Algebra 1

Welcome to the Algebra I page; it will be properly renamed very soon.  Here is our syllabus, including the weekly flow for our second semester:

Syllabus: Algebra I

Time Frame: One year

Course Overview: This one-year algebra course will have four major topics:

Linear Functions; Quadratic Functions; Systems of Equations; Inequalities and Absolute Value. Prior to tackling linear equations, students will work with expressions and equation basics.  Other essential mathematical concepts, such as Order of Operations, Integers (operations with negative numbers), Cartesian coordinates, Fractions, Exponents (4 operations with scientific notation, negative but not fractional exponents), Proportions, and Radicals, will be covered via Number Talks (see Learning Outcomes below). 

Learning Outcomes:

Teaching style will be constructivist more than group lecture, with a goal of having students work on challenging problems that require strategizing and decision-making as opposed to simple algorithms, especially during class time. Homework will typically be more algorithmic.  Homework is checked by instructor for completion while students check answers with one another for accuracy, with difficult problems then being done on the board, preferably by a student or students.  One instructional tool to be used will be Number Talks, which typically involve a problem that can be done mentally, with students sharing their solution methods with the group verbally.  Research has indicated that Number Talks are an effective way to increase mathematical fluidity.  Notes are taken during discussion of the various methods employed by the class. Number Talk-style presentations that involve algebra problems will include pencil and paper calculations, as such problems typically are not conducive to mental math. Students must keep excellent notes in lieu of a text book.  Students will be held accountable for keeping their notebooks in order by at least monthly in class self-assessments during which they will grade what I list as necessary elements of their table of contents on a simple point scale.

Expressions and Equation Basics

  • Read algebraic expressions fluidly by breaking down long expressions into smaller parts & identifying their terms, factors, & coefficients.
  • Simplify and reorganize expressions into equivalent forms to solve problems.
  • Write equations for word problems.
  • Understand solving equations as a process of reasoning and explain the reasoning.
  • Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  • Use units to solve problems.

Linear Equations

  • Solve linear equations in one variable, including equations with coefficients represented by letters.
  • Define slope and use slope-intercept form algebraically and graphically. 
  • Rates, Races, Raises (pay) – Word problems must be emphasized until students can go from equation to graph and vice versa.
  • See linear relationships fluently three ways: words, graphs, equations. 

Quadratic Equations and Polynomial Basics

  • Understand the relationship between zeros (roots) and factors of polynomials.
  • Quickly and efficiently factor polynomials and expand products of polynomials.
  • Solve quadratic equations in one variable.
  • Classify numbers as rational or irrational.
  • Classify numbers as real or imaginary and use complex number notation.
  • Understand the difference between linear and quadratic functions in terms of rate of change and what kinds of situations they represent.

Systems of Equations

  • Solve systems of equations in two and three variables.
  • Rethink slopes of parallel lines.

 Linear Inequalities and Absolute Value

  • Solve linear inequalities in one variable.
  • Solve linear absolute-value equations in one variable.
  • Represent and solve equations and inequalities graphically.
  • Represent and solve quadratic equations and inequalities graphically (enrichment challenge).

2nd Semester Weekly Flow:

Dates

Topic/Theme(s)

Learning Goals in Focus

Assignments and/ or Activities

Jan. 3-6

Quadratics

Distributive Property from various rectangles, from whole number sides to variables (squares) to x and x + 1, …

- Slopes of parallel lines, story graphs
- # Talk: Subtracting Negative Integers
- Given squares with sides 1 cm, 2 cm, …5 cm, graph perimeter and area simultaneously; struggle with how to connect curve and try sides 1.5, 2.5 cm

Jan. 9-13

Quadratics

Multiplying polynomials from various rectangles
FOIL (multiplying binomials) and factoring of trinomials with positive values of  A, positive or negative values of B, and positive values of C

- Round Robin FOIL and Factoring
GIT, Solve It?
- # Talk: Multiplying Mixed Numbers
- y = (x + 2)(x + 3) T-charts & proceed to graphing
- Quadratic Equations Quiz #1 (+ A, ± B, + C) 

Jan. 18-20

Quadratics

FOIL and Factoring with ± C, and eventually ±A values
What are the roots of a quadratic equation?

- Understand the difference between linear and quadratic functions in terms of rate of change and what kinds of situations they represent. 

- Quadratic Equations Quiz #2 (+ A, ± B, ± C)

Jan. 23-27

Quadratics

Projectiles and Parabolic Curves – What are roots of more complicated quadratic equations?

NASA Challenge (lab exercise)
- Solve quadratics to find when projectile lands (or where if y = d in lieu of y=t) 0 = -ax^2 + bx + c 
HW: Projectile Word Problems
# Talk: Dividing Mixed Numbers 

Jan. 30-Feb. 3

Quadratics

Quadratic Formula

- Quadratic Equations Quiz #3 (± A, ± B, ± C)
- Graph quadratics using short cuts.
- # Talk: Exponents

Feb. 6-10

Quadratics

Imaginary Numbers

#Talk: Radicals

Feb. 13-17

Quadratics

Review of all Quadratics

- Quadratic Equations Exam
- #Talk: The sum of what three consecutive even integers is 372?
- #Talk: 4 operations with scientific notation, negative but not fractional exponents)

 

Mar. 5,7-9

Systems of Equations

Include discussion of which problems are solvable and why, i.e., one needs n equations for n unknowns.

- Start with word problem with two variables; proceed to methods other than guess/check/revise.
- Proceed to systems of equations with two linear equations in slope-intercept form and solve with graphing.
- Look at parallel lines again, now “unsolvable” via algebraic methods.

Mar. 12-16

Systems of Equations

Solving systems with two variables

- # Talk: multiplying larger numbers

- PBS site “Atom Builder”, starting with quarks (up and down until it balances).  Determine charges of up and down quarks from two unknowns.  Start with “quark dance” from AZ, which includes their cute names.

Mar. 19-23

Systems of Equations

Solving systems with three variables

Types of WP’s: #legs and feet (in a box), money, rates (intersection of lines).
Systems of Equations Quiz
# Talk: Proportional relationships (algebraic cross multiplication: HW packet)

Mar. 27-30

Absolute Value

Manipulating equations as well as graphing
Predict graph of

Absolute value of x: What is it? How far can we go on ¼ tank of gas on I-91?  Graph it via T – chart. Students create and solve other such application problems.

Apr. 2-6

Absolute Value

Graphing of various absolute value equations.

T-charts of various absolute value functions followed by their shortcuts.

Apr. 9-13

Inequalities

Solve linear inequalities in one variable.

 

- Review graphing of slope-intercept form.
- Represent and solve equations and inequalities graphically.

Apr. 16-20

Inequalities

Experimenting with various points, find rules for graphing shortcuts for quadratic inequalities (in class challenge)

Explore graphs of linear inequalities.

Apr. 30-May 4

Inequalities

Linear programming

- Graph linear inequalities from word problems.
- Graph quadratic inequalities.

May 7-8

Review of First Semester

Order of Operations, Solving for a Variable, Fractions, Linear Equations

- Review and do self-assessment of notebooks.
- Do all problems in study guide

May 14-18

Review of Year

Quadratics, Systems of Equations, Absolute Value, Exponents, Radicals

- Review and do self-assessment of notebooks.
- Do all problems in study guide

May 21-25

FINAL

Review and Fermi Problems

Annual Cumulative Exam

1st Semester Weekly Flow:

Dates

Topic/Theme(s)

Learning Goals in Focus

Assignments and/ or Activities

Assessment

Sep. 6-9

Getting back into math

- Introduce daily mindfulness practices.

- Introduce Number talks with note taking.

 

 

- # Talk #1: adding whole numbers

- # Talk #2: subtracting whole numbers as well as adding or subtracting with negatives.

- Kuta Order of Operations (#1-5 or 10)

Diagnostic Exam

(including students’ affective comments)

Sep. 12-16

Basics of expressions and equations

- Table work with solving for a variable

- Read algebraic expressions fluidly by breaking down long expressions into smaller parts and identifying their terms, factors, and coefficients.

 

- # Talk #3: Subtracting Negative Integers

- # Talk #4: Solving for a Variable

 

- PBA 1.1, 1.3, 2.1, 2.3, 2.5, 1.10, 1.9 - - Journal entry re: How class is working for you personally and what you think of # Talks.

Warm-ups

Sep. 19-23

Simplify and reorganize expressions into equivalent forms to solve problems.

Fluidity of numbers and algebraic expressions

- PBA 4.1, 6.1

- Cartesian coordinates, statements into algebraic expressions

- Write equations for word problems.

- # Talk #5: The sum of what three consecutive integers is 54?

Quiz on order of ops, signed integers, (solving for a variable)

Sep. 26-30

Increase complexity of algebraic expressions and word problems.

Construction of word problems

- - #Table #6: The sum of what three consecutive even integers is 372?

- Complex word problems from GIT (Wodjah cluster)

- Carousel development of complex word problems

Warm-ups

Oct. 3-6

Complex word problems

 

- Understand solving equations as a process of reasoning and explain the reasoning.

- Use units to solve problems.

- # Talk #7: multiplying larger numbers

- Revise complex problems after feedback from other class and Ron.

- Complex multi-step word problem exchanges

 

(AZ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Quiz on expressions and word problems

Oct. 10-14

Linear Functions

Fermi problems

- What is slope?

- In naming first and second points for calculating slope, does order matter?  - - What are subscripts?

- # Talk #8: Division of Integers

- Slope measurement (Windshields)

- Slope Packet

Warm-ups

Oct. 17-19

Linear functions

- Coordinate graphs, graphing lines from equations using T-charts (input/output tables)

- #Talk#9: Order of Operations

- Slope Packet

- Story graphs of position function – Round Robin exchange of written story followed by graph followed by story based on reading graph….

- Warm-ups

- Quiz on Cartesian Coordinate Plane

Oct. 24-28

Linear functions

- interpreting a linear graph

 

- #Talk #10: Cartesian coordinates

- Finish slope Packet

Quiz on graphing lines using T-charts

Oct. 31-Nov. 4

Arithmetic: Fractions

Algebra: Linear functions

y = mx + b with slope and y-intercept as points of reference for graphing

- #Talk #11: Comparing and Ordering Fractions

- Slope Posters

Warm-ups

Nov. 14-18

Arithmetic: Fractions

Algebra: Linear functions

- manipulating equations

- solving equations

- # Talk #12: Mixed Numbers and Improper Fractions

 

Quiz on graphing lines using shortcut  of y = mx +b

Nov. 15-19

Arithmetic: Fractions

Algebra: Linear functions

- slopes of parallel lines

- # Talk #13: Multiplying Fractions

 

Warm-ups

Nov. 28-Dec. 2

Arithmetic: Fractions

Algebra: Linear functions

Slope word problems

- # Talk #14: Multiplying Mixed Numbers

 

Quiz on operations with fractions

Dec. 5-9

Arithmetic: Fractions

Algebra: Linear functions

Rates, Races, Raises (pay) – Word problems must be emphasized until students can go from equation to graph and vice versa.

- # Talk #15: Dividing Fractions Perhaps little physics experiments with data exhibiting linear relationships

Warm-ups

Dec. 12-16

Semester Review

Review of semester’s material, especially algebraic expressions and slope

Complete Semester Exam Study Guide.

Cumulative Semester Exam

Dec. 19-21

Proportional relationships

Compare and contrast linear and non-linear relationships.

- # Talk #16: Dividing Mixed Numbers

- End quarter with graphing perimeters of rectangles and touching on graphing their areas as transition to quadratics.

Warm-ups