Download Bringing Standards Together for Understanding

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.
What is the area of each?
16
3
12
4
24
2
1
48
What is the area?
9
8
3
4
9
What is the area?
4
9
+
4
Area = 72 + 32 + 27 + 12
= 143
8 8
9
4
A = 72
A = 32
+
3
3
Using
FOIL
8
Area = length • width
= (8 + 3)(9 + 4)
= 11 • 13
= 143
A = (8 + 3)(9 + 4)
= 72 + 32 + 27 + 12
first
= 143
outside inside
last
3
8
A = 27
9
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
x2
1
1
x2 + 2x + 3
What is the Area?
Area = x2
+
5x
+
4
+
4
1
x2 + 5x + 4 = (x + 1)(x + 4)
+
x
x
Area = (x + 1)(x + 4)
What is the Area?
A
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 Rectangles Sm. Squa res
x2
x
1Υs
1
2
Lg. Rectangle
Possible?
1
Yes
x2 + 2x + 1 = (x + 1)(x + 1)
Area
1
3
2
Area
2
Area
4
2
Drawing
Worksheets 3 & 4
Additional student worksheets are
provided for your use in connection to
these concepts.
Related to Graphing
x2 + 3x + 2 = (x + 1)(x + 2)
2
x2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
x = –1, –2
–2
–1
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
Excellent
Communicates complete understanding
3
Satisfactory
Communicates clear understanding
2
Partial
Evidence of conceptual understanding
1
Minimal
Minimal understanding
0
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.