Bringing Standards Together for Understanding

A Model Unit: Area Models for Multiplying and Factoring Presented by Dr. Dianne DeMille and Connie Hughes from the TASEL-M project http://taselm.fullerton.edu

Consider the Standards

• Algebra I – 10.0 Students add, subtract, multiply, and divide monomials and polynomials. . . .

– 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. . . .

– 14.0 Students solve a quadratic equation by factoring or completing the square.

– 21.0 Students graph quadratic functions and know that their roots are the

x

-intercepts.

Consider the Standards

• Grade 7 – NS 1.2 Add, subtract, multiply, and divide rational numbers and take positive rational numbers to whole-number powers.

– AF 1.3 Simplify numerical expressions by applying properties of rational numbers and justify the process used.

– MS 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures . . . – MR 2.2 Apply strategies and results from simpler problems to more complex problems.

– MR 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

1

What is the area of each?

16 3 4 12 24 2 48

What is the area?

9 4 8 3

9

What is the area?

4 9 + 4 Area = 72 + 32 + 27 + 12 = 143 9 4 8 8 + 3 3 Using FOIL Area = length • width = (8 + 3)(9 + 4) = 11 • 13 = 143 A = (8 + 3)(9 + 4) = 72 + 32 + 27 + 12 first outside inside last = 143 8 3 A = 72 A = 27 9 8 A = 32 3 A = 12 4

Area Models Worksheet 1

• Practice worksheet for finding the area – Find the area of each part and add – Find the area of the larger rectangle formed • Page 2 – Students are asked to explain why the two areas are equal

Write an Expression for this Group of Algebra Tiles

x x x

2 1 1

x

2 + 2x + 3

What is the Area?

Area = x

2 + 5x + 4

x

+ 4

x

2 + 5x + 4 = (x + 1)(x + 4) Area = (x + 1)(x + 4)

A

What is the Area?

B

• What are the pieces that make up each larger rectangle?

• What are the dimensions of each larger rectangle?

Area Models Worksheet 2 Lg. Squa res

x

2 Rectangles x Sm. Squa res 1’s Lg. Rectangle Possible? Area Area

1 1 2 1

x

2 + 2

x

+ 1 = (

x

+ 1)(

x

+ 1) Yes 3 2

Drawing Area

2 4 2

Worksheets 3 & 4

Additional student worksheets are provided for your use in connection to these concepts.

Related to Graphing

x

2 + 3

x

+ 2 = (

x

+ 1)(

x

+ 2)

x

2 + 3

x

+ 2 = 0 (

x

+ 1)(

x

+ 2) = 0

x

= –1, –2 –2 –1 2

Worksheet 5 Written Response

For a complete response: clearly explain your thinking, label any figures you draw, identify formulas you use, and make clear where the numbers come from in your work.

You have a rectangular yard that is 8 feet long and 6 feet wide. You decide you want to add to the same number of feet to each dimension to get an area 32 square feet more than the area of the original rectangle. By how many feet will you need to increase each dimension?

Rubric - 4 points

4 3 2 1 0 Excellent

Communicates complete understanding

Satisfactory

Communicates clear understanding

Partial

Evidence of conceptual understanding

Minimal

Minimal understanding

No Response

Looking at Student Work

• Discuss with a partner the qualities you see that determined the score points assigned to each paper.

• What would need to happen in the classroom to help all students get a score of “3” or “4”?

• How can you use written work with your students to help you understand their thinking?

Using Written Response Items With Your Students

• • •

The statement of the question should be explicit and clear. The extent to which students are to

– –

discuss their reasoning and results should be explicit provide examples, counterexamples, or generalizations should also be clearly stated When choosing items others have written, some edits may need to be made to achieve these guidelines.

Wrap Up

• What we presented is standards that should not be taught as independent lessons.

• This is an example of what a conceptual package might include.

• It would be a unit of instruction for the conceptual package that can be covered in less time than trying to cover these same standards as the textbook presents in multiple sections.