Dear Class,
I felt as though we competed in a marathon of mathematics
last weekend. For a large class I was extremely pleased to see that everyone
was extremely engaged and embraced the tape diagrams and double number lines. I
think as we work through using these models we need to keep aware of the fact
that the cells in the Tape Diagrams actually represent a variable used in
algebra but by working with the models before the variables, students
will/should be able to transition from the arithmetic to the algebra. The power
of these models is that they allow students to visualize a situation…not just
teach them to solve one problem.
Using the Tape Diagram in a part-to-whole measurement
problem is effectively portrayed using the Cuisenaire Rods. It doesn’t matter
which rod represents the whole as long as the shorter rods model the ratio
distribution. Students in the younger grades should be familiar with the Rods
if their teachers used Investigations or modeled the mathematics they were
teaching. So, although they may be new to some of us they are not so new to
students.
The comparative model which lists one ratio portion under
another is extremely powerful when there are more than two comparisons being
made. What is really nice about the models is that they work regardless of how
many comparisons are being made and each additional comparison does not need to
be taught separately. I was observing in one classroom when the students
completed comparing two ratios but when the next problem introduced three
ratios, the teacher stopped the lesson and said, “I will show you these
tomorrow.” My reaction was shock because there is no difference in how we
operate on two, three or more ratio comparisons and when students understand
the multiplicative underpinnings the number does not matter.
I personally like the double number lines for percent
problems and rate problems. They plus the graphical representation are
extremely powerful for introducing ratio as slope…an important concept in
algebra. But, the bottom line is all the models are algebraic in nature and
students who are comfortable representing the problems using the models
transition well to representing them algebraically as well.
Hope you all enjoyed the course as much as I did teaching
it.
Anne
