Reading Journal Week 5

1.      Click here to see our Grade 7 Unit Draft (Draft #2). (This and subsequent drafts are also listed under "Unit Plan" in "Cognition/Instructional Strategies/Planning" section).

2.      Read Chapter 5

3.       Recall our activity from the first day of class, wherein you were asked to find all possible noncongruent squares on the 11x11 geoboard (dot paper).

a.       How can this activity be morphed into a lesson that is designed to lead students to achieve the following construct-a-concept objective:  "The student geometrically distinguishes between square roots of perfect squares and square roots of non-perfect-square whole numbers." (Be creative; there's no "correct" answer, of course.) 

One idea for how to morph this activity into a construct-a-concept lesson is to allow students to try to solve question one, then either make sure each student has found or knows how to find all possible squares. Next I would present them with a tasksheet that has some different groupings of various squares where they are prompted to find groups of similar objects (the side length or area could be written in, or you could pass out rulers and let the students measure, or if you just want them to visualize the idea geometrically when dealing with a geoboard you wouldn’t need measurements), where they are categorizing specifics by their attributes (This is similar to Case 5.6). The point of this is for the students to distinguish nonexamples from examples of perfect squares, and from this, to find and describe attributes of perfect squares. Once students have done this, you could directly teach the connection between roots and square roots, and then allow them to revisit the previous activity, this time with a focus on finding and describing attributes of the square roots of perfect squares.

b.       How can this activity be morphed into a lesson that is designed to lead students to achieve the following discover-a-relationship objective:  "The student explains why, for every right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides." (Be creative; there's no "correct" answer, of course.)

                        I believe this lesson needs to be a little bit more guided than the lesson above. Depending on the time I had to carry out the lesson, I think I would still have students participate in the activity of question 1 on the tasksheet because there are applicable insights that could be gained from this activity. Then I would probably present students with a tasksheet consisting of questions like the ones included in question D on the “Geoboard Squares” tasksheet, as shown below:

A.     In the figure at right, find the following in terms of a and b.

a.     The side of the outside square

b.     The area of the outside square

c.      The area of each triangle

d.     The area of the inside square

e.     The side of the inside square

Previous to these general questions, however, I would put some examples with actual numbers on the tasksheet so that students could begin making more tangible connections between the lengths of the sides of right triangles. With these specific examples, I would also include more guided about specific relationships between numbers, but with the general case I would leave the questions more open to allow them to practice inductively reasoning.

c.     How are these two objectives different? Elaborate. 

These two objectives are different because, in the first case, your purpose is to lead students to discover for themselves attributes of a concept and to create a definition for that concept. In the second case, your purpose is to lead students to draw connections between two things, and to be able to describe the relationship between those things.

4.       Design a lesson and mini-experiment for a Construct- a-Concept objective from the Unit Draft you developed last week, and modify the Unit Plan if there is a need. For more details consult Activity 5.4 p.189, Activity 5.9 p.190, Synthesis Activity #3 p.206.

Click here to view the lesson plan (can also be found in "Cognition/Instructional Strategies/Planning" section.)

Click here to view the mini-experiment (can also be found in "Assessment"section.)

5.       In your journal describe the stages of construct-a-concept lesson.

The following are the four stages of construct-a-concept lessons:

         The first stage is “Sorting and Categorizing”. In this stage, students are given an activity to complete individually (with guidance from the teacher) that involves them classifying and categorizing specifics, thereby creating concepts.

         The second stage is “Reflecting and Explaining”. In this stage, students verbalize their logic behind the categories that they created. As the teacher, you help them to communicate effectively, and stimulate further ideas on the topic.

         The third stage is “Generalizing and Articulating”. In this stage, students further develop their categorizing through identifying specific attributes of the given concept. They will also verbalize the attributes of the concept. The students may develop an informal definition of the concept as well.

         The fourth and final stage is “Verifying and Refining”. During this stage, the definition of the concept is tested, revised, and/or verified through examining how examples and nonexamples hold up to the definition. Stages 1-4 are repeated as the teacher deems necessary.

        

6.     Having completed Geometry Lab 8.5 we started on first day of class (include it in appropriate folder of your portfolio) discuss how question 1 differs from Discussion A-D, and D in particular.

The following are question 1 and questions A-D from the “Geoboard Squares” activity:

1.                    There are 33 different-size squares on an 11x11 geoboard. With the hlelp of your neighbors, do the following:

a.     Find all the squares.

b.     Sketch each square on dot paper, indicating its area and the length of its side.

B.     How can you make sure that two sides of a geoboard square really form a right angle?

C.     How can you organize your search so as to make sure you find all the squares?

D.    Is it possible to find squares that have the same area, but different orientations?

E.     In the figure at right, find the following in terms of a and b.

a.     The side of the outside square

b.     The area of the outside square

c.      The area of each triangle

d.     The area of the inside square

e.     The side of the inside square

Question 1 differs from Discussion A-D in that question 1 is a much more open question that allows students to work through the problem in their own way. In answering question 1, students are allowed enough freedom that they can practice and refine their inductive reasoning. On the other hand, questions A-D guide students through steps of how to solve question 1; they could be viewed as hints to answering the main problem (question 1). Questions A-D, especially question D, do not allow students to reason their way through a challenging problem; they are more so a test of algorithmic skills rather than problem-solving skills.