Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Teaching Implementation

This unit will try to teach students how to translate word problems into the general abstract equation: ax + b = cx + d (Eq. 1), where a, b, c, and d are numbers supplied in the problem and x is the quantity to be found. Problems of this type were studied in our YNI seminar. The unit will include two sets of problems. The constants a, b, c, and d will be integers in the first set. Using the Rules of Arithmetic and the Principles of Equality (presented in the Appendix), the solution of this equation, which applies to all the problems in the set, can be found to be:

Deriving this equation will be a capstone part of my unit and will be carefully presented to my students as follows:

Subtracting b from both sides of the equation

ax + b = cx + d

yields the equation

ax + b - b = cx + d - b,

which simplifies using the Additive Identity rule, b - b = 0, to ax = cx + d - b

Next, subtracting cx from both sides of this equation and using the Commutative and Associative Rules on the right-hand side to put the -cx next to the cx gives the equation

ax - cx = cx - cx + d - b

Again, using the Additive Identity rule, cx - cx = 0, we obtain:

ax - cx = d - b

Using the Distributive and the Commutative Rule of Multiplication, we know that ax - cx = x(a-c). Thus, we have,

x(a-c) = d - b

Dividing both sides by (a - c), we have,

Moreover, using the Reciprocal Identity: , yields the final solution as:

The unit is scaffolded into four progressive lessons, expanding the algebraic concepts. First, the students will work with a set of real-life word problems that only involve constants (a, b, c, and d) that are integers. After the students become familiar with the generation of the corresponding equations, I first want them to find the solutions one at a time. Then toward the end of the unit, I will make the striking and important point that if you work symbolically, there is a universal formula for the solution, equation (2).

Next, the students will work with a second set of word problems, now involving constants that are rational numbers. Another version of equation 1 represents this second set of word problems, specially designed to have rational numbers in it, in which two of them share the same denominator: Eq. 3. This equation has a solution involving the division of fractions,

This can be justified directly, using the general symbolic solution given above, or rederived using the rule for computation with fractions. At this point, the division of fractions will be needed to finalize the problems.  Mathematically, fraction division can be presented as an algorithmic procedure that can be easily taught and learned by “invert and multiply.” The reasons for “invert and multiply” can be, and ideally would be, explained using the Rules of Arithmetic. It should be valuable for students to know that division by a whole number n is the same as multiplication by 1/n, and vice versa. Also, multiplication by a/b is the same as multiplication by 1/b followed by multiplication by a, according to the Associative Rule. (Or things can be done in the opposite order because of commutativity.)

Sharp and Adams recommend that students who can construct personal knowledge using various resources, i.e., pictures, symbols, and words, can improve their understanding and ability to communicate solutions (Sharp 2002). Building on previous knowledge and extending to the division of fractions, this unit aims to build confidence and a deeper understanding of mathematical procedures. This is very important as early teaching should provide realistic situations that enable students to build on their existing knowledge base (Streefland 1991).

The unit then will develop and test many different methods to teach the division of fractions which enrich the understanding of what the mathematical operations mean, following the guidance of several studies which have been conducted to verify the best approaches to teach students the invert-and-multiply algorithm (Elashhab 1978, Silvia 1983). These studies and descriptions of practical lessons rigorously guided students toward the invert-and-multiply algorithm using algebraic (Chabe 1963) and pictorial representations (Elashhab 1978; Silvia 1983) for instruction.