Fraction Builder

I submitted Fraction Builder to iTunes yesterday and it was approved in less than an hour!  It is so amazing to me what great service Apple offers to developers.  I certainly appreciate the way that the App Store provides access to my work to students and teachers all over the world.


In most of my math apps I try to make them icon-based so that language skills are not critical for working with concepts.  Fraction Builder has a series of icons across the top of the screen.  Tapping each icon results in a specific function.

For example, tapping the red dice generates a random fraction.  Denominators range from 1 to 12.  Once the denominator is set, numerators can be any number that results in a proper fraction.


When the question mark is tapped, a question is generated and displayed on a moveable note.  To answer the question students drag number tiles to make the fraction.  Usually it is best to start with the denominator.  If using this app in a classroom, the teacher should explain that the denominator represents the number of equal parts.  For this example, the student would first slide the three tile to the denominator position.  Next, the student should slide a one to the numerator position.  Once the numerator and denominator have been properly set, the students should tap the check mark.  When this icon is tapped the app compares the students answer to the correct answer for the question.


A scoreboard displays student progress as they work with the app.  It shows the topic, number of questions attempted and the percent correct.  Quiz questions are based on three main topic areas:

• Naming Fractions

• Equivalent Fractions

• Comparing Fractions

Various other functions are performed when other icons are tapped.  This chart explains the other functions.notebook

I hope that teachers will let me know if they are interested in evaluating this app for use at their school.   I have a limited number of download codes for FREE evaluation copies of this app.   For some students learning about fractions is difficult.  It is my hope that this colorful interactive app will help them in their journey to master fraction concepts.


Just in Time for Halloween

What could be more fun that having your monster be chased by three other monsters?  Well that’s the theme of my latest math game.title

Ziggy, Spot and Pete are the three monsters that you want to avoid.  Harry is your monster and is controlled by tapping arrows on the iPad version or swiping the Siri Remote on the tvOS version.  The tvOS version also supports the use of a game controller so Harry can be moved using a compatible gaming device.


The object of the game is to move Harry to the correct answer for a math problem shown at the bottom of the screen.  Anytime that  Ziggy, Spot or Pete run into Harry as they bounce around the screen, Harry loses a unit of energy.  Fortunately when Harry gets an answer correct he gets a unit of energy back.  The goal is to score ten points before running out of energy.



Teachers and parents will like the wide range of problems.  Problems are randomly generated so the game is different every time it is played.  Skills are organized and defined using the Common Core Standards.  Look Out! Monsters! is a fast-paced game specifically designed to help kids get quicker at doing mental arithmetic.  A limited number of free promotional codes are available by request.  The app is offered exclusively by the iTunes App Store and sells for $0.99.  Please visit our website for more information about this and other apps for education.

Skills Chart

Operations and Algebraic Thinking Add within 20 Integers from 1 to 10 
Operations and Algebraic Thinking Add within 20. Integers from 5 to 20
Operations and Algebraic Thinking Add within 20. Integers from 10 to 20
Operations and Algebraic Thinking Add within 20. Integers from 1 to 20 
Operations and Algebraic Thinking Subtract within 20. Integers from 1 to 10
Operations and Algebraic Thinking Subtract within 20. Integers from 1 to 20 
Operations and Algebraic Thinking Determine the unknown whole number in an addition equation. Integers from 1 to 20
Numbers and Operations in Base Ten Add within 100, including adding a two-digit number and a one-digit number. Integers from 1 to 100 and 1 to 10
Numbers and Operations in Base Ten Given a two digit number mentally find 10 more or 10 less. Integers from 10 to 100 and 10
Numbers and Operations in Base Ten Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences) Integers from 10 to 90
Operations and Algebraic Thinking Fluently add and subtract within 20 using mental strategies. Integers from 0 to 20
Numbers and Operations in Base Ten Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Integers from 0 to 100
Numbers and Operations in Base Ten Add up to four two-digit numbers using strategies based on place value and properties of operations. Number Range: Integers from 0 to 1000
Numbers and Operations in Base Ten Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. Number Range: Integers from 10 to 100 and 100 to 900
Operations and Algebraic Thinking Fluently multiply within 100. Factors from 0 to 10
Operations and Algebraic Thinking Fluently divide within 100. Divisors and Quotients from 0 to 10
Number and Operations in Base Ten Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Dividends from 10 to 1000 and Divisors of 10 or 100
Operations and Algebraic Thinking Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Integers 1 to 10
Expressions and Equations Evaluate expressions at specific values of their variables. Integers 1 to 10
Expressions and Equations Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. Integers 1 to 10


Classroom Spinners


Teachers need lots of tools in their arsenal when it comes to teaching any subject, especially math.  The use of a spinner is a great way to introduce math concepts related to probability.  The Classroom Spinners app provides teachers with six different types of spinners that can be configured to have sections with color or no color.  The sections can be labelled with numbers or letter.  The sections can also be unlabelled.

Tapping the info button brings up a screen explaining the various functions of the app.


The Classroom Spinners screen shown below is configured to have six sections.  Each section has a different color and the sections are labelled with letters.  Many different investigations of probability theory can be explored using this setup.


For example, here are a few questions to ask the students:

  1.  What is the probability of landing on a vowel?
  2. What is the probability of landing on A,B, or C?
  3. If the last five spins were A,A,C,A,A, what is the probability of the next spin being the letter A?
  4. What is the probability of getting an B on the first spin and E on the second?
  5. If you spin 10 times about how many times would you expect the letter D to occur?

Tapping the sigma icon will display a table of the Experimental Outcomes:


In this experiment the letter C was randomly selected 4 times.  The Theoretical Probability for any given letter is 16.7%.  What do you think will happen to the Experimental Outcome for the letter C if 90 more spins are performed?  Theoretically, how many times will C occur if  the experiment is run 1,000 times?

Classroom Spinners is a useful tool for teachers to have in their tool bag.   Check it out and let me know what you think.







Star Maze

Star Maze extends my series of visual problem solving challenges and is for iOS and tvOS devices.  Like Slip Sliders, Critter MatesThe Bird Puzzle, The Menagerie and The Hungry Rat, Star Maze is based on a grid where the object is to control a character’s movement to solve a visual challenge.  In Star Maze the characters position can be switched with a blocker and this feature is critical to solving some of the puzzles.  The character can move in any of the four directions, left, down, right or up but only stops when it encounters a wall or a target (star).


Working to solve the Star Maze puzzles helps to develop logical thinking skills because the sequence in which the moves are made effects how many moves it takes to solve the puzzle.  The goal is to collect all of the stars in the least number of moves.  Star Maze differs from Slip Sliders because in Star Maze the character can switch positions with a blocker.  A blocker is a fixed wall in the maze that can only be moved by changing positions with the character.   In the screen shot show above the owl is the character and the black square is the blocker.


Obviously by moving the owl straight up you can capture the first star, but now what?  Moving right and down doesn’t help much, but what about switching the character and the blocker.  Tap the switch icon and now the puzzle board looks like this:


Now by moving straight up again, another star can be captured.

Solving the Star Maze puzzles can be fun and relaxing.  There is an Apple TV version so you can play Star Maze on your big screen TV with the whole family in the comfort of your living room. Star Maze is also available for iOS devices in the iPad family.  Try it out and let me know what you think?

If you like puzzles you might like Star Maze.   For $0.99 you get the iPad and Apple TV versions.  For more information about the Star Maze app, please visit our website. The app is available at iTunes.

Slip Sliders

Slip Sliders is the latest in my series of visual puzzles for iOS and tvOS devices.  Slip Sliders joins Critter Mates, The Bird Puzzle, The Menagerie and The Hungry Rat as visual puzzles designed to help develop logical thinking skills.

I’ve always enjoyed puzzles and so having the opportunity to design them is particularly rewarding.  For Slip Sliders, I needed a set of icons that could be paired up as part of solving the puzzle. Using Adobe Illustrator and Photoshop I created a set of birds.  It works great to draw the images large (512px x 512px) and then shrink them to the size of the icons.  By the way, I liked the birds so much I made a couple of coffee cups using the images.  The icons slide based on a particular set of rules.  They move in the one direction (left, down, right or up) until they hit an obstacle.  One of the tricky parts about solving the Slip Sliders’ puzzles is that sometimes you will need to create the obstacles in order to be able to change the direction that the slider can move.

Initially the puzzles are fairly simple and a solution can be achieved with just a few moves. Show below is how the first puzzle looks on an Apple TV.  Obviously the solution is simple, just move each of the birds on the left to the right and they will find a mate.  The birds on the left are sliders and therefore they can be moved.  The birds on the right stay in a fixed position and are the targets.

Slip Sliders Screen 1

In the second puzzle things get a little more difficult.  The birds cannot just simply be moved down because the first two need to switch columns.  I like this type of problem because the are multiple solutions.  If you are a teacher and use problems like this with your students you will find that one benefit is that several students can share different answers and it is not just the first student who finds a solution that gets the reinforcement.

Slip Sliders Screen 2.png

As you progress through the puzzles by tapping the right arrow, the puzzles get even more challenging.  For example, in the puzzles shown below from an iPad screen, how are you going to get the slider to stop on the red bird show in approximately the middle of the screen?

Slip Sliders Screen 3

If you like puzzles you might like Slip Sliders.   For $0.99 you get the iPad and Apple TV versions.  For more information about the Slip Sliders app, please visit our website. The app is available at iTunes.


Critter Mates


You know the feeling when you get that “aha” moment and want to run through the streets shouting “Eureka!”. What is it about solving puzzles that is so satisfying?

There is some evidence that early people had an interest in problem solving. Major breakthroughs like the invention of the wheel and learning to control fire happened rarely, but finding solutions to small problems probably occurred on a regular basis.

Recently researchers have investigated the urge to solve puzzles and have attempted to explain the brain mechanisms involved. Different types of puzzles require different skills.  Crossword puzzle solvers obviously need to have a good vocabulary, memory and pattern recognizing ability.  While much more research is need, there does seem to be some evidence to suggest that a high level of experience with some times of puzzles, crosswords, for example, in older subjects helps to attenuate the negative effects of age on memory and some perceptual speed tasks.  Other types of puzzles require spatial reasoning skills.  For example, Rubik’s Cubes, pentominoes, tangrams, soma cubes and other similar puzzles are involve spatial reasoning as well as logical sequencing skills.  Critter Mates is a little bit like a crossword puzzle in that the puzzle format is a grid.  Logical sequential skills are also involved.

In children with brain damage research has shown positive effects from puzzle solving activities.  In particular children who had a weak visual symbol memory that impaired their ability to spell and read words benefitted from unscrambling puzzles where mixed-up letters were paired with pictures.  But knowing just how these mechanism work has yet to be explained.

One thing that many educators know is that problem solving skills can be taught.  Some of the methodologies that are part of a comprehensive problem solving curriculum can be taught in relation to puzzles like Critter Mates.  Many educators employ a five-stage model to teaching problem solving:

  1. Understand the problem.
  2. Describe any barriers.
  3. Identify various solutions.
    1. Create visual images.
    2. Guesstimate.
    3. Create a table.
    4. Use manipulatives.
    5. Work backward.
    6. Look for a pattern.
    7. Create a systematic list.
  4. Try out a solution.
    1. Keep accurate records.
    2. Persist with a strategy until is proven to not work.
    3. Monitor the steps.
    4. Step away from the problem for a while.
  5. Evaluate the results.

Some people like to solve puzzles and others like to design them. Which one are you? Math ability and music seem to be related to problem solving ability. So if you have talent in these areas you might also be a great puzzle solver or creator.  In many cases puzzle solving ability requires synthesizing several different pieces of information and the ability to see patterns which are also skills that are fundamental to mathematics and music.  Activities like solving the Critter Mates puzzles may help to develop or strengthen these skills.


When I designed the Critter Mates I wanted to create a problem solving challenge where kids would be motivated to solve visual puzzles using ‘critters’ as a virtual manipulative.  Understanding the puzzle is fairly simple.  The goal is to move all the critters so that they are next to each other.  One way to approach the problem might be to work backward.  Visualize the solution and then try to figure out the steps needed to get to the solution.  The Critter Mates puzzle definitely has barriers.  The critters need to navigate the walls and will often get logically stuck. Keeping notes as an accurate record of what has been tried might help to avoid pitfalls in future attempts.  I hope that you will find that Critter Mates is intrinsically motivating and useful in teaching kids to be better problem solvers.


For more information about Critter Mates app for iOS and tvOS please visit our website.  The app is available at iTunes for $0.99

Coordinate Geometry

When you begin the study of Coordinate Geometry it is hard to not appreciate the great genius of Rene Descartes.  In math circles Descartes is known as the father of coordinate geometry (also known as analytic geometry).  Coordinate geometry merges concepts from algebra and geometry and has many applications. Today the ideas of coordinate geometry are widely used in physics and engineering.  An understanding of coordinate geometry is essential for success in the study of other fields of geometry, for example, algebraic, differential, discrete and computational geometry.

When I designed the iPad app, Coordinate Geometry, I wanted to create a highly interactive environment for learning the basic ideas.  Usually teachers begin with an explanation of the coordinate system.  In the design of the app I used a metaphor of a table top with a slightly transparent grid.  The center of the grid is the origin and ‘origin’ is one of the many vocabulary words that a teacher would want students to learn in reference to coordinate geometry. From the origin, the grid lines are typically numbered with positive integers to the right and up.  The grid lines to the left and also below the origin are labeled with negative integers.


The horizontal line passing through the origin is the x-axis and the vertical line is the y-axis.  This divides the coordinate plane into four sections.  Conceptually the x and y axes are infinite. Traditionally the sections are labeled with Roman numerals.  Tapping the little dog-ear icon at the bottom right of the grid labels the sections according to the standard mathematical conventions.


Points are referenced using a coordinate pair where the corresponding position on the x-axis is given first and then the y-axis with the two numbers separated by a comma and enclosed within parentheses. If the x and y coordinates are both positive the point lies in Quadrant I.  Likewise if x is negative, and y is positive, the point lies in Quadrant II.

At the bottom left of the screen are three tools, the line, the rectangle and the circle.  By selecting one of these tools, students can draw figures on the grid.  Because Coordinate Geometry is designed as an open-ended teaching resource, teachers can direct the students to complete various task just be given oral or written instructions.  For example:

  1. Draw a line segment from (-6,-6) to (6,6).  Which special point does this line pass through?  (origin)
  2. Draw a square on the grid.  List the coordinates of the vertices.
  3. Draw a circle with the center at (3,3) and a radius of 4 units.

In addition to the many teacher-designed lessons that can be carried out using Coordinate Geometry, there are built-in lessons that can be accessed by tapping the lesson icon.


I wanted to also include some self-checking challenges for students.  When the question icon is tapped, the app automatically displays a quiz question. The quiz questions are randomly generated so a unique set of questions is presented each time the quiz is accessed.


The main topics that can be explored with the Coordinate Geometry app are:

  • Coordinate Plane
  • x and y axes
  • Origin (0,0)
  • Coordinate Pairs
  • Quadrants
  • Points, Lines and Planes
  • Slope and y-intercept

One of the great things about having a sister who is a mathematician is that she will write books to supplement my apps.  Be sure to check out the iBooks Math Fun Coordinate Geometry 5  and Math Fun Coordinate Geometry 6.  Both books are available from the iTunes iBooks Store.



2 Kids – 1 iPad, Lots of Frogs…

Hop to It Math

In an ideal world schools would be able to provide an iPad for every student, but in many cases a one-to-one ratio is just not feasible.  I have developed several learning apps that are designed to be used by two students at the same time.  Hop to It Math is one game designed to help kids improve math skills and can be used with two students simultaneously.



The board is layed out like a traditional board game where players move markers along a path to reach a goal.  In this case the markers are frogs and the goal is the finish line.

The game begins by displaying a problem.  Below the problem a calculator style number pad to use for entering the answer.


A wide range of problems can be selected by the players:

Level 1:
Addition – Whole Numbers (1-10)
Subtraction – Whole Numbers (1-18)
Multiplication – Facts (1-5)
Division – Facts (up to 25 ÷ 5)

Level 2:
Addition – Whole Numbers (0-20)
Subtraction – Whole Numbers (1-40)
Multiplication – Facts (0-12)
Division – Facts (up to 144 ÷ 12)

After a correct answer is entered, a spinner appears on the screen.  The player taps the spinner and the player’s frog hops to the next position.

Hop to It Math - tvOS

The Apple TV version of the game (shown above) offers a 1920 x 1080 pixels version of the game.  By swiping the Siri Remove players enter answers, operate the spinner and move their frog.  My goal in developing this game is to provide elementary students with a fun way to practice and improve math skills.


Another two player game that is based on the frog theme is FrogGo.  FrogGo is designed to support S.T.E.M objectives and introduce student to basic programming concepts. Science, Technology, Engineering and Mathematics are critically important 21st Century skills and I wanted to support these objectives by providing a fun learning experience.

Six commands are available to control the player’s frog:  Forward (FD), Right  90° (RT), Left 90° (LT), Jump (JMP), Double Move (2X) and Triple Move (3X).  The goal is to program your frog to move and capture the rewards that are randomly placed on the game board.  Players need to avoid moving out of bounds.


The scorecards are used to track the number of rewards each player has accumulated.  The winner is the player who has accumulated the most rewards at the end of the game.  Computational thinking is an essential process that is used in most careers and it can be introduced at a very early age with tools like FrogGo. In FrogGo each player’s turn begins with analyzing a problem, breaking it down into components parts or steps and then describing a path to a solution using algorithmic thinking. A programming sequence is defined by choosing from a list of functions. Sequences of commands are successful if they lead to one of three goals (flies, stars or trophies). Programs that don’t work need to be ‘debugged’ by the player.  I hope that FrogGo will provide teachers with a way to interest kids in learning more about computer programming and realize that designing computer games can be even more fun than playing them.

Math Bingo

If you’ve ever had the joyful experience of trying to teach a four-year-old how to play Tic-Tac-Toe you hopefully have witnessed the excitement that a child feels when they suddenly learn that games involve strategy.  Instantly games become more interesting.  Now there is a reason to hide your cards when playing Old Maid.  Math Bingo is a bit like Tic-Tac-Toe in that players can develop a strategy.  Regular Bingo involves calling out randomly balls,  B-13, I-24, and so on.  Each number only appears once on a typical bingo card.  Math Bingo involves a strategy because the same answer might occur a couple of times on the card and therefore the player must strategically decide which answer to pick.

Math BIngo 1

Math Bingo is designed for iPad and Apple TV.  It is a two player game.  On the iPad version players sit opposite each other so the playing surface looks something like this.  The bingo cards are rotated 180° so the orientation is correct for each user.


When designing the Apple TV version of Math Bingo, I thought it would be fun for kids to use slightly scary looking critters as markers in the game, so play begins after each player has selected which marker to use.

Math BIngo 2.png

Using the settings option, players can select the skill level that they would like to practice. The app provides 20 levels, all keyed to the Common Core Standards.  Level 1 is beginning addition facts so it’s a good place for K-1 students to start.

Math BIngo 3

When they are ready each player taps the Go! icon and the game begins.  The app will generate a set of 25 problems at the skill level selected using the settings option.  Every game is unique.

Here, two addition problems are presented 9+5 and 9+2.  During the game each player will be given the same problems but not in the same order.  Players take turns and move the yellow highlight to designate their answer choice.  When using the Apple TV version, the highlight is moved by swiping the Siri Remote or using a joystick.  On the iPad the player just taps the screen.  The goal is to select an appropriate answer with a strategy of trying to get 5 markers in a line.

Math BIngo 4.png

When the correct answer is chosen the game marker appears and an audible reward is given.  Play continues until one player gets a BINGO, five markers in a horizontal, vertical or diagonal line.  Let me know if you would like a free download code so you try this app to see if it would be useful to your students.

Math BIngo 5

And, oh, by the way if you are interested in a fun way to introduce Tic-Tac-Toe try my Tic-Tac-Tarantula for iPad, especially if you are into creepy crawly things.  Tic-Tac-Tarantula is a one or two player game — match wits against the computer or play with a friend.


Happy Pi Day 3.14.16

Well, this is it, the big one.  Pi Day 3.14.16 won’t happen again for 100 years.  I hope you had some mathematical fun today or at least had a piece of pie.  I’ve been working on a simulation of Napier’s Bones for iOS.  The iPad app is functionally similar to Napier’s calculating tool which, of course, in the history of computing devices, was one of the earliest calculators.   The app is designed using a table top or wooden box metaphor.


Rods are picked up by tapping a number on the left and dragging it into position on the right.  Each rod is etched with the multiplication table for the specific number. I am hoping that using the app will help kids better understand the meaning of multiplication.

A slider moves up and down highlighting the multiplier and showing how the partial products are used to generate the product.

The app also includes a bit of historical information about John Napier, the Scottish mathematician who lived in Edinburg from 1550 to 1617.


I included a quiz function so that the app will challenge kids to use the device to find the answers to multiplication problems.  It starts out with two-digit number multiplied by single digit number.  Problems then get progressively more challenging.


If all goes well the app should be available from the Apple iTunes store soon, hopefully by April Fools Day.😀