Emily's articles

Hi everyone,

I have some in depth summaries about my articles that I can email if you would like to see them. However, here are the main points of the three:

ETTY WANDA AND THE HAVE A HEART PROBLEM

*Two quantities (perimeter and area) cannot be compared.
* The way the numbers change when units are changed is different depending on what type of quantity you are using.
*Some students use hands-on manipulatives while others use numbers and formulas
* Scaling
*How to detect bogus questions in a textbook or in other cases.

DEVELOPING AN AREA FORMULA FOR A CIRLCE USING GOLDILOCKS

A circle drawn on graph paper with a radius of five squares has an area of 78 squares counting both partial and whole squares.

A square drawn around the circle has an area of 10 x 10 0r 100 squares. When divided into four smaller squares, the total area is 4 x 5 x 5 or 100 squares.

A smaller shape inscribed into the circle is dived into two traingles, each with and area of 25 squares or 2 x 5 x 5.

4 x 5 x 5 is too big
2 x 5 x 5 is too small
3 x 5 x 5 must be just right

the radius is 5 so the formula for a circle must be 3 x r x r or 3 x r squared. This equals 75, very close to first number of 78. When the tree is made 3.14, the number will be even closer.

FOSTERING COMMUNICATION ABOUT MEASURING AREA IN A TRANSITIONAL LANGUAGE CLASS


A math lesson was taught in a fourth grade classroom to determine the best way to emphasize the learners’ conceptual understanding (explanations, justifications, and representations) as opposed to their knowledge of procedures in math.

First Lesson:
The students were asked to find as many polygons as possible that had an area of 4 square units, using the geoboards. This showed that square units were flexible and could be sued to measure many shapes, even if partial squares were used. Before they stared, however, the class worked together to define polygons and square units so that all students were in the same page.

Students mostly struggled with the fact that they were dealing with square units but the shapes did not have to be squares. Also, a shape could look like a square (a diamond) but not use all four units. Still was especially hard for English language learners.

Second Lesson:
The students shared their polygon ideas and justifications with other students. Most were able to justify and explain polygons using whole or half square units, but not square units of other sizes. Many students discussed their polygons to the whole class. At this time, the student not only explained but answered questions and stimulated class discussion.

Third Lesson:
The students drew an interesting polygon on geoboard dot paper and on the back, wrote a paragraph about how they found the area of the polygon. The polygon could have any sized area. Before writing, however, they had to explain their procedure to at least two other people (especially helpful for English language learners).

However, in the end of the lesson, it was clear that the students needed more practice in using the square unit way of measuring the area because many did not use this method.