The Academy of the Most Blessed Sacrament ALGEBRA I CURRICULUM OUTLINE

(last updated August 9, 2019)                    (177 days)

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

I

REAL NUMBERS AND ALGEBRAIC EXPRESSIONS

 

 

 

 

first trimester – 15 days

 

 

 

AI1d1

Variables and Grouping Symbols

(parentheses, brackets and fraction bars; easy oops with substitution)

1.1, 1.2

1.2

1.1, 1.2

AI1d2

Translate Words into Symbols

(also, complete word phrase when given additional information) 

1.4

1.1

1.3

AI1d3

Classifying, Ordering, and Rewriting Real Numbers

(integer, real number, whole number defined; graphing and ordering real numbers on a number line;  rewriting repeating decimals as fractions)

1.8

0.2 part 1

2.1

AI1d4

Simplifying and Estimating Square Roots

(rounding square roots to nearest integer)

1.8

0.2 part 2

2.7

AI1d5

Opposites and Absolute Values

(simple sums, differences, comparisons, simple absolute value equations)

1.9

 

 

AI1d5

Adding and Subtracting Real Numbers

(no algebraic expressions yet)

 

0.3, 0.4

2.2, 2.3

AI1d5

Multiplying and Dividing Real Numbers

(no algebraic expressions yet; find a number between any two rational numbers)

 

0.3, 0.5

2.4, 2.6

AI2d1

Basic Assumptions 

(3 properties of addition and multiplication, including closure; simple addition of whole numbers, decimals, and mixed numbers;  simple simplification of algebraic expressions, addition and multiplication only, no combining like terms)

2.1

 

 

AI2d2

Adding and Subtracting Algebraic Expressions

(addition and subtraction of positive and negative integers, decimals and fractions, simplification of algebraic expressions, evaluation with substitution of integers, algebraic expressions that contain opposite of sum and difference; no combining like terms)

2.2, 2.3, 2.4

 

 

AI2d3

Properties of Numbers

(The RuST properties, substitution property, four addition properties, five multiplication properties; use properties to justify each step when simplifying or evaluating an expression)

 

1.3

 

AI2d4

Distributive Property

(use as short cut; simplifying algebraic expressions)

2.5

1.4

2.5

AI2d5

Rules for Multiplication

(includes algebraic expressions with combining like terms;  integers only)

2.6

1.4

 

AI2d6

Reciprocal of a Real Number

(simplify algebraic expressions, including fractions and distributive property) 

2.8

 

 

AI2d6

Dividing Real Numbers

(simplify algebraic expressions;  all involve integers - - no fractions or decimals)

2.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

II

EQUATIONS

 

 

 

 

first trimester – 18 days

 

 

 

AI1d1

Equations

(solve simple equations and word problems over given domain.  Introduce “is element of” symbol)

1.3

1.5

 

AI1d2

Transforming Equations: Addition and Subtraction

(integers;  opposite)

3.1

2.2

3.1

AI1d3

Transforming Equations:  Multiplication and Division

(includes simple fractions and mixed numbers)

3.2

2.2

3.1

AI1d4

Using Several Transformations

(simple two-step problems; fractional expressions; distributive property)   

3.3

2.3

3.2, 3.3

AI1d5

Translate Sentences and Problems Into Equations

(includes quantities grouped in parentheses, formulas, defining variable with “let”; include two-step translations with sketch)

1.5, 1.6

2.1

1.4

AI1d6

A Problem Solving Plan

(5 step Polya-type plan; answers given as choice of 3; solving problems involving geometry)

1.7, 3.4

 

1.5

AI1d7

Problem Solving:  Consecutive Integers and Diagrams

(word problems)

2.7

2.3

 

AI2d1

Equations with Variable on Both Sides

(includes fractional expressions and distributive property;  identity; no solution)

3.5

2.4

3.4

AI2d2

Absolute Value Equations

(includes real-world distance problems and tolerance problems)

 

2.5

 

AI2d3

The Fraction Destroyer

(solving linear equations more efficiently by removing fractions)

supplement

 

 

AI2d4

Transforming Formulas

(solving literal equations; stating restrictions;  Be careful, some problems may require factoring)   (Carter includes dimensional analysis, but we will postpone this to the unit on rates and ratios)

4.7

2.8

3.8

AI2d5

Problem Solving Using Charts

(rate problems – calories per fruit, rides per ticket, etc.)

3.6

 

 

AI2d6

Cost, Income, and Value Problems - Introductory

(using charts; includes dimes and nickels, profit as selling price minus buying price)

3.7

 

 

AI2d7

Cost, Income, and Value Problems - Challenging

(also, more challenging consecutive integer problems)

3.7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

III

RATES, RATIOS AND PERCENTS

 

 

 

 

first trimester – 14 days

 

 

 

AI1d1

Ratios

(convert simple measurements (days and weeks, etc) to reduce ratio to simplest form;  ratios of perimeters and areas of rectangles; real-world problems;  ratio of x to y)

7.1

2.6

3.5

AI1d2

Proportions

(Solve with cross-multiplication;  some numerators and denominators are binomials; some have variables on both sides)

7.2

part 1

2.6

3.5, 3.6

AI1d3

Real-world Problems with Proportions

(Include classic forestry ratio problems;  also, more challenging “find ratio of x to y” problems;  confirming that a pair of ratios are proportional)

7.2

part 2

2.6

 

AI1d4

The Mean Proportional

(also, using the mean proportional to find the missing number in a geometric sequence)

7.2

part 3

2.6

 

AI1d5

Percents

(convert among fractions, percents, and decimals; solve basic percentage word problems;;  Dolciani omits the percent proportion; Carter covers the percent proportion.)

7.5

 part 1

0.6

 

AI1d6

Real-world Problems with Percents

(solve percentage real-world problems with the percent proportion: solve equations using the decimal destroyer)

7.5

 part 2

0.6

3.7

AI2d1

Percent Change Problems

7.6

2.7

post 3.7

AI2d2

Compound Proportions

supplement

 

 

AI2d3

Dimensional Analysis

(include unit rates;  conversions between customary and metric units of measure)

7.3 extra

2.8

 

AI2d4

Mixture Problems

(“raisins, nuts and mixture”, “dilute 70% acid to 40% acid”, with charts)

7.7

2.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

IV

GRAPHING LINEAR EQUATIONS

 

 

 

 

second trimester – 19 days

 

 

 

AI1d1

Equations in Two Variables

(determine whether an ordered pair is a solution of an equation in two variables; isolate a given variable;  use guess-and-check to find a list of ordered-pair solutions)

8.1

 

 

AI1d2

Points, Lines, and Their Graphs

(plot points on coordinate plane; graph a line by finding its x and y intercepts)

8.2

3.1

4.1

AI1d3

Slope of a Line

(use slope formula when given coordinates of two points; find any two coordinates when given equation;  draw line on coordinate plane when given point and slope, using “rise over run”; identify “no slope” and “slope equals zero”.  Carter also has exercises in finding a missing coordinate.)   

8.3

part 1

3.3

4.4

AI1d4

Linear Equations

(recognizing linear equations, converting equations to standard form, finding a missing coordinate, determining whether three points are collinear)    

8.3

part 2

3.3

4.2

AI2d1

The Slope-Intercept Form of a Linear Equation

(graph equations written in slope-intercept form;  transform equation to slope-intercept form and graph; determine whether pair of lines whose equations are given are parallel or perpendicular.)

8.4

4.1. 4.4

4.5

AI2d2

Determining the Equation of a Line

(find equation of line when given slope and y-intercept;  when given slope and coordinates of one point;  when given two points;  when given other descriptions)

8.5

part 1

4.2

5.1, 5.2

AI2d3

Point-Slope Form

(Write equation in point-slope form when given slope and coordinates of one point, and convert to slope-intercept form or standard form.)

 

4.3

5.3, 5.4

AI2d4

Determining Equation of a Line from a Mix of Information

8.5

part 2

 

 

AI3d1

Relations

(four ways to represent a relation)

8.6

1.6

 

AI3d2

Functions

(determining whether a relation is a function;  equations as functions;  simple function notation)

8.7

1.7 part 1

1.6, 1.7

AI3d3

Composite Functions

(function notation with simple expressions within function, composite functions; finding the zero of a function)

8.7

1.7 part 2

 

AI3d4

Inverse Functions

(finding the inverse of a relation;  finding and graphing the inverse of a linear function)

 

4.7

 

AI3d5

Direct Variation

(finding the constant of variation;  include real-world problems)

8.9

3.4

4.6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

V

EXPONENTS AND POLYNOMIALS

 

 

 

 

second trimester – 23 days

 

 

 

AI1d1

Exponents

(write multiplication expressions with variables in exponential form; evaluate exponential expressions by substituting, including negative bases; find area of rectangle with variable expressions as lengths, evaluate exponential expressions with and without parentheses, evaluate with substitution)

4.1

7.1

 

AI1d2

Adding and Subtracting Polynomials

(horizontal and vertical; expressions with multiple variables and variety of powers; vocabulary: degree of variable, of monomial, and of polynomial)

4.2

8.1

9.1

AI1d3

Real-World Problems Adding and Subtracting Polynomials

(solve equations with polynomials, where 2nd degree terms drop out; solve real-world problems involving geometry and consecutive integers)

4.2

 

 

AI1d4

Multiplying Monomials

(product of power rule; all fractions simplify to integers;  also, sums of products of monomials; advanced problems:  exponents are polynomials)

4.3

7.1

8.1

AI1d5

Real-World Problems Multiplying Monomials

(find perimeters and surface areas of geometric figures by multiplying and adding monomials;  simplify expressions that include exponents with variables)

4.3

 

 

AI2d1

Powers of Monomials

(power of product rule; power of power rule; evaluation of expressions using substitution;  advanced problems:  exponents contain variables)

4.4

7.1

 

AI2d2

Multiplying Polynomials by Monomials

(distributive property; distribute, then combine like terms; solve equations in which the powers greater than one conveniently drop out)

4.5

8.2

9.2

AI2d3

Multiplying Polynomials by Polynomials

(horizontally and vertically;  FOIL;  product of binomials and trinomials; rewrite product in standard form;  advanced problems:  solve equations)

4.6, 5.4

8.3

9.2

AI2d4

Special Products

(recognize patterns:  binomials squared;  product of sum and difference)

 

8.4

9.3

AI3d1

Division Properties of Exponents: Quotient of Powers

(include products of fractions containing variables with exponents;  include problems that involve finding the missing factor)

5.2, 6.2 part 1

7.2 pt 1a

8.2

AI3d2

Division Properties of Exponents: Powers of Quotients

(postpone Dolciani “level B” factoring problems to later unit)

6.2

part 2

7.2 pt 1b

8.2

AI3d3

Negative Exponents

(evaluate expressions with negative exponents and exponents of zero, rewrite expressions with positive exponents)

7.9 

7.2 pt 2

8.3

AI3d4

Scientific Notation

(convert between standard form and scientific notation;  rewrite numbers in expanded notation with powers of ten;  multiply and divide numbers written in scientific notation)

7.10

7.4

8.4

AI4d1

Rate-Time-Distance Problems

(motion in opposite directions; motion in same direction; use a chart.)

4.8

2.9

 

AI4d2

Area Problems

(flower garden surrounded by walkway, for example.  Use Polya’s method;  quadratic portions of equation usually cancel each other out.) 

4.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

VI

FACTORING POLYNOMIALS

 

 

 

 

second trimester – 22 days

 

 

 

AI1d1

GCF, and Monomial Factors of Polynomials

(Review GCF of algebraic expressions;  then, factor a polynomial by finding the GCF of the terms, and divide polynomial by monomial by finding the GCF of the polynomial and factoring the numerator.)

5.1 and 5.3

8.5 part 1

 

AI1d2

Differences of Two Perfect Squares

(factoring D2PS;  also, factoring with a combination of GCF and D2PS)

5.5

part 1

8.8 part 1

9.7

AI1d3

The Zero Product Property – Basic Problems

(solving equations when the polynomial has already been factored;  multiple roots;  isolating the zero)

5.12 part 1

8.5 part 2

9.4

AI1d4

The Zero Product Property – Factoring First

(solving equations by factoring a polynomial and then applying the zero product property)

5.12 part 2

8.8 part 2

 

AI1d4

Advanced Cases of D2PS

(expressions within parentheses;  exponents with variables)

5.5

part 2

 

 

AI2d1

Perfect Square Trinomials

(factoring PST)

5.6

part 1

8.9

9.7

AI2d2

Advanced Cases of PST

(expressions within parentheses, and combos with GCF or D2PS)

5.6

part 2

 

 

AI2d3

Factoring Trinomials with Leading Coefficient of One

(“reverse FOIL”;  both “c positive” and “c negative”)

5.7 and 5.8

8.6 part 1

9.5

AI2d4

Solving Equations by Factoring Trinomials

(Use “reverse FOIL” to factor, then use zero product property to solve.)

 

8.6 part 2

9.5

AI2d5

Special Problems Related to Factoring

(Problems with geometry;  finding all possible coefficients that make a trinomial factorable)

5.7 and 5.8

advanced

 

 

AI3d1

Factoring with the Box Method

(factoring trinomials with a leading coefficient other than one with the “box” method)

5.9

8.7

9.6

AI3d2

Factoring by Grouping

(factoring trinomials with a leading coefficient other than one by grouping)

5.10 part 1

8.5 part 2

 

AI3d3

Solving Equations Using Several Methods of Factoring

(factor polynomials with a variety of characteristics, then use the zero product property to solve the equation)

supplemental

 

 

AI3d4

Advanced Factoring by Grouping

(factoring four-term polynomials by grouping)

5.10 part 2

 

 

AI3d5

Using Several Methods of Factoring

(factoring polynomials with a variety of characteristics)

5.11

 

 

AI3d6

Advanced Factoring of Trinomials

(factoring when “variable” is a binomial instead)

5.8 and 5.9

advanced

 

 

AI3d7

Using Factoring to Solve Real-World Problems

(number puzzle problems and geometry problems)

5.13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

VII

ALGEBRAIC FRACTIONS

 

 

 

 

third trimester – 14 days

 

 

 

AI1d1

Simplify Algebraic Fractions by Factoring

(simplify algebraic fraction by factoring numerator and denominator;  includes factoring out a negative one to eliminate opposites.  Advanced problems involving equation solving.)

6.1

11.3

12.4

AI1d2

Simplify Products of Fractions by Factoring

6.2

part 2+

11.4

12.5

AI1d3

Simplify Algebraic Quotients by Factoring

(advanced problems involve combinations of multiplication and division)

6.3

11.4

 

AI1d4

Adding and Subtracting Algebraic Fractions

(add and subtract algebraic fractions, including those with numerators or denominators with products of binomials and monomials, by factoring the polynomials and finding the least common denominator)

6.4 and 6.5

11.6

12.6

AI2d1

Adding and Subtracting Mixed Expressions

(rewriting sums and differences of fractions and monomials as one algebraic fraction)

6.6

11.7

 

AI2d2

Polynomial  Long Division

(supplement with determining whether a binomial is a factor of a four-term polynomial, and solving factoring problems with partial solutions)

6.7

11.5

12.3

AI2d3

Simplify Complex Fractions by Factoring

6.7 extra

11.7

post 12.5

AI2d4

Equations with Fractional Coefficients

(using fraction destroyer;  include real-world problems)

7.3

 

 

AI2d5

Fractional Equations

(using fraction destroyer with variables in denominator)

7.4

11.8

12.7

AI2d6

Work Problems

(using work rate and tables)

7.8

11.8

 

 

 

 

 

 

 

 

 

 

 

VIII

SYSTEMS OF LINEAR EQUATIONS

 

 

 

 

third trimester – 7 days

 

 

 

AI1d1

The Graphing Method

(include “no solution” and “infinitely many solutions”) 

9.1

6.1

7.1, 7.5

AI1d2

The Substitution Method

(include “no solution” and “infinitely many solutions”)

9.2

6.2

7.2, 7.5

AI1d3

 The Addition-or-Subtraction Method

9.4

6.3

7.3

AI1d4

 Multiplication with the Addition-or-Subtraction Method

9.5

6.4

7.4

AI1d5

Solving Problems with Two Variables

(real world problems)

9.3

6.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

IX

INEQUALITIES

 

 

 

 

third trimester – 10 days

 

 

 

AI1d1

Solving Inequalities

(reversing inequality symbol; graphing solution set; solving real-world problems involving “no more than”, “at least”.)

10.2

5.1 and 5.2

6.1, 6.2

AI1d2

Solving Multi-Step Inequalities

10.2

5.3

6.3

AI1d3

Solving Combined Inequalities

(conjunctions, disjunctions, and graphing)

10.4

5.4

6.4

AI1d4

Absolute Value Equations

(“absolute value” as “distance”; graphing solution sets; review of topic in unit 2)

10.5 part 1

2.5

6.5

AI1d5

Absolute Value with “Less Than” Symbol

(“absolute value” as “distance”; graphing solution sets)

10.5, 10.6

5.5

6.6

AI1d6

Absolute Value with “Greater Than” Symbol

10.5, 10.6

5.5

6.6

AI1d7

Graphing Linear Inequalities in Two Variables

(on coordinate plane;  include situations with one variable only)

10.7

5.6

6.7

AI1d8

Graphing Systems of Linear Inequalities

(on coordinate plane;  shading and double shading)

10.8

6.6

7.6

 

 

 

 

 

 

 

 

 

 

X

RADICAL EXPRESSIONS AND EQUATIONS

 

 

 

 

third trimester – 10 days

 

 

 

AI1d1

Properties of Rational  Numbers

(density property of rational numbers; find number half way between two fractions;  state whether fraction increases or decreases as variable increases)

11.1

0.2

 

AI1d1

Decimal Forms of Rational Numbers

(converting between fractions and terminating or repeating decimals)

11.2

 

 

AI1d2

Rational Square Roots
(product property of square roots;  quotient property of square roots; finding square root using prime factorization; square root of fractions and decimals; “plus or minus”)

11.3

10.2

11.2

AI1d2

Irrational Square Roots

(simplify by finding hidden squares through prime factorization; using calculator or estimating from table of square root values)

11.4

10.2

11.2

AI1d3

Square Roots of Algebraic Expressions

(simplify by finding even powers of variables;  when to use absolute value symbol; factoring a polynomial under a radical; solving an equation by isolating a variable squared and using the definition of square root)

11.5

10.2

11.2

AI1d4

Multiplying, Dividing, and Simplifying Radicals

(attributes of “simplest form”; rationalizing the denominator;  using distributive property with radicals)

11.7

10.3

11.2

AI1d5

Adding and Subtracting Radicals

(“like” and “unlike” radicals; simplifying to create “like” radicals)

11.8

10.3

11.2

AI1d6

Multiplication of Binomials Containing Radicals

(FOIL binomials containing radicals;  use conjugates to rationalize the denominator)

11.9

10.3

 

AI1d7

Simple Radical Equations

(isolate the radical, square both sides, and check the solution)

11.10

10.4

11.3

AI1d8

Rational Exponents and nth Roots

(converting between radicals with indexes other than two, and fractions as exponents;  solving exponential equations;  not in Academy curriculum)

11.10 extra

7.3

post 8.3

Unit

Topic

Dolciani

Carter

Larson

 

 

 

 

 

XI

QUADRATIC FUNCTIONS

 

 

 

 

third trimester – 10 days

 

 

 

AI1d1

Quadratic Equations with Perfect Squares

(express solution in simplest radical form)

12.1

 

10.4

AI1d2

Completing the Square

(express solution in simplest radical form)

12.2

9.4

10.5

AI1d3

The Quadratic Formula

(express solution in simplest radical form;  estimate solution)

12.3

9.5

10.6

AI1d4

Graphing a Quadratic Equation

(find line of symmetry;  find vertex; use symmetry for short cut) 

8.8

9.1

10.1, 10.2

AI1d5

Solve the Quadratic Equation by Graphing

(find the companion function;  find the roots of the function) 

 

9.1

10.3

AI1d6

Quadratic Equations: The Discriminant

(use discriminant to count roots;  does vertex lie above, on, or below x-axis) 

12.4

 

10.7

AI1d7

Method of Solution

(solve variety of quadratic equations by selecting the best method)

12.5

9.5 part 2

 

AI1d8

Solving Problems Involving Quadratic Equation

(real-world problems, mostly involving geometry)

12.6

 

 

 

 

 

 

 

XII

SPECIAL FUNCTIONS AND OTHER TOPICS

 

 

 

 

third trimester – 15 days

 

 

 

AI1d1

Factor Sums and Differences of Perfect Cubes

(hide topic in intro, and reveal as a surprise)

5.13 extra

 

 

AI1d2

Graphing Quadratic Inequalities in Two Variables

 

9.2 extend

 

AI1d3

Interpreting Graphs of Functions

(intercepts; symmetry; extrema; end behavior;  they don’t include domain and range, but these should be included)

 

1.8

 

AI1d4

Solve a Quadratic Inequality in One Variable

(intercepts; symmetry; extrema; end behavior;  they don’t include domain and range, but these should be included)

12.6 extra

 

 

AI1d5

Attributes of Graphs of  Quadratic Functions

(group quadratic functions into families;  predict shape of graph of quadratic function)

 

9.3 explore

10.8

AI1d6

Transformations of Graphs of Quadratic Functions

(translations, dilations, and reflections)

 

9.3

 

AI1d7

Exponential Functions

(graph exponential functions; identify data that display exponential behavior)

 

7.5

8.5

AI1d8

Growth and Decay

(real-world problems)

 

7.6

8.6

AI1d9

Special Functions

(absolute value functions; piece-wise functions; step functions)

 

9.7

 

AI1d10

Writing Equations To Model Data

(linear, quadratic, and exponential functions)

 

9.6

 

AI1d11

Square Root Functions

 

10.1

11.1

AI1d12

Inverse Variation

8.10

11.1

12.1

AI1d13

Rational Functions

 

11.2

12.2