Advanced Common Core Math Explorations: Fractions
The Advanced Common Core Math Explorations series creates and nurtures mathematical adventurers! Students engage in challenging activities that deepen and extend their understanding of concepts from the Common Core State Standards for Mathematics. They stretch their mathematical imaginations to the limit as they solve puzzles, create stories, and explore fraction-related concepts from the mathematics of ancient Greece to the outer reaches of infinity. Each activity comes with extensive support for teachers including learning goals, discussion guides, detailed solutions, and suggestions for extending the investigations. There is also a free supplemental ebook offering strategies for motivation, assessment, parent communication, and suggestions for using the materials in different learning environments.
Exploration 1: Sharing and Grouping
It's one thing for a teacher to tell you about the connection between division and fractions and then for you to apply to it answer story problems. It's another thing for you to create the stories yourself and use pictures to show why your strategies make sense. That is one of the many new challenges you will face in this activity. You will even discover what it means to have 'fractions in your fractions'!
Exploration 2: Fraction Puzzlers
These problems will get your brain puzzling 'til your "puzzler is sore," to quote Dr. Suess. The usual process of adding fractions is turned upside down as you find the fractions that you need to make a number. There are many answers to these problems, but they're still a real challenge to find.
byrdseed.tv has a video to accompany this exploration!
Exploration 3: Working Together
In algebra classes, you learn all sorts of rules for solving special kinds of problems such as mixture problems, consecutive number problems, work problems, etc. But you don't need to be in an algebra class in order to figure these things out! In this activity, you use your knowledge of fractions along with a little creativity and imagination to discover your own rules and patterns for the traditional algebra "work problems." And you will probably even learn some algebra along the way!
Exploration 4: Fractions Forever
What if someone told you that they knew how to add an infinite number of fractions together? Impossible, right? Yes and no... It all depends on what you mean by adding! In math, you can use logical thinking to discover that all sorts of seemingly impossible things are not impossible after all.
Exploration 5: Visualizing Fraction Multiplication
To most people, the rules for multiplying fractions seem easier than the rules for adding and subtracting them. But where do these rules come from—and why do they work? There are actually some pretty challenging ideas lurking behind these simple rules, but by drawing a few pictures and using some common sense and logical reasoning, you can make sense of the rules for yourself and expand your mathematical thinking power.
Exploration 6: Undo it!
Have you heard the old saying "Mine is not to reason why. Just invert and multiply"? That is not the philosophy of this exploration. You will discover that there is much more to reciprocals that just flipping a fraction upside-down—and the familiar rule for dividing fractions is just one of many strategies. A true mathematician likes to have many ways of thinking about things!
Exploration 7: Sum-Product Pairs
Have you ever noticed that 2 times 2 is equal to the same thing as 2 plus 2? Are there other numbers like this? Well, 0 does the same thing, but that's not so surprising, because 0 is a pretty special number. But what about other numbers? In this activity, you will discover beautiful patterns within patterns as you explore this question. And if you're adventurous enough, you may even be able to use some algebra to prove your discoveries!
Exploration 8: Unit Fraction Hunt
This activity is a personal favorite of mine and my students. Your task is to complete a grid that shows how to make a unit fraction (a fraction with 1 in the numerator and a counting number in the denominator) using copies of whole numbers. Some cells in the grid are easy. Some take some thinking. And others take some incredible mathematical detective work! This is a great activity for the whole class to work on together, and it may take most of the school year to solve the whole thing!
Exploration 9: Continued Fractions
Math students are taught some decimals go on forever. But what about fractions? It turns out that they can do this trick, too! In fact, decimals did not even "exist" in ancient times (before the invention of place value). People in those times came up with pretty creative ways to represent numbers, and you will learn about one of them in this exploration.
This website from Dr. Ron Knott is the place where I learned about using rectangle diagrams to represent continued fractions. It contains a thorough introduction to the topic, many more questions to explore, and a continued fraction calculator!