# All Labs

What if Math Labs are spreadsheet-based single-concept lessons. Each is designed to use functions and to develop functional thinking. In the STEM tradition they approach math as an experimental science.

### 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… Continue

### 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… Continue

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… Continue

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… Continue

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… Continue

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… Continue

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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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? Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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.… Continue

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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### Division and Ratio

We can make a division table just like we made a multiplication table. Division is surprisingly our most important operation in terms of most of… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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?… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### Interest

Which form of interest, simple or compound, is the fairest? If you were buying a house or a car which would you rather have, which… Continue

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… Continue

### 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… Continue

### 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… Continue

### Inverse of Linear Functions

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

### Inverse Variation

The variables in most of the functions we are used to working with vary directly, as one goes up the other goes up. What do… Continue

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… Continue

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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### Magic Number (Infinite Series)

What if you made a series of unit fractions with the odd numbers as their denominator, added the first two together, subtracted the next one,… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### Phone

Use spreadsheets to compare popular phone carriers and the plans they offer. Continue

### 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… Continue

### 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. Continue

### 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.… Continue

### 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… Continue

### 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… Continue

### Powerball

You just won the Lottery worth \$600,000,000. You have a choice between taking it as a lump sum of \$376.9 million or in yearly payments… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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. Continue

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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### Subtraction Tables

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

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### 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… Continue

### What is x?

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

### 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… Continue