### Absolute Value Functions

The absolute value (ABS) function is the positive value of a number or quantity. Its graph has a very peculiar shape, a V. Since absolute values are always positive we can think of them as distances on a numberline. Read more

### Absolute Value Polynomials

What would you guess the absolute value of a polynomial function would look like? Try some here. I think you will be very surprised. Can you explain what you see? Read more

### Adding

Adding to 10 is but on example of a large number of spreadsheets that could be developed to practice addition. We give students two number lines (number bars) to 10, and have them copy a number from each to build a pattern of all the sums to 10. We suggest… Read more

### Adding Machine

Multiplication is often thought of as repeated addition. By building a times table by using a repeated addition rule you will have a fun practice of the multiplication facts, and you will learn to build rules in spreadsheets. Rules let you program spreadsheets to do remarkable things like make the… Read more

### Addition Patterns

Now that students can build addition tables, they should look for the patterns in them. We introduce them to a variety of things they might want to look for. We also suggest that some students may want to try to make a subtraction table. We have another Lab for making… Read more

### Addition Table

From here on students can go in most any order they choose. They can start with addition or with multiplication. They should imagine themselves on an elevator able to go up or down anytime they want, to find interesting floors to explore. NOTE: The addition table, like the other tables,… Read more

### Addressing

We have been using cell addresses informally until now, but now we can be more formal and explicit. Different spreadsheets have different types of address bars, but all use the same format, letters for columns and numbers for rows with letters first and numbers second. We introduce this on the… Read more

### Air Pollution

This chart recently came out from the American Lung Association about air pollution in the United States. How would you present this data to Congress to get them to deal with this serious problem? Read more

### Associativity

Parentheses are not only important in paper math, they are critical in spreadsheets. To make sure that terms are handled properly by spreadsheets, we have to be sure we use parentheses to write our formulas so that there is no ambiguity. Associativity gives us the principle behind this grouping, for… Read more

### Battleship

Spreadsheets can be a great place for you to build your own games. Ryan has built one of his early favorites, Battleship, where you learn graphing as you try to sink your opponent’s battleships. Making your own games can be great fun. Try it. Read more

### Birthday

In a class of 23 students, the chances are fifty-fifty that two of them will have the same birthday. Now that may sound impossible since there are 365 different possibilities, but we can use probability to see that it is true. Read more

### Build a House

Spreadsheets with their natural grid make a great, though not entirely flexible, platform for architectural design and for working with shapes. Build a house introduces students to using spreadsheets to create floor plans and to measure area. Most spreadsheets have amazing graphic flexibility. You may want to encourage students to… Read more

### Build a Times Table

Students are tasked to build a times table in just two steps. They have to learn to use absolute as well as relative addressing to do, and the Lab takes them through using them. We encourage students to work with just a row or a column rather than with the… Read more

### CO2 Growth

Spreadsheets offer us a nearly unlimited ability to develop and learn from case studies using real world data. We will focus mainly on climate change which is an area rich in possibilities and of great interest to students. In this case study we look at the production of carbon dioxide… Read more

### Coffee Money

Are you a coffee drinker? Small daily expenses like coffee can quickly add up. By using spreadsheets to organize cost, you can monitor how much you’re really spending. Read more

### Coin Problems

Suppose Briley has 10 coins in quarters and dimes and has a total of $1.45. How many of each coin does she have? Read more

### Common Denominators

We can use these proportions to compare two ratios with different denominators by finding a denominator that their proportions have in common. Thus the common denominator of 2/3 and 3/4 is 12. We then can use the common denominator to add/subtract and divide common ratios (fractions). This approach to division… Read more

### Commutativity

The symmetry of the multiplication table around the square numbers diagonal we call commutativity or the commutative property. It means that in a 12 by 12 multiplication table we need only learn 72 or so facts and not 144. It also means that the square numbers are not the only… Read more

### Composition of Functions

One of the most powerful aspects of the mathematics of functions is our ability to treat them as abstract quantities (essentially numbers) and then combine them with standard operations. But with functions we can go further and develop a new operation we call composition or taking a function of a… Read more

### Counting By

Counting-By introduces multiplication. Counting-by or skip-counting is, we believe, the best way to help students build their multiplication facts, and though they will live in an age of ubiquitous spreadsheets and calculators, they still need to have mastered their multiplication facts to do any interesting math in their heads. Here… Read more

### Credit Cards

How much are your credit cards costing you? If you are like most of us, they are very expensive because we have credit card debt. This Lab will help you to see what your credit cards are costing you and help you to keep track and pay off your credit… Read more

### Decimal Addition

Decimal addition is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Subtraction and Decimal Multiplication and Decimal Division.… Read more

### Decimal Division

Decimal Division is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Subtraction and Decimal Multiplication.… Read more

### Decimal Multiplication

Decimal Multiplication is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Subtraction and Decimal Division.… Read more

### Decimal Subtraction

Decimal subtraction is one of a series of Labs to help you understand and learn to use decimals. I think you will find it fun because it uses random numbers and it will help you visualize decimal operations. Take a look at Decimal Addition and Decimal Multiplication and Decimal Division.… Read more

### Decimals and Percents

Ratios can be written in a wide variety of different way: as fractions, as decimals, and as percents.,with a colon, with a slash, as a fraction and even as a baseball batting average. Here we compare a decimal ratio and a percent by building decimal and percent tables in the… Read more

### Distributivity

The distributive property turns out to be central to a surprising variety of important mathematics. One of the most valuable is to use it to break products into two pieces to make them easier to compute. Thus 5*6+5*10 is easier to solve in your head than 5*16. Here again we… Read more

### Division and Ratio

We can make a division table just like we made a multiplication table. Spreadsheets will usually turn such tables into decimal quotients. Excel will allow you to see these quotients as fractions. We will use Concatenate to keep show the quotient as a ratio and not as a decimal so… Read more

### Drawing Triangles

Though spreadsheets lets you put geometric shapes on the screen, those shapes are not connected with the cells and cannot be changed by using different values. We wanted to make geometric shapes that we could control and change by changing parameters. This is our first try. You can learn to… Read more

### Enigma Machine

Spreadsheets are great for creating secret codes and for breaking them. During World War II the German military used a machine they called Enigma to send coded messages. In a box about the size of a typewriter, wheels with letters on them were spun around to encode or to decode… Read more

### Equivalent Fractions

Fractions are ratios that’s why we call them rational numbers (ratio numbers). If you think about fractions as ratios, how does this help you to understand them? Read more

### Exploring Triangles

Create, manipulate, and explore triangles in this live interactive spreadsheet. On the surface, a triangle is being drawn on a graph, but the real magic lies in the formulas used to make this spreadsheet work. Read more

### Exponential Functions

What if you created a function where the exponent is a variable? As you might expect, this would be called an exponential function. When you hear someone talk about “growing exponentially,” they are talking about an exponential function. Exponential functions have some very interesting patterns. Read more

### Factor Pairs

Multiplying creates products, factoring separates a product into the numbers that make it up. We thus start with the table and then look at the axes to find the factor pairs that make the product. Once again we focus on the patterns in the times table so that you can… Read more

### Factor Table

Spreadsheets always automatically perform the operations you ask them to do. But sometimes we want to see the process. We can make spreadsheets show us that in several ways. Here we show the factors and have students build a times table showing the factors by using a special formula called… Read more

### Fibonacci’s Sequence

Fibonacci, the nickname given the great medieval mathematician Leonardo of Pisa, is connected in most of our minds with the Fibonacci sequence. Spreadsheets make wonderful tools for creating such sequences. This one is amazingly simple. Just select a cell, any cell and write a formula in that cell that adds… Read more

### Gas Money

If you commute to work or school, you understand how expensive driving can be. Choose a car in this interactive spreadsheet as it breaks down the cost of driving, using MPG, distance traveled, and cost of gas. Read more

### GDP

The GDP or Gross Domestic Product of a country is one measure of its wealth. What can this data tell us about the U.S economy, about how far the nation has come in the past 87 years? Are we getting wealthier? Are each of us really wealthier after inflation? What… Read more

### Graphing

Graphs as we know them were first invented by Galileo. They are powerful images of functions. I will introduce you to graphs by letting you graph different functions represented by a table. Fill in each table visualize its graph. Read more

### Hindu Algebra Problem

This problem is typical of the earliest algebra problems that likely came out of India. It is interesting historically, and it is the kind of problem students are still taught to solve today. We can do it very differently using spreadsheets. Read more

### Hit Streak

In 1941, Joe DiMaggio got a hit in 56 straight games. This record has never been beaten. Some say it is the greatest record in all of sports. You can develop a simulation of DiMaggio’s streak to see why it is considered so difficult to beat. And you can decide… Read more

### How Many Times

How many of the numbers from 1 to 100 are in the times table? All, Most, Less than half? I think you will find in this exploration of the relationship between the multiplication table products and the whole numbers as fascinating as I have. It is one of my favorite… Read more

### Hundreds Table Patterns

We introduce students to more complex patterns and rules in hundreds tables. In particular, we have students look at diagonal patterns and develop rules to fill them. This spreadsheet hundreds table practice is designed to build numbersense, the primary building block for a strong math education. We encourage students and… Read more

### Introducing Spreadsheets

In introducing Spreadsheets we want you to learn to build a numberline by using a rule (a formula). We begin with a simple rule that you can copy and paste into the entire numberline. Then we want to add an input from another cell into the rule to give you a chance to change… Read more

### Introducing Subtraction

What if you built a rule that would enable you to count backwards? How would it be the opposite of addition? What would happen if it goes past zero? Read more

### Inverse Functions

What does a function look like when we interchange the inputs and outputs, that is make the x-axis the y-axis and vice versa. Read more

### Inverse of a Function

Spreadsheets make it very easy to switch axes and add graphs. They enable students to play with what may have been difficult and abstract concepts like the inverse of a function. You may want to approach the inverse of a function by challenging students to fill in a table of… Read more

### Lease or Buy

If you are in the market for a new car, you often have a decision to make. Do you want to buy the car or lease it. I have, over the years, developed a simple model to help me decide. I call it my rule of 6. Read more

### Lemonade Stand

Manage a business. The Lemonade Stand has been a business simulation developed in many different versions. We believe this is the first time that it has been created as a spreadsheet simulation. If you are a young entrepreneur who wants to learn how a business works and how you can use spreadsheets… Read more

### Lights Out

This is one of those math puzzles that come up in contests but which turn out to be quite interesting mathematically. Imagine a long hallway with lights in the ceiling, all on and each controlled by its own chain. A long line of people (as many as there are lights)… Read more

### Linear Functions

Linear functions are the most important family of functions. They pervade our everyday lives and our work. Their graph is a line, and their general form is f(x)=mx+b where m is the slope of the function and b is the y-intercept, the value where the line crosses the y-axis. This… Read more

### Lissajous Figures

We often see Lissajous figures in old sci-fi movies because they are so cool. As you play with them I think you will find them as fascinating as I do. They created those figures by graphing points as a function of a third variable (the parameter). We can do the… Read more

### Make a Hundreds Table

This is the graduation exercise for the basic use of spreadsheets. We combine rules and addressing to have students build their own hundreds table in the fewest steps. There are many ways of doing this and students can be as creative and exploratory as they want. Nor should they feel… Read more

### Making Fractions

Spreadsheet math generally focuses on ratio and proportion to develop the concept of fraction. But fractions is such a big problem in today’s curriculum that it seemed only right to use the power of spreadsheets to help students and teachers to gain some fraction-sense. This is a very simple spreadsheet… Read more

### Margin vs Markup

How should you figure out your profit? If you have an item that you are selling, should you price it at 25% over its cost to you, or should you price it so that your margin is 25%. Many people think that their profit will be the same either way.… Read more

### Mixture Problems

How many gallons of a 70% alcohol solution must be added to 30 gallons of a 10% alcohol solution in order to produce a new mixture, a 20% alcohol solution? Read more

### Moore’s Law

It was one of the most amazing visions of the future ever made. In 1965 Gordon Moore, one of the founders of Intel, proposed a law governing the future of computing. He originally proposed that the number of components on a chip would double every year. Later he revised that… Read more

### More Number Lines

We use rules to build new numberlines. For example we can start in the middle and go both forward or backward using adding and subtracting rules. You can even generate and experiment with negative numbers by subtracting below 0. As you build numberlines on spreadsheets you are building them in your mind. And by thinking of numberlines in terms of rules you… Read more

### Motion Problems

George is in New York and Martha is in Washington. They leave at the same time and follow the same road to meet each other on the way. The distance between New York and Washington is 229 miles. George has a fast horse and averages 16 miles/hr. Martha has a… Read more

### Multiplying Integers

We have made a big deal of the times table and of other tables.Now we extend the times table to negative numbers and thus to all 4 quadrants of the real number space. We hope to build student intuition about this space and to gain a spatial sense of graphing… Read more

### Napoleon’s Pyramid

When Napoleon conquered Egypt in 1798 he went to the Great Pyramid of Giza. While his men climbed to its peak, he figured out that “There is stone enough to build a wall 3 meters high and 1/3 meter thick around the whole of France.” Was Napoleon right? How and… Read more

### Normal Distribution

Most museums with math exhibits have a Pascal’s triangle made up of pegs with balls falling down between them and bouncing off of them. One of the things they want to show is probability and the Normal or Bell curve produced by these balls as they fall down most of… Read more

### Number Lines

Number Lines introduce students to functional thinking and the use of formulas in spreadsheets. For younger students we call these formulas “rules” and ask students to build a variety of number lines using rules. For example they can build a whole number line by creating a rule that adds 1… Read more

### Number Patterns

Use numberlines and spreadsheet rules to explore the amazing patterns we find in our whole numbers. Did you know that you can get the odd numbers by subtracting the square numbers? I won’t give away any other secrets, but I know you will find in these patterns some wonders. And… Read more

### Number Series

Spreadsheets make it easy for us to explore patterns in the whole numbers. This Lab does that and helps you learn the basics of spreadsheets like cell addressing, copy and pasting, and making rules. It is designed for every learner including young students. Read more

### Odd Times

How many of the products in a 12 by 12 times table are odd numbers? This is a question we rarely ask in paper-based math classrooms, yet it is an important and a very interesting question. We ask students to explore it, learning to Show and Hide rows and columns… Read more

### Parametric Equations

Parametric equations are powerful tools to model projectile motions and to graph things that are not functions like circle or ellipses. The x and y coordinates are defined as two separate functions with a common independent variable often labelled “t”. Read more

### Parentheses and Pi

Parentheses are very important in spreadsheets because like all programming, spreadsheet formulas have to be very specific. A big formula, especially one like Viete’s approximation of pi, likely will require us to think both in parentheses and in creating formulas that naturally build a series. This one is quite interesting… Read more

### Pascal’s Triangle

Another famous pattern, Pascal’s triangle, is easy to construct and explore on spreadsheets. Create a formula for any cell that adds the two cells in a row (horizontal) above it. This pattern is like Fibonacci’s in that both are the addition of two cells, but Pascal’s is spatially different and… Read more

### Pennies to Heaven

Pennies to Heaven is a Fermi Problem, basically a “headmath” experiment. Fermi Problems, originally developed by Enrico Fermi, one of the greatest experimental and theoretical physicists of the 20th century, are real-world estimation problems. So we ask, “If we had a stack of pennies as tall as the Empire State… Read more

### Personal Budget

How much money do you spend? How much money can you save? We all need to know these things, but you can’t know them until you build and track a budget. Here is your template for a personal budget to track your income and your expenses and figure out what… Read more

### Peter’s Taxi

There are a wide variety of financial literacy problems. This is the kind of problem that appears on many tests. It does not ask you directly to find the cost of a taxi ride which includes both a fixed and variable amount. See if you can use this Lab to… Read more

### Place Value

Our number system inherited from India and from the Medieval Arab world enables us to use just 10 symbols to write any number we can imagine. Students learn in this spreadsheet to enter numbers, to compare compact and expanded forms of those symbols and to add units to any number.… Read more

### Place Value: Decimals

We take our place value generator to decimals to help students see the simplicity of the place value pattern going right as well as left. Read more

### Place Value: Thousands

We extend the place value generator to 100’s of thousands to show you how the pattern of 1’s, 10’s, 100’s, continues to 1,000’s, 10,000’s, 100,000’s. Enter numbers and watch the expanded and compact forms of place value change. Pay special attention to using text units and take a look at… Read more

### Polynomial Functions

Polynomial functions are not limited to the highest term and while that term is most important in determining the shape of its graph, additional terms play a role. Try out additional terms to see how they affect the shapes of the graph. Focus on the patterns! Read more

### Power Functions

Adding an exponent, sometimes referred to as a ‘power’, to the input variable of a linear function that passes through the origin creates a power function. Changing the parameters of these functions reveal some important and interesting patterns. Read more

### Powers of Ten

Picturing exponential growth, powers of 10, can be hard for any of us to imagine. The spreadsheet has the flexibility to enable us to explore the powers of 10 and to get a visual image of them. We can see the difference in shape between odd and even powers, and get a… Read more

### Prime Numbers

The prime numbers are among the most fascinating objects in all of mathematics. While we can generate them, we do not know or understand their pattern. Yet, they have some fascinating patterns that we can easily see like the twin primes. We found on the Web a Conditional Formatting formula… Read more

### Probability: Flipping a Coin

That probability is multiplicative is not an easy concept for many of us. Using the spreadsheet with our ability to make tables and to cut and paste can make this important concept much more transparent. We look forward to your thoughts about what we have done. Read more

### Products as Areas

Using the times table, students can see that products are always rectangles, and that they represent the area of that rectangle. They should explore the times table by playing with these rectangles whose sides are the factor of the products. Read more

### Projectile Motion

Let’s do a little target practice with this spreadsheet projectile simulator, which will map out the flight path of an arrow shooting toward a target. Read more

### Quadratic Functions

What does each of the coefficients do? How does it change the graph of the parabola. What does a do, what does c do and a question still rarely asked, what does changing b do to change the graph. To see what b does more clearly we have you add… Read more

### Rate of Growth

We look at world population over the past 60+ years and ask whether the earth’s population is growing faster or slower today. Is it out of control and something we should all worry about or are we getting it under control? This is another problem directly related to climate change… Read more

### Ratio and Proportion

We think about ratio tables in terms of motion. Move up 2 and over 1, or move up 1 and over 2. In this way we build proportional patterns. By coloring the cells we land on like knights in a chess table, we can see the proportions of different ratios.… Read more

### Rows and Columns

We use the hundreds table to introduce rows and columns and focus students on seeing the patterns in these tables. Again and again we go back to making rules and using rules to ask and answer questions. For example, what rule would you make to fill in a column on… Read more

### Rule of 72

The rule of 72 is an old banker’s rule of thumb to find out how long it will take to double your money at different interest rates. Financial literacy has become an increasingly important topic for K-12 education and we believe spreadsheets and headmath or mental estimation should be central… Read more

### Shapes

Shapes introduces student to changing the colors in cells and to changing the shapes of cells by dragging the column or row separators in the address axes. Students can use spreadsheets as drawing tools and can create some wonderful pictures with them. Spreadsheets can thus be tools for visualizing mathematics… Read more

### Sierpinski Fractals

Fractals are a new 21st century mathematics. They are patterns that repeat themselves at various scales. This one is based on the odd numbers in Pascal’s triangle. We learn to create it easily by using Conditional Formatting which enables us to color cells or text based on a quantitative relationship.… Read more

### Similar Triangles

Scatterplot graphs enable us to build shapes using spreadsheets and to practice transformational geometry. They are surprisingly flexible tools. And since they depend upon a table of value and that table can have both fixed numbers and rules, we can not only build shapes but change them and watch the… Read more

### Sine Function

Spreadsheets are not limited to algebraic functions, they can also display trigonometric functions. We are modeling the sine function, but you can try any of the trig functions by going to the Formulas menu and choosing it. We have the 3 most significant parameters of the sine function, to control… Read more

### Solar System

When I was young I loved to play with planetary data, to explore their patterns, to learn more about astronomy, and to deal with large numbers. Spreadsheets make it much easier to study the solar system and to find relationships between the planets that are fascinating and unexpected. In the… Read more

### Solving Equations

Typical algebra courses start with equations and solving equations and then move to graphing and functions. We start with functions and use them to solve equations. We treat an equation as the equality of two functions, graph each one and then look at their intersection. This is a powerful way… Read more

### Solving Equations Digitally

This Lab introduces a method for solving or estimating the solution to an equation digitally that can be applied to many types of equations. This Functional Thinking approach reduces the need to remember a variety of rules and procedures. It is 1 of 3 Labs on this topic. Read more

### String Challenge

Strings need not begin and end on axes that are at right angles to each other which we call Cartesian. It is quite interesting that Descartes himself did not use axes at right angles. We consider this a challenge because students have to figure out how to move both the… Read more

### String Diagrams

The usual way to make string diagrams using rubber bands or yarn on a board with nails does not allow much exploration. Mary Boole meant them as exercises in visualization. Building these diagrams using spreadsheets not only shows their versatility and capability for artistic expression, it helps students get used… Read more

### Subtraction Tables

What would a subtraction table look like? How would its pattern be different from an addition or multiplication table? Is subtraction commutative? Read more

### Sudoku Challenge

Have you ever played Sudoku? It is fun and challenging. You have to find the numbers from 1 to 9 in each cell so that that all of the numbers appear only once in every row, column, and grid square. Ryan added a sweet wrinkle to the traditional Sudoku game,… Read more

### Syracuse Problem

I built a Lab for you to play with the Syracuse Problem and to learn to use spreadsheets to play with like problems in fun and interesting ways. The Syracuse Problem is a simple one. Pick a number, any whole number. If it is even divide it by 2, if… Read more

### Systems of Equations

Solving systems of equations sometimes called simultaneous equations with graphs is simply a matter of finding out where they intersect. One of the most valuable things students can learn is to be able to visualize linear equations and systems of equations so that they can tell the quadrant where the… Read more

### The Chessboard

We take that great old problem of the inventor of chess and the ruler of India and use it to see how powers of 2 grow in size. We start out with a chessboard and look at doubling each successive number. Then we seek a method of representing this doubling… Read more

### The Magic Rectangle

Multiplication tables have some wonderful and quite surprising patterns. This is one of them. Draw any rectangle in a multiplication table and you will find that the products of opposite corners are equal. For example a rectangle around a full 12 by 12 table will be 1*144 and 12*12. Try… Read more

### The Square Numbers

The square numbers form an interesting pattern on the times (multiplication) table. They run along a diagonal from 1 to the top right of the table separating the table into two halves. This is the first step in looking at patterns in the multiplication table. Students build a new square… Read more

### Triangular Numbers

1, 3, 6, 10… are called the triangular numbers because they can be stacked up to form a triangle. They are very interesting numbers, and they form a very interesting pattern when graphed. Can you guess the next triangular number? Can you guess the shape of the graph of the… Read more

### What is x?

What is “x”? Or how do we represent variables and functions on spreadsheets? Download What is x? Read more

### Work Problems

Suppose Tom can paint the entire fence in twelve hours, and Huck takes eight hours. How long would it take the two of them together to paint the fence? Read more