Springfield Blog #2

What a whirlwind class we had Fri evening and Sat. I hope the clarification on how to set up a tape diagram and the manner in which one is used to illustrate comparisons, and comparisons when values are added or subtracted resulting in revised ratios is clear. It was satisfying to see how well you, as a group, took to illustrating computation of fractions on the Cartesian coordinate plane. How many of you have used this in your classrooms? How successful has it been? Do you use the common denominator method …actually anything is better than giving them a “cutesy saying” or a rule to follow. We really need to be at the fore front of improving the way in which we teach mathematics. I was listening to the news this morning and the newscaster was reporting on a survey about how well mathematics is being taught to their children. 83% of the respondents said the teaching is deplorable and their children do  not understand why any of the procedures work. That is evident in my own teaching and coaching when I visit classrooms and children are doing rote procedures. When I ask why, most answer, “that’s what my teacher told me to do!”

I keep thinking about the problem with the ratio of girls to boys being 2:1 and if 30% of the girls leave how many boys must be added to have a 1:1 ratio. Have any of you tried to graph it since class? I hope so and challenge you to but suggest you don’t label the x-axis in percents…we did see a great example of how this looks (no cheating and asking your fellow two students who did it this way) how to model it. (no names to protect the innocent) Also, try to solve some of this week’s homework using the Cartesian plane…in fact I will insist upon it!

I am also curious about what you thought about TAU versus PI as the value for the ratio for the circumference to either the diameter or the radius and why TAU is so much more accurate.

Curious to hear about your experiences doing Gulliver.

I look forward to reading your blogs.

Anne

Springfield Ratio, Rate, and Proportion Blog #1

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