Pioneers in Math and Science

New apps for iOS and tvOS Devices:

Aristotle • Ancient Greek Philosopher
Rachel Carson – Environmental Scientists
Bonaventura Cavalieri • Mathematician
Nicolaus Copernicus • Astronomer
Marie Curie – Chemist & Physicist
Charles Darwin  • The Science of Evolution
René Descartes • Philosopher • Mathematician
Albert Einstein • Theoretical Physicist
Euclid • The Father of Geometry
Leonhard Euler • Mathematician
Michael Faraday • Chemist & Physicist
Fibonacci • Mathematician
Leonardo Fibonacci • Mathematician
Galileo Galilei • The Father of Modern Science
Carl Gauss • Physicist • Mathematician • Astronomer
Johannes Kepler • Astronomer & Mathematician
Sofia Kovalevskaya • Mathematician
Ada Lovelace • The First Coder
James Maxwell – Physicist • Mathematician
Lise Meitner – Nuclear Physicist
Gregor Mendel • Botanist
Isaac Newton – Mathematician • Physicist • Astronomer
Blaise Pascal – Mathematician • Scientist • Philosopher
Louis Pasteur – • Microbiologist • Chemist
Cecilia Payne – Stellar Astronomer • Astrophysicist
Pythagoras • Mathematician & Philosopher
Ernest Rutherford • Nuclear Physicist

It is reported that Newton once said , “If I have seen further than others, it is by standing upon the shoulders of Giants”.  The Pioneers in Math and Science series of apps for iOS and tvOS devices explores the lives and contributions of several giants in astronomy, physics, philosophy, biology and mathematics who lived before and after Newton.


In Galileo Galilei • The Father of Modern Science, a timeline feature is used to introduce students to the fascinating life of Galileo.  It discusses his early education and explains that as a young man Galileo’s father moves the family to Florence which at the time was a major center for artistic, literary and scientific studies.


The timeline feature is interactive and students can request more information about any of the highlighted years but tapping the year.  The arrows at either end of the time allow students to scroll through the time period.

Illustrated information screens pop up to help students learn about Galileo’s accomplishments.  Students will find it inspirational that many of the things that fascinated Galileo as a young man became a major focus of his studies and lead to mathematical and scientific breakthroughs for him later in his life.


In Isaac Newton – Mathematician • Physicist • Astronomer, students explore the many Newton’s achievements including those related to the nature of white light.  Newton discovered that sunlight was actually made up of many different colors.  He should this by holding up a prism near a window in his laboratory and projecting a spectrum of colors.


René Descartes • Philosopher • Mathematician presents the work of the man who linked algebra and geometry and laid the foundation for modern mathematics.


Descartes work in philosophy formed the basis for the modern scientific method.  He wrote about his skepticism regarding what he could experience through his senses and encourage people to seek knowledge through reason, mathematics and science.

In Charles Darwin • The Science of Evolution a map of the world is used to show the voyage of exploration of the HMS Beagle.  This five year journey around the world gave Darwin the information he needed to develop his Theory of Evolution.Screen2.png

Each of the apps in this series feature a quiz so that student can demonstrate comprehension of the reading material and other special features such as a speech where the text from the diagrams, quizzes or scrolls can be read aloud to students.


Galileo Galilei • The Father of Modern ScienceIsaac Newton – Mathematician • Physicist • AstronomerRené Descartes • Philosopher • Mathematician and Charles Darwin  • The Science of Evolution and other  apps in this series are available with a volume discount for educational institutions. These apps can be purchased worldwide exclusively through the Apple App Store.  The apps are also available through Apple’s volume purchase program. Schools get a significant discount when purchasing multiple copies of any of these apps. Contact Apple Education for more information about the volume purchase program.  Please visit Ventura Educational Systems’ website for more information about this and other iOS and tvOS apps for education.


Math Decoder

Most people who are involved in education and have studied educational psychology know the name, Jean Piaget.  Piaget’s Theory of Cognitive Development, defined and described stages of development: sensorimotor, pre-operational, concrete operational and formal operational.  Many math curriculum materials are designed to fit with these developmental stages that roughly correspond to age groups:  sensorimotor (0-2), pre-operation stage (2 to 7), concrete operational (7 to 11), and formal operational (11+ and adulthood).  The availability of computers and educational software may have shifted some of these age groupings toward the younger ages since very young children are often using iPads and other devices to explore and learn concepts with assistance of highly interactive educational games and experiences.  Apps can be very engaging and often create multimedia learning environments that motivate students through positive reinforcements such as praise, points, stars, trophies or virtual stickers.

The manipulative approach to teaching math is based on the cognitive development theories of Piaget, Bruner and others.  Software designed to support these theories, for example my Hands-On Math Series, brings the manipulative approach to the computer screen by creating interactive representations of mathematical objects.  Math Decoder is designed in a similar way so that the student is able to move numbered tokens into place to show his or her answer.

Math Decoder presents random numbers in a grid that is similar to the familiar tic-tac-toe board.  Problems are given using shapes that reference the position of the numbers in the grid.  To give an answer students must determine which numbers are referenced by the shapes and then perform the arithmetic operation.


Using Math Decoder helps build mental math skills.  It involves abstract thinking because the symbols are used to represent numbers in way that is similar to how variables are used in algebra.  I designed Math Decoder to be a quick exercise that teachers and parents can assign to students when they have a few minutes of time.  Challenge students to earn a trophy.  It’s fun!

Math Decoder is available with a volume discount for educational institutions. Algebra Portal AR can be purchased worldwide exclusively through the Apple App Store.  It also available through Apple’s volume purchase program. Schools get a significant discount when purchasing multiple copies of Algebra Portal AR. Contact Apple Education for more information about the volume purchase program.  Please visit Ventura Educational Systems’ website for more information about this and other iOS and tvOS apps for education.

Algebra Portal AR

Algebra Portal AR

Apple’s new ARKit has opened the door to new kinds of app experiences.  My new app, Algebra Portal AR, creates virtual rooms filled with information about important terms and key concepts related to the Algebra I curriculum.  What is really exciting about the Algebra Portal is students stand up, walk around and experience a totally immersive environment that is filled with illustrated examples that are used to explain some of the fundamental ideas of algebra.


The floor Algebra Portal is a standard Cartesian plane with the x and y axes labelled.  Teachers could develop lesson experiences that involve students walking into the portal and then moving to a specific (xy) position to learn about a particular topic from the algebra curriculum.

When a wall inside the Algebra Portal is tapped a menu of terms is displayed.  If a term is selected detailed information about the topic is displayed with examples.algebra_portal_4

Students and teachers will be amazed when they create virtual rooms filled with information about key terms and concepts from the middle school and high school algebra curriculum.

Explore algebra in a new way with the Algebra Portal AR. Use the camera view on your iPad to detect a flat surface and then tap the screen. Instantly a portal opens to a world filled with information about the terms and concepts that are part of the algebra curriculum.


In an nutshell, to use the app, students walk around in the augmented reality algebra portal where they can tap on the walls and other objects in the rooms to get more information and to hear Siri read aloud information about the selected item. Students can scroll through an extensive list of terms related to the information on the walls of the rooms, walls and objects in the portal.

Algebra Portal AR creates virtual rooms that are approximately 4 meters by 7 meters. It is best used in a well-lit open space, perhaps in a school’s auditorium or multi-purpose room. If the Algebra Portal sounds interesting to you, you might also like the Geometry Portal AR.

Algebra Portal AR is available with a volume discount for educational institutions. Algebra Portal AR can be purchased worldwide exclusively through the Apple App Store.  It also available through Apple’s volume purchase program. Schools get a significant discount when purchasing multiple copies of Algebra Portal AR. Contact Apple Education for more information about the volume purchase program.  Please visit Ventura Educational Systems’ website for more information about this and other iOS and tvOS apps for education.


Target 10 – Like a Word Search but with Numbers


It is great fun when you have precocious grandchildren who have creative and inventive minds.  One of my grandkids is really into word search puzzles.  He loves finding the names of states, or capitals, or astronomy words, or anything else that might be the theme of a word search puzzle.  But the gears in that creative mind started spinning one day and he created a search type of puzzle with numbers.  The gist of the idea was to write an array of random numbers on graph paper and then pick a random target number and try to find numbers in the array that add up to the target number.  This concept gave birth to an iOS and tvOS app named Target 10.


The screen above shows the main game board for Target 10.  The object of Target 10 is to earn 10 stars by finding sets of numbers in the grid that add up to the target number.

Target 10 offers four levels. On an iPad simply tap the Settings icon and then from a screen similar to the one show below, choose a level.  If you are using Apple TV you swipe to get to the Settings Icon and then swipe and click to choose a level.


Automatically a new puzzle is generated at the level you selected.  Study the puzzle to find chains of numbers that add up to the target number.  There’s a fun little twist.  The position of any two numbers can be switched.  When you can’t find any more sets of numbers to make the target sum, try using the Switch icon to move numbers.  To do a switch tap the Switch icon and then navigate to the two numbers whose positions you want to switch. If you are still stuck, you can tap the New Board icon for a new grid.

Switch Icon                                                            New Board Icon

icon_switch            icon_dice

Continue finding sets of numbers that add up to the target number until you have earned 10 stars.


In designing this app I wanted to motivate elementary age students to get quicker at mental math.  It is my intention that by using Target 10 on your iOS device or Apple TV you will be able to engage students problem solving and strategic thinking activities that they will enjoy. The tvOS version of Target 10 offers the same great features as the iOS version, and additionally can be controlled using tvOS compatible game controllers so maybe the kids will think they are playing a video game.

Target 10 is available with a volume discount for educational institutions.  Target 10 is available worldwide exclusively through the Apple App Store.  It also available through Apple’s volume purchase program. Schools get a significant discount when purchasing multiple copies of Target 10. Contact Apple Education for more information about the volume purchase program.  Please visit Ventura Educational Systems’ website for more information about this and other iOS and tvOS apps for education.

More Widgets

Once I got started designing the Math Widgets series I realized it was going to be a lot of fun.  The study of mathematics is an immensely broad topic and for kids who are really into math it is wonderful if a teacher can provide enrichment experiences that introduce new ideas and concepts that go beyond “what’s on the test”.  The Math Widgets series is a collection of tools with these kids and teachers in mind.


Math Widgets III explores some special concepts in number theory.  For example, the Multibase Abacus piques one’s interest in alternative number systems.  What if we only had 8 fingers instead of 10?  Would we use a base 8 system?  With the Multibase Abacus students can investigate base 8 and any other base from 2 to 10.


On the Multibase Abacus, numbers are represented by tapping beads to add or subtract values from the columns of various place value systems.  In the example below we are showing a base 5 system.  Starting on the right we have the ones place labeled with 50.  Moving to the left we have 5’s, 25’s, 125’s, 625’s etc.  Tapping one of the column labels results in a bubble showing the value of the place in base 10.  In the quiz mode students are challenged to represent randomly selected numbers in randomly selected bases.  Quite a few skills come into play when trying to answer the questions posed by the Multibase Abacus, including: estimation, addition, subtraction, multiplication and using exponents.


Clock Arithmetic is another topic in math that is perfect for an enriched curriculum.  Whenever we divide two integers there is always a remainder (sometimes it’s zero).  Clock Arithmetic is a widget that encourages student’s to explore this concept which is known as Modular Arithmetic.  In programming languages there is a special operator used to signify the modulus function.  If a=7 and b=3 then in computer code a%b returns 1 since one is the remainder when 7 is divided by 3.  Sometimes the word ‘mod’ is used instead of the “%” symbol to describe this calculation, for example 15 mod 12 is equal to 3.  In the example below, tapping the 2 is the correct response.


Once in a while, but rarely, you might find a newspaper article about a mathematician solving a problem or finding a proof that has been elusive for hundreds of years.  For example, the conjecture  that no three positive integers ab, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.  First conjectured by Pierre de Fermat in 1637, this theorem had been the focus of study by mathematician for centuries and was finally proven in 1995 by Andrew Wiles.  Goldbach’s Conjecture has never been proven. It was first posed in a letter from Goldbach to Euler in 1742.  The conjecture posits that every even integer greater than 2 can be expressed as the sum of two primes.  Use the Goldbach’s Conjecture widget to explore this idea by finding the primes whose sum is the given even numbers.


The Hexagon Arithmetic Widget is a six number system that uses emojis as the symbols.  Counting moves in a clockwise direction around the hexagon.  The object is to complete the addition table using logic to figure out the pattern.


Math Widgets I, II and III are offered exclusively by the iTunes App Store and sell for $1.99 each.  Educational discounts are available for schools.  Please visit our website for more information about these and other apps for education.


Math Widgets & Math Widgets II

There are two definitions for the term widget.  First, a widget is a small gadget or mechanical device that performs a useful function. In this context, the term is often applied to a device where the actual name is unknown or unspecified.  Second, a widget is an application, or component of an interface, that enables a user to do something special, for example perform a useful function or access special information.  Math Widgets fit both definitions.  The Math Widgets apps are collections of math tools that provide opportunities to explore a variety of important math concepts.  Some of the concepts extend the K-8 math curriculum laterally and therefore these apps are particularly useful for teachers who are looking to provide enrichment.

Let’s take a look and the first Math Widgets app.  This app is available for both iPad and Apple TV and includes four widgets: Slide Rule, Fraction Action, Integers and Coordinate Grid.

screen_1With the advent of calculators and computers, obviously the need for a slide rule has diminished, if not, virtually vanished, but what fun for kids to learn the basic idea of a slide rule by manipulating a virtual slide rule on their iPad or Apple TV.  In addition to representing the meaning of two fundamental math concepts: adding and subtracting,  the slide rule gives teachers an opportunity to discuss some of other historical tools used to help people do math for example the abacus or Napier’s Bones.  (See Abacus Deluxe and Napier’s Bones)

screen_2The operation of the Slide Rule widget is straightforward. The widget presents a problem in the middle of the screen and challenges the student to show the answer using the slide rule.  Addition and subtraction problems are presented.

screen_4The Integers widget uses a number line to help students understand operations with positive and negative numbers.  A problem is presented and the student is challenged to slide the marker to show the answer.  When slid in a positive direction a blue bar appears on the number line.  A red bar is used to show movement in the negative direction.  By using this app student will develop a better understanding of basic operations with integers.

screen_3Fraction Action provides an interactive widget for learning about equivalent fractions.  A fraction is randomly selected and displayed as a numerator over a denominator. It is also shown as parts of a circle.  The challenge for the student is to move the indicator along the fraction ruler to select the equivalent fraction.

screen_5The Coordinate Grid widget is designed to help students learn to locate points on a standard Cartesian plane.  The x and y axes and the quadrants are labelled.  The student is given a coordinate pair and is challenged to find the corresponding point on grid by moving a slider.

The first Math Widgets app seemed so useful that I thought I would do another one so I developed Math Widgets II.


Math Widgets II has 4 widgets:  Multibase Chart, Arrow Math, Tinker Totals and Peg Puzzle Party.  While studying number systems other than Base 10 might not be a critical part of the standard elementary curriculum, it is a fun enrichment idea for teachers who are looking to extend the curriculum for certain students. Multibase Chart provides an interesting experience in representing numbers using different bases.

screen_2In this example the computer has challenged the student to represent 19 in Base 8.  The red slider has be moved to isolate all the possible two digit numbers in Base 8.  Since 2 x 8 + 3 = 19, the correct answer is 23.   The correct answer is located at column D, row C.  Tapping this cell results in a positive reinforcement message and increase in score.

screen_3Success in mathematics and many other areas of study involves the skill of being able to follow a specific set of instructions.  Arrow Math provides an opportunity to practice two skills, the ability to carefully follow a set of instructions and also the meaning of an inverse operation.  Arrow Math provides teachers with a visual way to talk about inverse operations which are important in the study of mathematics.  When using this widget ask students questions such as ‘What is the inverse of moving to the right?’ or ‘What is the inverse of moving down and left?’.

screen_4The Tinker Totals widget creates number puzzles where the object is to arrange number given into the cells of a pattern so that the numbers along each line add to the same sum.

screen_5Peg Puzzle Party is a logical thinking puzzle where the challenge is to end up with the least number of pegs on the board.  A move consist of pick a peg and jumping over another peg to land in an open space.  When a peg is selected the available moves are highlighted in green.  Peg Puzzle Party is a fun way to exercise your brain and can be used to help students develop strategic thinking skills.

Math Widgets and Math Widgets II are offered exclusively by the iTunes App Store and sells for $1.99.  Please visit our website for more information about these and other apps for education.


Pete the Penguin’s Math Game

titlePractice, Practice, Practice!  When it comes to building basic math skills most teachers will agree that kids need a lot of practice.  The app for iPad and Apple TV creates an interactive learning environment where student’s get to practice basic math skills. One of four skills can be selected: addition, subtraction, multiplication or division. Within each skill there are five levels of difficulty.

The goal is to get 10 problems correct for each skill and for each level.


To begin the student selects a skill and then a marker to move along the trail. Sound effects and positive auditory feedback encourages the students to do their best.

• Single-Digit Addition
• Adding Multiples of 10
• Addition – Two-Digit
• Addition – Three-Digit Plus Two-Digit
• Make a 10 Addition Practice

• Single-Digit Subtraction
• Subtracting 10
• Subtraction – Two-Digit Subtraction (No Regrouping)
• Subtraction -Two-Digit (with Regrouping)
•Three-Digit Minus a Two-Digit Number

• Multiplication Facts (0-9)
• Multiplication Facts (5-12)
• Multiplication by 10 or 100
• Multiplication – Two-Digit
• Multiplication – Three-Digit by One-Digit

• Division Facts (0-9)
• Division Facts (5-12)
• Division by 10 or 100
• Division -Two-Digit by Single-Digit
• Division – Three-Digit by Single-Digit

I hope teachers will enjoy using Pete the Penguin’s Math Game with their students.  Teachers should 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.   I hope that elementary students enjoy practicing basic math skills with this app and that this colorful interactive app will help them in their journey to become great problem solvers.

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.