Learning by Doing

Day, July 20, 2015

Thinking Tools

Algebra / Fractions

“When I listen, I hear.  When I see, I remember.  But when I do, then I understand.”

Chinese Proverb

Today, we dealt with feedbacks, thinking tools and working with visual models for fractions and algebra.  Some of the lessons I have absorbed and their implication to my practices in the classroom.

Fractions can be represented using arrays. 

Most of us are used to solve fractions using the least common denominator for addition and subtraction.  For multiplication, we directly multiply both numerators to calculate the product-numerator and do the same with the denominators.  Then, simplify result to it’s lowest term.  For division, by multiplying reciprocals.  Manipulatives are helpful to visualize how parts are related to the whole or to other parts.  Today, I realize it is easier to visualize these operations using arrays.  Fraction problems can be solved without the use of the least common denominator, cross-multiplication, and reciprocals.

If I am to teach my students on how to solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate and proportions, I will definitely deliver instructions using arrays.  And I will prefer students to learn multiplication using the Area Method.

The Area Method is a better approach to Multiplication.

The standard algorithm is perhaps the fastest method to multiply and demonstrate the distributive property of multiplication. The Area method does the same, plus the latter promotes understanding and encourage the development of mathematical thinking. 

In practice, I would foster the use of the Area or Box method because it represents a rectangle or a square.  The area of a rectangle is the simplest: “length times width”.   A visual representation of multiplication is easier to comprehend than the numeric symbols. It also supports the important ability to estimate answers.  Further, it facilitates students’ understanding on how to multiply expressions and solving equations involving polynomials and in learning the higher level mathematics.  The understanding of the Box Method also facilitates comprehension of the utility of the Algebra Tiles. 

The use of Algebra Tiles is experiential  learning.

The reason I have hated the Algebra Tiles before  is I because I do not understand how it works.  Today,  I have learned  that is a powerful thinking tool.  This manipulative can help learners to develop concepts related to integers, algebraic expressions, equations, and polynomials.

These tiles are: big blue square (+x2 tiles); big red square tiles (-x2 tiles); rectangular green (+x tiles); rectangular red (-x tiles); small yellow (+1 tiles); small red tiles (–1 tiles) are used for negative integers (Bettye C. Hall).  How they are used to simplify, to solve, to expand, to factor, to complete the square is intuitive and exciting.

The use of these thinking tools will drastically change the way I manage my math class and hopefully translate to a better mathematical thinking for my students.