Algebra 1 Unit Plan (Tier 1& 2)

Algebra 1 Unit 1 – Tier 1September 8th – October 2nd

Tier 1 Unit 1: Quantitative Relationships, Graphs, and Functions September 8th – October 5th

Contents

Unit Overview

Calendar

Assessment Framework

Scope and Sequence

Lesson Analysis

Ideal Math Block

Sample Lesson Plan

Supplemental Material

Multiple Representations

Unit Authentic Assessment

PARCC Sample Assessment Items

Unit Assessment Question Bank

Additional Resources

Appendix A – Acronyms

Curriculum Map

Unit Overview

Unit 1: Quantitative Relationships, Graphs, and Functions
Essential Questions
In what ways can we manipulate an algebraic equation to find the value of an unknown quantity?
How do variables help you model real-world situations and solve equations?
How can you determine if something is a mathematical function?
How can we use mathematical models to describe physical relationships?
How can we use different tools and representations to solve problems?
How can the same linear relationship be represented in multiple ways?
Enduring Understandings
By applying mathematical properties, a linear equation can be manipulated to produce many different but equivalent forms. These manipulations can lead to solution for the unknown value.
Units can be used to describe and explain steps and solutions of problems that model a real world scenario.
Functions can be categorized into function families, each with their own algebraic and graphical characteristics.
There are often two quantities that change in problem situations. One of these quantities depends on the other, making it the dependent quantity and the other the independent quantity
A mathematical function is a relation between a set of inputs (values of the domain) and outputs (values of the range) in which one element of the domain is assigned to exactly one element of the range.
A linear relationship is one where the dependent quantity is changing at a constant rate per unit of the independent quantity.
A Linear function can be represented in multiple ways and can be used to model and solve problems in a real world context.
(NJSLS/CCSS)
1)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.
2)A.REI.3:Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
3)F.IF.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Iffis a function andxis an element of its domain, thenf(x) denotes the output off corresponding to the inputx. The graph offis the graph of the equationy=f(x).
4)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.
5)N.Q.2: Define appropriate quantities for the purpose of descriptive modeling.
6)N.Q.3:Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
7)A.REI.10:Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
8)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.
9)A.CED.2:Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
10)F.IF.2:Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
11)F.IF.5:Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*
12)F.IF.7a: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Graph linear and quadratic functions and show intercepts, maxima, and minima.
13)F.LE.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
14)A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
M : Major Content S: Supporting Content A : Additional Content

21st Century Career Ready Practice

Calendar

Please complete the pacing calendar based on the suggested pacing on page 7.

September 2016
Sun / Mon / Tue / Wed / Thu / Fri / Sat
1 / 2 / 3
4 / 5 / 6 / 7 / 8 / 9 / 10
11 / 12 / 13 / 14 / 15 / 16 / 17
18 / 19 / 20 / 21 / 22 / 23 / 24
25 / 26 / 27 / 28 / 29 / 30
October2016
Sun / Mon / Tue / Wed / Thu / Fri / Sat
1
2 / 3 / 4 / 5 / 6 / 7 / 8
9 / 10 / 11 / 12 / 13 / 14 / 15
16 / 17 / 18 / 19 / 20 / 21 / 22
23 / 24 / 25 / 26 / 27 / 28 / 29

Assessment Framework

Assessment / CCSS / Estimated Time / Date / Format / Graded
Diagnostic/Readiness Assessment
CL Chapter 1 Pretest #’s 1-6
CL Chapter 2 Pretest #’s 1-5 / A.CED.1, A.CED.2, A.REI.1, A.REI.3, F.IF.1, F.IF.2, N.Q.1, N.Q.2, F.IF.7, F.LE.1b / ½ Block / After Lesson 2 / Individual / No
Assessment Checkup #1
CL Chapter 1 End of Chapter Test #’s 1, 2, 5, 8, 10 / F.IF.1, F.IF.2, N.Q.1, N.Q.2, F.IF.7, F.LE.1b / ½ Block / After Lesson 9 / Individual / Yes
Assessment Checkup #2
CL Chapter 2 End of Chapter Test
#’s 1-6 / A.CED.1, A.CED.2, A.REI.1, A.REI.3, F.IF.2, N.Q.1, N.Q.2 / ½ Block / After Lesson 13 / Individual / Yes
Performance Task / N.Q.1, A.CED.1, A.CED.2, F.LE.1, A.REI.3 / ½ Block / After Lesson 13 / Individual, pair, or group / Yes
Unit 1 Assessment / A.SSE.1a, A.CED.1, A.CED.2, A.REI.1, A.REI.3, A.REI.10, F.IF.1, F.IF.2, F.IF.5, N.Q.1, N.Q.2, N.Q.3, F.IF.7, F.LE.1b / 1 Block / After whole unit completed / Individual / Yes
Assessment check points (exit tickets) / Varies by lesson / 5-10 minutes / Everyday / Individual / Varies

Scope and Sequence

Overview
Lesson / Topic / Suggesting Pacing
1 / Input – output tables / Intro to functions / 1 day
2 / Mathematical functions / 1 day
3 / Independent vs. dependent quantities / 1 day
4 / Domain/range and discrete/continuous graphs / 1 day
5 / Function notation and recognizing function families / 1 day
6 / Solving linear equations (justifying with mathematical reason) / 1 - 1 ½ days
7 / Modeling a linear situation / 2 days
8 / Analyzing linear functions / 2 days
9 / Solving linear inequalities / 2 days
10 / Performance task / 1 day
11 / Review / 1 day
Summary:
14days on new content (9 lessons/topics)
2task days
1 review day
1 test day
1 flex day
19 days in Unit 1

Lesson Analysis

Lesson 1: Input-output tables & intro to functions
Objectives

After given a provided strategy and visual representation of an input-output table, students will be able to find the rule for a given set of inputs and outputs, with at least ___ out of ___ answered correctly on an exit ticket.
Through an investigation of input-output relationships, students will be able to provide an informal definition of a function and identify when a “rule” cannot be created for a given set of inputs and outputs, with at least ___ out of ___ answered correctly on an exit ticket.

Focused Mathematical Practices

MP 2: Reason abstractly and quantitatively (when explaining why a certain input-output table cannot have a rule)
MP 6: Attend to precision (use correct vocabulary and require students to do the same)
MP 8: Look for and express regularity in repeated reasoning

Vocabulary

Inputs, outputs, function

Common Misconceptions

Some students may struggle with just coming up with a rule. Resort to the strategy provided, and in the case of linear examples, have them look for patterns in an output column (this only works when the inputs go up by a consecutive amount). For these students, focus on input-output tables that result in a linear rule.
Students will often see different inputs result in the same output and think it is not a function. Use a numerical example with a rule where this might happen to help explain this (i.e. x2). You can also use a vending machine and remote control as examples. In the case of a remote control, no button will result in two channels or functions, that’s impossible! However, a “last channel” button and “5” could both bring you to the same channel or station.

Lesson Clarifications

Suggested outline

AM 3.5 #’s 1 and 2 in the Opener
AM 3.5: #’s 1, 2, 4, 5, 6 in the Core Activity
AM 3.5: # 5 in the Consolidation Activity
AM 6.1: #’s 1-3 in the Opener
AM 6.1: #’s 1-2 in the Core Activity (here students should be applying the mathematical definition of a function to the vending machine example)

Input/output tables can be strategically selected or modified in order to prevent too much time spent on coming up with a rule.

(NJSLS/CCSS) / Concepts
What students will know / Skills
What students will be able to do / Material/
Resource / Suggested Pacing / Assessment Check Point
A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. / Review

An algebraic rule using mathematical operations may exist to explain the relationship between a set of inputs and outputs.
An algebraic rule that exists for a set of inputs and outputs should work for each input-output pair. This rule can also be used to find additional input-output pairs.

New / Review

Write a rule that models a relationship between a set of inputs and outputs (may be review or new depending on the level of difficulty)

New

Write a rule that models a relationship between a set of inputs and outputs (may be review or new depending on the level of difficulty)

/ AM 3.5
AM 6.1 / 1 day / AR 5.1
F.IF.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Iffis a function andxis an element of its domain, thenf(x) denotes the output off corresponding to the inputx. The graph offis the graph of the equationy=f(x). / Review

A set of inputs and outputs may or may not represent a mathematical function.

New

If a set of inputs and outputs represents a mathematical function, it is because each input value is assigned to exactly one output value.

/ Review

Determine whether something is a function or not

New

Explain why something is or is not a function

Lesson 2: Mathematical functions
Objectives

After discussion of a vending machine example, students will be able to explain why something is a function or not, with at least ___ out of ___ answered correctly on an exit ticket.
Given a set of relations in multiple forms (table and graph), students will be able to describe how you can determine if something is a function specifically by looking at a table or a graph, with at least ___ out of ___ answered correctly on an exit ticket.

Focused Mathematical Practices

MP 2: Reason abstractly and quantitatively (when explaining why something is a function or not AND when describing how to determine if something is a function from a table and graph)
MP 6: Attend to precision (use correct vocabulary and require students to do the same)

Vocabulary

Inputs, outputs, function

Common Misconceptions

Students will often see different inputs result in the same output and think it is not a function. Use a numerical example with a rule where this might happen to help explain this (i.e. x2). You can also use a vending machine and remote control as examples. In the case of a remote control, no button will result in two channels or functions, that’s impossible! However, a “last channel” button and “5” could both bring you to the same channel or station.
Some students may think that something is or is not a function for reasons unrelated to the definition of a function. For example, a student may think that there needs to be an obvious pattern/relationship to be a function (i.e. linear) or that it must form a straight line to be a function. Validate that their thinking is intuitively meaningful, but refer these students to the definition of a function, or provide them with an example that IS a function but does not align their reasoning.

Lesson Clarifications

If time is an issue, only focus on the Core Activity of AM 6.1 (vending machine example). Strive to complete all of AM 6.2.

CCSS / Concepts
What students will know / Skills
What students will be able to do / Material/
Resource / Suggested Pacing / Assessment Check Point
F.IF.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Iffis a function andxis an element of its domain, thenf(x) denotes the output off corresponding to the inputx. The graph offis the graph of the equationy=f(x). / Review

A set of inputs and outputs may or may not represent a mathematical function.

New

If a set of inputs and outputs represents a mathematical function, it is because each input value (a value from the domain) is assigned to exactly one output value (a value in the range).
By looking at a table, you can see if something is a function by making sure each input is mapped to exactly one output.
By looking at a graph, you can see if something is a function by making sure each x-value only corresponds to 1 y-value (a graph cannot exists in more than one location for any vertical line)

/ Review

Determine whether something is a function or not

New

Use the mathematical language from the definition of a function to explain why something is or is not a function
Generalize how to determine if something is a function or not based upon the information given (table or graph).

/ AM 6.1
AM 6.2 / AR 6.1
Lesson 3: Independent vs. dependent quantities
Objectives

Given a set of problem situation descriptions, students will be able to identify the independent and dependent quantities and units, and label them correctly on a coordinate plane with at least ____ out of ____ parts answered correctly on an exit ticket.
Given a set of problem situation descriptions, students will be able to recognize how to represent something that is changing at a constant rate with at least ____ out of ____ parts answered correctly on an exit ticket.

Focused Mathematical Practices

MP 1: Make sense of problem and persevere in solving them
MP 4: Model with mathematics
MP 6: Attend to precision (use correct vocabulary and require students to do the same)

Vocabulary

Independent quantity/variable, dependent quantity/variable, constant rate of change

Common Misconceptions

Some students may struggle with working through a problem situation. Provide students with structure for persevering (i.e. step #1 is to find the variable or changing quantities, step #2 is to determine which is dependent and independent).
Some students may identify key information (i.e. “increases at a rate of 10 gallons per minute) as a variable quantity. Preemptively explain the difference between something that is constant and a quantity that is unknown/changing in a problem situation.
For students who struggle with correctly identifying the independent/dependent quantities, try using one of these questions as a strategy.

“Which quantity depends on the other? Does ____ depend on ____?”
“Which quantify would you input/choose in order to determine the outcome of the other?”

Lesson Clarifications
CCSS / Concepts
What students will know / Skills
What students will be able to do / Material/
Resource / Suggested Pacing / Assessment Check Point
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.
N.Q.2: Define appropriate quantities for the purpose of descriptive modeling. / Review

There are often two quantities (each with their own units) that change in problem situations.

New

When one quantity depends on the other, it is said to be the dependent quantity, and is used to label the y-axis.
When one quantity is used to affect the outcome of another, it is said to be independent quantity, and is used to label the x-axis.

/ New

Identify variable quantities and units given a problem situation
Identify which variable quantity is independent and which is dependent given a problem situation
Label the x-axis of a coordinate plane with the independent quantity (including units) and the y-axis with the dependent quantity (including units)

/ CL ST 1.1 / 1 day / CL SP 1.1
(#’s 2, 7, 15)
F.LE.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. / Review

Graphs can be used to model problem situations.
Some problem situations have quantities that change at a constant rate per unit. Graphs that model these situations are straight (linear) lines.

/ Review

Correctly match graphs with their problem situations that model a constant rate of change (linear)

New

Problem solving strategies

/ CL ST 1.1
Lesson 4: Domain/ range and discrete/continuous graphs
Objectives

Given a set of continuous and discrete graphs and problem situations, students will be able to determine the domain of a function (with and without a context) by answering at least ___ out of ___ questions correctly on an exit ticket.

Focused Mathematical Practices

MP 3: Construct viable arguments and critique the reasoning of others (when determining the domain from a problem situation)
MP 6: Attend to precision (use correct vocabulary and require students to do the same)

Vocabulary

Relation, Vertical Line Test, continuous graph, discrete graph, function, domain, range

Common Misconceptions

If students struggle with understanding the concept of “domain”, try referring to it (in the beginning) as “allowable inputs”.
For students who struggle with choosing a domain, provide contexts that are simple and ask them to think about which values would make sense. (i.e. “Does -5 make sense as an input when talking about the number of miles driven? Does 12.5 make sense when talking about how many t-shirts can be ordered?”)

Lesson Clarifications

This lesson is significantly modified from how it was intended to be used in the Carnegie Learning Curriculum. Please refer to the Sample Lesson Plan in this Unit Plan for further information.