# Math Building BlocksIntroduction

Math Building Blocks is a representation based on prime numbers and prime factorizations. Blocks of different colors represent different prime numbers. Joining two or more blocks represents multiplication. The blocks make it easy to visualize, analyze, and connect concepts across middle school math standards—everything from factors, multiples, GCFs, and LCMs to fraction computation, properties of exponents and radicals, and algebra.

### The organization of the book

Each chapter deals with a different broad math topic. Each section within a chapter begins with a page of images representing one aspect of the topic followed by a set of teacher's notes containing observations, questions, and predictions for discussion.

### Using the images

Show a page of images to your students and ask what they notice, wonder, and predict. Use the teacher's notes to guide the discussion. Ask students to create and discuss their own images to represent new examples.

Use the images from Chapter 1 early in the year in order to introduce the model. Use images from later chapters throughout the school year as they apply to the topic at hand.

Once students are familiar with the blocks, you may use the representation to support teaching and learning at any time. Since it is often not practical to show the colors, simply draw the blocks and write the prime numbers inside of them.

### Strengths of the Math Building Blocks representation

• Math Building Blocks reinforce the meaning of prime numbers as building blocks of the natural numbers, or as some people put it, "the atoms of arithmetic."
• Math Building Blocks make difficult multiplication concepts much easier to visualize by making multiplication look like addition!
• Math Building Blocks help students to: (1) move beyond memorized procedures (such as using factor trees), (2) compare prime factorizations of different numbers, and (3) see the usefulness of prime factorizations throughout the curriculum.
• Math Building Blocks give students a visual language to support reasoning and communication about multiplicative number concepts.
• Math Building Blocks help students to discover rules for multiplying and dividing fractions, properties of radicals, etc.
• Math Building Blocks encourage students to observe, question, and predict.
• Math Building Blocks create appealing images that make abstract concepts more approachable.
• Math Building Blocks provide a natural bridge between ideas in arithmetic and algebra.

### Limitations of the Math Building Blocks representation

• Because the Math Building Blocks represent multiplication as joining, students will sometimes interpret multiplication as addition and division as subtraction. You may address this issue by remaining alert to it and asking students to verbalize the distinctions. The chapters on fractions provide a visual prompt to separate addition from multiplication.
• Math Building Blocks is an algebraic model. It represents numbers by their properties, not their size. Students may need to be reminded not to use the size of a diagram as a cue to the size of the number it represents. (For example, 64 has a larger-looking diagram in the model than 65, even though 64 < 65.)
• Some students may be unable to distinguish colors in the images. You may address this issue by offering images using textures or alternate symbols in place of colors. (See the More Resources page in this book). In some cases, it may be helpful to have students collaborate with others who can see the colors more easily. Once students have seen the initial images early in Chapter 1, you may begin writing numbers in the blocks (with or without the colors).