# 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.

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.

# 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.

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.

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?

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.

# 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
• 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.

# Hands-On Math Hundreds Chart

When I was teaching elementary math in Southern California one of my favorite teaching aids was the hundreds chart.  I would burn up my photocopying budget reproducing hundreds charts so my students could color patterns showing the multiples of 2, 3, 5 or 7.  It really comes as no surprise to me that Hands-On Math Hundreds Chart is one of my favorite apps.  No longer is necessary to provides students with paper hundreds charts.  Using the app students just pick a color and a marker and then away they go marking patterns on the chart.

The interface is intuitive.  The markers function like objects and can picked up and moved.  To complete remove an object simply drag it off the chart.

A Teaching Tool

So what can you do with a hundreds chart?  Which concepts can you teach?  One of the first activities to do with young students is skip counting.  Pick a color and a type of marker.  To count by 3’s, for example, begin by tapping the cell labelled ‘3’ and then continue to 6, 9, 12, 15, etc.  Eventually a pattern begins to emerge.  Perhaps some of your students will discover that the pattern for 3 creates diagonal lines.  Encourage students to use math vocabulary in describing their hundreds chart explorations.

But more can be done with a hundreds chart.  Marking patterns to show multiples of a given number is a great way to practice and learn multiplication tables and it is easy for teachers to check students work with just a glance at the iPad screen.  But there must be more that can be done with a hundreds chart.

Least Common Multiple

The reason the Hands-On Math Hundreds Chart app has 8 colors and 6 shapes is so that overlapping patterns can be created on the chart.  When kids start overlapping patterns things get really interesting, mathematically speaking.  Understanding Least Common Multiple, in my mind, is an essential skill for success in mathematics.  Using Least Common Multiple (LCM) students can reduce fractions to simplest terms.  LCM is the smallest positive integer that is evenly divisible by two other numbers, a and b.

Let’s define a as 3 and b as 5 (which also happen to be two prime numbers, but more on that later).  Here’s what the chart will look like after a pattern for 3 has been marked with a red square.

Now let’s overlay the pattern for 5 with a blue circle.  Things are starting to get interesting…

At this point a teacher would want to ask the students to find all the numbers that got both marks, the red square and the blue circle.  I would recommend that the students mark these numbers with a yellow highlight (transparent yellow square).

Wow! Look at that the yellow highlighted numbers also make a pattern.  This numbers are the common multiples of 3 and 5.  Now study the chart and find the smallest number in this set.  15 is the LCM of 3 and 5.

Find the LCM of other numbers.  Just tap the eraser to completely erase the chart. Oh, and remember, if you only want to remove one marker just slide it off the chart.

A Classic Lesson

The first 4 prime numbers are 2, 3, 5, and 7.  In mathematics, the sieve of Eratosthenes is a pattern that reveals the prime numbers.  To investigate the pattern on the Hands-On Math Hundreds Chart, begin by marking the pattern for the number 2 and then we will mark 3, 5, and 7.  The procedure involves skipping the first number in the series and then marking out all the multiples.  So, skip 2 and then more out 4, 6, 8, 10, etc.  The chart should look something like this:

Use a different color for each number.  Now mark out the pattern for 3, but remember to not mark 3 since it is also prime.  Continue by marking the patterns for 5 and 7.  If a number is already marked just skip it.  Since the next prime is 11 and 11² is 121 so not on the chart and we don’t need it for this experiment.

When the marking procedures are finished, the chart should look like this:

The numbers that did not get marked are prime numbers.  All the marked numbers are composite numbers.

Get Creative!

There are lots of ways to make math fun using Hands-On Math Hundreds Chart.  Make up your own problem sets so that when the problems are answered correctly a picture is created.  For more information about this app click here.