 The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

 Term Tiles and Binomial Expansions

 Choose A Point of View

 1a. How does one square a binomial? Cube a binomial? 2a. What are the coefficients of a binomial squared? Cubed? Raised to the 4th power? 3a. When are we ever going to use this stuff? 1b. What does it mean to square a binomial? Cube a binomial? 2b. Why is the sum of the coefficients of a binomial raised to the nth power the same as 2n? 3b. What patterns do you see in binomial expansions and what causes each pattern?

Which set of questions do you prefer to answer, the first that focuses on what or the second that focuses on why?

Don't bother choosing, answer both. Use both points of view. Let the how explain why. Think not just "What is the pattern?" but also "Why is that the pattern?"

It all starts with multiplication.

 Warm Up Material

Qu. 1.   Expand (x-y)², the square of the difference between two numbers.
Qu. 2.  Expand (x-y)³, the cube of the difference between two numbers.   Thus far, all material has been review, warm up. 1a.   How does one square a binomial? Cube a binomial? 1b.   What does it mean to square a binomial? Cube a binomial?  To square a binomial, multiply it by itself. To square a binomial means make a square with the binomial as an edge, or, find the area of that square. To cube a binomial, multiply it by its square. To cube a binomial means make a cube with the binomial as an edge, or, find the volume of that cube.

 New Material

New material begins now.

3a.   When are we ever going to use this?
Binomial expansion and Pascal's Triangle are extremely important for two reasons.
• Pattern recognition, expression, and verification IS the "job" of mathematics. Techniques, skills, insights, and the mathematics learned here are LIFE SKILLS not just stuff used to learn the next new topic.

• Recognizing an expanded binomial, factoring it, is a skill required in precalc, calc I, calc II, calc III in polynomial and rational number expression and graphing, in finding maximums and minimums and extrema, and in Taylor and McLauren expansions.
3b.   What patterns do you see in binomial expansions and what causes each pattern?

An extensive collection of the patterns is found on the page "Binomial Expansion Presented Symbolically and Pascal's Triangle," but, it is not meant to be an exhaustive list. Students should be encouraged to seek, express, and verify other patterns.

An extensive discussion on combinations is found on the page "Factorials, Permutations, and Combinations -- Ways of Counting Sets of Stuff" and additional brief comments are made below.
 Take Your Pick on Focus and Method.

Select 1, 2, 3, or 4 of the next 4 questions. Questions 3 and 5 pose the question in the more abstract form. Questions 4 and 6, suggest using a graphic summary. One might pose question 3, then use 4 as a hint, then pose question 5. Asked alone, Question 6 prompts visual evaluation and summary.

 Qu. 3. "Why is the sum of the coefficients of a binomial raised to the nth power the same as 2n?" Qu. 4. Given a larger print copy of the graphic below, linked here, answer the question: "Why is the sum of the coefficients of a binomial raised to the nth power the same as 2n?" Qu. 5. Seek, express, and try to verify some patterns. Qu. 6. Given a print copy of the graphic below, seek, express, and try to verify some patterns. Click on the graphic for a larger copy.

 Seek, express, and try to verify some patterns. Click on the graphic for a larger copy. 2a.   What are the coefficients of a binomial squared? Cubed? Raised to the 4th power?   2b.   Why is the sum of the coefficients of a binomial raised to the nth power the same as 2n?

2a. What are the coefficients of a binomial squared? Cubed? Raised to the 4th power? 2b.   Why is the sum of the coefficients of a binomial raised to the nth power the same as 2n?

Each time a binomial is raised to the next highest power, the number of multiplications, and therefore terms, is doubled. The doubling function, 2x, is needed. 3b.   What patterns do you see in binomial expansions and what causes each pattern?
• Every term -- as every tile, token, and fold-able manipulative in an expansion -- has the same degree.

• The number of different kinds of terms -- as different kinds of tiles, tokens, or fold-able manipulatives in an expansion -- is one more than the degree.

A square, 2nd degree, has 3 terms, 3 different kinds of tiles.

A cube, 3rd, degree, has 4 tokens (or fold-able manipulatives).

• The coefficient of a term states the number of identical products, order doesn't count, though the foldable manipulative or colored picture illustrates different products where order or orientation does count.

The coefficient counts the number in the group, hence, combinations is idea for this purpose. • The degree of a term indicates the "purity" with respect to x and y.

In an expansion of the nth degree:

xn and yn are the "purest."

x(n-1)y1 and x1y(n-1) are the next purest, and so on.

those in the middle (for even degrees) equally mix x and y.   www.termtiles.com, Unit 44   © 2008, A. Azzolino