See cover page for notes on pacing guide structure. 2010-2011 Pre-Algebra (including Advanced) Pacing Guide Big Idea(s): Algebra is distinguished from arithmetic by the systematic use of symbols for values. Writing and evaluating expressions with algebraic notation follows the same rules/properties as in arithmetic expressions. These expressions and equations can be used to model real world situations. 8/23-27 (5 days) Essential Question Set #1 How are algebraic expressions simplified? How do the basic properties help us understand simplifying expressions? What mathematical tools do we use to help us describe and understand the world? Concepts/Skills/Standards -Writing expressions -Evaluating expressions -Drawing a diagram or chart -Using Formulas Standards: AF1.1; AF1.2(PWR) Expressions can be simplified as a step to solving equations or inequalities. 1 First Day Policies and Procedures 1-1 Variables and Expressions 1-2 Order of Operations 1-3 Evaluate Expressions Big Idea(s): The rational numbers include integers and fractions. These numbers can be manipulated, represented on the number line and computed in the manner known to whole numbers. Essential Question Set #2 Optional Diagnostic: MDTP How is the rational number line made? Written Response item: Number How does computing with all rational numbers differ from Line problem (p. 36) Also just the whole numbers. How is it the same? appropriate for chapter 5. 1-4 Integers and Concepts/Skills/Standards -Computing with all integers Absolute Value -Absolute value 1-5 Add Integers -Using models to demonstrate computing with integers 1-6 Subtract Integers Standards: NS1.2(PWR) 1-9 Multiplying and ------------------------------------------------------------------------ Dividing Integers Essential Question Set #3 Enrichment 1-4 Absolute How do we use math to help us describe and understand Value Equations patterns? Supplemental Activity: Concepts/Skills/Standards Math Behind The Trick. -Making conjectures Family Math for Middle -Writing expressions School AF1.1, 1.2, 1.3, 1.4 -Drawing a diagram or chart -Using Formulas 1-7 Inductive reasoning Standards: AF1.1; AF1.4 1-8 Finding a Pattern 8/30-9/10 (9 days)including Chapter 1 review & Test Big Idea(s):
One applies the basic properties to simplify expressions. The addition and multiplication properties for equations are used to solve equations completely. 9/13-17 (5 days) 9/20-29 (8 days) Essential Question Set #4 How do the associative and commutative properties help us in mathematics? How does the distributive property add to our methods for simplifying an expression? Simplifying variable expressions involves which properties of mathematics? How can you model the simplification of an algebraic expression? Concepts/Skills/Standards -Commutative property -Associative property -Distributive property -Variables and expressions Standards: AF1.3(PWR); AF1.4 Essential Question Set #5 How do the addition and multiplication properties of equations help us in mathematics? How are equations represented in real contexts? What is a formula? Concepts/Skills/Standards - Identifying the basic properties -Solving equations -Evaluating formulas Standards: AF4.0(PWR); AF1.2(PWR) 2 2-1 Properties of Numbers 2-2 Distributive property 2-3 Simplify Variables 2-4 Variables & Expressions 2-5 Solving Eq: (+/ ) 2-6 Solving Eq: (x/ ) 2-7 Reasoning strategy: Try, Test, Revise And 3-4 Using Formulas 3-5 Eq w/ Decimals 3-6 Eq w/ Decimals Big Idea(s): Solving, simplifying and graphing inequalities utilizes properties and methods similar to that for equations. The solution is a set of values for which the inequality is true. 9/30-10/6 (5 days) Essential Question Set #6 How does the addition property for inequalities help us solve inequalities? How does the multiplication property for inequalities help us solve inequalities? Why and how does the graph of an inequality differ from that of an equality? How are inequalities represented in real contexts? Concepts/Skills/Standards -Graphing inequalities -Solving one-step inequalities Standards: AF4.0(PWR) 2-8 inequalities and their graphs 2-9 Solving 1-step inequalities by adding or subtracting 2-10 Solving one-step inequalities by multiplying or dividing Enrichment 2-8 Conjunction & Disjunction Big Idea(s): The fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product (up to ordering of the terms) of prime numbers. Exponents allow the
representation of the product of the same factor more than once as a base and a power. Multiplying or dividing numbers with the same base written in exponential form requires the addition or subtraction of the exponents only. 10/7-29 (17 days) Includes Benchmark Review and Test - 2 Parent Conference days 10/15-16 Essential Question Set #7 How does factorization provide more than one way to represent a number? How is prime factorization different than factorization? What is the greatest common factor of any two numbers? What does x y mean? How do you define a rational number? /Standards -Factoring a number into factors -Factoring a number into primes Standards: NS2.3(PWR) Essential Question Set #8 What is the product of x a x b? What is the quotient of x a x b? How do you write 342 and 0.352 in scientific notation? -using the rules for exponents -Convert numbers between standard and scientific notation 3 Support topics (4-1 to 4-5): 4-1 Divisibility and factors 4-2 Exponents 4-3 Prime factorizations and GCF 4-4 Simplifying Fractions 4-5 Reasoning Strategy: Account for all possibilities 4-7 Exponents and Multiplication 4-8 Exponents and Division 4-9 Scientific Notation 5-9 Powers and products of quotients. Chapter 4 Review and Test Standards: NS1.1,NS2.1; AF2.2 Benchmark #1: Essential Question Sets #1-8. Suggested Window: October 28-29/ Due in Edusoft November 6 Trimester 1 ends November 12 Big Idea(s): The rational numbers include whole numbers, the set of integers, fractions and decimals. All rational numbers can be expressed as the ratio of an integer and a whole number. Factorization is used to create equivalent fractions. Equivalent fractions are used to compare, simplify and to add fractions with unlike denominators. 11/1 to 12 (9 days) Essential Question Set #9 How is comparing fractions different than comparing whole numbers? How does factorization help us simplify fractions? How does one add fractions? Why are some harder to add than others? How is dividing fractions different than multiplying fractions? /Standards - Factoring whole numbers. - Adding/subtracting fractions & mixed Numbers - multiplying/dividing Fractions & mixed Numbers - Solving equations Standards: NS1.1; NS1.2(PWR); NS1.3; NS1.4; NS2.2(PWR); MG1.1; 4-6 Rational Numbers 5-1 Comparing and ordering Fractions 5-2 Fractions And Decimals 5-3 Adding and Subtracting Fractions 5-4 Multiplying and Dividing Fractions Supportive topic: 5-5 Using Customary units of Measurement Optional Diagnostic: MDTP Written Response item: Number Line problem (p. 36-38) Big Idea(s): Solving equations with fractions utilizes the same algebraic rules as with whole numbers.
5-7 Solving Equations by Adding or Subtracting Fractions 5-8 Solving Equations by Multiplying fractions 11/15 to 19 (5 days) Include Ch 5 Review & Test Essential Question Set #10 What are the steps to solving an equation that contains an addition expression? What are the steps to solving an equation that contains a fractional coefficient? Concepts/Skills - Solving equations -Solving equations containing fractions Standards: AF4.0(PWR); NS1.2(PWR) Big Idea(s): Proportionality is a mathematical relation between two quantities. Two equivalent ratios form a proportion. Proportions can be used to solve for unknown quantities. Application of proportion utilize formulas and geometric relationships to describe, model and design to scale. 11/29 to 12/3 (5 days) Essential Question Set #11 What is a unit rate? What does it mean to be a proportion? How does knowing a rate allow us to find one quantity of the rate if we know the other? How is proportionality applied to geometry? How does one apply proportions to scale drawings? -Rates and ratio -Apply rates to solve problems -Solving proportions -Solving similar figures -Solving scale problems Standards: AF4.2(PWR); MG1.3(PWR) Supportive topic: 6-1 Ratios and Rates -------------------------------- 6-1A Converting between measurement systems 6-2 Proportions 6-3 Similar Figures and Scale Drawings Extension Topics: 6-3A Dilations 6-4 Probability Probability is discussed more deeply in chapter 12. Big Idea(s): Percents are a special proportion where the ratio (relative size) is always described as a part of 100. Percents can be applied to many everyday uses such as percent of change, simple interest and tips. 4
12/6 to 12/17 (10 days) Includes Review and Ch. 6 test days Essential Question Set #12 What does a percent of a quantity mean? How do you convert from a fraction or decimal part of a quantity to percent? How do you find the percent of a quantity? How do we use proportions to solve for unknown amounts in percent problems? -Percent -Converting between decimals, fractions, percents -Using proportions to solve for percents or missing amounts when percent is known -Using equations to solve for parts or percents. -Convert between fractions, decimals, percents -Finding percent of a quantity -Solving percent equations Standards: NS1.3; AF 1.1 Essential Question Set #13 What does percent of change mean and how is it calculated? How do you find simple interest? How do you figure the tip at a restaurant? -Calculating percent of change -Calculating simple interest -Calculating the tip. Standards: NS1.7(PWR) 6-5 Fractions Decimals, Percents, oh my! 6-6 Proportions and Percents 6-7 Percent and Equations Diagnostic: MDTP Written Response item: Cake problem (p. 24-26) 6-8 Percent of Change 6-9 Markup and Discount 6-10 Making a Table for Reasoning Big Idea(s): Solving two-step equations requires application of the basic properties for numbers and equations to isolate the variable and leave its coefficient as 1. The solution can be checked in the original equation. 5
1/3 to 1/21 (14 days)includes Review and Ch. 7 test days Essential Question Set #14 How do you know if an equation is a one or two-step equation? How do you decide which property to use first when solving a two-step equation? Describe the type of applied problems that are represented by two-step equations. -Solving two-step equations -writing two-step equations to solve real life problems -use logical reasoning to solve problems -using the properties to transform formulas Standards: AF1.3; AF4.1; AF4.2 Essential Question Set #15 In what ways are solving inequalities like solving equations? In what ways are solving inequalities unlike solving equations? -Solving two-step inequalities -writing two-step inequalities to solve real life problems Standards: AF1.3; AF4.1; AF4.2 7-1 Solving 2-Step eq. 7-2 Solving multi-step eq. 7-3 Solving equations with Fractions and Decimals 7-4 Equations and Reasoning 7-5 Solve equations with Variable on Both Sides 7-7 Transforming Formulas 7-8 Simple and Compound Interest 7-6 Solving 2-Step inequalities 7-6A Solving Compound 2- Step inequalities 7-8A Spreadsheet activity Multi-step application: Density (D=m/v) Big Idea(s): An equation in two variables describes a relationship. In a function, each member of the domain is paired with one member of the range. A function can be represented in words, algebraically or in a graph. 6
1/24 to 2/11 (15 days) Incl. Review + Ch. 8 test Essential Question Set #16 What is the difference between domain and range? How does an equation in two variables differ from an equation in one variable? What makes a function a linear function? What does the slope of a line represent? -Identifying a relation that is a function -Finding solutions to an equation in two variables given domain values. -Graphing linear equations. -Identifying slope and y-intercept on a graph and in a linear equation. -use logical reasoning to solve problems Standards: AF1.5; AF3.3(PWR); AF3.4(PWR); AF4.2(PWR) Essential Question Set #17 How does the graph of a nonlinear function differ from a linear function s graph? -Graphing nonlinear functions Standards:AF3.1; AF3.2 8-1A Relating graphs to events 8-1 Relations and Functions 8-2 Equations with 2 Variables 8-2A Direct Variation 8-3 Slope and y-intercept 8-4 Writing Rules for Linear Functions Extension Topics: 8-5 Scatter Plots (in pacing guide in April) 8-6 Solve By Graphing -- Note: AF1.5 is only covered in 8-1A AF3.4 Is covered only in 8-2A 13-2 Graphing Nonlinear Functions (especially problem 22) Extension Activity: 13-3 Growth and Decay Big Idea(s): In plane geometry, the basic elements of the 2-dimensional plane are points, lines and figures constructed of lines, both straight and curved. Parallel lines in a plane do not intersect. Intersecting lines form angles. Angles can be classified by degrees. Polygons are enclosed figures formed by line segments in a plane. 2/14 to 2/25 (8 days) Incl. Review + Ch. 9 Test Days Essential Question Set #18 How does the number of vertices relate to the number of diagonals in a polyhedron? What are the possible ways three planes can intersect in space? (MG3.6) What would a two-dimensional design look like that could be folded to make a cylinder, cone or prism (one design per shape)? Under what conditions would you be able to prove two geometric figures are congruent? -Identifying basic plane figures. -Measuring angles Standards:MG3.4; MG3.5; MG3.6(PWR)-Not fully covered. Support Topics : 9-1 Points, Lines and Planes 9-2 Drawing and Measuring Angles 9-4Draw a Diagram 9-5 Congruence 7
Essential Question Set #19 What distinguishes a parallelogram from other quadrilaterals? What distinguishes a rectangle from other parallelograms? What distinguishes a square from a rectangle? What do you call an equilateral parallelogram that is not a square? Will it call you a funny name, too? -Identifying and classifying polygons Standards: (4 th and 5 th grade standards) Three-dimensional measure of capacity is called volume. 8 Support Topic 9-3 Classifying Polygons Big Idea(s): The set of all points equidistant from a point in a plane is called a circle. Pi is the ratio of circumference to diameter. Pi is about 3.14, rounded. Pi is useful in circle area and circumference formulas, A=πr 2 ; C=πd. This section is to be used as a review of prerequisite knowledge, if needed. 2/28 to 3/1 (2 days) Essential Question Set #20 How do you find the area and perimeter of a circle? How is pi different than other measures used in area and perimeter formulas? What is a chord? What is the longest chord of a circle? -Parts of a circle -Area and circumference of a circle Standards: MG2.1(Power) 9-6 Circles Circles are not specifically mentioned in the 7 th grade standards but are part of the skill set needed for working with cylinders and can be considered among the common geometric figures. Extension Topics: Big Idea(s): Geometric construction is the method of drawing plane figures with only a compass and straight edge. A translation is a transformation that moves all the points in a geometric figure the same distance and the same direction, keeping the shape intact while moving position. Insert into Pacing, if used. Essential Question Set #21 (Extension) What is a geometric construction? What is a translation? Is there symmetry when you reflect a geometric figure? -Construction with compass and straight edge -Transformations on the Cartesian plane. Standards: MG3.1; MG3.2 Extension Topics: 9-7 Constructions 9-8 Translations 9-9 Symmetry and Reflections Benchmark #2: Essential Question Sets #9-18 & 20. Suggested Window: March 2-4/ Due in Edusoft March 11 Trimester 2 ends March 4 Big Idea(s): One, two and three-dimensional figures can be described through their measure. One-dimensional measure is called length (including perimeter and circumference). Two-dimensional measure is called area.
3/7 to 3/18 (10 days) Incl. Review + Ch. 10 Test Days Essential Question Set #22 How do you find the area of quadrilaterals? Why is there a ½ in the formula for the area of trapezoids and triangles? How do you find the area of circles? What is the difference between surface area and area? What is the basic formula for volumes of polyhedrons? -Finding area and perimeter of 2-D shapes -Finding surface area and volume of 3-D shapes -Making a 2-D model that can be folded to be a specific 3-D figure. Standards: MG2.1(PWR); MG2.3 (partially covered) (If extension 10-8): MG3.5 Not covered: MG3.6 10-1 Area: Parallelograms 10-2 Area: Triangles and Trapezoids 10-3 Area: Circles 10-5 Surface Area: Prisms, Cylinders 10-6 Surface Area: Pyramids Cones, Spheres 10-7 Volume: Prisms and Cylinders Extended Topics: 10-8 Make a Model 10-9 Volume: Pyramids, Cones, Spheres Also: Enrichment 10-3 to 10-9 (for advanced) Supplement: MG3.6 Big Idea(s): When the lengths of a figure are multiplied by a scale factor, the area is multiplied by the square of that factor and the volume is multiplied by the cube of that factor. Insert into Pacing, if used. Essential Question Set #23 (Option: Move to be in place of Set #29- After CST) What happens to the area of a polygon if the length of each side is multiplied by a number, resulting in a similar and larger polygon? What happens to the volume of a polyhedron if all lengths are increased by multiplying by the same number? -Finding area and perimeter of 2-D shapes -Finding surface area and volume of 3-D shapes -Making a 2-D model that can be folded to be a specific 3-D figure. Standards: MG2.3 (MG2.4 - enrichment 3-7 only) Not clearly in text. This is an extended topic that is used as deeper analysis of ratio, proportion and geometric figures. MG 2.3 specifically mentions this concept. Enrichment 3-7 Square It 1 square cm = square mm? Big Idea(s): Extracting the square root of a number is the inverse of finding the square of a number. The irrational numbers include the non-perfect square roots and they can be represented by nonterminating, non-repeating decimals. 9
3/21 (1 day) Essential Question Set #24 What is the length of the side of a square with an area of 81 square units? 24 square units? In between what two whole numbers is the square root of 30? 95? -Taking the square root of a number. -Squaring a number Standards: NS1.4(Key, not PWR); NS2.4 11-1 Square Roots and Irrational Numbers Big Idea(s): The Pythagorean theorem states that for a right triangle, the sum of the squares of the two legs are equal to the square of the hypotenuse. The Pythagorean Theorem can be used to find the unknown length of sides of a triangle given the measure of the other two or in some cases one other length. 3/22 to 3/25 (4 days) Essential Question Set #25 What relationship does the Pythagorean Theorem describe? How do we use the Pythagorean theorem to find unstated dimensions of a triangle? -The Pythagorean Theorem -Using the Pythagorean Theorem -(Extended) The midpoint Formula -(Extended) The distance Formula Standards: MG3.3(PWR); MG3.2 (if cover midpoint & distant formula) 11-2 Pythagorean Theorem Extended Topics: 11-2A The Pythagorean Theorem and Circles 11-3 Distance and Midpoint Formulas Big Idea(s): Data can be displayed in various ways to convey information. 10
Essential Question Set #26 How does a frequency table help to organize data? What does a line plot help us to see? A histogram is used for what type of data? What do Box-and-Whisker-Plots help us to see? How are scatter plots used and what is a line of best fit? -Frequency Tables -Line plots -Histograms -Upper quartile, lower quartile, the minimum and maximum of a data set Standards: SDAP1.1; SDAP1.2; SDAP 1.3 3/28 to 4/1 (5 days) 11 12-1/12-1A Frequency Tables, Line Plots, and Histograms. 12-2/12-2A Box-and-whisker Plots 8-5 Scatter Plots (unless covered in January as extension) Big Idea(s): Application of mathematical ideas requires looking for patterns, accessing prior knowledge and communicating ideas using standard methods as well as self-devised methods. 4/4 to 5/6 (15 days) Essential Question Set #27 (Reasoning) When I approach a problem, how do I access prior learning if memory is only partially helpful? How can a problem be stated algebraically? What relationships do I see in this problem and how does this imply a strategy? Will estimation help me see if my answer is reasonable? Does my answer express the solution clearly? -Analyzing problems -Checking for reasonableness -Determining how to break a problem into smaller parts Standards: MR1.0; MR2.0; MR3.0 Problem Solving and Review of essential standards. Various sections will be reviewed as determined by site planning. Emphasis on problem solving and independent student thinking. Diagnostic: MDTP Written Response item: Bent Wire problem (p. 39-41) Area/perimeter, quadrilaterals & Circles April 4- June 8 Three possible areas of focus for during and Post-CST are 1. Application of Mathematical Skills. See suggested focus below. 2. Extension Topics. See topics below. 3. Review/Remediation. Use benchmark #3 as a formative diagnostic. Differentiation in small groups for individual needs is recommended. Topics are listed as suggestions. Order and pacing is to be site determined. Application/Extension Topics --------CST Window is April 4-15-------- ---Spring Break is April 18-29--- Big Idea(s): Theoretical probability is the likelihood of an event occurring based on possibility spaces using mathematical counting theory. Experimental probability is the likelihood of an event occurring based on possibility spaces derived from repeated trials.
5/9 to 5/20 (10 days) Essential Question Set 28 (Optional Extension) Extension topics (for both How is experimental probability different than classes): theoretical probability? 12-4 Counting Outcomes How are the number of possible choices or outcomes and theoretical probability from a single trial counted? 12-5 Independent and dependent events How is a tree diagram used? 12-6 Permutations and What is the difference between an independent and a combinations dependent event? 12-7 Experimental How are the possible outcomes counted for dependent Probability events? 12-8 Random samples and How are the possible outcomes counted for surveys independent events? -Counting Outcomes and theoretical probability -Independent and dependent events - Permutations and combinations -Making and reading Tree diagrams Standards: 6 th Grade standards Big Idea(s): Patterns can be described algebraically. This description can be used to predict specific terms of a pattern. Polynomials are algebraic expressions with one or more terms. Computation with polynomials generalize the rules of arithmetic computation. Sums and products of polynomials are polynomials. Essential Question Set 29 (Optional Extension) (option: replace with Question set #23) 5/23 to 6/8 (11 days) Includes Benchmark #3 How can a pattern be described with algebra? What does the graph of a nonlinear function look like? What is a polynomial? How do I add polynomials? How do I multiply polynomials? -Identifying a pattern -Writing algebraic expressions for patterns -Graphing nonlinear functions -Computing with polynomials (combining like terms, the distributive property) Standards: AF1.1; AF1.2(PWR); AF1.3 (PWR); AF3.1 Diagnostic: MDTP Written Response item: Growth patterns - Island problem (p. 30-32). Benchmark 3: Essential Question Sets #22-26 June 6 Due Edusoft June 8 Trimester 3 ends June 8 Note: Assignments may have been given from sections 13-1 and 13-3 earlier in the year. Topics are still relevant. Assignments may come from other sources. Suggested: Activities that compare linear and nonlinear growth. 13-1 Patterns and sequences 13-2 Graphing Nonlinear Functions 13-3 Growth and Decay 13-4 Polynomials 13-5 Adding and Subtracting Polynomials 13-6 Multiplying a polynomial by a polynomial Application of Skills: Problems involving rates such comparative cost problems can be solved using algebraic analysis. For first year Algebra students, the focus will be on problem solving 12
strategies and communication of method and solution through algebra. Simple system of linear equation problems are ideal for giving students practice using algebra and graphing to communicate methods and results of problem solving. Site determined Essential Question Set (Application of Math) How is math used to solve applied problems? How are equations and inequalities used to solve various types of applied problems? In what ways do models, such as drawings or physical models help us in problem solving or understanding a concept? How does a solution in algebra translate into a graphic representation? -Applying algebra to word problems -Accessing notes and other text-based examples for use in problem solving -Using models to solve problems -Using graphs to explain a solution or solve a problem Standards: M.R. Standards Site Planned 13