Foundations of Algebra I Unit 3 Equations
BY THE END OF THIS UNIT:
CORE CONTENT
Cluster Title: Interpret the Structure of ExpressionsStandard A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.
Concepts and Skills to Master
· Given an expression, identify the terms, bases, exponents, coefficients, and factors.
· Determine the real world context of the variables in an expression.
· Identify the individual factors of a given term within an expression.
· Explain the context of different parts of a formula.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Understand the meaning of symbols indicating mathematical operations, implied operations (e.g. 2x), the meaning of exponents, and grouping symbols.
Academic Vocabulary
Exponents, factors, terms, bases, coefficients, expression
Suggested Instructional Strategies / Resources
· Given a word problem and a formula have students examine the structure and explain the context of different parts of the formula.
· Design a game around identifying terms, bases, exponents, coefficients, and factors.
· Create formulas based on context / · Textbook Correlation: 1-1, 1-2, 1-7, 3-7, 4-5, 5-3, 5-4, 7-7, 8-5, 8-6, 8-7, 8-8, 9-5, CC-2, CC-10
· MARS Concept Development Lesson:
Sorting Equations and Identities
Sample Formative Assessment Tasks
Skill-based task
Consider the formula Surface Area = 2B + Ph
a. What are the terms of this formula?
b. What are the coefficients? / Problem Task
Interpret the expression: 5 – 3(x – y)2. Explain the output values possible.
CORE CONTENT
Cluster Title: Create equations that describe numbers or relationshipsStandard A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Concepts and Skills to Master
· Create one-variable linear equations and inequalities from contextual situations (stories).
· Create one-variable exponential equations and inequalities from contextual situations (stories).
· Solve and interpret the solution to multi-step linear equations and inequalities in context.
· Use properties of exponents to solve and interpret the solution to exponential equations and inequalities in context.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Understand and use inverse operations to isolate variables and solve equations.
· Efficiently use order of operations
· Understand notation for inequalities
· Understand and use properties of exponents
Academic Vocabulary
Greater than, less than, at most, at least, =, < ,>, ≤ ,≥ , no more than, no less than
Suggested Instructional Strategies
· Convert contextual information into mathematical notation.
· Use story contexts to create linear and exponential equations and inequalities / Resources
· Textbook Correlation: 1-8, 2-1, 2-2, 2-3, 2-4, 2-5, 2-7, 2-8, 3-2, 3-3, 3-4, 3-6, 3-7, 9-3, 9-4, 9-5, 9-6, 11-5
· MARS Apprentice Tasks:
Functions
Multiplying Cells
Printing Tickets
· MARS Expert Tasks:
Fearless Frames
Skeleton Tower
Best Buy Tickets
Sample Formative Assessment Tasks
Skill-based task
Juan pays $52.35 a month for his cable bill and an additional $1.99 for each streamed movie. If his last cable bill was $68.27, how many movies did Juan watch? / Problem Task
Juan pays $52.35 a month for his cable bill and an additional $1.99 for each streamed movie. Gail pays $40.32 a month for her cable bill and an additional $2.59 for each streamed movie. Who has the better deal? Justify your choice.
CORE CONTENT
Cluster Title: Create equations that describe numbers or relationshipsStandard A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axe with labels and scales.
Concepts and Skills to Master
· Write and graph an equation to represent a linear relationship
· Write and graph an equation to represent an exponential relationship
· Model a data set using an equation
· Choose the best form of an equation to model linear and exponential functions
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Graph points
· Choose appropriate scales and label a graph
· Understand slope as a rate of change of one quantity in relation to another quantity
Academic Vocabulary
Variable, dependent variable, independent variable, domain, range, scale
Suggested Instructional Strategies
· Use story contexts to create linear and exponential graphs.
· Use technology to explore a variety of linear and exponential graphs.
· Use data sets to generate linear and exponential graphs and equations / Resources
· Textbook Correlation: 1-9, 4-5, 5-2, 5-3, 5-4, 5-5, 7-6, 7-7, 9-1, 9-2, 10-5, 11-6, 11-7, CB 11-7
· MARS Apprentice Tasks:
Functions
Multiplying Cells
Printing Tickets
· MARS Expert Tasks:
Fearless Frames
Skeleton Tower
Best Buy Tickets
Sample Formative Assessment Tasks
Skill-based task
Write and graph an equation that models the cost of buying and running an air conditioner with a purchase price of $250 which costs $0.38/hr to run. / Problem Task
Jeanette can invest $2000 at 3% interest compounded annually or she can invest $1500 at 3.2% interest compounded annually. Which is the better investment and why?
CORE CONTENT
Cluster Title: Create equations that describe numbers or relationshipsStandard A.CED.4: Rearrange formulas to highlight a quantity of interest
Concepts and Skills to Master
· Extend the concepts used in solving numerical equations to rearranging multi-variable formulas or literal equations to solve for a specific variable.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Recognize variables as representing quantities in context
· Solve multi-step equations
Academic Vocabulary
Constant, variable, formula, literal equation
Suggested Instructional Strategies
· Use formulas for a variety of disciplines such as physics, chemistry, or sports to explore the advantages of different formats of the same formula. / Resources
· Textbook Correlation: 2-5
Sample Formative Assessment Tasks
Skill-based task
I = Prt Solve for r. / Problem Task
Paul just arrived in England and heard the temperature in degrees Celsius. He remembers that . How will Paul find the temperature in Fahrenheit?
CORE CONTENT
Cluster Title: Understand solving equations as a process of reasoning and explain the reasoningStandard A.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Concepts and Skills to Master
· Understand, apply, and explain the results of using inverse operations
· Justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity. Justifications may include the associative, commutative, and division properties, combining like terms, multiplication by 1, etc.
· Use the names of the properties and common sense explanations to explain the steps in solving an equation
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Use order of operations
· Simplify expressions using properties of algebra
Academic Vocabulary
Constant, coefficient, properties of operations and properties of equality, like terms, variable, evaluate, justify, viable
Suggested Instructional Strategies
· Have students share different ways of solving equations that lead to the same answer.
· Find and analyze mistakes in student work samples
· Partner problems: one student solves, the other writes reasons why steps work.
· Introduce a two-column proof as a way of organizing justifications / Resources
· Textbook Correlation: 2-2, 2-3, 2-4, 2-5, 9-4, 9-5
· Introduction lesson on using manipulatives to solve equations
· Lesson on solving equations with variables on both sides with manipulatives
· Solving Linear Equations in One Variable Formative Assessment Lesson
· Steps for Solving Equations Formative Assessment lesson
· Links to more equation activities
Sample Formative Assessment Tasks
Skill-based task
Justify the equation solution by writing the property or reason why each step works.
3x + 7 = 12
3x + 7 – 7 = 12 – 7
3x + 0 = 5
3x = 5
(3x)(1/3) = (5)(1/3)
1x = 5/3
x = 5/3
/ Problem Task
When Sally picks any number between 1 and 20, doubles it, adds 6, divides by 2 and subtracts 3, she always gets the number she started with. Why? Evaluate and use algebraic evidence to support your conclusion.
CORE CONTENT
Cluster Title: Reason quantitatively and use units to solve problemsStandard N.Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Concepts and Skills to Master
· Select and use appropriate units of measurement for problems with and without context
· Given a graph draw conclusions and make inferences
· Choose appropriate scales to create linear and exponential graphs
· Determine from the labels on a graph what the units of the rate of change are (e.g. gallons per hour)
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Know how various attributes are reasonably measured
Academic Vocabulary
Scale, units of measurement
Suggested Instructional Strategies / Resources
· Explore a variety of examples of measurements used in graphs
· Construct graphs using a variety of data sets / · Textbook Correlation: 2-5, CB 2-5, 2-6, 2-7, 4-4, 5-7, 12-2, 12-4
Sample Formative Assessment Tasks
Skill-based task
What is the area of strip of wall that is 48 inches by 10 yards? / Problem Task
Your college savings fund has $1800 in it and you plan to spend $30 a week. What would be an appropriate viewing window and scale to see the remaining balance each week until the money is gone? Explain.
CORE CONTENT
Cluster Title: Reason quantitatively and use units to solve problemsStandard N.Q.2: Define appropriate quantities for the purpose of descriptive modeling
Concepts and Skills to Master
· Choose appropriate measures and units for problem situations
· Create a relationship among different units (i.e. feet per second, bacteria per hour, miles per gallon)
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Compute unit rates associated with ratios of fractions
· Recognize and calculate basic conversions (e.g. 3 feet = 1 yard)
Academic Vocabulary
Units rates, modeling, quantity, unit conversion, proportion, ratio
Suggested Instructional Strategies / Resources
· Integrate this objective into problem solving throughout the curriculum.
· Place an emphasis on relationships between two different units (e.g. dollars per hour, pressure over altitude, calories per gram) / · Textbook Correlation: 2-6, 3-3, 4-5, 5-2, 5-5, 12-3, CC-7
Sample Formative Assessment Tasks
Skill-based task
How would you measure the rate at which a bathtub fills? Justify your answer. / Problem Task
(re-)Create a scenario you have encountered involving two changing quantities and determine appropriate units to describe the relationship between the quantities.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.