The students who will be enrolled in a transitional math ... · 3 The students who will be enrolled...
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School District U‐46 Elgin, Illinois
Instructional Council Proposal Summary
High School Transitional Math
Proposal/Recommendation for Action
The purpose of this proposal is to recommend the adoption of three new high school transitional
mathematics courses, Transition to College Algebra, Transition to Quantitative Literacy and Statistics,
and Transition to Technical Math.
The Postsecondary and Workforce Readiness Act establishes a new statewide system for high school
seniors who are lacking a mathematical foundation from their previous education. Transitional math
instruction provides students with the mathematical knowledge and skills to meet their individualized
college and career goals and to be successful in college‐level math courses.
The proposed courses titles and descriptions are as follows:
Transition to College Algebra Grade Level: 12 Prerequisites: Completion of math graduation requirements and at least one of the following criteria:
B or better in Algebra 1 or a higher math course Math GPA of 2.5 or higher
Course Description: The Transition to College Algebra course is for students with career goals that require advanced algebraic skills. Successful completion of the course guarantees student placement into College Algebra or its equivalent at any Illinois community college and select universities. The main emphasis of the course is the understanding of functions (linear, polynomial, rational, radical, and exponential) and how they naturally arise through problem solving and authentic modeling situations. Essential algebraic topics include simplifying expressions, solving equations, and graphing functions, which will be explored deeply, allowing students to address any deficits.
Transition to Quantitative Literacy and Statistics Grade Level: 12 Prerequisites: Completion of math graduation requirements Course Description: The Transition to Quantitative Literacy and Statistics course is intended for students whose career goals do not involve occupations relating to College Algebra or Technical Math, as well as those students who have not yet selected a career goal. Successful completion of this course guarantees student placement into a credit‐bearing general education mathematics course or its equivalent at any Illinois community college and select universities. Essential topics include numeracy, algebra, and functions and modeling. At least one additional topic will be chosen from the following list: systems of equations and
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inequalities, probability and statistics, and proportional reasoning. This course is focused on attaining competency in general statistics, data analysis, quantitative literacy, and problem solving. Transition to Technical Math Grade Level: 12 Prerequisites: Completion of math graduation requirements and concurrent or prior enrollment in technical coursework. Content: The Transition to Technical Math course is for students who have career goals involving occupations in technical fields that do not require advanced algebraic or statistical skills. Successful completion of this course guarantees student placement into a credit‐bearing postsecondary mathematics course required for a community college career and technical education program. The mathematics in this course emphasizes the application of mathematics within career settings.
Rationale
Elgin Community College reports that 62.4% of high school graduates who enroll at their school have to
be placed into remedial mathematics courses, based upon the college’s placement exam. When looking
specifically at students from U‐46, approximantely 65% of our students are placed into a remedial math
course. This percentage has been fairly consistent for the past 12 years.
Year Total # of HS Grads at ECC
% in Remedial
# of U46 Students
Percent in
Remedial
2006 957 71.2% 537 68.1%
2007 927 67.7% 471 64.1%
2008 1,013 65.7% 497 64.9%
2009 1,136 65.9% 619 65.9%
2010 1,309 62.7% 715 62.8%
2011 1,296 62.9% 663 64.8%
2012 1,334 61.1% 728 62.6%
2013 1,296 58.3% 717 66.3%
2014 1,111 55.8% 593 61.5%
2015 1,154 56.2% 598 60.3%
2016 1,124 59.3% 571 65.0%
2017 1,037 62.2% 619 68.4%
Average 1,141 62.4% 611 64.6%
The use of transitional math courses during students’ senior year of high school will reduce the
remediation rates for our students. The transitional courses will also bridge the gap for students who
often opt out of math in their senior year, reducing their chances of needing remedial coursework. Upon
successful completion of a transitional math course in high school, students will receive guaranteed
placement into a credit bearing math course at any Illinois community college.
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The students who will be enrolled in a transitional math course will be those do not demonstrate college
readiness, as measured by multiple means including standardized test scores and course grades. The
transitional math courses are designed to address gaps in understanding by working on bigger problems,
emphasizing problem‐based learning, projects, communication, and integration of concepts, not just
skill acquisition. Contexts will be authentic, and whenever possible, apply to the student’s college or
career path. This approach will be motivating and engaging, and also sets the stage for the types of
problems a student will be exposed to when they reach college.
Description of Procedures
In May 2018, a call to committee was sent out to all teachers to participate in the curriculum process.
All applicants were accepted and include:
Joann Annable ‐ Elgin High School
Greg Anthony – Bartlett High School
Alex Camacho ‐ Elgin High School
Kari Foerster ‐ Elgin High School
William Hice – ROE/Dream Academy
Kenneth Kater ‐ Central School Programs
Liam Keigher ‐ Larkin High School
Jordan Kimbro ‐ Elgin High School
Alma Miho ‐ Bartlett High School
Maureen Toth ‐ Elgin High School
Kristen Wilmot ‐ South Elgin High School
Kameron Matthis – District Math Coach
Amy Ingente ‐ Math Coordinator
Within this group of teachers, all programs were represented, including Special Education, Dual
Language, and the District’s alternative high school programs.
The committee met in June 2018 to develop the curriculum frameworks for the three transitions courses
identified in the Postsecondary and Workforce Readiness Act, aligning the curriculum to the State
recommended competencies and policies.
In June 2018, a sub‐group comprised of 5 teachers from the committee and 2 District administrators
also began meeting with Elgin Community College and the feeder high schools to develop a
memorandum of understanding and discuss portability requirements. These agreements will establish
expectations for each school partner and guarantees placement at any Illinois community college into
the appropriate math course(s) upon successful completion of the course.
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Description of Recommendation
The Postsecondary and Workforce Readiness Act defines benchmarks for projected readiness in college‐
level math. The Transitional Math Recommended Competencies and Policies document states the
following:
A high school junior who has successfully completed state math graduation requirements and
meets at least two of the following criteria is projected to be ready for college level coursework
in mathematics when arriving at a postsecondary institution in Illinois. This determination is
conditional based on enrollment in a senior year of math.
• B or better in Algebra 2 • C or better in a course higher than Algebra 2 • GPA ≥ 3.0 • Standardized Assessment: Math SAT or PSAT ≥ 530 or Math ACT > 22 • Placement test score (such as ALEKS, Accuplacer, Compass, local placement instrument, etc.) into college‐level math at the partner community college after taking their placement exam • PARCC math score of 4 or 5 • Teacher and/or advisor recommendation of college‐level math in the senior year
A high school junior who has successfully completed state math graduation requirements but
has not met at least two of the college‐level math projected readiness criteria will be projected
as NOT ready for college‐level math and will be given transitional math opportunities in relation
to their current math achievement and career interests. A student should consult with a teacher
and/or advisor to determine the appropriate transitional math pathway.
Transition to College Algebra
The Transition to College Algebra course is for students with career goals that require the application of
calculus or advanced algebraic skills. Essential algebraic topics are included in this transition course so
that they can be worked on deeply, allowing students to address any deficits. Students will simplify
expressions, solve equations, and graph functions in the following function families: Linear; Polynomial;
Rational; Radical; Exponential. Successful completion of the course guarantees student placement into
a College Algebra community college mathematics course.
By the end of this course, students will:
1. apply, analyze, and evaluate the characteristics of functions in mathematical and authentic
problem solving situations.
2. simplify expressions, solve equations, and graph functions from the linear, polynomial, rational,
and radical function families in mathematical and authentic problem solving situations.
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3. use their understanding of exponential functions of the form 𝑓 𝑥 𝐶𝑏 , for some constants
𝑏 0 and 𝐶, in mathematical and authentic problem solving situations.
4. create, solve, and reason with systems of equations and inequalities in mathematical and
authentic problem solving situations.
Transition to Quantitative Literacy and Statistics
The Transition to Quantitative Literacy and Statistics course is focused on attaining competency in
general statistics, data analysis, quantitative literacy, and problem solving. This pathway is intended for
students whose career goals do not involve occupations relating to College Algebra or Technical Math,
as well as those students who have not yet selected a career goal. Successful completion of this course
guarantees student placement into a credit‐bearing general education community college mathematics
course, including general education statistics, general education mathematics, quantitative literacy, or
elementary math modeling.
By the end of this course, students will:
1. apply, analyze, and evaluate the characteristics of numbers in authentic modeling and problem
solving situations.
2. Perform operations on numbers and make use of those operations in authentic modeling and
problem solving situations.
3. propose various alternatives, determine reasonableness, and then select optimal estimates to
justify solutions.
4. demonstrate understanding of the characteristics of variables and expressions and apply this
knowledge in authentic modeling and problem solving situations.
5. perform operations on expressions in authentic modeling and problem solving situations.
6. create, solve, and reason with equations and inequalities in the context of authentic modeling
and problem solving situations.
7. apply, analyze and evaluate the characteristics of functions in authentic modeling and problem
solving situations.
8. build and use functions, including linear, nonlinear, and geometric models in authentic modeling
and problem solving situations.
9. evaluate mathematical models and explain the limitations of those models.
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Transition to Technical Math
The Transition to Technical Math course is for students who are enrolled in a Technical course during
their senior year and have career goals involving occupations in technical fields that do not require the
application of calculus, advanced algebraic, or advanced statistical skills. The mathematics in this course
emphasizes the application of mathematics within career settings. Successful completion of this course
guarantees student placement into a credit‐bearing postsecondary mathematics course required for a
community college career and technical education program.
The technical math competencies that follow are considered the core skills and contexts for this
transitional course. However, due to the highly varied career paths that exist in this pathway, it is
recommended to also include additional topics and contexts authentic to the career path.
By the end of this course, students will:
1. use their understanding of operations with real numbers in authentic contexts.
2. perform unit conversions using dimensional analysis and proportions in both the standard and
metric systems and between both systems in authentic contexts.
3. use their understanding of exponents and radicals of real numbers to calculate quantities in
formulas and be able to explain the results.
4. use their understanding of graphs and charts in order to interpret them in contextualized
workplace scenarios.
5. use their understanding of geometry to find and analyze parameters of geometric figures in
authentic contexts.
6. use their understanding of geometry to correctly measure and apply the parts of geometric
figures in authentic contexts.
7. use their understanding of geometry to analyze authentic applications involving right triangles.
8. use algebra to analyze authentic contexts that involve linear equations and inequalities.
9. represent perimeter, volume, and area as a function of a single variable in authentic contexts.
10. apply formulas to solve problems in authentic contexts.
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Recommendations for Implementation
2018‐2019
Continue collaboration with Elgin Community College and surrounding High School Districts to
o Complete the Memorandum of Understanding and Portability Documentation
o Develop common assessments
o Select resources to support the curriculum
January 2019 – Begin registration for the three transitional math courses
Summer 2019 – Provide professional development for teachers about the new curriculum
2019‐2020
Full Implementation
Ongoing professional development
Professional development will be provided for teachers during summer 2019 and on the District
Collaboration Days. These sessions will include a description of the new curriculum, as well as best
practices to support students. The District Math Coach will also make classroom visits and provide
ongoing support as it is needed throughout the school year.
Evaluation
STAR Math will be used to benchmark and progress monitor students’ growth. District Common
Assessment Data will be reviewed to ensure students’ proficiency in the course competencies. Cohort
data will also reviewed to monitor long term performance in subsequent math courses at Elgin
Community College.
Appendix
Curriculum Documents
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Transition to Quantitative Literacy and Statistics ‐ Numeracy & Proportional Reasoning
Stage 1 Desired Results
ESTABLISHED GOALS QL‐N1. Students can apply, analyze, and evaluate the characteristics of numbers in authentic modeling and problem solving situations. QL‐N2. Students can perform operations on numbers and make use of those operations in authentic modeling and problem solving situations. QL‐N3. Students can propose various alternatives, determine reasonableness, and then select optimal estimates to justify solutions.
Transfer
Students will be able to independently use their learning to…
Examine an authentic problem/situation involving numbers to initiate and execute a plan.
Evaluate and explain the reasonableness of a solution.
Meaning
UNDERSTANDINGSStudents will understand that…
Estimation builds logical understanding and is an essential part of problem solving to justify solutions
Fractions, decimals and percentages may represent the same value
Part‐whole relationships exist in real life contexts
Demonstrating measurement sense includes predicting, estimating and solving problems with appropriate units
Organizing numbers helps analyze data and make decisions
ESSENTIAL QUESTIONSStudents will keep considering…
How do we know when we have found a solution?
How do we determine if an answer is reasonable?
What are all the ways to represent a number?
How do we best represent part and whole relationships?
How can we summarize data to make decisions?
What units are most appropriate for this solution?
Acquisition
Students will know…
Operations with whole numbers, integers, fractions, and decimals
The effect of operations on numbers in words and symbols
Part‐whole representations
Place values and numbers written in scientific notation
Mean, median and mode
Various data displays
Estimation strategies
Students will be skilled at…
Basic computations and estimations without a calculator
Justifying estimates
Applying quantitative reasoning
Converting between units
Converting between fractions, decimals and percentages
Interpreting data displays
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
QL‐N1 abf
5.NF 6.NS NQ.1 NQ.2
In authentic modeling and problem solving situations(no calculator – keep numbers reasonable and do mental math)
Convert between words, numbers, and operations (contextual, not algebraic equations)
Explain the effect of operations (eg. tripling, half, squaring, fraction operations, etc.)
Apply mathematical properties in numeric and algebraic contexts (eg. 4+n‐2+5=11+3+5, 4 more than twice her age)
Demonstrate measurement sense that includes o Predicting using appropriate units (eg. in the thousands) o Estimating using appropriate units (eg. 2000) o Solving problems using appropriate units (eg. 2143)
Performance Taskhttps://robertkaplinsky.com/work/stars‐in‐the‐universe/ Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
QL‐N2
7.NS.3 7.RP.3
In authentic modeling and problem solving situations(no calculator – keep numbers reasonable and do mental math)
Use basic arithmetic operations without a calculator to solve problems with
o whole numbers o integers o fractions o decimals
Solve problems involving part‐whole relationships and rates
QL‐N3 4.OA.3
In authentic modeling and problem solving situations
Assess the reasonableness of an answer using mental computation and estimation strategies, including rounding
Justify estimates o Propose various estimates o Select optimal estimates
QL‐N1ef 8.EE.3‐4
In authentic modeling and problem solving situations
Use magnitude in the contexts of o Place values (ten times larger/smaller) o Numbers written in scientific notation o Fractions (with negative exponents written as positive exponents
in the denominator)
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Transition to Quantitative Literacy and Statistics ‐ Algebra
Stage 1 Desired Results
ESTABLISHED GOALS QL‐A1. Students can demonstrate understanding of the characteristics of variables and expressions and apply this knowledge in authentic modeling and problem solving situations. QL‐A2. Students can perform operations on expressions in authentic modeling and problem solving situations. QL‐A3. Students can create, solve, and reason with equations and inequalities in the context of authentic modeling and problem solving situations.
Transfer
Students will be able to independently use their learning to…
Examine an authentic problem/situation and use algebraic reasoning to initiate and execute a plan.
Evaluate and explain the reasonableness of a solution.
Meaning
UNDERSTANDINGSStudents will understand that…
Variables and expressions represent quantities in real world situations
There is a relationship between zeros and factors in real world situations
Equations and inequalities describe relationships in real world situations
Solving equations and inequalities builds logical understanding and is an essential part of problem solving to justify solutions in real world situations
Changing variable values has an effect on an algebraic relationship in real world situations
ESSENTIAL QUESTIONSStudents will keep considering…
What is an appropriate method to solve a problem?
How can variables be used to represent quantities in real world situations?
How can I translate an authentic context to an algebraic expression?
What are the differences between expressions, equations, and inequalities? When are they used in in real world situations?
Why is it helpful to rewrite expressions in equivalent forms?
Acquisition
Students will know…
Variables represent real world quantities or attributes
Parts of expressions, including terms, factors and coefficients
Zeros and factors of polynomials
Expressions, equations, and inequalities
Students will be skilled at…
Developing and solving expressions, equations and inequalities
Comparing and contrasting equations and expressions
Justifying reasoning while solving equations
Performing operations on polynomials
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Creating equivalent forms of expressions
Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
QL‐A1 A.CED.1 A.CED.3 N.Q.2 6.EE.2 A.SSE.1 A.SSE.2 A.SSE.3
In authentic modeling and problem solving situations
Use variables to accurately represent quantities or attributes
Predict and confirm the effect that changes in variable values have in an algebraic relationship
Interpret parts of expressions o Terms o Factors o Coefficients
Write expressions and/or rewrite expressions in equivalent forms to solve problems
Performance Task:Defined STEM: Computer Manufacturer Defined STEM: Food Truck Entrepreneur Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
QL‐A2 A.APR.1 A.APR.2
In authentic modeling and problem solving situations
Perform arithmetic operations (addition, subtraction, multiplication) on polynomials in authentic tasks
Demonstrate the relationship between zeros and factors of polynomials
QL‐A3 A.CED.1 F.IF.9 A.REI.1
In authentic modeling and problem solving situations
Create equations and inequalities that describe numbers or relationships
Compare and contrast expressions and equations
Use and justify reasoning while solving equations
Develop and solve equations and inequalities in one variable
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Transition to Quantitative Literacy and Statistics ‐ Functions
Stage 1 Desired Results
ESTABLISHED GOALS QL‐FM1. Students can apply, analyze and evaluate the characteristics of functions in authentic modeling and problem solving situations. QL‐FM2. Students can build and use functions including linear, nonlinear, and geometric models in authentic modeling and problem solving situations. QL‐FM3. Students can evaluate mathematical models and explain the limitations of those models.
Transfer
Students will be able to independently use their learning to…
Apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems.
Evaluate and explain the appropriateness of a model.
Meaning
UNDERSTANDINGSStudents will understand that…
Authentic problem/situation can be modeled in multiple ways (descriptions, tables, graphs, and equations)
A function models a relationship between quantities in real world situations
Changing variable values has an effect on an algebraic relationship in real world situations
ESSENTIAL QUESTIONSStudents will keep considering…
What model best represents this situation?
How do I know if this is the best representation?
How do I know that I have chosen an appropriate function?
How do I know if I need more than one function? How do I know when I need to combine functions?
What are the limitations of the chosen models?
Acquisition
Students will know…
Linear, polynomial, exponential, and logarithmic functions
Functions have a single output
Functions can be built from existing functions
Different representations of functions (descriptions, tables, graphs, and equations)
Characteristics of functions
Relationship between variables
Variables represent real world quantities or attributes
Students will be skilled at…
Graphical analysis
Modeling with geometry
Modeling with linear and non‐linear functions
Compare models
Interpreting functions
Identifying the reasonableness of a model
Describing relationships
Transferring between different representations of functions
Using models to solve problems
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
QL‐FM1
N.Q.2 A.CED.1 F.BF.1 F.IF.2 F.IF.8 F.IF.1 F.IF.4 F.IF.7 F.IF.9
In authentic modeling and problem solving situations, which may include linear, polynomial, exponential, logarithmic functions
Use variables to represent quantities or attributes
Predict and confirm the effect that changes in variable values have in an algebraic relationship
Identify if a relation is a function
Identify what the parts of the function represent in a given situation
Identify and analyze functions using different representations (descriptions, tables, graphs, and equations)
o Maximum / Minimum o Increasing / Decreasing o Intercepts o Domain / Range o End behavior (if appropriate)
Performance Task:Defined STEM: Automobile Efficiency Expert Defined STEM: Earth Scientist Defined STEM: Wind Energy Specialist Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
QL‐FM2
A.CED.1 A.CED.2 F.BF.1 F.LE.1 F.LE.2 A.SSE.1 G.MG
In authentic modeling and problem solving situations
Represent common types of functions using words, algebraic symbols, graphs, and tables. Types of functions include:
o Linear o Polynomial o Exponential o Logarithmic
Translate a context into a mathematical representation
Translate a mathematical representation into a context
For linear, polynomial, exponential and logarithmic functions o Build a function that models a relationship between two quantities o Build new functions from existing functions o Compare two functions o Use created functions to solve problems
Apply geometric concepts in modeling situations (eg. volume, perimeter, similarity, Pythagorean Theorem, etc.)
QL‐FM3
S.ID.6 In authentic modeling and problem solving situations
Identify the reasonableness of a linear model for given data and consider alternative models
Explain limitations of models in real‐world scenarios or relationships
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Transition to Quantitative Literacy ‐ Probability and Statistics
Stage 1 Desired Results
Established Goals: Interpreting categorical and quantitative data
Summarize, represent and interpret data on a single count or measurement variable
Summarize, represent and interpret data on two categorical and quantitative variables
Interpret linear models Make inference san justify conclusions from data Conditional probability and the rules of probability
Understand independence and conditional probability and use them to interpret data
Use the rules of probability to compute probabilities of compound events
Using probability to make decisions
Calculate expected values and use them to solve problems
Use probability to evaluate outcomes of decisions
Transfer
Students will be able to independently use their learning to…
Make predictions and decisions on real world events based on sampling, statistics and probability
Meaning
UNDERSTANDINGS Students will understand that…
Data can be represented in multiple ways
Data can be organized and analyzed to solve problems and make decisions
Data can be interpreted many ways
Probability can help predict outcomes
Probability can be analyzed and used to make fair decisions in real‐world situations
Statistics and probability are used in everyday life
ESSENTIAL QUESTIONSStudents will keep considering…
How can I tell if a representation or interpretation of data is misleading?
What is the best way to represent a set of data?
What is the relationship between data sets? Is there a correlation?
What predictions can be made based on the patterns I see/knowledge of past events in the data set?
Acquisition
Students will know…
Scatterplot, two‐way frequency table, histogram, boxplot, circle graph, dot plot, pictograph,
Measures of central tendency
Shape, center, spread
Correlation coefficient
Causation
Sample space
Independent and conditional probability
Probability rules
Students will be skilled at…
Using technology to represent and analyze data
Representing data with multiple models
Analyzing, comparing and interpreting data
Fitting a function to data
Recognizing possible associations and trends
Use sample data to make inferences about a population
Evaluate reports based on data
Describing sample space
Determining independent and conditional probabilities
Using probability to analyze and make fair decisions
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
S.ID.2 S.ID.2 S.ID.5 S.ID.6 S.ID.8
In authentic modeling and problem solving situations
Represent and analyze data to solve problems, make decisions, and
predictions using:
o Scatterplot
o Two‐way frequency table
o Histogram
o Boxplot
o Circle graph
o Dot plot
Analyze data using the following statistical concepts
o Measures of central tendency
o Shape, center, spread
o Correlation coefficient
Performance Task: Defined STEM: Recycling Wins, Defined STEM: Sabermetrician: Baseball Stats Defined STEM: Tunnel Team Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
S.CP.1 S.CP.2 S.CP.3 S.CP.5 S.CP.6 S.CP.7 S.CP.8 S.MD.6 S.MD.7
Use probability to predict outcomes and make fair decisions in real‐
world situations
o Sample space
o Independent vs. dependent vs. mutually exclusive events
o Independent probability
o Conditional probability
o Probability rules (and/or)
Performance Task: Defined STEM: Carnival Game Designer Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
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Transition to College Algebra ‐ Linear Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A2. Students can simplify expressions, solve equations, and graph functions from the linear, polynomial, rational, and radical function families in mathematical and authentic problem solving situations. CA‐A4. Students can create, solve, and reason with systems of equations and inequalities in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to…
Use appropriate linear mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
A linear function can be used to model real world situations
There is a relationship between graphs and equations
The equation of a line can be represented in multiple ways
The solution to an inequality can be represented in multiple ways
The solution to a system can be found using various methods
Systems of equations and inequalities can be used to model real world situations
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know if I found all the solutions?
Which form of the equation is the best representation for this situation?
What is the relationship between parallel and perpendicular lines?
Which method is best to solve this system of equations?
Acquisition
Students will know…
The equation of a line in various forms
The relationship between a function and its graph
Linear equations and inequalities
Systems of equations
Domain and range
Students will be skilled at…
Rewriting and analyzing equations to reveal key information
Create linear equations and inequalities to model authentic real world situations
Solving equations and inequalities
Solving systems graphically and algebraically
Graphing linear equations and
inequalities
Justifying and explain their mathematical thinking
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Task: Defined STEM: Dream Job Defined STEM: Hydroelectric Engineer: The Xayaburi Dam Project And The People Of Laos Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A2 F.IF.4 A.SSE.4 8.EE.5 8.EE.6 F.IF.7 A.CED.2 8.EE.8 A.REI.3 G.GPE.5
In mathematical and authentic problem solving situations
Identify dependent and independent variables in linear relationships to model authentic situations
Explain the relationship between lines and their equations
Graph a line using slope‐intercept form of the linear equation
Determine the equation of a line from its graph
Determine the equation of a line from point‐slope formula
Use graphs of lines to identify solutions to linear equations
Solve linear inequalities o Writing solution sets in interval notation o Graphing solution sets on number lines o Interpreting solutions in context
Use and explain slope relationships for parallel and perpendicular lines
CA‐A4 A.REI.6 A.CED.2 A.CED.4 A.REI.11 A.CED.3 A.REI.2
In mathematical and authentic problem solving situations
Solve and create models for systems of linear equations using both graphical and algebraic methods (limit to two equations and two variables, without matrices)
Use linear inequalities and systems of linear inequalities in two unknowns to create models
Graphically identify solutions sets to linear inequalities or systems of inequalities
Performance Task:Basketball Math Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
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Transition to College Algebra ‐ Exponential Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A3. Students can use their understanding of exponential functions of the form f(x) = C bx, for some constants b > 0 and C, in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to… Use appropriate exponential mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
An exponential function can be used to model real world situations
There is a relationship between graphs and equations
Exponential functions behave differently from other functions
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know my solution is reasonable and complete?
How do the properties of this function differ from the properties of another?
Acquisition
Students will know…
Exponential models
Graphs of exponential functions
Domain and range
Students will be skilled at…
Creating exponential models
Solving exponential models
Justifying the function they used to model an authentic real world situation
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Task Defined STEM: Exponential Decay Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A3 F.LE.1 F.IF.7 F.BF.4 F.BF.5 F.LE.4
In mathematical and authentic problem solving situations
Create and solve simple models involving exponential equations
Distinguish exponential rate of change from other rates of change
Graph and recognize the graph of exponential functions of the form f(x) = a∙bx
Solve exponential equations numerically and algebraically
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Transition to College Algebra ‐ Quadratic Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A2. Students can simplify expressions, solve equations, and graph functions from the linear, polynomial, rational, and radical function families in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to… Use appropriate quadratic mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
A quadratic function can be used to model real world situations
There is a relationship between graphs and equations
Quadratic functions behave differently from other functions
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know my solution is reasonable and complete?
How do the properties of this function differ from the properties of another?
Acquisition
Students will know…
Quadratic models
Graphs of quadratic functions
The relationship between zeros and factors
Domain and range
Students will be skilled at…
Creating quadratic functions
Solving quadratic equations and inequalities
Graphing quadratic functions (by hand and with technology)
Rewriting quadratic functions to reveal key features
Justifying the function they used to model an authentic real world situation
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Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Tasks: Defined STEM: Rocket Golf Course Designer Defined STEM: Architect Using Ancient Design Symbols Defined STEM: Grain Farmer Defined STEM: Green Construction Defined STEM: Regional Transportation & Railroad Director Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A2 A.CED.1 A.SSE.2 A.REI.4 F.IF.7 F.IF.8
In mathematical and authentic problem solving situations.
Create and solve models involving quadratic equations.
Apply standard factoring techniques to quadratics, including identifying non‐factorable expressions
Solve quadratic equations by o Factoring o Completing the square o Quadratic Formula
Graph quadratic functions
Determine the quadratic function from a graph
22
Transition to College Algebra ‐ Polynomial Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A2. Students can simplify expressions, solve equations, and graph functions from the linear, polynomial, rational, and radical function families in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to… Use appropriate polynomial mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
A polynomial function can be used to model real world situations
There is a relationship between graphs and equations
Polynomial functions behave differently from other functions
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know my solution is reasonable and complete?
How do the properties of this function differ from the properties of another?
Acquisition
Students will know…
Polynomial models
Graphs of polynomial functions
The relationship between zeros and factors
Domain and range
Students will be skilled at…
Creating polynomial functions
Graphing polynomial functions
Solving polynomial equations and inequalities
Rewriting polynomial functions to reveal key features
Justifying the function they used to model an authentic real world situation
23
Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Tasks: Defined STEM: Slow Those Trucks Down! Defined STEM: Automotive Finance Manager Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A2 A.CED.1 A.SSE.2 A.APR.2 A.APR.3 A.SSE.3 A.APR.6
In mathematical and authentic problem solving situations
Create and solve models involving polynomial equations.
Apply standard factoring techniques to polynomials, including identifying non‐factorable expressions
Understand the relationship between zeros and factors of a polynomial of degree 2 and higher
Solve polynomial equations and inequalities of degree 2 and higher
Analyze graphs of polynomial functions using technology o Zeros o y‐intercept o Turning points (i.e., relative maximum/minimum) o Domain and range
24
Transition to College Algebra ‐ Radical Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A2. Students can simplify expressions, solve equations, and graph functions from the linear, polynomial, rational, and radical function families in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to… Use appropriate radical mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
A radical function can be used to model real world situations
There is a relationship between graphs and equations
Radical functions behave differently from other functions
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know my solution is reasonable and complete?
How do the properties of this function differ from the properties of another?
Acquisition
Students will know…
Radical equations
Rules of exponents
Graphs of radical functions
Domain and range
Students will be skilled at…
Converting between radical and rational exponents
Simplifying radical and rational exponents
Solving radical equations
Graphing radical equations
25
Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Task: Captain Red Ickle’s Booty Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A2 A.REI.2 F.IF.7 N.RN.2 N.RN.1 A.SSE.3
In mathematical and authentic problem solving situations
Create and solve models involving radical equations
Convert between radical and rational exponent notation
Simplify expressions involving radical and rational exponents using appropriate exponent rules
Solve equations involving radical expressions
Analyze graphs of radical functions using technology o Zeros o y‐intercept o Domain and range
26
Transition to College Algebra ‐ Rational Functions
Stage 1 Desired Results
ESTABLISHED GOALS CA‐A1. Students can apply, analyze, and evaluate the characteristics of functions in mathematical and authentic problem solving situations CA‐A2. Students can simplify expressions, solve equations, and graph functions from the linear, polynomial, rational, and radical function families in mathematical and authentic problem solving situations.
Transfer
Students will be able to independently use their learning to… Use appropriate rational mathematical models to solve real life applications
Meaning
UNDERSTANDINGS Students will understand that…
A rational function can be used to model real world situations
There is a relationship between graphs and equations
Rational functions behave differently from other functions
ESSENTIAL QUESTIONS
What function models this situation and what are the key features?
How do I know my solution is reasonable and complete?
How do the properties of this function differ from the properties of another?
Acquisition
Students will know…
Rational equations
Graphs of rational functions
Domain and range
Students will be skilled at…
Simplifying rational expressions
Solving rational equations and inequalities
Create rational functions to model an authentic situation
Graphing rational functions
27
Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
CA‐A1 F.IF.1 F.IF.2 F.IF.4 A.SSE.1 F.BF.1 F.IF.5 F.BF.3 F.IF.9
For functions in mathematical and authentic problem solving situations
Explain the concept of a function and use function notation
Interpret the dependent and independent variables in the context of functions
Create and interpret expressions for functions, including selecting appropriate domains for these functions
Understand the relationship between a function and its graph
Find the domain and range of a function
Analyze functions using different representations (graphic, numeric, algebraic, verbal)
Performance Task: Defined STEM: Dream Job Salary Other Evidence
Quizzes and Tests
Short answer to prompts
Observations and dialogues
Formative checks for understanding
CA‐A2 A.REI.2 A.APR.7
In mathematical and authentic problem solving situations
Create and solve models involving rational equations
Simplify rational expressions
Solve rational equations
Solve rational inequalities algebraically
Analyze graphs of rational functions using technology o Zeros o y‐intercept o Asymptotes o Domain
28
Transition to Technical Math ‐ Number Systems & Basic Algebra
Stage 1 Desired Results
ESTABLISHED GOALS TM‐NS1. Students can use their understanding of operations with real numbers in authentic contexts. TM‐NS2. Students can perform unit conversions using dimensional analysis and proportions in both the standard and metric systems and between both systems in authentic contexts. TM‐NS3. Students can use their understanding of exponents and radicals of real numbers in order to calculate quantities in formulas and be able to explain the results. TM‐NS4. Students can use their understanding of graphs and charts in order to interpret them in contextualized workplace scenarios. TM‐BA1. Students can use algebra to analyze authentic contexts that involve linear equations and inequalities. TM‐BA3. Students can apply formulas to solve problems in authentic contexts.
Transfer
Students will be able to independently use their learning to…
Work with measurement, estimation, formulas to solve authentic applications in their field of study.
Engage in using mathematical models to solve real world problems through effective and accurate use of mathematical notations, vocabulary, and reasoning in their field of study.
Meaning
UNDERSTANDINGSStudents will understand that…
Computing unit rates are associated with ratios and proportional relationships
Mental computations, estimation and rounding strategies will help with assessing reasonableness of answers and efficiency in career fields
Using visual models/artifacts will help draw and justify conclusions in career fields
Non‐perfect square and cube roots are irrational
Linear equations, linear inequalities, and formulas can be used to solve problems in their field of study
ESSENTIAL QUESTIONS
How do I know my answer is reasonable for the context of the problem?
How do I know I am using appropriate units?
How do I know I am reading and interpreting a model correctly?
When is it appropriate to use estimation strategies? When do I need to have a more precise answer?
How do I know I chose an appropriate method/model for this situation?
How do linear equations and inequalities represent this situation?
What formula(s) can be used in this situation?
Acquisition
Students will know…
Proportions
Graph and chart displays
Square roots and cube roots are inverse operations of numbers that are squared or cubed respectively
Estimating strategies
Unit conversions with proportions
Equation/inequality/formulas
Radicals and integer exponents
Fractions, numbers, and percentages
Students will be skilled at…
Analyzing proportional relationships
Analyzing graphs and charts
Justifying decisions based on data
Converting between measurements
Computing basic computations mentally using the properties of real numbers
Applying properties of operations to linear expressions
Solving equations, inequalities, and formulas
Using estimating strategies
Converting units of measure
Assessing the reasonableness of their solution
How to decide what formulas are appropriate
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Stage 2 ‐ Evidence PWR CCSS Proficiency Criteria Assessment Evidence
TM‐NS1 7.RP.2 7.RP.3 6.RP.2 6.RP.3 7.EE.3 8.NS.2
In authentic contexts
Use proportional relationships to solve problems
Compute unit rates in like or different units of o Fractions o Decimals o Percents o Ratios of lengths o Ratios of areas
Apply properties of operations to calculate values using numbers in any form
Convert numbers between forms as appropriate
Assess the reasonableness of answers using mental computation, estimation and rounding strategies
Use rational approximations of irrational numbers to compare the size of irrational numbers and estimate the value of expressions (e.g., π/2 is slightly larger than 1.5)
Performance Task:Defined Stem: Fuel Efficient Cars Defined Stem: Marketing Facial Products Defined Stem: School Store Defined STEM: Food Truck Entrepreneur Defined STEM: Nutritionist Defined STEM: Baker Defined STEM: Catering Company Other Evidence • Quizzes and Tests • Short answer to prompts • Observations and dialogues • Formative checks for understanding
TM‐NS2 S.MD.1 6.RP.3
In authentic contexts
Convert like measurement units within a given measurement system (e.g., inches to feet) in solving authentic multistep problems
Convert like measurement units between systems (e.g., inches to centimeters) in solving authentic multistep problems
Use ratio reasoning (dimensional analysis) to convert measurement units including, but not limited to, distances and rates
Manipulate and transform units appropriately when multiplying or dividing quantities
TM‐NS3 6.EE.2 A.APR.1 8.EE.2 8.NS.1
For formulas in authentic problems, including exponential and radical formulas
Evaluate expressions at specific values for their variables
Perform arithmetic operations, including those involving whole‐number exponents, using order of operations
Work with radicals and integer exponents, applying rules of exponents
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number
Evaluate square roots of small perfect squares and cube roots of small perfect cubes
Know that square roots and cubed roots of non‐perfect squares and cubes are irrational and understand what irrational numbers are
TM‐NS4 S.ID.1 In contextualized workplace scenarios
30
S.ID.2 7.SP.2 5.OA.3 F.LE.1 S.IC.B
Draw and justify conclusions from graphics such as o order forms o bar charts o pie charts o diagrams o flow charts o maps o dashboards
From graphs and charts, identify and interpret o Trends o Patterns o Relationships
Identify types of graphs that best represent a given set of data, including o Bar graphs
o Pie graphs
o Line graphs
Make and justify decisions based on data.
TM‐BA1 6.EE.3 7.EE.1 A.REI.3 A.CED.1 A.CED.2
In authentic contexts that involve linear equations and inequalities
Add, subtract, factor, and expand linear expressions with rational coefficients
Solve linear equations and inequalities in one variable
Use linear equations to model authentic contexts
TM‐BA3 6.EE.1 6.EE.2 N.Q.1 N.Q.2 A.CED.4
In authentic contexts:
Evaluate expressions at specific values for their variables
Analyze and use units to solve multistep problems and justify the solution
Choose and interpret units in formulas
Apply appropriate formulas
31
Transition to Technical Math ‐ Geometry
Stage 1 Desired Results
ESTABLISHED GOALS TM‐G1. Students can use their understanding of geometry to find and analyze parameters of geometric figures in authentic contexts. TM‐G2. Students can use their understanding of geometry to correctly measure and apply the parts of geometric figures in authentic contexts. TM‐G3. Students can use their understanding of geometry to analyze authentic applications involving right triangles. TM‐BA2. Represent perimeter, volume, and area as a function of a single variable in authentic contexts. TM‐BA3. Students can apply formulas to solve problems in authentic contexts.
Transfer
Students will be able to independently use their learning to…
Apply geometric concepts to their field of study
Understand spatial relationships in real world situations
Meaning
UNDERSTANDINGS Students will understand that…
Figures can be measured by perimeter, area, and volume
Angle relationships, Pythagorean theorem, and ratios can be used to find unknown measurements in their field of study
Similar figures can be used to create scale models
Coordinate grids can be used to represent values for a given situation
Changing the value of one quantity effects the value of another
ESSENTIAL QUESTIONS
What geometric concepts apply to this real world situation? What information do I need to know to solve this problem?
Why are scale models helpful in real world situations?
How do I know my solution is appropriate for the situation?
How precise does my solution need to be?
Acquisition
Students will know…
Area, perimeter and volume formulas
Units/scales of measure
Scale drawings
Coordinate grid
Pythagorean Theorem
Right triangle ratios (sine, cosine, tangent, and inverses)
Supplementary, complementary, vertical, adjacent, corresponding, alternate interior, alternate exterior angles
perimeter, circumference, diagonals, diameter
Students will be skilled at…
Evaluate expressions, that arise from formulas in real world situations
Accurately measuring parts of geometric figures
Creating geometric models
Creating equations from real world situations
Graphing coordinates/figures
Converting units of measure
Use the relationship between two quantities to solve problems in real world situations
32
Stage 2 ‐ Evidence
PWR CCSS Proficiency Criteria Assessment Evidence
TM‐G1 6.G1‐4 7.G.4 7.G.6 8.G.9
In authentic contexts:Use perimeter, area, and volume formulas to calculate measurements of geometric figures
Performance Task:Defined STEM: Design Architect: Building a Fun Zone Defined STEM: Astronomer Design a Telescope Defined STEM: Crime Scene Investigation Other Evidence
Quizzes and Tests Short answer to prompts Observations and dialogues Formative checks for understanding
TM‐G2 7.G.5 7.G.1 6.NS.8 2.MD.1 3.MD.3 3.MD.8 4.MD.6 6.NS.8 5.G.2
For geometric figures in authentic contexts
Use facts about angles to solve for an unknown angle o Supplementary o Complementary o Vertical o Adjacent o Corresponding o Alternate interior o Alternate exterior
Accurately measure parts of geometric figures using the correct measurement tool o Sides o Perimeter/Circumference o Diagonals o Diameter o Angles
Solve problems involving scale drawings of geometric figures including o Computing lengths o Computing areas o Reproducing a scale drawing at a different scale
Graph and interpret points in the coordinate plane
TM‐G3 8.G.7 G.SRT.8
In authentic applications involving right triangles
Use the Pythagorean Theorem to solve for a side of right triangles
Use trig ratios and their inverses to solve for unknown sides and angles in right triangles
TM‐BA2 G.GMD.3 6.EE.9 A.CED.4
In authentic contexts involving perimeter, volume and area
Use variables to represent two quantities involving geometric figures that change in relationship to one another
Write an equation to express one quantity in terms of the other quantity (independent and dependent variable)
Rearrange formulas to highlight a quantity of interest
TM‐BA3 6.EE.1 6.EE.2
In authentic contexts:
Evaluate expressions at specific values for their variables
33
N.Q.1 N.Q.2 A.CED.4
Analyze and use units to solve multistep problems and justify the solution
Choose and interpret units in formulas
Apply appropriate formulas