Worksheets for Workshop #7

About the Workshop

What is the theme of the workshop?
President Clinton's education agenda calls for all American students to be competent in algebra by the end of the eighth grade. This can only be accomplished if algebraic concepts are introduced and developed throughout the grades, beginning in kindergarten.

Whom do we see? What happens in the videoclips?We will see students from kindergarten through eigth grade using variables, studying relationships, exploring multiple representations, and making generalizations, all of which are at the heart of algebra.

What issues does this workshop address?Algebra is not the meaningless abstraction nor the symbolic manipulation that math leaders are pushing to incorporate into the early curricula. Nor is it what we studied it in high school. What, then, constitues algebra in the primary and middle grades? The answer to this question is the main issue of this workshop.

What teaching strategy does this workshop offer?A variety of teaching strategies will be modeled and discussed. Of note is the integration of graphing calculators in the teaching and learning of algebra. We will also see students working alone, in pairs, and in small groups and using a variety of manipulatives.

To which NCTM Standards does this workshop relate?This workshop will stress number sense and spatial sense (Standards 6 and 9 in the K-4 content standards). The activities we will explore involve finding patterns and exploring relationships (Standard 13 in the K-4 standards, Standard 8 in the 5-8 standards), and measurement (Standard 10 in the K-4 standards, Standard 13 in the 5-8 standards). The four process standards (Mathematics as Problem Solving, Communication, Reasoning, and Making Connections) will also figure into this workshop.

Suggested Classroom Activities and Strategies

Developing Algebraic Reasoning Through Literature (K-2)
As Jerry Kincaid does in Teaching Math K-4, you can read Pat Hutchins' The Doorbell Rang (or a similar story) to your students, and then help them discover the relationship between the number of children and number of cookies each child gets. Encourage your students to use any of a variety of forms of representation to express this relationship (e.g., concrete, verbal, numerical, tabular, pictorial, graphical, or symbolic).

Suggested Strategies This activity can be done with the entire class. Invite a volunteer to come up in front of the class and model the solution. This first solution may be in the form of a sentence or picture. Thereafter, invite others to give alternative representations for the same relationship (assuming it is correct). Explore multiple representations with the class (concrete, numerical, tabular, graphical, symbolic), pointing out the similarities and differences among them.

What Comes Next? (Grades 2-3)
Give students a variety of sequences (e.g., numerical, color, shape, texture). Provide at least the first five terms of each sequence, and have students figure out the following three terms. Then concentrate on numerical sequences. Ask students the following:

(a) Do they see the "change" from one term to the next? Can they articulate it?
(b) Can they find the 10th, 20th, 50th, 100th term?
(c) Can they generalize for the nth term in their own way (using either words, manipulatives, or symbols)?

Suggested Strategies This activity could begin as a whole-class activity for one or two sequences. After stressing what is important (i.e., looking at the "change" from term to term, devising generalization techniques, etc.), have students make up their own sequences and quiz their peers. It is important, every now and then, for students to be problem writers, not simply problem solvers. They invest more energy if the problem is their own creation. Also, creating a good problem can be more involved and challenging than solving one.

Discovering the Meaning of Pi (grades 4-8)
Arrange your class in small groups. Give a circular object to each group (e.g., a bottle cap, a soda can, a trash can, a flower pot). Have them measure the diameter (D) and the circumference (C). Have them also compute the ratio C/D for their object. When all the groups have completed the task, have them complete the following chart together:

Ask students to verbalize their findings. Encourage them to express the relationship between C and D symbolically.

The Cube Counting Problem
Have students build 2x2x2, 3x3x3, 4x4x4, and 5x5x5 cubes with multilink or snap cubes. Then have students mark the outside of the cubes thus formed (they could paint or color the outside of the cubes, or could simply use a washable marker to dot each and every visible face). When the faces have been marked, have the students take the cubes apart.

1. In each case, how many cubes have three faces painted? Two faces? One? Zero?
2. Can students discover the patterns in each of the four cases? Can they express the patterns algebraically?
3. Can students make sense of these algebraic expressions by relating them to the geometric structure of the cubes?

Suggested Strategies This activity (as well as the previous one) could be assigned to small groups of students. Divide the tasks beforehand. When the groups are done, have group reporters take turns sharing their work with the rest of the class. Encourage the "listening" groups to be attentive to the presentations by requiring an "intelligent question" of them. Finally, conclude by articulating the "big ideas."

Post-Workshop Questions

1. Many people, young and old, think of algebra as that "obscure, abstract, meaningless subject one studies in high school." How did you feel about algebra before this workshop? Do you feel differently about algebra now? How will that difference affect your math lessons?

2. What do you think of the use of technology in the teaching and learning of mathematics in general? Of algebra in particular? Have your thoughts about the issue changed since participating in this workshop series?

Suggested Resources

Lappan, Glenda, et. al. Variables and Patterns. Palo Alto, CA: Dale Seymour Publications, 1997.

Laycock, Mary. Algebra in the Concrete. Hayward, CA: Activities Resources Co., Inc., 1997.

Mathematics Teaching in the Middle School. NCTM. February, 1997.

National Council of Teachers of Mathematics. Algebra for Everyone. Reston, VA: NCTM, 1990. (Book and Videotape).

National Council of Teachers and Mathematics. The Ideas of Algebra, K­12. Reston, VA: NCTM, 1988. (Yearbook).

National Council of Teachers of Mathematics. Patterns: K-4 Addenda Series. Reston, VA: NCTM, 1993.

National Council of Teachers of Mathematics. Patterns and Relationships: 5-8 Addenda Series. Reston, VA: NCTM, 1991.

Teaching Children Mathematics. NCTM. February, 1997.

Pre-Workshop Assignment for Workshop #8

Let n denote the row number, beginning with:

n = 0 for row "1", and then

n = 1 for row "1 1",

n = 2 for row "1 2 1",

n = 3 for row "1 3 3 1", and so on.

Questions:

1. Add up the numbers in each row. Do these sums form a pattern? Can you express the nth sum in terms of n, the row number?

2. If you are given a certain row of Pascal's Triangle, how can you derive the row beneath it?

3. Can you find the famous Fibonacci sequence lurking within Pascal's Triangle?

4. Can you find the consecutive powers of 11, starting with 110, inside Pascal's Triangle? What happens after 114? Can you explain?

5. Notice the positions of even numbers within the triangle. Is there a pattern?

6. What other numerical patterns can you find?