Bringing Standards Together for Understanding

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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.e
du
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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.

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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.

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What is the area of each?
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What is the area?
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What is the area?
Area 72 32 27 12 143
Area length width (8 3)(9 4)
11 13 143
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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

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Write an Expression for this Group of Algebra
Tiles
x2 2x 3
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What is the Area?
Area x2 5x 4
x2 5x 4 (x 1)(x 4)
Area (x 1)(x 4)
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What is the Area?
A
B

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

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Area Models Worksheet 2
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Worksheets 3 4

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

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Related to Graphing
x2 3x 2 (x 1)(x 2)
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Worksheet 5Written 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?

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Rubric - 4 points

• Excellent
• Communicates complete understanding
• 3 Satisfactory
• Communicates clear understanding
• 2 Partial
• Evidence of conceptual understanding
• 1 Minimal
• Minimal understanding
• 0 No Response

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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?

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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.

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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.