OMLO, Saskatchewan Education,

Regina, Canada S4P 3V7

In a recent article, *On My Mind - Conceptual Understanding and
Computational Skills in School Mathematics*, Scavo and Conroy consider the
following problem:

Two numbers are in the ratio of 2 to 5. One number is 21 more than the other. What are the two numbers?

The authors look at one student's solution:

5 - 2 = 3; 21/3 = 7; 2*7 = 14 and 5* 7 = 35

and wonder as to whether or not to give full marks to the student.

Scavo and Conroy present the following algebraic verification of the student's method:

"We want to solve the equations

given constants a, b and d with b not 0. From the first
equation we have,

for some unknown constant c.

The constant c corresponds to the `magic' number
7 in the solution. From the right-hand side we see that

for some unknown constant c. The equations yield

provided that b - a is not 0. Substituting we obtain

which solves ..." our problem.

I would like to offer an alternate, visual, demonstration of the student's method that I believe demonstrates conceptual understanding and computational skills in a way that the idea will remain with the problem solver, as opposed to the algebraic method given which I believe often does not.

Two numbers are in the ratio of 2 to 5 (think of two parts and five parts), and one number is 21 more than the other |

and thus

i.e. the parts have size 7 and the numbers are 14 and 35. |

So, quite simply, one number has two parts; another has five parts; their difference of three parts has size 21; each part has size 7 and ... .

Of course, in general if two numbers are in the ratio of a to b and one is d more than the other:

and each of the (b - a) parts has size d/(b - a) so that the numbers are ad/(b - a) and bd/(b - a). My experience tells me that students (particularly in middle years) readily understand the logic of this approach and furthermore such problem solving skills remain with them.

**Reference**

Scavo, T. R. and Conroy, N. K., *On My Mind - Conceptual Understanding and
Computational Skills in School Mathematics*, Mathematics, Teaching in the
Middle School, NCTM Vol. 1, No. 9, 1996, 684-686.

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