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Tiling Learning Task

Mark Henderson
Mathematics I: Algebra/Geometry/Statistics
Starr’s Mill High School


In Unit 2, students develop skills in adding, subtracting, multiplying, and dividing elementary polynomial, rational, and radical expressions. By using algebraic expressions to represent quantities in context, students understand algebraic rules as general statements about operations on real numbers. The focus of the unit is the development of students’ abilities to read and write the symbolically intensive language of algebra. Students will apply and extend all of the Grade 6 – 8 standards related to writing algebraic expressions, performing operations with algebraic expressions, and working with relationships between variable quantities. Students are assumed to have a deep understanding of linear relationships between variable quantities. Students should understand how to find the area of triangles, rectangles, squares, and trapezoids and know the Pythagorean theorem and need to have worked extensively with operations on integers, rational numbers, and square roots of nonnegative integers as indicated in the Grade 6 – 8 standards for Number and Operations.

By engaging in the tasks of this unit, students will experience opportunities to apply the basic function concepts of domain, range, rule of correspondence, and to interpret graphs of functions learned in Unit 1 of Mathematics I. The unit tasks begin with intensive work in writing linear and quadratic expressions to represent quantities in a real-world context. The initial focus is the development of students’ abilities to read and write meaningful statements using the language of algebra. In the investigation of operations on polynomials, the special products of standard MM1A2 are studied as product formulas and interpreted extensively through area models of multiplication. Factoring polynomials and solving quadratic equations by factoring are covered in a later unit, but the work with products and introduction of the zero factor property are designed to build a solid foundation for these later topics.

The work with rational and radical expressions is grounded in work with real-world situations that show applications of working with such expressions. Students need to practice computational skills, but extensive applications are needed to demonstrate that the skills have important applicability in modeling and understanding the world around us. The work with rational expressions includes this specific task’s consideration of the application to calculating average speeds.

Students are given many opportunities to use geometric reasoning to justify algebraic equivalence and to understand algebraic rules as statements about real number operations. Early in their study of algebra, students may have difficulty grasping the full content of abstract algebraic statements. Many of the questions in the tasks of this unit are intended to guide students to see how a geometric or other relationship from a physical context is represented by an algebraic expression. So, although some of the questions may seem very simple, it is important that they not be skipped. Gaining the ability to see all the information in an abstract algebraic statement takes time and lots of practice. Algebra is the language that allows us to make general statements about the behavior of the numbers and gaining facility with this language is essential for every educated citizen of the twenty-first century. Throughout this unit, it is important to:
  • require students to explain how their algebraic expressions, formulas, and equations represent the geometric or other physical situation with which they are working.
  • encourage students to come up with many different algebraic expressions for the same quantity, to use tables and graphs to verify expressions are equivalent, and to use algebraic properties to verify algebraic equivalence.
  • make conjectures about relationships between operations on real numbers and then give geometric and algebraic explanations of why the relationship always holds or use a counterexample to show that the conjecture is false.
The launching task for the unit, Tiling, provides a guided discovery for the following:
For each positive integer k, the expression 2k – 1 gives the k –th odd positive integer.
for each positive integer k, the expression 2k gives the k –th even positive integer.
For each positive integer k, the expression k(k + 1) gives a positive integer because one of k or k + 1 is even.
The k-th triangular number, denoted by Tk, is defined to be the sum of the first k positive integers.

so the sum of the first k positive integers is equal

Every square number is the sum of two consecutive triangular numbers; in particular, if Sk denotes the k-th square number k2, then Sk = Tk-1 + Tk-1 + Tk.

As the task progresses, students get lots of practice writing algebraic expressions to reflect geometric relationships they have observed. They also write several equations. They write equations that provide a computational formula, such as the equation above, but they also write equations that express equivalence between different expressions, such as the equation in (6). To be successful in working with the rules for adding, subtracting, multiplying, dividing, and factoring algebraic expressions, students develop this latter understanding of the use of equals. The developmental focus of the task is work with equivalent algebraic expressions. The concept of solving quadratic equations is also foreshadowed. Thus, it can serve to launch a unit focused on traditional algebraic manipulation with polynomials, rational expressions, and radical expressions and an introduction to solving quadratic equations using factoring and taking square roots.

Every square number is the sum of two consecutive triangular numbers; in particular, if Sk denotes the k-th square number k2, then Sk = Tk-1 + Tk-1 + Tk.

Latasha and Mario are high school juniors who worked as counselors at a day camp last summer. One of the art projects for the campers involved making designs from colored one-square-inch tiles. As the students worked enthusiastically making their designs, Mario noticed one student making a diamond-shaped design and wondered how big a design, with the same pattern, that could be made if all 5000 tiles available were used. Later in the afternoon, as he and Latasha were putting away materials after the children had left, he mentioned the idea to Latasha. She replied that she saw an interesting design too and wondered if he were talking about the same design. At this point, they stopped cleaning up and got out the tiles to show each other the designs they had in mind.

Mario presented the design that interested him as a sequence of figures as follows:

GPS Addressed
MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions.

Video Information
Use these questions to guide your thinking about some of the important teacher ideas in the lesson featured in the video clip.
  1. What kinds of questions does the teacher ask to promote students’ problem solving?
  2. How is the teacher gauging students’ current understandings and building from those understandings?
  3. Consider the GPS standards listed with this video.
  4. What makes this lesson different from lessons you have taught on this topic?

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