Classroom Spinners

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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.

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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.

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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:

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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).

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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.

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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:

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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.

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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?

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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.

 

Line Design (a.k.a. String Art)

In general, one of my goals in designing the Hands-On Math apps was to create ways to simulate learning experiences that students would have if using actual manipulatives.  Simulated manipulatives can offer additional learning opportunities.

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Special features, in the Line Design app, take the learning opportunities beyond what can be down with physical manipulates.  For example, a grid with the x and y axes labelled can be added to the image.

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Line Design can be used for exploring the relationship between geometry and art.  In addition to calling line designs, string art, sometimes it is referred to as aesthenometry, Using the term, aesthenometry, refers more to the aesthetic beauty of the designs.

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It is always fun for students to use boards, nails and various colors of embroidery thread to make string artworks to hang on their wall.  The process involves lots of careful planning, measuring and some materials (boards, nails and string).  The Line Design app can be used to plan the project and to provide help in choosing the colors and design patterns to be used.

Personally my favorite pattern to use for instructional purposes is the circle.  When the circle design is selected a slider appears.  Using the slider the number of points on the circle can be set to any number between 3 and 36.

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There are lots of mathematically interesting explorations that are available in this mode.  Here are a couple of suggestions:

  1. With the number of points set to 12, what shape will result if lines are draw to connect every other point?
  2. What shape results when the number of points is set to 10 and every other point is connected.  What about connecting every fourth point.
  3. What’s the difference between setting the number of points to a prime number versus a composite number?
  4. Set the number of points to 24.  Explore connecting every 4, 6, or 8 point.

The learning environment gets even richer if the circular protractor is added to the iPad display.  In this mode students can measure angles. Here we have the number of points set to 36.  The screen shows a 30° angle.  Two legs of the triangle are radii of the same circle and therefore congruent.  What is the measure of the other two angles?

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Election 2016

With the presidential election coming up in November of this year, I thought it would be beneficial for teachers to have an app that helps to explain the election process and the meaning of the Electoral College.  The world’s attention is focused on the U.S. Election since the outcome will have a profound effect all countries.  It is important for U.S. citizens to understand how the process works.

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The app has historical data from 1976 to present and by tapping icons the user can manipulate the U.S. map to show which states cast their votes for the Democratic candidates and which voted Republican in previous elections.  It’s interesting to tap through the years and watch the map change dynamically.  At any point the user can tap the issues icon to display  a short list of the major issues of the day.

If desired, as the user taps through the presidential election years the symbol for the winning party can be displayed.

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One of the best features of the app is the Electoral Vote scorecard for the 2016 election.  Here the user can predict which states they think will be red Republican States and which will be blue Democratic states.  User can test different scenarios to see how their favorite candidate might win.  Will 2016 be a landslide or a narrow victory.  Only time will tell…

Get your students involved in the political process and help them better understand the Electoral College using Election 2016.

For more information about Election 2016 app for iOS please visit our website. The app is available at iTunes for $0.99

Critter Mates

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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.

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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.

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

The Menagerie

The dictionary defines a menagerie as ‘a collection of wild animals kept in captivity for exhibition’, so it seemed to be an appropriate name for this logic game where the object is to get all the critters on a 6 x 6 grid into locked cages.

title When you tap a critter all the available moves are shown with green lights.  Critters can only be moved once and the number of spaces that each can be moved is indicated.  The critters in each puzzle are randomly picked so each puzzle provides a unique experience.  Enjoy hours of fun solving these challenging puzzles.

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The critters range from bears to butterflies and will hopefully be appealing to children.  There are 11 puzzles.  Critters can be moved 1, 2, 3 or 4 spaces but only once.  It’s a challenge to find a way to get all the critters safely locked in the menagerie.

Thinking logically is such an important skill for kids to develop.  Puzzles and games offer an opportunity to develop logical thinking skills.  I think it would be good for kids to work in small groups to try to solve The Menagerie puzzles which is why I developed it for both iPad and Apple TV.

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The Apple TV version of The Menagerie works with both the Siri Remote and compatible bluetooth game controllers.  Hopefully teachers and parents will encourage the players to discuss possible moves and explain the logic that they used to decide on a move.

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Each puzzle has about 20 to 30 critters.  With more critters the puzzle gets a little more challenging.  If you like word or number games that are played on a grid, you will also like The Menagerie.

It has always been interesting to me that grids just seem to be a natural way to organize information.  Are our brains predisposed to information arranged in rows and columns? Spreadsheets, for example, use a grid where rows and columns are labelled with letters and numbers to identify cells. When the streets of a town are numbered in one direction and labelled alphabetically in the perpendicular direction , it always seems easier for me to find my way around.  Recently, brain researchers  studied a group of human participants.  The experiment used a virtual reality environment with the subjects lying in a fMRI brain scanner.  The research showed that specialized grid cells in the brain have a spatially-organized firing pattern.  These grid cells function to create a spatial map in the brain.  The research suggests that internal grids or maps are created in our brains to help organize and understand new environments.

Scientists at the University College of London Institute of Cognitive Neuroscience have shown that specialized neurons play a role in spatial memory.  Specialized neurons that are being called ‘grid cells’ have been identified through rat and mice studies and seem to represent a mechanism for how animals locate themselves in their environment.  It is almost like having a brain cell based GPS mechanism.  In animals, and now possibly humans, neuron firing patterns show up as geometrically regular grids.

So it seems obvious that the critters in The Menageries should be arranged in a 6 x 6 grid and it just seems to be an appropriate way to layout a logic puzzle.  I guess after millions of years of evolution our brains just like it that way!

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