why we call our lessons Labs. It gives business today the power that it gave to science that created the scientific revolution. Functional thinking represents inputs and outputs with tables and graphs. Functional thinking lets you be creative and ask, always ask, “What if…”

### Getting Started

If you or our students are new to spreadsheets we have several getting started Labs for you. Though spreadsheets may look scary to some, they are surprisingly simple and really enjoyable if you know a few things. Check out our Getting Started page to find our introductions to spreadsheets. We think you will find spreadsheets very friendly after you have tried these.

We have not built a management or grading system into What if Math. You can make assignments directly from our website, let students choose what they want to work on, or save our spreadsheets on your LMS for assignment. We believe that students are creative and are motivated when they have choice. We suggest a portfolio grading system collecting student spreadsheets in folders, to make them sharable, and to encourage students to present and to perform them.

### Graphing

Spreadsheets give us a great variety of graphing (charting) tools. We try to introduce you to all of these in a variety of ways in What if Math. As you think about graphing from a functional thinking perspective, remember that generally the input is on the horizontal or x-axis (called the abscissa), and that the output is on the vertical or y-axis (called the ordinate). Though this is not always the case for certain kinds of charts like horizontal bar graphs or pie charts, it can help you connect functional thinking to most graphing.

### Handmath

Handmath, what we do with our fingers with a pencil on paper, or with keys on a phone, calculator, or keyboard is exact and based on algorithms. It should give us an accurate answer through a process we are taught to follow carefully. Calculators and especially computers using spreadsheets make most all of the handmath we learn in school, process like the long division algorithm designed for rapid and accurate calculation on paper, obsolete. We no longer need to memorize or even learn to use those algorithms, for either the computer will automatically do them or do their own, or we can look them up.

Headmath, what we do in our heads to solve problems, is approximate and based on tricks. Headmath requires numbersense and conceptual understanding. It uses estimation to make a good guess of an answer. We need to learn to use headmath all of the time, even when we have a calculator or a spreadsheet open. Headmath tells us that we have not made a mistake, that we have the right vision of the problem, that our model is the right one. Headmath is intuitive and it takes lots of practice. Unfortunately in our schools today the primary concentration is on handmath because that is what we test. We need to focus much more on mental or headmath.

### Labs

We call our spreadsheet lessons “Labs”. They are experiments, explorations, and opportunities to ask “What if…” We generally plan on them taking 20 minutes to an hour. But they are all open ended and nearly all end with “What if…” questions that we hope you will explore into any depth and breadth you want. Some may take you on a long path. Math is endlessly fascinating and spreadsheets are wonderful laboratories to play, to invent, to think out-of-the-box, and just to have fun learning.

Link sheets present content using the Rule of Four — combining pictorial, table, graph, and symbolic representations to enable students of all learning styles to understand a concept or a problem. Spreadsheets are natural Link Sheets tools. Most of our content reflects these multiple representations and we strongly encourage you to use them.

### Numbersense

We usually think of numbersense as something we want children to have as they enter school; knowing their numbers, that number and quantity are connected, and that size and number are linked. But numbersense is much bigger than that. It is the deep understanding that mathematics is patternmaking and that the patterns of numbers can help us not only compute, but to solve problems. The connection between the odd numbers and the square numbers are such a pattern, and that numbersense can enable us to understand calculus.

### Platform

What if Math spreadsheets are (.xlsx) files. You can use them with any of the standard spreadsheet programs like Excel, Google Sheets, or Numbers, on, smartphones, computers or tablets. We have tried to make them generic so that they are completely functional on all platforms. Just choose a Lab from the Lab cards on our Web-based home screen and click on the link. Be sure you open your spreadsheet in your spreadsheet program. You are welcome to use these files yourself, with your class, to link to them, and to share them.

### Programming

Spreadsheets are business programming platforms. They give us the same thrill that writing computer code does. As a student builds a spreadsheet, tests it out, and makes it do what they want it to do, they are programming. For advanced students spreadsheets have most of the tools they will find in programming languages and they will learn much of the methodology needed to understand computing, programming, and the Internet by working with them.

### Reinventing Mathematics

The word fraction appears in the Common Core Standards 211 times. The word spreadsheet appears only 8 times. Our children spend as much as a year and a half mastering the multiplication and division algorithms and another year and a half mastering the fraction operations. All too many never do master them, and none will have to use them as adults. Less than 20% of all of our students will actually master and use the math they are now required to learn. It is time we reinvented mathematics for the 21st century.

### Search

We have tagged all of our Labs so you can easily find them from our search box or from the Lab Tags at the bottom of our screens. You may just want to browse our cards on our home screen. If you don’t find what you are looking for please Contact Us and tell us what kind of Lab you want. We will try to develop it for you.

We seek to make our spreadsheets work on any of the major programs, Microsoft’s Excel, Google’s Sheets, or Apple’s Numbers. We created them on Excel. They do not use macros or any special programming. If you find any problem with a spreadsheet please send us a note.

### STEM/STEAM

What if Math is:

• Science – because we treat learning math as a scientific experiment
• Technology – because we use the foremost business technology program, the spreadsheet, to enable every student to learn math and functional thinking
• Engineering – because we treat functions as machines which have inputs, rules and outputs.
• Art – because we are focused on pattern and design infusing art in everything we do.
• Math – because we want every student to learn to use and to love mathematics.

### Students

What if Math is a new way for you to learn. Our lessons are Labs designed for you to use on your own or in a small group. Your laboratory is a spreadsheet, most any spreadsheet on your computer, tablet, or cell. Our Labs are designed for real spreadsheets, the kind used today by business, science, and even mathematicians, for you to learn not only math, problem solving, and functional thinking but how to be a maven on spreadsheets.

### Student-Centered

We strive to make this truly student-centered. This means to us that students must seek answers to most of the questions they have, to work with fellow students, to use the web, or to keep trying until they solve the problem. Using spreadsheets to learn math is a new idea, teachers should not be expected to know how to solve these problems! Students should be challenged to figure or find it out!

### Suggestions

We would love to hear from you. We would love your suggestions for new Labs, for improving the Labs we have put up, and for different ways of using spreadsheets to reinvent math education. We hope to enable others to develop Labs in the future, but we are not yet equipped to vet or edit outside submissions. So for the time being send us your ideas.

### Sustainablearning

Sustainablearning, Inc. located in Cambridge, MA is a 501(c)(3) non-profit corporation whose mission is the reinvention of math education. Sustainablearning was formed by Art Bardige and Larry Reeves in 2012. What if Math is a product of Sustainablearning.

### Symbols

We use * for multiplication and / for division throughout as the standard operation signs used in spreadsheets and computer programs. We believe that these signs should replace the traditional ones which are either unnecessary or confusing.

### Teachers

If you are a teacher you are welcome to use spreadsheet math with your classes. We recommend you try these spreadsheet Labs out with those students who are outpacing the class, can work on their own, and need something more to do. What if Math is so different from the common math curriculum that doing this math will not interfere with the math you are teaching. As you become more familiar with What if Math you may choose to extend it to other students in your classes and to use it with your whole class as Problem-Based-Learning or as interactive whiteboard examples to help students visualize concepts.

### Times Table

You may be wondering why we changed the vertical direction of the times table, why we go up when the standard school times table starts with 1 on the top and goes down. Going up on the vertical axis joins the times table to graphing so that students get used to this all important pattern. Now the factors are ordered pairs and commutativity is obvious as are the square numbers on the diagonal.

### Undo

Undo: Spreadsheets have a magic commands “UNDO” that lets you go back if you made a mistake. If you have really gotten yourself in trouble that you don’t think you can undo, then reload the file. Yes it is ok to do that. You can’t break either you computer or your spreadsheet. Spreadsheets also enable you to copy and paste functions to make it easy to build models.

### Variables

Descartes, one of the inventors of functional thinking, was the first to treat variables in our modern way. He introduced many of our standard symbols and divided mathematical objects into constants to be labeled with letters from the beginning of the alphabet and variables to be labeled with letters from the end of the alphabet. This is the reason we most often call a variable x, though y and z are perfectly fine. And today we often choose other letters. With spreadsheets we can choose cell addresses and even cell names.

### What if Math

What if Math
is a reinvention of our mathematics curriculum. Today’s K-12 curriculum was defined in the year 1202 by Leonardo of Pisa to give merchants and traders a better system for doing their math. It is obsolete. Our computers do the algorithms for us, and business is no longer primarily about solving equations. We use spreadsheets and functional thinking. What if Math is built entirely on spreadsheets, designed to enable students to learn math on a functional foundation and to learn to use spreadsheets and functional thinking that is demanded by 21st century business.