Algebra 1 Unit 3: Systems of Equations - wikispaces.netwysiwyg.cmswiki.wikispaces.net/file/view/Math I Unit 4... · Web viewInclude equations arising from linear and quadratic functions,

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.

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BY THE END OF THIS UNIT:

Enduring understanding (Big Idea): Students will understand that manipulation of polynomials will enable them to model and analyze real-world non-linear situations.

Essential Questions:1. How are the properties of real numbers related to polynomials?2. How can two algebraic expressions that appear to be different be the same?3. What are the characteristics of a quadratic function? 4. How can you use functions to model real-world situations?

Students will know…

The Quadratic Formula

Rules of exponents – (focus on negative, zero and fractional exponents

Types of Factoring (excluding sum or difference of cubes) Special products

Vocabulary: Greater than, less than, at most, at least, =, < ,>, ≤ ,≥ , no more than, no less than, Variable, dependent variable, independent variable, domain, range, scale, constant, variable, formula, literal equation, exponents, factors, terms, bases, coefficients, expression, polynomial, trinomial, binomial, linear terms, quadratic terms, scale, units of measurement, units rates, modeling, quantity, unit conversion, proportion, ratio, precision, accuracy, exponent, rational, radical, constant, coefficient, properties of operations and properties of equality, like terms, variable, evaluate, justify, viable, completing the Square, Square Roots, factoring, perfect-square trinomial, zeros, solutions

Students will be able to…A.CED.1: I can write equations in one variable and apply them to the real world. I can write inequalities in one variable and apply them to the real world.A.CED.2: I can write/create an equation with 2 or more variables.I can create a coordinate plane using appropriate labels and scales.I can graph an equation on a coordinate plane with 2 or more variables.I can represent/ interpret/ identify relationships between quantities from equations and graphs.A.CED.3: I can represent constraints by equations or inequalities.I can represent constraints by systems of equations/inequalities.I can interpret solutions as viable or nonviable solutions graphically.A.CED.4: I can rearrange/rewrite formulas to solve for a given variable, using the same steps to solve equations.A.SSE.1a: I can identify the parts of an expression, including, its terms, its factors, and its coefficients.I can interpret the parts of an expression, including, its terms, its factors, and its coefficients.A.SSE.1b: I can interpret a complex expression by dissecting it into individual parts.A.SSE.2: I can recognize the patterns of special expressions (i.e. difference of perfect squares or sum of perfect cubes) I can rewrite special expressions. A.SSE.3a: I can factor a quadratic expression.I can identify (reveal) the zeros of a quadratic expression.A.SSE.3b: I can complete the square of a quadratic expression. I can find the maximum or minimum value of a quadratic function.A.SSE.3c: I can rewrite exponential expressions by using properties of exponents.I can rewrite exponential functions by using properties of exponents.A.APR.1: I can add polynomials. I can subtract polynomials. I can multiply polynomialsA.REI.1: I can explain each step in solving equations using properties of equality. I can give (provide) a reasonable (viable) explanation for each step. A.REI.3: I can solve equations with one variable, including equations with coefficients. I can solve inequalities with one variable, including inequalities with coefficients.A.REI.4a: I can solve a quadratic equation by completing the square. I can derive the quadratic formula by completing the square. I can write a quadratic equation as a binomial square.A.REI.4b: I can solve quadratic equations using square roots. I can solve quadratic equations by completing the square. I can solve quadratic equations by factoring. I can solve quadratic equations by inspection. I can identify which method to use to solve a quadratic equation.N.Q.1: I can identify the units in problems. I can include the units when setting up and solving the problem. I can interpret my answer, including the appropriate units. I can choose an appropriate scale with units for the graphs or data displays.N.Q.2: I can choose appropriate units to write an equation for a real world application.N.Q.3: I can identify variable quantities. I can choose a level of accuracy based on the problem situation.N.RN.1: I can apply properties of integer exponents. I can define nth root of a to the nth power as a to the m/n power. I can apply properties to rational exponents.N.RN.2:I can write expressions using radical notation. I can write expressions using rational exponent notation.

Unit ResourcesLearning Task:Concept Byte: Using Models to FactorPerformance Task:Ch 8 of Alg. 1 textbook pg 67 – “Pull-it-all-together” tasks #1, 2, and/or 3Ch 9 of Alg. 1 textbook pg 151 – “Pull-it-all-together” task #1, 2, CCSS Included:A.CED.1, A.CED.2, A.CED.3, A.CED.4, A.SSE.1, A.SSE.2, A.SSE.3, A.APR.1, A.REI.1, A.REI.3, A.REI.4, N.Q.1, N.Q.2, N.Q.3, N.RN.1, N.RN.2Released Test Questions:3, 5, 8, 13, 16, 20, 33, 38Algebra I Project Binder:Pages 1 - 119


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CORE CONTENTCluster 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 TEACHERSCritical 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 VocabularyGreater than, less than, at most, at least, =, < ,>, ≤ ,≥ , no more than, no less thanSuggested Instructional Strategies:

Convert contextual information into mathematical notation.

Use story contexts to create linear and exponential equations and inequalities

NCDPI Unpacking:A-CED.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include linear and exponential functions. At this level, focus on linear and exponential functions.

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, 7-1, 7-3, 7-4, 7-5, 9-3, 9-4, 9-5, 9-6, 11-5

MARS Apprentice Tasks:FunctionsMultiplying CellsPrinting Tickets

MARS Expert Tasks: Fearless Frames Skeleton Tower Best Buy Tickets

Skill-based task

Juan pays $52.35 a month for his cable bill and an

Problem Task

Juan pays $52.35 a month for his cable bill and an additional

http://map.mathshell.org/materials/tasks.php?taskid=286&subpage=expert



http://map.mathshell.org/materials/tasks.php?taskid=260&subpage=apprentice




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additional $1.99 for each streamed movie. If his last cable bill was $68.27, how many movies did Juan watch?

$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.

Two boys, Shawn and Curtis, went for a walk. Shawn began walking 20 seconds earlier than Curtis.

Shawn walked at a speed of 5 feet per second

Curtis walked at a speed of 6 feet per second

For how many seconds had Shawn been walking at the moment when the two boys had walked exactly the same distance?

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandard: A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph


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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 TEACHERSCritical 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 VocabularyVariable, dependent variable, independent variable, domain, range, scaleSuggested 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

NCDPI Unpacking:A-CED.2 Given a contextual situation, write equations in two variables that represent the relationship that existsbetween the quantities. Also graph the equation with appropriate labels and scales. Make sure students areexposed to a variety of equations arising from the functions they have studied. At this level, focus on linear, exponential and quadratic equations. Limit to situations that involve evaluating exponential functions for integer inputs.

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:

FunctionsMultiplying CellsPrinting Tickets

MARS Expert Tasks: Fearless Frames Skeleton Tower Best Buy Tickets

Sample Assessment TasksSkill-based taskWrite 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 TaskJeanette 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?








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Two times Antonio’s age plus three times Sarah’s age equals 34. Sarah’s age is also five times Antonio’s age. How old is Sarah?

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandard: 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 const constraints on combinations of different foods.Concepts and Skills to Master:

Determine whether a point is a solution to an equation or inequality


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Determine whether a solution has meaning in a real-world context Write and use a system of equations and/or inequalities to solve a real-world problem Recognize that the equations and inequalities represent the constraints of the problem. Use the Objective

Equations and the Corner Principle to determine the solution to the problem. (Linear Programming)SUPPORTS FOR TEACHERSCritical Background Knowledge

Ability to read and write inequality symbols Ability to graph equations and inequalities on the coordinate plane

Academic VocabularyConstraint, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤ , inequality, viableSuggested Instructional Strategies:

Use a story context such as those found in linear programming problems to write and graph equations with constraints.

Analyze real-world problems connected to student interest

NCDPI Unpacking:A-CED.3 Use constraints which are situations that are restricted to develop equations and inequalities and systems of equations or inequalities. Describe the solutions in context and explain why any particular one would be the optimal solution. Limit to linear equations and inequalities.

Resources: Textbook Correlation: 6-5, 6-6, CC-7 MARS Concept Development Lessons:

Defining Regions using InequalitiesSolving Linear Equations in Two Variables

MARS Problem Solving Lesson: Optimization problems

Sample Assessment TasksSkill-based taskGiven y ≤ 2x + 1 and y > x – 3 find a point that:

a. Satisfies bothb. Satisfies one, but not the otherc. Satisfies neither

Problem TaskIced tea costs $1.50 a glass and lemonade costs $2.00. If you have $12, what can you buy? Justify your answer using multiple representation.

Paul sells chocolate chip cookies and peanut butter cookies. Baking a batch of chocolate chip cookies

http://map.mathshell.org/materials/lessons.php?taskid=207&subpage=problem

http://map.mathshell.org/materials/lessons.php?taskid=209&subpage=concept



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takes 1.75 cups of flour and 2 eggs. Baking a batch of peanut butter cookies

takes 1.25 cups of flour and 1 egg. Paul has 10 cups of flour and 12 eggs. He makes $4 profit per batch of chocolate

chip cookies He makes $2 profit per batch of peanut butter cookies

How many batches of peanut butter cookies should Paul make to maximize his profit?

A 1B 2C 5D 8

All American Bats, produces two different quality wooden baseball bats, the Aaron Bat and the DiMaggio Bat. The Aaron Bat takes 8 hours to trim and 2 hours to finish it. It has a profit of $16. The DiMaggio Bat takes 5 hours to trim and 5 hours to finish it, but it has a profit of $27. The total time available per day for trimming is 80 hours and 50 hours for finishing. What is the maximum profit the All American Bats can make each day?

A $270B $296C $366D $432

CORE CONTENTCluster Title: Create equations that describe numbers or relationshipsStandard: A.CED.4: Rearrange formulas to highlight a quantity of interestConcepts 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.


Recognize variables as representing quantities in context Solve multi-step equations

Academic VocabularyConstant, variable, formula, literal equation


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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

NCDPI Unpacking:A-CED.4 Solve multi-variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations. Limit to formulas which are linear in the variable of interest or to formulas involving squared or cubed variables.

Resources: Textbook Correlation: 2-5

Sample Assessment TasksSkill-based task

I = Prt Solve for r.

Which equation is 5x + 4y = 9x + 8 correctly solved for x?

A

B

C

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?


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D

Energy and mass are related by the formula E = mc2

m is the mass of the object c is the speed of light

Which equation finds m, given E and c?

AB

C

D

CORE CONTENTCluster 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 coefficientsb. 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.



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Understand the meaning of symbols indicating mathematical operations, implied operations (e.g. 2x), the meaning of exponents, and grouping symbols.

Academic VocabularyExponents, factors, terms, bases, coefficients, expressionSuggested Instructional Strategies:

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

NCDPI Unpacking:A-SSE.1a. Students manipulate the terms, factors, and coefficients in difficult expressions to explain the meaning of the individual parts of the expression. Use them to make sense of the multiple factors and terms of the expression. For example, consider the expression 10,000(1.055)5. This expression can be viewed as the product of 10,000 and 1.055 raised to the 5th power. 10,000 could represent the initial amount of money I have invested in an account. The exponent tells me that I have invested this amount of money for 5 years. The base of 1.055 can be rewritten as (1 + 0.055), revealing the growth rate of 5.5% per year. At this level, limit to linear expressions, exponential expressions with integer exponents, and quadratic expressions.A-SSE.1b Students group together parts of an expression to reveal underlying structure. For example, consider theexpression 4000p – 250p2 that represents income from a concert where p is the price per ticket. The equivalent factored form, p(4000 – 250p) shows that the income can be interpreted as the price times the number of people in attendance based on the price charged. At this level, limit to linear expressions, exponential expressions with integer exponents, and quadratic expressions.


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


Consider the formula Surface Area = 2B + Pha. 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.

Ex. The expression 20(4x) + 500 represents the cost in dollars of the materials and labor needed to build a squarefence with side length x feet around a playground. Interpret the constants and coefficients of the expression incontext.



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CORE CONTENTCluster Title: Interpret the Structure of ExpressionsStandard: A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).Concepts and Skills to Master:

Rewrite an algebraic expression in different forms such as factoring or combining like terms Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference

of cubes, or a combination of methods to factor completely. Simplify expressions including combining like terms, using the distributive property and other operations with

polynomials.SUPPORTS FOR TEACHERSCritical Background Knowledge

Clear and concrete understanding of the distributive property and factoring polynomialsAcademic Vocabulary


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Suggested Instructional Strategies: Use algeblocks or algebra tiles to explore structure

of expressions.

NCDPI Unpacking:A.SSE.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.

Resources: Textbook Correlation: 5-3, 5-4, 5-5, 8-7, 8-8, CC-

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Ex. Expand the expression 2(x – 1)2 - 4 to show that it is a quadratic expression of the ax2 + bx +c.

Which expression is equivalent to t2 – 36?

A (t – 6)(t – 6)

B (t + 6)(t – 6)

C (t – 12)(t – 3)

D (t – 12)(t + 3)

Problem Task

Interior DesignA square rug has an area of 49x2 -56x + 16. A second square rug has an area of 16x2 + 24x + 9. What is an expression that represents the difference of the areas of the rugs. Show two different ways to find the solution.


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CORE CONTENTCluster Title: Write expressions in equivalent forms to solve problemsStandard: A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it definesb. Complete the square in a quadratic expression to reveal the maximum or minimum values of the function it

defines.c. Use the properties of exponents to transform expressions for exponential functions.

Concepts and Skills to Master: Given a quadratic function explain the meaning of the zeros of the function. e.g. if f(x) = (x – c)(x – a), then f(a) = 0

and f(c) = 0 Given a quadratic expression, explain the meaning of the zeros graphically. e.g. for an expression (x – c)(x – a), a

and c correspond to the x-intercepts (if a and c are real numbers). Write the vertex form of a quadratic expression by completing the square. Use the vertex form to find the maximum or minimum of a quadratic function and explain the meaning of the

vertex.SUPPORTS FOR TEACHERSCritical Background KnowledgeStudents should have a good understanding of simple factoring techniques to this point. Students should also understand about the GCF and how to identify or determine the GCF.


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Academic Vocabulary

Suggested Instructional Strategies: Teachers should be familiar with most all aspects

of trinomial factoring. Teachers should be particularly well versed in factoring using the GCF (greatest common factor). A brief review of factoring with algebra tiles and GCF would greatly help the teacher with this lesson.

Use algeblocks or algebra tiles to explore the meaning of “completing the square” and discover the algorithm

NCDPI Unpacking:A-SSE.3a Students factor quadratic expressions and find the zeros of the quadratic function they represent. Zeroes are the x-values that yield a y-value of 0. Students should also explain the meaning of the zeros as they relate to the problem. For example, if the expression x2 – 4x + 3 represents the path of a ball that is thrown from one person to another, then the expression (x – 1)(x – 3) represents its equivalent factored form. The zeros of the function, (x –1)(x – 3) = y would be x = 1 and x = 3, because an x-value of 1 or 3 would cause the value of the function to equal 0. This also indicates the ball was thrown after 1 second of holding the ball, and caught by the other person 2 seconds later. At this level, limit to quadratic expressions of the form ax2 + bx + c.

Resources: Textbook Correlation: 7-7, CC-15


It is given that x2 + 10x + 24 = (x + 5)

2 + k. Find the value

of k.

What is the smallest of 3 consecutive positive integers if the product of the smaller two integers is 5 less than 5 times the largest integer?

Problem Task

The larger leg of a right triangle is 3 cm longer than its smaller leg. The hypotenuse is 6 cm longer than the smaller leg. How many centimeters long is the smaller leg?


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CORE CONTENTCluster Title: Perform Arithmetic Operations on PolynomialsStandard: A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Concepts and Skills to Master:

Look for and make use of the structure of addition and subtraction extended to linear and quadratic polynomials Understand the concept of like terms and closure Identify and make use of the structure of special products of linear binomial expressions.


Distributive Property Addition, subtraction and multiplication of integers

Academic VocabularyPolynomial, trinomial, binomial, linear terms, quadratic termsSuggested Instructional Strategies:

NCDPI Unpacking:

A-APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial. Polynomials are not closed under division because in some cases the result is a

Resources: Textbook Correlation: 8-1, 8-2, 8-3, 8-4 MARS Concept Development Lesson:

Interpreting Algebraic Expressions



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rational expression rather than a polynomial. At this level, limit to addition and subtraction of quadratics and multiplication of linear expressions.

A-APR.1 Add, subtract, and multiply polynomials. At this level, limit to addition and subtraction of quadratics and multiplication of linear expressions.


Which expression is equivalent to r2 + 2(rw – x)

2 when r = -

3 and x = -1?

A 18w2 + 12 w + 11

B 18w2 – 12w + 11

C 99w2 + 66w + 11

D 99w2 – 66w + 11

Problem Task

The area of a trapezoid is found using the formula

, where A is the area, h is the height, and

b1 and b2 are the lengths of the bases.

What is the area of the above trapezoid?

A A = 4x + 2B A = 4x + 8C A = 2x

2 + 4x – 21

D A = 2x2 + 8x – 42


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CORE CONTENTCluster 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)


Know how various attributes are reasonably measuredAcademic VocabularyScale, units of measurementSuggested Instructional Strategies:

Explore a variety of examples of measurements used in graphs Construct graphs using a variety of data sets

NCDPI Unpacking:N-Q.1 Use units as a tool to help solve multi-step problems. For example, students should use the units assigned to quantities in a problem to help identify which variable they correspond to in a formula. Students should also analyze units to determine which operations to use when solving a problem. Given the speed in mph and time traveled in hours, what is the distance

Resources: Textbook Correlation: 2-5, CB 2-5, 2-

6, 2-7, 4-4, 5-7, 12-2, 12-4


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traveled? From looking at the units, we can determine that we must multiply mph times hours to get an answer expressed in miles: (mi/hr)(hr) = mi (Note that knowledge of the distance formula is not required to determine the need to multiply in this case.)N-Q.1 Based on the type of quantities represented by variables in a formula, choose the appropriate units to express the variables and interpret the meaning of the units in the context of the relationships that the formula describes.N-Q.1 When given a graph or data display read and interpret the scale and origin. When creating a graph or data display, choose a scale that is appropriate for viewing the features of a graph or data display. Understand that using larger values for the tick marks on the scale effectively “zooms out” from the graph and choosing smaller values “zooms in.” Understand that the viewing window does not necessarily show the x- or y-axis, but the apparent axes are parallel to the x- and y-axes. Hence, the intersection of the apparent axes in the viewing window may not be the origin. Also be aware that apparent intercepts may not correspond to the actual x- or y-intercepts of the graph of afunction.

Sample Assessment TasksSkill-based taskWhat is the area of strip of wall that is 48 inches by 10 yards?

A radio-controlled car traveled 30 feet across the classroom in 1.6 seconds. Use appropriate quantities from the box to determine how fast the car traveled in miles per hour._____________x_______________x_____________x_____________=

Problem TaskYour 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.

Ex. When finding the area of a circle using the formula ! ! !!!, which unit of measure would be appropriate forthe radius?a. square feetb. inchesc. cubic yardsd. pounds

Ex. Based on your answer to the previous question, what units would the area be measured in?

Ex. What scale would be appropriate for making a histogram of the following data that describes the level of lead in the blood of children (in micrograms per deciliter) who were exposed to lead from their parents’ workplace?


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Alex walked 1 mile in 15 minutes. Sally walked 3520 yards in 24 minutes. In miles per hour, how much faster did Sally walk than Alex? (Note: 1 mile = 1760 yards)

10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 23, 24, 25, 27, 31, 34, 34, 35, 35, 36, 37, 38, 39, 39, 41, 43, 44, 45, 48, 49,62, 73

CORE CONTENTCluster Title: Reason quantitatively and use units to solve problemsStandard: N.Q.2: Define appropriate quantities for the purpose of descriptive modelingConcepts 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)


Compute unit rates associated with ratios of fractions Recognize and calculate basic conversions (e.g. 3 feet = 1 yard)

Academic VocabularyUnits rates, modeling, quantity, unit conversion, proportion, ratioSuggested Instructional Strategies:

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)

NCDPI Unpacking:

N-Q.2 Define the appropriate quantities to describe the characteristics of interest for a population. For example, ifyou want to describe how dangerous the roads are, you


CC-7


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may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to report the number of exits on that stretch of interstate to describe the level of danger.

Sample Assessment TasksSkill-based taskHow would you measure the rate at which a bathtub fills? Justify your answer.

Ex. What quantities could you use to describe the best city in North Carolina?

Ex. What quantities could you use to describe how good a basketball player is?

Problem TaskRecreate a scenario you have encountered involving two changing quantities and determine appropriate units to describe the relationship between the quantities.


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CORE CONTENTCluster Title: Reason quantitatively and use units to solve problemsStandard: N.Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantitiesConcepts and Skills to Master:

Determine whether whole numbers, fractions, or decimals are most appropriate Determine the appropriate power of ten to reasonably measure a quantity Determine the resulting accuracy in calculations Determine what level of rounding should be used in a problem situation


Understand basic units of measurements and their relationship to one another (e.g. a foot is smaller than a yard) Understand how to properly round and estimate

Academic VocabularyPrecision, accuracySuggested Instructional Strategies:

Discuss misconceptions in resulting calculations involving measurement, e.g. you cannot increase accuracy through calculation, only through more accurate measurement

Compare the difference between rounding at different places in a calculation and discuss which yields the best result

Discuss how mathematicians maintain precision through the representations that they use.

NCDPI Unpacking:

N-Q.3 Understand that the tool used determines the level of accuracy that can be reported for a measurement. For example, when using a ruler, you can only legitimately report accuracy to the nearest division. If I use a ruler that has centimeter divisions

Resources: Textbook Correlation: 2-10, 9-5, 9-6, CC-7

MARS Apprentice Task (N.Q.1 through 3):Leaky Faucet

MARS Apprentice Task (N.Q.1 through 3): Yogurt




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to measure the length of my pencil, I can only report its length to the nearest centimeter.


Ex. What is the accuracy of a ruler with 16 divisions per inch?

Problem Task


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CORE CONTENTCluster Title: Extend the properties of exponents to rational exponentsStandard: 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 = 51/3*3 to hold so (51/3)3must equal 5Concepts and Skills to Master:

Understand that the properties of integer exponents extend to rational exponents.SUPPORTS FOR TEACHERSCritical Background Knowledge

Properties of integer exponentsAcademic VocabularyExponent, rational, radicalSuggested Instructional Strategies:

Students can investigate by considering patterns such as (34) = 31 and ( ) = 31 to arrive at a conjecture.

NCDPI Unpacking:N-RN.1 In order to understand the meaning of rational exponents, students can initially investigate them byconsidering a pattern such as:

Resources: Textbook Correlation: CC-8


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Sample Assessment TasksSkill-based task Problem Task


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CORE CONTENTCluster Title: Extend the properties of exponents to rational exponentsStandard: N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.Concepts and Skills to Master:

Rewrite numbers with rational exponents in radical form. Rewrite numbers in radical form as number with rational exponents.


Properties of integer exponentsAcademic VocabularyExponent, rational, radicalSuggested Instructional Strategies:

NCDPI Unpacking:

N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational exponents. At this level, focus on fractional exponents with a numerator of 1.

Resources: Textbook Correlation: CC-9


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Which expression is equivalent to ?

A

B

C

D

Which expression is equivalent to ?

A

B

C

D

Problem Task


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CORE CONTENTCluster 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 equationSUPPORTS FOR TEACHERSCritical Background Knowledge

Use order of operations Simplify expressions using properties of algebra

Academic VocabularyConstant, coefficient, properties of operations and properties of equality, like terms, variable, evaluate, justify, viableSuggested 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

NCDPI Unpacking:

A-REI.1 Relate the concept of equality to the concrete representation of the balance of two equal quantities.Properties of equality are ways of transforming equations while still maintaining equality/balance. Assuming anequation has a solution, construct a convincing argument that justifies each step in the solution process withmathematical properties.

Resources: Textbook Correlation: 2-2, 2-3, 2-4, 2-5, 9-4, 9-5


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Sample Assessment TasksSkill-based taskJustify the equation solution by writing the property or reason why each step works.

3x + 7 = 123x + 7 – 7 = 12 – 73x + 0 = 53x = 5(3x)(1/3) = (5)(1/3)1x = 5/3x = 5/3

MULTIPLE CHOICE: Dave correctly solved the equation below using 4 steps.

Given: 5(3x + 2) – 7x = 34

Step 1: 15x + 10 – 7x = 34

Step 2: 8x + 10 = 34

Step 3: 8x = 24

Step 4: x = 3

Between which 2 steps did Dave use the Subtraction Property of Equality?

A Between Given and Step 1 B Between Steps 1 and 2 C Between Steps 2 and 3 D Between Steps 3 and 4

Ex. Solve 5(x+3)-3x=55 for x. Use mathematical properties to justify each step in the process.

Problem TaskWhen 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.


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CORE CONTENTCluster Title: Solve equations and inequalities in one variableStandard: A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Concepts and Skills to Master:

Write equations in equivalent forms to solve problems. Analyze and solve literal equations for a specified variable. Understand and apply the properties of inequalities. Verify that a given number or variable is a solution to the equation or inequality. Interpret the solution of an inequality in real terms.


Solving linear equationsAcademic VocabularyProperties of inequality as interpreted in table 5 of the CCSS glossary page 90Suggested Instructional Strategies:

Solve specified variables, using common formulas used in science, economics, or other disciplines

Examine and prove why dividing or multiplying by a negative reverses the inequality sign.

Use applications from a variety of disciplines to motivate solving linear equations and inequalities

NCDPI Unpacking:

A-REI.3 Solve linear equations in one variable, including coefficients represented by letters.A-REI.3 Solve linear inequalities in one variable, including coefficients represented by letters.


2-8, 3-2, 3-3, 3-4, 3-5, 3-6


Solve 2(x + 4) – 3 ≥ 4x – 2

Problem TaskThe perimeter of a rectangle is given by P = 2W + 2L. Solve for W and restate in words the meaning of this new formula


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A school purchases boxes of candy bars. Each box contains 50 candy bars Each box costs $30

How much does the school have to charge for each candy bar to make a profit of $10 per box?

A $0.40B $0.50C $0.80D $1.25

Ex. Solve, Ax +B =C for x. What are the specific restrictions on A?

Ex. What is the difference between solving an equation and simplifying an expression?

in terms of the meaning of the other variables.

Ex. Grandma’s house is 20 miles away and Johnny wants to know how long it will take to get there using various modes of transportation.

a. Model this situation with an equation where time is a function of rate in miles per hour.b. For each mode of transportation listed below, determine the time it would take to get to Grandma’s.

Mode of Transportation Rate of Travel in mph Time of Travel hrs.bike 12mphcar 55mphwalking 4mph

Ex. A parking garage charges $1 for the first half-hour and $0.60 for each additional half-hour or portion thereof. If you have only $6.00 in cash, write an inequality and solve it to find how long you can park.

Ex. Compare solving an inequality in one variable to solving an equation in one variable, also compare solving a linear inequality to solving a linear equation.


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CORE CONTENTCluster Title: Solve equations and inequalities in one variableStandard: A.REI.4:a. Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.Concepts and Skills to Master:

Use the method of completing the square where the leading coefficient of x2 is one (ax2 + bx + c where a=1) to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions.

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.


Square Root Perfect Square Trinomials Factoring Quadratic Equations Zeros, Roots, Solutions

Academic VocabularyProperties of inequality as interpreted in table 5 of the CCSS glossary page 90Suggested Instructional Strategies:Use Algebra Tiles to develop the understanding of what is meant by “completing the square”

Resources: Textbook Correlation: 9-5 Using Algebra tiles to complete the square Video of Completing the Square

Completing the square lesson plan

http://www.acoe.org/acoe/files/EdServices/Completing%20the%20Square%20LessonV7PDF.pdf

http://www.youtube.com/watch?v=GyCuj1hx_zc

http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm


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Solve the following by completing the square

x2 + 6x – 7 = 0

Problem Task

Documents

Algebra 1 Unit 3: Systems of Equations - wikispaces.netwysiwyg.cmswiki.wikispaces.net/file/view/Math I Unit 4... · Web viewInclude equations arising from linear and quadratic functions,