Mathematics – Algebra 1
Unit 3: Linear Functions
Second Grading Period - Weeks 1- 3 (14 days) CURRICULUM OVERVIEW
Big Idea / Unit RationaleUnderstand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; analyze change in various contexts. NCTM / The student should understand:
· the importance of a solution of an equation in two variables.
· the conversion of variables in an equation into the slope-intercept form to represent solutions for real world situations.
· the various effects of the rate of change
TEKS / TEKS Specificity - Intended Outcome
Concepts / (A.1)Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to:
A.1(C) describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations;
A.1(D) represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
(A.2)Foundations for functions. The student uses the properties and attributes of functions. The student is expected to:
A.2 (B)identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete;
A.2(C) interpret situations in terms of given graphs or creates situations that fit given graphs.
(A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to:
A.4(C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.
(A.5)Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations.
The student is expected to: / ” I CAN” statements highlighted in yellow should be displayed for students.
I can:
· describe functional relationships for a given situation (A.1C)
· write equations to answer questions related to a given situation (A.1C)
· represent linear relationships using concrete models, tables, graphs, diagrams, verbal descriptions, and equations (A.1D)
· identify domains and ranges and their reasonableness for given situations (A.2B)
· interpret situations that fit a given graphs (A.2C)
· create situations that fit a given graph (A.2C)
· connect equation notation with function notation (A.4C)
· determine the domain and range for linear functions in a given situation (A.5B)
· use, translate and make connections among algebraic, tabular, graphical or verbal descriptions of linear functions (A.5C)
· develop the concept of slope as a rate of change (A.6A)
· determine slopes from graphs, tables, and algebraic representations (A.6A)
· interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs (A.6B)
· predict the effects of changes in m and b on the graph of y = mx + b (A.6C)
· graph and write equations of lines given characteristics such as two points (A.6D)
· graph and write equations of lines given characteristics such as a point and a slope, (A.6D)
· graph and write equations of lines given characteristics such as a slope and
· yintercept (A.6D)
· determine the intercepts of the graphs of linear functions from the graph, table or equation (A.6E)
· determine the intercepts of the graphs of linear functions from the graph, table or
A.5(B)determine the domain and range for linear functions in given situations
A.5(C)use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. / · equation (A.6E)
· determine the zeros of a linear functions from the graph, table or equation (A.6E)
· predict the effects of changing slope and y-intercept in an applied situation (A.6F)
· formulate linear equations for a given situations to solve the problem A.7(A)
· use the different methods for solving linear equations (A.7B)
· determine reasonableness of solutions of linear equations A.7(C)
(A.6)Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to:
A.6(A)develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations;
A.6(B)interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs;
A.6(C)investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b;
A.6 (D) graph and write equations of lines given characteristics such as two points, a point and slope, or a slope and y-intercept;
A.6(E)determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations;
A.6(F) interpret and predict the effects of changing slope and y-intercept in applied situations.
(A.7)Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:
A.7(A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;
A.7 (B)investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities;
A.7 (C) interpret and determine the reasonableness of solutions to linear equations and inequalities.
Evidence of Learning
· Given a functional relationship, the student will be able to write an equation and represent it using concrete models at least 80% of the time.
· Given a functional relationship, the student will be able to determine, identify and describe the domain and range at least 80% of the time.
· Given multiple representations the student will be able to determine the effects of changing slope and y-intercept at least 80% of the time.
· Given two points, a point and a slope, or a slope and y-intercept the student will be able to graph or write equations of lines at least 80% of the time.
· Given the different methods for solving linear equations, the student will be able to determine reasonableness at least 80% of the time.
Mathematics – Algebra 1
Unit 3: Linear Functions
Second Grading Period - Weeks 1- 3 (14 days) CURRICULUM GUIDE
Essential Questions / Essential Pre-requisite SkillsWhat are the steps in writing an equation of a line in slope-intercept form?
How do you graph linear equations?
How can you tell if a function is discrete or continuous?
How do you use intercepts to graph equations?
How do you find the slope of a line and interpret slope as rate or change?
How do you find the equation of a line given two points, a point and a
slope, or a slope and y-intercept? / Students should know how to:
· locate and name points on a coordinate plane using ordered pairs of rational numbers (8.7D)
· make conjectures from patterns(8.16A)
· identify the properties and attributes of domain and range (A.2A.)
· identify different correlations as a trend (A.2D)
· describe the effect of rate of change ((A.6A)
· find specific function values (A.4A)
The Teaching Plan
Instructional Model & Teacher Directions
The teacher will… / So students can…
Day 1 - Solving for “Y”
Engage:
· Play a number game with students.
· There are 5 steps.
1st write down the year they were born;
2nd write down the year a significant event happened in their
lives;
3rd write down how many years ago that event
occurred;
4th write down their age (or what their age will be
that year even if their birthday has not yet occurred);
finally add up all of the numbers.
· Ask how many students got 4016 (for 2008, for any other year value should be 2 * calendar year)
Explore:
· Encourage students to figure out why they all got the same answer.
Explain:
· Explain that they key to knowing the right value in the number game is the balance in numbers. The birth year was balanced out by the age and the event date was balanced out by the years since the date occurred.
· Tell students that when they solve and equation for a variable they must keep a balance in numbers. “What happens to one side of the equality happens to the other side of the equality”.
Elaborate:
· Write the equation y + 2x = 5 on the board as an example and go through the steps to solve the equation for y.
· Be explicit in writing detailed steps, with cues, that students can follow.
· Practice other examples as necessary, allowing students to ask questions.
· Have students go to the board or work in groups to practice.
Monitor and facilitate students as they continue to complete the “Solving for Y Practice” practice sheet problems # 1-18. Solving for y teacher notes / Day 1 - Solving for “Y”
Engage:
· write down the specific numbers as the instructor directs
Explore:
· share their answers and explain how they came to a predictable result
Explain:
· actively listening and responding to teacher prompts
Elaborate:
· take notes over example y + 2x = 5 and write down the steps as the teacher goes over them (A.1D)
· ask questions about the steps, for understanding.
· work in groups discussing problems #1-18 on the “Solving for Y Practice” worksheet (A.1D)
Day 2 - Solving for “Y”
Explain:
· Explain that problems 1-18 involve only a single step. Model examples of problems that involve multi-steps.
· Explain the steps to 5y + 15x = 25 using cues, develop a series of steps that students can follow and practice.
Explore:
· Monitor and facilitate students as they work on “Solving for Y Practice” problems #19-30 in pairs.
Elaborate:
· Introduce the concept “finding a solution of an equation in two variables”. Answer the question “Which ordered pair is a solution of 4x – y = 6?” p. 215 Section 4.2 McDougal Littell example 1.
· Allow for practice p. 219 #3-10
Evaluate: Monitor student work
Extension:
· Teach students how to graph linear equations p. 215-221
Include vocabulary and introduce Horizontal and Vertical
Lines.
· Practice on problems 11-25 p. 219
· Explain reasonable domains and ranges example 4 p. 217
· Assign problems 26-31, p. 219 and discuss #35 and #36
· Explain “4.2 Graphing Linear Equation” using a graphing calculator p. 222 / Day 2 - Solving for “Y”
Explain:
· write down examples and steps, asking questions for better understanding(A.4B)
Explore:
· work in pairs to solve problems #19-30 on “Solving for Y Practice” (A.4B)
Elaborate:
· set up each ordered pair as a solution of the given equations follow the teachers demonstration (A.7B)
· practice p. 219 #3-10 (A.7B, A.7C))
Evaluate:
· ask questions for better understanding.
Extension:
· take notes on graphing linear equations and define vocabulary.
· practice graphing linear equations, 11-25 p. 219 (A.5C)
· write down the example on finding reasonable domains and ranges for given problems example 4 p. 219(A.2.B, A.5B)
· practice problems 26-31 p. 219 discuss reasonable domains for #35 and 36 p. 220 (A.2B, A.5.B, A.5C)
· use a graphing calculator to solve problems p. 222 (A.7B)
Day 3 Slope Intercept Form
Engage:
· Show changes in real world visual structures, in the form of pictures or short video clips.
Explore:
· Hand out “Slope Intercept Form Notes”, have students graph lines, observe the changes in slope and the role of the “m” in y= mx + b.
Explain:
· Explain the function of slope, and the role of “m” in the slope intercept form.
Elaborate:
· Pass out the “Slope Intercept Form Assignment”
Evaluate:
· Monitor and facilitate student work / Day 3 Slope Intercept Form
Engage:
· respond to teacher prompts discussing “What is slope, and where does it appear in the real world?” (A.6A, A.6B)
Explore:
· write on the “Slope Intercept Form Notes” worksheet. (A.6C)
· Observe the changes in the slope and the role of “m” in y=mx+b. (A.6C)
Explain:
· ask questions about slope and the “m” in the slope intercept form. (A.6C)
Elaborate:
· work on the “Slope Intercept Form Assignment” (A.6B)
Evaluate:
· ask questions on the assignment.
Day 4 Finding Slope Using Formulas
Engage:
· Materials: ruler and stacks of books
· McDougal Littell Section 4.4 Activity “Slopes of Lines” p. 234
Explore:
· Hand out “Finding Slope Notes” to students.
· Monitor and facilitate students as they explore the impact of changing the slope and the y-intercept.
Explain
· Go over solutions to the notes worksheet with students and discuss what they have learned
· Have students begin work on “Finding Slope Assignment”
Elaborate:
· Work on Section 4.4 in McDougal Littell, “given two points find the slope using the slope formula.”
· Assign problems 8-16 page 239, “find the slope of the line that passes through the points”.
Evaluate:
· Assign problems 19-23 page 240, real world situations. / Day 4 Finding Slope Using Formulas
Engage:
· observe the run of the ramp as the slope in the activity increases and decreases.
Explore:
· complete the notes worksheet, pay attention to the impact of the changing slope and y-intercept. (A.6A, A.6B, A.6C)
Explain:
· go over solutions to the notes worksheet and share what you have learned
· begin working on “Finding Slope Assignment” (A.6A, A.6B, A.6C)
Elaborate:
· work Section 4.4, “given two points find the slope using the slope formula.” (A.6A)
· work problems 8-16 page 239, “find the slope of the line that passes through the points”. (A.6A)
Evaluate:
· work on problems 19-23 page 240, real world situations. (A.6A, A.6B)
Day 5 - Discovering Slope and Y-Intercept From an Equation/Identifying Parallel Lines