Mathematics Pacing Guide
Time Frame:7 Weeks – September-OctoberAlgebra I
Unit 1: Relationships between Quantities and Reasoning with Equations
Standards for Mathematical Practice / Literacy Standards1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning / RST.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
RST. 4. Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.
RST. 7. Translate quantitative or technical informationexpressed in words in a text into visual form(e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.
WHST.2. Write informative/explanatory texts, including the narration of historical events, scientific procedures/ experiments, or technical processes.
d. Use precise language and domain-specific vocabulary to manage the complexity of the topic and convey a style appropriate to the discipline and context as well as to the expertise of likely readers.
WHST.6 Use technology, including the Internet, to produce, publish, and update individual or shared writing products, taking advantage of technology’s capacity to link to other information and to display information flexibly and dynamically.
WHST. 7. Conduct short as well as more sustained research projects to answer a question (including a self-generated question) or solve a problem; narrow or broaden the inquiry when appropriate; synthesize multiple sources on the subject, demonstrating understanding of the subject under investigation.
WHST. 10. Write routinely over extended time frames (time for reflection and revision) and shorter time frames (a single sitting or a day or two) for a range of discipline-specific tasks, purposes, and audience.
SL.4Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task.
SL.5 Make strategic use of digital media (e.g., textual, graphical, audio, visual, and interactive elements) in presentations to enhance understanding of findings, reasoning, and evidence and to add interest.
Common Core / Essential Questions / Assessments / Vocabulary / Resources
Reason quantitatively and use units to solve problems
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.
N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities
Note: Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions. / How can tables, graphs, and rules relating variablesbe used to answer real-world and mathematical questions about the relationships between variables?
How does the shape of a graph, the patterns in a table, the parts of the rule or verbal description give clues about the way the variables are related to one another?
What are the advantages and disadvantages of using graphs, tables, descriptions, and algebraic rules to demonstrate the relationship between two variables?
What is a function?
How do you recognize a function in various representations?
Given two variables, how do you decide which is the independent variable and which is the dependent variable? / Before:
Pre-test on simplifying expressions and equations
Entry slip
During:
Cross Totals: In this task, you investigate a number puzzle.
Interpreting Distance-Time Graphs:This lesson unit is intended to help assess how well students are able to interpret distance–time graphs, and in particular, to help you identify students who have difficulties interpreting distance–time graphs as if they are pictures of situations rather than abstract representations of them and relating speeds to slopes of these graphs.
Patchwork: In this task, students investigate number patterns to find a rule, or a formula, that will help Kate figure out the number of squares she needs for cushions of different sizes.
After:
Interpreting Functions:A set of two short questions, asking students to interpret and use distance time graphs.
Sorting Functions: Match graphs with an equation, a table and a rule.
Post-test / absolute value
additive identity
additive inverse
algebraic expression
associative property
base
coefficient
commutative property
compare and contrast common patterns
conditional statement
connections between multiple representations
constant term
converse of a conditional
counterexample
dependent variable
distributive property
domain
equation
equivalent expressions
evaluate
examine relationships between variables
exponent
formula
function
if-then statement
independent variable
inequality
input
integers
irrational number
patterns of change from a variety of function families
like terms
literal equation
multiplicative Identity
multiplicative inverse
open sentence
opposites
order of operations
output
perfect
power
proportion
quadrants
radicand
range
rate
ratio
rational number
real number
square
square root
terms
unit rate
variable
verbal model
whole numbers / Domain and Range - Graphically!: This demo is designed to help students use graphical representations of functions to determine the domain and range.
Determining Functions Using Regression: This unit guides students though activities that ask students to collect data. Then, they use technology to find functions that best describe a data collected. After analyzing the data, the student should be able to determine a best type of function to describe the trend.
All In the Family:Students use a dynamic geometry applet to conjecture about therelationshipsbetween characteristics of a square: side length, diagonal length, perimeter, and area. Graphs are used to representfunctionalrelationshipsbetween two characteristics, such as diagonal length as afunctionof perimeter. This lesson helps students deepen their understanding of basicfunctions(e.g., linear, quadratic, square root) and their knowledge of the measures of a square.
Algebra for All Resources: The Shapes of Algebra: In the shapes of Algebra, groups of students graph 4 different functions and identify which function is not like the others. In the process they explore features of tables and graphs of various function families. The activity concludes with a practice section in which students use a variety of representations to identify function families. The online version of the activity includes several automated practice applets to support graphing and identifying function families.
Movement with Functions: These investigations use movement to reinforce the concepts of linear functions and systems of equations. Multiple representations are used throughout, along with tools such as motion detectors and remote-controlled cars. Students explore how position, speed, and varying motion are reflected in graphs, tables, and algebraic equations.
Texas Instruments
Distance-Time Graphs (TI-84 andCBR™):CBR™ and Graphing Calculators allow a conceptual understanding of distance-time graphs. Students walk using a CBR to record their walk, then analyze the graphs produced.
Articles from National Council of Teachers of Mathematics (
Articles available as free downloads toNCTMmembers, or for a fee to non-members.
Fernandez, M.L. (2006). Teaching about Functions through Motion in Real Time. MathematicsTeacher, 99(6), 430-437. Retrieved December 23, 2011 from
White, A. and Van Dyke, F. (2006). Habits in the Classroom. Mathematics Teacher, 100(4), 270-274. Retrieved February 29, 2012 from
Rider, R. (2007). Shifting from Traditional to Nontraditional Teaching Practices Using Multiple Representations. Mathematics Teacher, 100(7), 494-500. Retrieved February 29, 2012 from
Hartter, B. (2009). A Function or Not a Function? That is the Question. Mathematics Teacher, 103(3), 200. Retrieved on March 7, 2012 from
Barnes, J. and Jaqua, K. (2011). Algebra Aerobics. Mathematics Teacher, 105(2), 97. Retrieved March 9, 2012 from
Van Dyke, F. (2003). Using Graphs to Introduce Functions. MathematicsTeacher,96(2), 126. Retrieved on March 7, 2012 from
Choike, J. (2000). Teaching Strategies for “Algebra for All”. Mathematics Teacher, 93(7), 556. Retrieved on August 29, 2012 from
Interpret the structure of expressions
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 theproduct of P and a factor not depending on P.
Note: Limit to linear expressions and to exponential expressions with integer exponents. / Sorting Functions:
This site has additional resources for teachers and students:
Create equations that describe numbers or relationships
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.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints oncombinations of different foods.
A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’slaw V = IR to highlight resistance R.
*Instructional
Note: Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.
Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas which are linear in the variable of interest.
Understand solving equations as a process of reasoning and explain the reasoning
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.
Note: Students should focus on and master
A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Algebra II.
Solve equations and inequalities in one variable
A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x=125 or 2x=1/16
Mathematics Pacing Guide
Time Frame: 7 Weeks –November - DecemberAlgebra I
Unit 2: Linear and Exponential Relationships
Standards for Mathematical Practice / Literacy Standards1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning / RST.1 Cite specific textual evidence to support analysis of science and technical texts, attending to the precise details of explanations or descriptions.
RST.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
RST. 4. Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.
RST. 7. Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.
RST. 9. Compare and contrast findings presented in a text to those from other sources (including their own experiments), noting when the findings support or contradict previous explanations or accounts.
WHST.2. Write informative/explanatory texts, including the narration of historical events, scientific procedures/ experiments, or technical processes.
d. Use precise language and domain-specific vocabulary to manage the complexity of the topic and convey a style appropriate to the discipline and context as well as to the expertise of likely readers.
WHST.4 Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.
WHST.5 Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach, focusing on addressing what is most significant for a specific purpose and audience.
WHST.6 Use technology, including the Internet, to produce, publish, and update individual or shared writing products, taking advantage of technology’s capacity to link to other information and to display information flexibly and dynamically.
WHST. 8. Gather relevant information from multiple authoritative print and digital sources, using advanced searches effectively; assess the usefulness of each source in answering the research question; integrate information into the text selectively to maintain the flow of ideas, avoiding plagiarism and following a standard format for citation.
WHST. 10. Write routinely over extended time frames (time for reflection and revision) and shorter time frames (a single sitting or a day or two) for a range of discipline-specific tasks, purposes, and audience.
SL. 4 Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task.
SL.5 Make strategic use of digital media (e.g., textual, graphical, audio, visual, and interactive elements) in presentations to enhance understanding of findings, reasoning, and evidence and to add interest.
Common Core / Essential Questions / Assessments / Vocabulary / Resources
Extend the properties of exponents to rational exponents
N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Note: In implementing the standards in curriculum, these standards should occur before discussing exponential functions with continuous domains.
Use properties of rational and irrational numbers
N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Note: Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2. / How can you recognize linear change from a graph,equation, table, or real-world situation?
Given a set of bivariate data that appears to have a linear pattern in a scatter plot, how do you find a least squares regression line? What is the meaning of the slope and y-intercept of that line in terms of the problem situation?
What are some methods to solve an absolute value equation?
Given equations of two lines, how can you determine whether the lines are parallel, perpendicular, intersecting or concurrent?
Describethe similarities and differences between exponential and linear functions.
What similarities do all tables of exponential functions share?
How doesthe value of a in theequationf(x)=a(bx) affect the graph and table of an exponential function?
How do you recognizeexponential growth or decay from a graph, function rule, table,or real-world situation?
What are the similarities and differences between quadratic, exponential, and linear functions? / Before:
Pre-test on linear and exponential functions
Entry slip
During:
Interpreting Algebraic Expressions: This lesson unit is intended to help you assess how well students are able to translate between words, symbols, tables, and area representations of algebraic expressions. It will help you to identify and support students who have difficulty in recognizing the order of algebraic operations, recognizing equivalent expressions, and understanding the distributive laws of multiplication and division over addition (expansion of parentheses).
Linear Functions - Ford and Ferrariwith correspondingLinear Functions Formative Assessment
After:
Race Car Activity
Post-test / asymptotic behavior
compare and contrast exponential functions with linear functions
compare and contrast quadratic functions with other function families
connect solutions of equations to real-world situations
constant of variation
definitions and standard rules for exponents
direct variation
equivalent linear expressions
exponential patterns - explicit form (i.e., f(x) = a(bx))
exponential patterns - recursive form
family of functions
function notation
linear equation
linear function
linear patterns – explicit form
linear patterns – recursive form
modeling exponential functions (key applications: scientific notation, half-life, compound interest)
modeling linear functions
multiple representations
multiplicative rate of change
parallel lines
parent function
patterns of change in exponential functions (growth and decay)
perpendicular lines
point-slope form
quadrants
rate of change
slope
slope-intercept form
solution
solve equations and inequalities
standard form
systems of equations
x-intercept
y =a + bx or y=mx + b
y-intercept
zero of a function / Got A Plan?:Compare a variety of cell phone plans in this activity.
The Line Runner: Explore the concept of rate of change using real world data, comparing the value and the steepness of the line as well as positive and negative rates.
Movement with Functions:This investigation uses a motion detector to help students understand graphs and equations. Students experience constant and variable rates of change and are challenged to consider graphs where no movements are possible to create them. Multiple representations are used throughout the lesson to allow students to build their fluency with in how graphs, tables, equations, and physical modeling are connected. The lesson also allows students to investigate multiple function types, including linear, exponential, quadratic, and piecewise.