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Mathematics and gesture

De
33 pages
  • expression écrite - matière potentielle : mathematicians
  • expression écrite
  • exposé - matière potentielle : the subject for the better part
Mathematics and Gesture SLSA Atlanta 2009 Mathematics and gesture. An odd couple: the supreme form of disembodied thinking and its crude corporeal antithesis. If mathematics engenders forms of thought which exceed that available to speech, gesture seems an atavistic return to simian calls and signals before speech. The antithesis is puzzling. Mathematics has many affiliations -- structural, aesthetic, experiential -- with music. Music is thoroughly gestural: literally in terms of its production through the playing of instruments, but also by the sonic gestures inherent in its composition and internal dynamics.
  • forms of thought
  • psychoanalytic understanding of unconscious interiority
  • language of sets
  • internal dynamics
  • arrows
  • gestures
  • mathematical entity
  • diagrams
  • sets
  • mathematics
  • 3 mathematics
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1 3. 5.2

Integrated Algebra Curriculum Map

Semester 1: Units 1, 2 and 3 Semester 2: Units 4, 5, 6, and 7
Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)
Unit 1: Relationships between Quantities and reasoning with Equations and Inequalities

1A: Number Theory and Operations (9 days instruction)
What makes Classifying Numbers/Comparing Numbers N.6 Evaluate expressions N.RN.3
something a
o

Classifying numbers according to which set involving factorials,
mathematical each number belongs absolute values, and
term?
o

Comparing and ordering real numbers exponential expressions
How do we Properties of Real Numbers N.1 Identify and apply the N.RN.3
translate
o

To understand and use the properties of properties of real numbers
written rational numbers. (closure, commutative,
information
o

To reason logically with given facts to a associative, distributive,
into conclusion identity, and inverse)
mathematical Distributive Property/ Combining Like Terms N.1 Identify and apply the
terms?
o

To understand and apply the distributive properties of real numbers
property. (closure, commutative,
associative, distributive,
identity, and inverse)
A.13 Add, subtract and
multiply monomials and
polynomials.
Adding and Subtracting Rational Expressions N.1 Identify and apply the
(Fractions) properties of real numbers
o

To evaluate expressions using the property (closure, commutative,
of numbers. associative, distributive,
o

To understand and apply addition and identity, and inverse)
subtraction of signed numbers
Multiplying and Dividing Rational Expressions N.1 Identify and apply the
o

To understand and apply the rules for properties of real numbers
multiplication and division of rational (closure, commutative,
numbers. associative, distributive,

4.2

2.1, 2.2

3.2

Diagnostic
Delayed
Test
Learners
Use algebra tiles
to simplify
expressions
Advanced
Learners
Have students
write complicated
expressions for
partners to solve
ELL’s
Should construct
math glossaries in
English and use
them during class
and on homework
assignments.

1

Essential
Question

2.1

2.1

Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)
identity, and inverse)

N.6 Evaluate expressions
involving factorials,
absolute values, and
exponential expressions
Evaluating Algebraic Expressions (PEMDAS) N.6 Evaluate expressions
o

Symbols in algebra involving factorials,
absolute values, and
o

Order of operations exponential expressions
o

Evaluating Algebraic Expressions
Evaluating Algebraic Expressions with A.5 Write algebraic
Parenthesis equations or inequalities
o

Order of operations with parentheses that represent a situation
(grouping symbols) N.6 Evaluate expressions
involving factorials,
absolute values, and
exponential expressions
Modeling verbal Relationships with Variables A.1 Translate a quantitative N.Q.2, A.SSE.1, 1.1
o

Translating English phrases to algebraic verbal phrase into an
expressions and equations algebraic expression
A.2 Write verbal

expressions that match
given
mathematical expressions
A.3 Distinguish the
difference between an
algebraic expression and an
algebraic equation
A.4 Translate verbal
sentences into mathematical
equations or inequalities
A.5 Write algebraic
equations or inequalities
that represent a situation
1B: Solving Equations (8 days instruction)
One-Step Equations A.5 Write algebraic A.CED.1, 3
equations or inequalities A.CED.3
that represent a situation
A.REI.1, A.REI.3
A.6 Analyze and solve
verbal problems whose

Delayed
Learners
Should be given
steps for solving
equations and
2

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)
One-Step Equations solution requires solving a A.CED.1, 3

allowed to use
linear equation or linear A.CED.3 during class until
What does it inequality in one variable.
A.REI.1, A.REI.3
memorized.

mean to A.25 Solve equations Use simple
balance an involving fractional examples initially,
equation? expressions. building to more
Two-Step Equations A.CED.1, 3.1

complicated ones.

o

To understand and solve 2-step equations. A.CED.3


o

Using deductive reasoning to justify steps in
A.REI.1, A.REI.3
What does it solving equations
Advanced
mean to solve
o

To write and solve equations that model real
Learners
an equation? world situations Have students
Solving Equations with Combining Like Terms N.1 Identify and apply the A.CED.1, 3.2

look for multiple
and Distributive Property properties of real numbers A.CED.3 ways of solving
o

To understand and apply the distributive (closure, commutative,
A.REI.1, A.REI.3
the equations.
property to solve equations. associative, distributive, Have them try and
o

To write and solve equations that model identity, and inverse) construct an
real world situations A.5 Write algebraic equation with no
equations or inequalities solution
that represent a situation

A.22 Solve all types of
linear equations in one
variable.
Solving Equations with Variables on Both A.5 Write algebraic A.CED.1, 3.3

Sides/ Identities and No Solution equations or inequalities A.CED.3
o

To understand how to solve equations with that represent a situation
A.REI.1, A.REI.3
variables on both sides. A.6 Analyze and solve
o

To write and solve equations that model verbal problems whose
real world situations solution requires solving a
linear equation or linear
inequality in one variable.
A.22 Solve all types of
linear equations in one
variable.
Solving Equations with Fractions A.25 Solve equations A.CED.1, Extra

ELL’s
involving fractional A.CED.3 Should construct
expressions.
A.REI.1, A.REI.3
math glossaries in
English and use
3

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

Transforming formula and solving literal A23 Solve literal equations A.CED.2, 3.4

them during class
equations for a given variable A.CED.3, and on homework
o

To transform formulas for particular A.CED.4, assignments.
variables
A.REI.1, A.REI.3


Consecutive Integer Problems A.6 Analyze and solve
A.REI.1
Extra

o

To write and solve equations that model verbal problems whose

real world situations involving consecutive solution requires solving a
integers, odd and even inclusive linear equation or linear

inequality in one variable.
Test 1

1C: Proportional Reasoning (8 days instruction)
What are Ratio and Rates, Unit Rates, Transforming A.26 Solve algebraic N.Q.1, N.Q.2, 3.4
Delayed
proportions Formulas proportions in one variable N.Q.3, A.CED.4,
Learners
and how are
o

To understand and apply ratio in solving Which result in linear or
A.REI.1, A.REI.3
Use a table to
they related to problems. quadratic equations. organize
rates, ratios,
o

To find unit rates and to convert rates. M.1 Calculate rates using information
scales and Solving Proportions appropriate units. N.Q.1, N.Q.2, 3.4

fractions?
o

To understand and solve algebraic M.2 Solve problems N.Q.3, A.CED.4,

involving conversions

How does an proportions within systems, given the
A.REI.1, A.REI.3


understanding relationship between the
Advanced
of percents and units.
Learners
proportions Similar Figures A.6 Analyze and solve N.Q.1, N.Q.2, 3.5 Have students
help us make
o

To understand and apply proportions to verbal problems whose N.Q.3, A.CED.4, write their own
better spending solve similarity problems solution requires solving a
A.REI.1, A.REI.3
proportion
decisions? Indirect Measurement and Scale Drawing linear equation or linear N.Q.1, N.Q.2, 3.5 problems

o

To investigate scale factors and ratios of inequality in one variable N.Q.3, A.CED.4,

perimeters, areas, and volume A.25 Solve equations
A.REI.1, A.REI.3

ELL’s

o

To apply proportions using scale drawings involving fractional Should construct
expressions. math glossaries in
to find distances. A.26 Solve algebraic English and use
proportions in one variable them during class
which result in linear or and on homework
quadratic equations assignments.

Motion Problems/ Distance-Rate-Time A.5 Write algebraic N.Q.1, N.Q.2, 3.6

o

To define one variable in terms of another. equations or inequalities N.Q.3, A.CED.4,
o

To translate, set-up and solve distance-rate-that represent a situation
A.REI.1, A.REI.3,

time problems F.IF.6
4

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

Geometric Figures G.1 Find the area and/or N.Q.1, N.Q.2, Activity Lab

o

Area and Perimeter of Composite Figures perimeter of figures N.Q.3, A.CED.4, (4.7, 10.9)
o

Surface Area and Volume of Rectangular composed of polygons and
A.REI.1, A.REI.3

Prisms and Cylinders circles or sectors of a circle
o

Circumference and Area of Circles G.2 Use formulas to
calculate volume and
surface area of rectangular
solids and cylinders
Review Percent/ Percent of Change N .5 Solve algebraic N.Q.1, N.Q.2, 3.7

o

To understand and apply the ratio of percent problems arising from N.Q.3, A.CED.4,
of change. situations that involve F.IF.6
o

To apply the percent proportion to solve fractions, decimals, percents
percent problems.

Percent Error / Relative Error M.3 Calculate the relative N.Q.1, N.Q.2, 3.7

o

To understand and apply the percent of error error in measuring square N.Q.3, A.CED.4,
in calculating area and volume. and cubic units, when there F.IF.6, G.GMD.3
is an error in the linear
measure

1D: Solving Inequalities (6 days instruction)
How do Simple Inequalities and Their Graphs A.4 Translate verbal A.CED.1, 4.1, 4.2, 4.3

inequalities
o

To find the solution to an inequality in one sentences into mathematical A.CED.3,
compare and variable. equations or inequalities
A.REI.1, A.REI.3

contrast to
o

To plot inequalities on a number line. A.6 Analyze and solve
equations?
o

Using the addition property to solve verbal problems whose
inequalities. solution requires solving a
Do we use linear equation or linear
different
o

Using the subtraction property to solve inequality in one variable
mathematical inequalities A.21 Determine whether a
procedures
o

Using multiplication and division to solve given value is a solution to a
when working inequalities.

given linear equation or
with linear inequality in one
inequalities? variable.
A.24 Solve linear
inequalities in one variable
G.6 Graph linear
inequalities
Multi-Step Inequalities A.5 Write algebraic A.CED.1, 4.4
o

Solving inequalities with one variable equations or inequalities A.CED.3,
that represent a situation
A.REI.1, A.REI.3

Delayed
Learners
Should be given
steps for solving
inequalities and
allowed to use
during class until
memorized

Advanced
Learners
Students should
discuss situations
that can be
represented by
inequalities and
determine
reasonable
solutions
5

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

Multi-Step Inequalities A.6 Analyze and solve A.CED.1, 4.4


o

Solving inequalities with one variable verbal problems whose A.CED.3,
ELL’s
solution requires solving a
A.REI.1, A.REI.3
Have students
Inequalities with Variables on Both Sides linear equation or linear A.CED.1, 4.4

explain the word
inequality in one variable A .CED.3, problems to a
A.24 Solve linear
A.REI.1, A.REI.3
partner

Compound Inequalities (and/or) inequalities in one variable A.CED.1, 4.5

o

Solving and graphing compound A.CED.3,

inequalities.
A.REI.1, A.REI.3

Test 2
Unit 2: Linear Relationships

2A: Relations and Functions (5 days instruction)
What makes a Writing a Function Rule A.5 Write algebraic A.CED.1, 1.4
relation a
o

Recognizing equations that describe a equations or inequalities A.CED.2, F.IF.1,
function? functional relationship. that represent a situation F.IF.2, F.IF.5,

o

Using tables to write function rules G.3 Determine when a F.BF.1a, b
Domain and Range of a Function relation is a function, by F.IF.1, F.IF.2, 1.4
How do we
o

Identifying independent (domain) and examining ordered pairs and F.IF.5, F.BF.1a, b
write and use dependent (range) variables. inspecting graphs of
function relations
notation? Relations and Functions G.3 Determine when a F.IF.1, F.IF.2, 5.2
o

To represent relations and functions using relation is a function, by F.IF.5, F.BF.1a, b
mapping diagrams, tables, and function examining ordered pairs and
rules inspecting graphs of
o

To determine when a relation is a function. relations
Function Notation/ Evaluating Functions/ A.5 Write algebraic A.CED.1, 5.3, 5.4
Function Rule equations or inequalities A.CED.2, F.IF.1,
o

Identifying and graphing functions, using a that represent a situation F.IF.2, F.IF.5,
table G.4 Identify and graph F.BF.1a, b
o

To understand and write function rules. linear quadratic (parabolic),
o

To determine the range of a function given absolute value, and
the domain and function rule exponential functions
o

Modeling real world situations using
function rules

Delayed
Learners
Have students
substitute
solutions into the
function rule to
check their
answers.
Advanced
Learners
Discuss why
specific variables
are graphed on the
x-axis and others
on the y-axis. Use
terms independent
& dependent
variable
ELL’s
Have students
explain the
terminology to a
partner

6

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

2B: Linear Relationships (9 days instruction)
What kinds of Direct Variation N.5 Solve algebraic A.CED.2, 5.5

Delayed
relationships
o

To model and solve equations of direct problems arising from A.CED.3,
Learners
are expressed variation, using the equation
y=kx
situations that involve A.CED.4, Have students use
by linear
o

To find the constant of variation given a fractions, decimals,
A.REI.10
, F.IF.4, each ordered pair
equations? table of values. percents, and F.IF.6, F.IF.7a, in a table to write

o

To determine whether or not a relationship proportionality/direct F.LE.5 equations rather
varies directly variation. than just one
How can A.5 Write algebraic

different equations or inequalities

graphing that represent a situation

choices affect Graphing a Line (Using a Table) A.CED.2, Extra


interpretation
o

Identifying and graphing functions, using a A.CED.3,

of the data? table A.CED.4,
Advanced

A.REI.10
, F.IF.4,
Learners
F.IF.6, F.IF.7a Arrange a
Finding the slope of a line/ Rate of Change A.32 Explain slope as a rate A.CED.2, 6.1

selection of
o

To understand slope as a rate of change of change between A.CED.3, equations in order
o

To find the slope of a line given two points dependent and independent A.CED.4, from steepest to
on the line variables.
A.REI.10,
F.IF.4, least steep
o

To determine the slope of a line based on A. 33. Determine the slope F.IF.6, F.IF.7a,
the graph of the line of a line, given the F.BF.3, F.LE.5
o

To identify the slopes of horizontal and coordinates of two points on
vertical lines the line.
Graphing a line using Slope, y-intercept Form A.34 Write the equation of a A.CED.2, 6.2, 6.3

ELL’s
o

To write the equation of a line given a point line, given it s slope and the A.CED.3,
on the line and its slope coordinates of a point on the A.CED.4,
o

To identify and graph linear functions. line
A.REI.10
, F.IF.4,
o

To investigate how changing the A.35 Write the equation of a F.IF.6, F.IF.7a,
coefficients of a function will affect its line, given the coordinates F.BF.3
graph. of 2 points on the line
A.37 Determine the slope of
o

To model real world problems involving a line, given its equation in
linear relationships any form.
7

Writing Equations of a Line
o

Given any point and a slope
o

Given any two points

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

G.5 Identify and generalize
how changing the
coefficients of a function
affects its graph
A. 34 Write the equation of A.CED.2, 6.4, 6.5
a line given its slope and the A.CED.3,
coordinates of a point on the A.CED.4,
line.
A.REI.10,
F.IF.4,
A.35 Write the equation of a F.IF.6, F.IF.7a,
line, given the coordinates F.BF.3
of 2 points on the line
A.A.39 Determine whether
a given point is on a line,
given the equation of the
line
Parallel Lines A.36 Write the equation of a A.CED.2, 6.6
o

To discover that parallel lines have equal line parallel to the
x-
or
y
- A.CED.3,
slopes axis. A.CED.4,
o

To determine if lines are parallel

based on

A.REI.10
, F.IF.4,
their graphs or equations F.IF.6, F.IF.7a,
o

To write equations of lines parallel to each F.BF.3, F.LE.5
other

2C: Systems of Equations (7 days instruction)
What does the Solving systems of equations graphically A.7 Analyze and solve A.CED.2, 7.1
intersection
o

To understand how to solve systems of verbal problems whose A.CED.3,
points of a equations graphically. solution requires solving A.CED.4, A.REI.6,
system of
o

To analyze special types of systems systems of linear equations
A.REI.10,
graphs tell us?
o

To understand how to solve a system using of 2 variables
A.REI.11

G.7 Graph and solve
a graphing calculator systems of linear equations
What and inequalities with
information rational coefficients in 2
does the variables
solution set of Solving systems of equations algebraically A.7 Analyze and solve A.CED.2, 7.2
a system of (substitution) verbal problems whose A.CED.3,
equations give
o

To understand how to solve a system by solution requires solving A.CED.4,
A.REI.6

us? using substitution systems of linear equations
of 2 variables

Test 3

Delayed
Learners
When solving
word problems
have students list
all the information
and what they
need to find
Advanced
Learners
Have students
8

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

Solving systems of equations algebraically A.10 Solve systems of 2 A.CED.2, 7.2

write a system of
(substitution) linear equations in 2 A.CED.3, three equations
o

To understand how to solve a system by variables algebraically A.CED.4,
A.REI.6
that form a
using substitution G.7 Graph and solve triangle
systems of linear equations

Solving systems of equations algebraically and inequalities with A.CED.2, 7.3


(elimination) rational coefficients in 2 A.CED.3,
ELL’s
o

To understand how to use addition or variables A.CED.4, Have students
subtraction to solve a system
A.REI.5, A.REI.6
explain the
o

To understand how to use multiplication to difference
solve a system. between intercept
Applications of linear systems A.7 Analyze and solve A.CED.2, 7.4

and intersect

o

To model real world applications that lead verbal problems whose A.CED.3,
to writing and solving systems of equations solution requires solving A.CED.4,
systems of linear equations
A.REI.5, A.REI.6,
of 2 variables
A.REI.10,
G.7 Graph and solve
A.REI.11

systems of linear equations
and inequalities with
rational coefficients in 2
variables

2D: Graphing Inequalities (4 days instruction)
Why does the Graphing and modeling linear inequalities A.6 Analyze and solve A.CED.3,
solution set for
o

To understand how to graph linear verbal problems whose
A.REI.10,
a system of inequalities. solution requires solving a
A.REI.12

inequalities
o

To determine where the solution set lies on linear equation or linear
differ from the the coordinate plane (above or below the inequality in one variable
solution set of line) A.21 Determine whether a
a system of given value is a solution to a
equations? given linear equation or
linear inequality in one
variable.
G.6 Graph linear
inequalities
Graphing systems of linear inequalities A.40 Determine whether a A.CED.3,
o

To understand and graph systems of linear given point is in the solution A
.REI.10,
inequalities ser of a system of
A.REI.12

o

To identify the solution set on the inequalities.

5.7

6.7

Delayed
Learners
Make sure that
students are using
a number of
points to verify a
solution
Advanced
Learners
Write their own
inequalities and
solve them
graphically
9

Essential Content/Skills
(with
suggested
days
Performance Indicators Common Core Resources Assessment Modifications
Question
instruction for each unit)
(Appendix B)
Standards
(Appendix A)
to instruction
(Appendix C) (*see footnote)

coordinate plane G.7 Graph and solve

ELL’s
o

To model real world problems that involve systems of linear equations

Emphasize the
writing and solving a system of linear and inequalities with difference
inequalities rational coefficients in 2

between the
variables

inequality

Test 4
symbols and what
they mean on a
graph
Unit 3: Descriptive Statistics and Data Analysis (9 days instruction)
Measures of Central Tendency S.4 Compare and contrast S.1D.2, S.ID.3 1.6

Delayed
How can we
o

To find mean, median, mode, and range the appropriateness of
Learners
use statistical
o

To represent data using a stem and leaf plot different measure of central Create their own
data to make
o

To interpret the data in a stem and leaf plot tendency for a given data list of numbers
informed set. where the median
decisions? S.16 Recognize how linear and mean are the
transformations of one-same
How can we variable data affect the

collect and data’s mean, median, mode,

present data in and range.

a way that Box and Whisker Plot S.5 Construct a histogram, N.Q.1, S.ID.1, NY2

Advanced
makes it
o

To understand the applications of box-and cumulative frequency S.1D.2, S.ID.3
Learners
accessible and whisker plots histogram, and a box-and-Describe a
useful?
o

To understand how the minimum, whisker plot, given a set of situation where
data the mean or the
maximum, and the 3 quartiles are used to S.6 Understand how the five median or the
construct a box-and-whisker plot. statistical summary mode would be
(minimum, maximum, and the best measure
the three quartiles) is used of central
to construct a box-and-tendency
whisker plot

S.9 Analyze and interpret

a

frequency distribution table
ELL’s
or histogram, a cumulative
frequency distribution table
or histogram, or a box-and-
whisker plot
S.11 Find the percentile
rank of an item in a data set
01