### INTRODUCTION

Intro to Building Numbers from Primes discusses the use of colored building blocks as a hands-on model to teach students about prime factorizations and related concepts. Reading this page will provide a good background for understanding the remaining activities, games and discussions.

### THE 100-BOARD

Intro to Building Numbers from Primes (above) shows the most direct approach for introducing students to the prime building blocks. If you would prefer a more challenging problem-based approach, try using the 100-Board!

In the 100 Board Activity, students are given a grid showing a picture (using the colored blocks) of the prime factorizations of the numbers 1 through 50 - but don’t tell them what it is! Instead, they look for patterns and try to figure out how to continue the grid themselves. There are so many patterns in the grid that some students will have success continuing it (or at least parts of it) even before they discover the fact that the blocks are showing prime factorizations!

This is a challenging activity. In most cases, it probably works best to have students work in small groups. Be sure to set aside at least two class days to give students plenty of time to think - and be prepared to offer occasional hints when needed.

Even if you’re not teaching the building blocks model to your whole class, the 100 Board Activity can make a great enrichment activity for students who need an extra challenge!

Materials:

### PRIME OUT!

Prime Out! is a web-based game in which you earn points by building numbers from primes - one step at a time. You will see how prime numbers are like the ‘atoms’ of arithmetic - the basic building blocks of the numbers we count with every day.

You can play Prime Out! in many different ways: using small or large numbers; by yourself or with a friend; as a leisurely game or a fast-paced timed game! No matter how you choose to play, you will have fun while strengthening your math knowledge and skills. But watch out - if you play often, you might just start thinking about numbers in a whole new way!

Prime Out! is based on a board game originally called “Build It!”. Materials for the original game are still available as well:

Build It Game Score Sheet (PDF)

### FACTOR SCRAMBLE

In this entertaining puzzle, every word has a special number associated with it. Students are given the number and asked to ferret out the matching word. Most students begin by using a trial and error process, but as they work they gradually discover that prime factorizations hold the key!

The Factor Scramble puzzle is a fun way to help students discover that *all* the factors of a number can be built from its prime factors.

Factor Scramble (Activity) (PDF)

### THE MYSTERY CODE

The Mystery Code (PDF) is a thought-provoking activity in which students solve and extend a number code based on exponents in prime factorizations - but they don’t know that’s what it is! There are some hints included in case you or your students are really baffled!

### EUCLID’S PROOF

One of the most famous proofs in all of mathematics is Euclid’s Proof that there are infinitely many prime numbers. This page shows how the colored building blocks can help students visualize this proof - or a slight variation of it - in order to make it easier to understand. There are a some questions at the end to encourage them to think more deeply about the proof.

### PRIME BUILDING BLOCKS AND SQUARE ROOTS

This page provides a discussion of some ideas for using a “building blocks model” to better understand radicals and rational exponents - with a focus on square roots. Teachers may be able to use some of the ideas explored here to develop lessons or activities to help students understand why a “one-half power” represents a square root, how to simplify radicals, and how to visualize properties of radicals in a natural way.

This page offers a discussion to illustrate a method of using building blocks to understand and visualize why the square root of 2 cannot be represented exactly as a ratio of counting numbers. Some questions/problems are provided at the end to help encourage a deeper understanding of the issues involved.

This page describes an extension of the building blocks model that incorporates reciprocals of prime numbers. In some ways, it is an analogue to the commonly used “+/- counters” model for the addition and subtraction of integers. Using this “reciprocal blocks” model, students and teachers can represent the simplification, multiplication and division of fractions in a new way. It amy also help students visualize the connection between negative exponents and reciprocals.

This contains a table that builds on the analogy suggested above between addition/subtraction models on one hand, and multiplication/division models on the other.