Process Standards in the High School Mathematics Classroom

Focus: Connections and Representations

Michael BollingTCTM – High School Breakout – 10.1.13

michael.bolling@doe.virginia.gov

Mathematical Connections

Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other.

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Mathematical Representations

Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations graphical, numerical, ⎯algebraic, verbal, and physical and recognize that ⎯representation is both a process and a product.

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Multiplication and Area

2 groups of 3

2 x 3 = 6

2 x 3

Concept of multiplication Connection to area

Area is 6 square units

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3

5

Multiplication and AreaMultiplying whole numbers – progression of complexity

8 x 10

10

8 12

23

8 groups of 10

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Multiplication and AreaMultiplying whole numbers

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2320 3

10

2

23

12

2 20 40 10 3 30

10 20 200 276“Partial Products”

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Multiplication and Area

Connection to Algebra I

x 3

2

( 3)( 2)x x x · x = x2

2 5 6x x

x

This will work for more than multiplying binomials! (unlike FOIL). This model is directly linked to use

of algebra tiles.

2 · 3 = 6

2 · x = 2x

3 · x = 3x

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Multiplication and Area

original warehouse

x

x

3

2

The sides of a square warehouse are increased by 2 meters and 3 meters as shown.

The area of the extended warehouse is 156 m2.

What was the side length of the original warehouse?

New Zealand Level 1 Algebra 1Asia-Pacific Economic Cooperation – Mathematics Assessment Database

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Multiplication and Area

original warehouse

30

50

x

x

The original warehouse measured 30 meters by 50 meters.

The owner would like to know the smallest length by which she would need to extend each side in order to have a total area of 2500 m2.

New Zealand Level 1 Algebra 1 (modified)Asia-Pacific Economic Cooperation – Mathematics Assessment Database

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Multiple Representations• 7.12 – represent relationships with tables, graphs,

rules, and words• 8.14 – make connections between any two

representations (tables, graphs, rules, and words)• A.7f – make connections between and among

multiple representations of functions (concrete, verbal, numeric, graphic, and algebraic)

• AFDA.4 - transfer between and analyze multiple representations of functions (algebraic formulas, graphs, tables, and words)

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Algebra I.7Relation or function?

Domain/range

Zeros

x- and y-intercepts

Function values for elements of the domain

Connections among representations

AFDA.1Continuity

Domain/range

Zeros

x- and y-intercepts

Function values for elements of the domain

Connections among representations (AFDA.4)

Local/absolute max/min

Intervals of inc/dec

End behaviors

Asymptotes

Algebra II.7Domain/range (includes discontinuous domains/ranges)

Zeros

x- and y-intercepts

Function values for elements of the domain

Connections among representations

Local/absolute max/min

Intervals of incr/decr

End behaviors

Asymptotes

Inverse functions

Composition of functions

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Rational Functions

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Geometric Constructions

Connections -

SOL 7.7 students learn properties of parallelograms, including that the diagonals of a rhombus bisect each other and are perpendicular.

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Geometric Constructions

Connections -

SOL 6.12 students learn to identify congruent polygons by their attributes.

SOL 7.6 students demonstrate knowledge of congruent polygons when learning about similar polygons

SOL G.6 students prove triangles congruent

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Coordinate Geometry

Connections -

SOL A.6 students determine the slope of a line

SOL 8.10 students learn about the Pythagorean Theorem, a direct connection to the distance formula

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Discussion• With which content could we do a better job

of facilitating connections or using multiple representations?

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