# Advanced Common Core Math Explorations: Numbers and Operations

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 investigate numeration systems of beings from another planet, create and solve stories and problems with extreme numbers, use place value to design their own divisibility tests, and play with a magical number line that multiplies numbers without a calculator. Each activity comes with extensive support for teachers including learning goals, discussion guides, detailed solutions, and suggestions for extending the investigation. 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: Triangle Sums

After you find your first solution to this classic problem, you discover that the fun has just begun! In "Triangle Sums," you learn about finding multiple strategies and solutions, making and testing conjectures, justifying conclusions, and creating new questions to explore.

Exploration 2: Torran Math

Deepen your understanding of place value by exploring the numeration system on the planet Torr, whose inhabitants have only two fingers on each hand!

Exploration 3: Number Line Magnifiers

Very small numbers can be hard to visualize. One way to imagine them is to put the number line "under a magnifying glass." Suddenly, the rules for rounding, adding, and subtracting decimals make sense!

Exploration 4: Million, Billion, Trillion...

Aren't those "illions" all pretty much the same? Solve real-world (and some not so real-world) problems as you explore large-number place value and learn how scientific notation simplifies the process of reading, writing, and calculating with numbers that boggle the imagination.

• Error Correction for Problem #6 (The Student Handout in the book is missing a table. However, the table does appear in the Solutions at the top of pg. 65.)

Exploration 5: Discovering Divisibility Tests

You may have learned that you can add the digits of a number to see if it is divisible by 3 or 9, but why does this 'trick' work— and why doesn't it work with other numbers? Place value holds the key. Once you understand this, you can create your own tests for other numbers.

Exploration 6: Visualizing Decimal Multiplication

How do you know where to put the decimal point when you multiply? By drawing pictures and creating stories, you never have to worry about "forgetting the rules" again. You can discover your own!

Exploration 7: Think Like a Torran!

Dig more deeply into the mysteries of Torran mathematics. Learn how the Torrans use their numeration system to add, subtract, multiply, and divide. You will even learn about the Torran version of decimals.

Exploration 8: Extreme Calculations

Many parts of our world are hidden from us because of their extreme size. In this activity, you use the power of math to delve into the mysteries of the very large and the very small.

Exploration 9: Multiplication Slide Rules

Before the days of calculators, people used special rulers to carry out complex computations. With the advent of handheld calculators and computers, we no longer need these tools, but it is still fascinating to watch them in action. By playing with them, we gain access to important concepts that hide in the background when the computers do all of the work!

Exploration 10: Factor Blocks and Radicals

Square roots and their properties will longer seem mysterious after you explore them using pictures of colored blocks in this activity. The images make abstract ideas visible as you discover and apply algebraic formulas and procedures without needing to look them up in a book.