Notes and Objectives

I.        Topic Concept 1: Proportions
A)     ( 11/1/2011 – 11/23/2011) - Proportional Reasoning
a.       Can identify a proportion and manipulate it to find a missing (value, quantity, percent, missing value given two similar shapes, or unit – given a unit rate) represented as a variable
i.      Proportion: Two ratios set equivalent to each other
1.       Ratio: Comparison of two quantities that uses division
a.       Representation of ratios
i.      a:b , a to b , a/b
ii.      Manipulation of ratios
1.       Ratios can be represented as Equivalent Fractions
a.       Simplified ratios or fractions can be multiplied by any number to find an equivalent form of the same value represented by the ratio or fraction
2.       Equivalent: Comparison of two or more values that are equal, the same, or similar to each other
a.       The idea of a balance
b.      2 pencils in the left hand has the same value or quantity as two pencils in the right hand
3.       Cross Product: The product of the means and the product of the extremes. In the proportion, a/b = c/d , ad = bc The a and d represent the means, b and c represent the extremes.
a.       Recognition that cross products are equal
b.      If the cross products are not equal, then the ratios do not represent a proportion
1. Can manipulate a proportion to find a missing value or quantity
i.      Solving for a variable, when a ratio is missing a value or quantity
1.       Variable: A letter or other symbol that can be replaced by any number or any other expression
a.       The missing quantity or symbol in the numerator or denominator of a given ratio that is set equivalent to each other
b.      Finding the value of the variable when using two ratios set equivalent to each other (using the idea that cross products are equal) – setting the expression of the (means) and( extremes) equal to each other
2.       Manipulating each expression of the equation on each side of the equal sign exactly the same way by combining like terms, and using inverse operation to each term by (multiplying, dividing, subtracting, or adding)
a.       Isolating the variable: to get the variable by itself on one side of the equation, creating a value or expression to the isolated variable
b.      Finding the percent of a ratio using part/whole is equal to %/100
c.       Finding a unit of two different units using the unit rate set equal to a similar ratio using the same units
d.      Finding the missing value or quantity of two equivalent ratios or fractions set equivalent to each other
1. Can recognize portions of two similar shapes to find a missing value of a proportional shape
i.      This will be addressed later when we cover shapes
ii.      Will continue with the similarities of Equations and the setup of Proportions

1. Introduction of Topic 2 Objective: Proportions vs Equation
A) 11/29/2011 – Objective: (Recognizing the similarities of Proportions and Equations)
a.       11/29/2011 Objective: Able to classify proportion and equation
i.      Transitional piece for Introduction into Equation
ii.      Introduction of Topic Concept 2 began with Review and Summary of Topic Concept 1 with a transition and introduction to the Analogy with the focus being similarities that there are similarities when classifying Proportions and Equations. With the daily objective being to understand that the cross products of Proportional Ratios create the expressions of an equation.
B) - The similarities of Proportions and Equations
iii.      The cross products in a proportion of the two ratios create an equation
1.       Identifying the parts of an equation and how we can manipulate it…
2.       Introductory Summary to what an equation is made up of
a.       The definition of an Equation – Two equivalent expressions separated by an equal sign
i.      *Review for proportion – two ratios set equivalent to each other
ii.      *Cross products of the means and extremes of two ratios create an expression
b.      The definition of an Expression – Is formed by variables combined with numbers and operations separated by terms
i.      Numerical expression – contains no variables
1.       Example –   25+5=35-5
ii.      Algebraic expression – contains variables
1.       Example -    25+v=7v-1v
2.       Ex. -     2v = v+v
c.       The definition of Terms - The numbers, variables, or products or quotients of numbers and variables that are added or subtracted in an algebraic expression
i.      Terms are separated by (+) and (-) signs
d.      The definition of Like terms – Terms that contain the same form of a variable
i.      Similar terms that can be combined
III.                12/1/2011 Topic Concept 3 Objective: How to dissect and manipulate an Equation
A) Objective: Dissection of an Equation and its parts
a) 11/29/2011 - Introduction and Summary to what an equation is, and it’s parts (Equation, Expression, terms, like terms, equivalency sign (=), variables, numbers, coefficients, separation of terms, combining like terms, balancing of an equation, solving for a variable, etc.)
1) Used as a schema builder to Introduction for new information
b) 11/30/2011 Objective: Using different symbols to identify like terms
i.      Pre-assessment – identifying like terms in differing expressions by creating different symbols that represent the idea of likeness among terms that can be combined
1.       Model the concepts that are to be done for students to identify like terms
a.       Worksheet (original / modified)
b.      Review in class
c)       12/1/2011 Objective: Recognition of Terms and Like Terms in an expression
1)      Instructional Strategies used
c.       Set objective
d.      Use of Advanced organizer
i.      Separating terms with symbols that represent like terms
ii.      Use of different colors representing like terms in an expression
e.      ISD
f.        Summarizing
g.       Cues / Questions
h.      Modification of Notes
2.       Notebook
a.       Journal 12/1/2011
i.      How can we identify terms in an expression?
1.       Students can (show examples, summarize, Use an invention of thought, anything that can be communicated on paper).
2.       Example: separated by (+) and (-) signs… / can identify by using different symbols
b.      Class Notes
i.      12/1/2011
ii.      Recognition of terms and like terms in an expression
1.       5x+7y+4=10+5-4x-9y
2.       Term, Expression, Equation, Like terms, coefficient, sign of term,
3.        Similarities and differences of a given term
4.       Recognizing like terms
3.       Notes that are handed out
i.      Advanced Organizer: Use of November objective Notes given on 11/29/2011
ii.      Summarized notes of lesson (modified) {highlighted key objectives}
iii.      Fill in the blank to summarized notes (differentiated)
1.       (Expression = Expression) = Equation
2.       Vocabulary:
3.       Copy attached to lesson
4.       Activities
i.      Using symbols such as circling, putting a box, or putting a triangle around terms that show the similarities among the like terms using a specific symbol for each individual like term.
ii.      Teacher directed lesson…
5.       Worksheets
a.       Main attached
b.      Modified attached
6.       Items to be put in Folder
a.       Summarized noted
b.      Worksheets
c.       Objective summary
d.      Handouts
e.      Vocabulary
f.        Anything that is for math that is loose from notebook
7.       Reinforcement
a.       Praise
b.      Check for accomplishing objectives for the day
c.       Reinforcement of Effort
d.      Providing Recognition
e.      Student Achievement Rubric (Checklist / (Areas to Improve, Areas did well)- w/explanation
8.       Assessments
a.       Worksheet
b.      Notebook entries
c.       Knowledge of objective
d.      Identify by defining vocabulary
e.      Observation
f.        Diagram an expression using symbols
9.       Materials
a.       Notes
b.      Handouts
c.       Vocabulary
e.      Objectives of lesson
f.        Notebook
g.       Folder
h.      Manipulatives
i.         Etc.
d)      12/2/2011 Objective: Combining Like terms – Simplifying an expression “Why we identify like terms”
i.      Introduction using what they know and the notes of Representation of the variable
1.       Representation of the variable – to find like terms
2.       Exponential form of the variable
3.       Variables strung to together
e)      12/5/2011 Combining like terms
i.      Instructional Strategies
1.       Notes
2.       Cues
3.       Review
4.       Cooperative Learning
5.       Practice
6.       Providing Feedback
7.       Using the ISD model
8.       Setting of Objective
ii.      Notebook – Journal / Notes
1.       Journal- 12/5/2011 Objective: Simplifying an expression
a.       Like Terms are identified by what in an expression?
i.      Thoughts
ii.      Representation of the variable s or constant
1.       How many of each term is represented by the coefficient of the represented variable s
2.       Notes- 12/5/2011
a.       Terms that have same representation of the variable
b.      The identification of this will give the ability to combine the like terms in an expression
c.       Combine by adding or subtracting the coefficient of the variable
d.      Review adding and subtracting positive and negative numbers
e.      Can use a calculator
i.      Using the positive or negative key on the calculator
f.        Negative and positive numbers are added and subtracted by the knowledge of the Absolute value
i.      Absolute value is determined by the distance from “0”
ii.      The absolute value of the positive or negative determines the sign of the answer
1.       -2 + 1 = ?
2.       Ex: l-2l = 2
l1l = 1
a.       2 – 1 = 1 but since the -2 is further from “0” on a number line (using the idea of Absolute value) it would carry the sign to the problem giving the answer to 2 – 1 = 1 to the answer of -1
iii.      Given the answers to a simplified expression
iv.      Find the representation to the given variable
1.       The answer is represented by what terms
2.       Example : 6x is represented by 4x and 2x
a.       The answer is represented by the like terms of 4x and 2x in which the coefficients were added together
iii.      Class activity in groups
1.       The answer of the given expression – represents which terms in the expression
2.       Problems (1-4) in groups on the Modified worksheet
iv.      Notes that are provided
v.      Worksheets
1.       Modified
2.       Regular
vi.      Folder items
1.       Vocabulary
2.       Worksheets
3.       Provided Notes
vii.      Materials
1.       Notes – Teacher made Provided and Lectured
2.       Worksheets –
a.       Modified
b.      Regular
3.       EdHelper
i.      Creation of activity
2.       Find the like terms that created the answer
3.       Recognition of Like Terms
a.       Create simplified answer
b.      Carry the same form of the variable
c.       These forms of the variable will be used in the activity that will be part of the lesson objective map using manipulatives to represent the variable
viii.      Reinforcement
1.       Praise
2.       Recognition of how to better their participation
a.       What they are doing…
b.      What they need to do to do better…
f)       12/6/2011 – Determining the representation of the variable for like terms
i.      Will be continuing the concept from 12/5/2011
1.       Journal-                12/6/2011
a.       What does a simplified expression represent?
i.      Thoughts
ii.      The combination of all the like terms
b.      12/8/2011
What have you noticed about like terms?
i.      You can combine them to create a simplified expression.
2.       Worksheet
a.       Given the answer of simplified answer to expression
i.      What does each simplified answer represent in the problem
ii.      What was done to get the simplified answer
b.      Modification were created for the assignment with one on one help
c.       Modifications included the use of an instructional strategy that was taught to the students before this lesson. This instructional strategy was then used as a modification. In sense highlighting terms in order to show comparisons.

g)      Objective 12/8/2011-12/9/2011 To recognize Vocabulary that describes an equation and commonly used math terms
a.       Vocabulary – definitions
i.      Reviewed
ii.      Discussed
b.      Vocabulary Quiz – 12/9/2011
i.      Matching
ii.      Modified
a.       Revised missed vocabulary on quiz by writing out the definition
h)      Objective 12/12/2011 – 12/16/2011
a.       12/12-12/13 Prepared for the Activity of
Modeling the Breakdown of an expression
i.      Reviewed Like Terms
ii.      Reviewed Vocabulary quiz
iii.      Revised like terms H.W. handed in for grade
iv.      Reviewed Symbols and substitution
v.      Inverse Operations
1.       Plus to minus
2.       Minus to plus
3.       Multiply to divide
4.       Divide to multiply
vi.      Absolute Value – The distance from “0” on a number line
vii.      Coefficient
viii.      Constant
ix.      Simplifying expressions
b.      12/14/2011 - 12/20/2011
i.      Begin Activity (The conceptual Model of the Breakdown of an Expression)
ii.      Start with Journal Reviewing the cancellation of numbers using manipulatives
1.       Make copies of Journal to hand out
2.       Have Journal written in Billy and Joe’s Journal
iii.      Helpful Hints for the Activity
1.       Subtract by cancelling
2.       If numbers are circled they are not included
3.       Inverse operations of + and – cancel each other out
4.       Need to identify the constants used by writing them on the line
by differentiating whether they are positive or negative
5.       Negative or minus signs distribute through the model displayed
6.       Number with the greatest absolute value when subtracting and adding negative numbers keeps the sign of the original
7.       – or + which ever is left takes the sign
8.       Write the simplified Expression in the order that each term is found from top to bottom on the Activity worksheet
9.       Count what is left and it will give you the simplified expression
10.   Show work by circling variables and numbers that cancel and determining the sign
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