Algebra I
Horizontal Alignment Planning Guide (HAPG)
Fifth Six-WeeksMathematics Algebra I
Fifth Six Weeks Horizontal Alignment Planning Guide (HAPG)
Learning Focus 5.1 – Patterns in a Quadratic Function (Part 2)Students will connect aspects of quadratic functions to their algebraic, tabular, graphical, and concrete representations. Students will apply binomial multiplication, factoring, and solving quadratics to the concepts of perimeter and area.
Key Concepts
· Multiplication / · Solutions / ·
· Measurement
· Distributive Property / ·
· Factoring / · Perimeter
· Pattern
· Perimeter
/ What is it we want all students to learn? / Objective Code Key:
EL – Content Area.Grade.Strand.Objective
HS – Course.Strand.Objective
è- Power Objective / T - Assessed on TAKS at grade level
(Obj) = TAKS objective tested
*(x) - Assessed on TAKS at specified grade level /
- Literacy Strategy / © Houston ISD – Curriculum
June 2008
Page 2 of 6
/ What is it we want all students to learn? / Objective Code Key:
Elementary – Content Area.GradeLevel.Objective
Secondary – Course.Objective / T - TAKS
(Obj) = TAKS objective tested
*(x) - Grade level Assessed on TAKS / ³ - ELPS (English Language Proficiency Standards)
· - Literacy Leads the Way Best Practices
è - Power Objective / © Houston ISD – Curriculum
DRAFT 2010 – 2011
Page 1 of 11
Algebra I
Horizontal Alignment Planning Guide (HAPG)
Fifth Six-WeeksMathematics Algebra I
Fifth Six Weeks Horizontal Alignment Planning Guide (HAPG)
HISD Objectives / Time Allocation / Assessment Connections / Instructional Considerations / Instructional Strategies / Resources /ALGI.4A
Find specific function values; add, subtract, multiply, or divide to simplify polynomial expressions; transform and solve equations including factoring as necessary in problem situations which are expressed in verbal, algebraic, or pictorial (algebra tiles) representations.
è ALGI.10A
Solve quadratic equations in applied settings using concrete models, tables, graphs, and algebraic methods including factoring and the quadratic formula. / TAKS(Obj. 5)
10th 2006 #42; 2004 #40
Exit 2006 #18; 2004 #42, 2003 #15 / Note: students may find multiplying binomials more visually acceptable if you switch the traditional “FOIL Method” from horizontal to vertical.
“Box Method”
X + / 2
X / X2 / 2x
+
3 / 3x / 6
Vertical Multiplication
X + 2
X + 3
3X + 6
X2 + 2X
X 2 + 5X + 6
Marzano (Homework and Practice)
Use Dizzy Diamonds to prepare students to look for sums, and differences, and products when factoring. This activity gives students an opportunity to explore possible patterns that will be used later in factoring.
For special products, students build models with tiles for multiplying and factoring and then relate to special patterns of multiplying and factoring. Refer to Find the Dimensions.
Use the box method to work backwards for factoring with a GCF. This connecting activity may be found on the second part of Multiplying and Factoring Polynomials Guide. Also see the Algebra I textbook p. 582 and 592 for “Factorization with Algebra Tiles.”
Identifying Similarities and Differences
Elaborate on the lessons by Ccoordinatinge how to finding the solutions with factoring and with the graph using the graphing calculator. See activities Solving Quadratic Functions by Factoring and Graphing and Two Ways to Solve Quadratic Equations. Students should demonstrate an understanding of all methods whether algebraic or graphical and make connection between the two representations.
To elaborate, in LTF page 274, work problems 1 and 2 with the students. This activity shows the symmetry of a graph, and how to solve quadratics using factoring.
Evaluate students’ ability to transfer concepts from numeric examples to algebraic examples with Guiding Questions concerning factoring (see Instructional Considerations column, page 2). / Algebra I, McDougal-Littell, 2006 (con’t):
· 9.3 Find Special Products of Polynomials pp. 569-574
· Investigating Algebra Activity: Factorization with Algebra Tiles pp. 582, 592
· 9.5 Factor x2 + bx + c pp. 583-589
· 9.6 Factor ax2 + bx + c pp. 593-598
· 9.7 Factor Special Products pp. 600-605
· 9.4 Solve Polynomial Equations in Factored Form pp. 575-580
· 10.3 Solve Quadratic Equations by Graphing pp. 643-648
Clarifying Activities:
These activities provide alternative opportunities to demonstrate algebraic concepts concretely, graphically, and algebraically.
· Dizzy Diamonds
· Find the Dimensions
· Multiplying and Factoring Polynomials Guide
· Solving Quadratic Functions by Factoring and Graphing
° Solving Quadratic Functions by Factoring and Graphing (Key)
· Two Ways to Solve Quadratic Equations.
Connecting Algebra I to Advance Placement Mathematics (LTF):
· Graphing Quadratic Functions pp. 274-278
· Quadratic Optimization pp. 264-266
è ALGI.9D
Analyze maximum or minimum points, direction of opening, symmetry, and x- and y-intercepts of graphs of quadratic functions and draw conclusions from the graph and analysis.
³ 4(G), 1(E), 2(G)
è ALGI.10A
Solve quadratic equations in applied settings using concrete models, tables, graphs, and algebraic methods including factoring and the quadratic formula.
³ 1(E), 2(E), 4(D)
è ALGI.10B
Make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function on graphs, tables, or algebraic expressions.
è ALGI(8.15A)
Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.
³ 1(B), 2(E), 3(D)
ALGI.1E
Interpret and make decisions, predictions, and critical judgments from functional relationships.
ALGI.4A
Find specific function values; add, subtract, multiply, or divide to simplify polynomial expressions; transform and solve equations including factoring as necessary in problem situations which are expressed in verbal , algebraic, or pictorial (algebra tiles) representations.
³ 5(G), 1(C) / 6 days
for this cluster of objectives:
Three
90-minute
lessons
Or
Six
45-minute lessons / TAKS (Obj. 5) Grade 10
2009, #50 (J) 2008, #2 (A)
2006, #22 (J)
Exit
2009, #58 (J) 2008, #2 (C)
2006, #43 (C)
TAKS(Obj. 5)
Grade 10 2008, #3 (B)
Exit
2009, #43 (B) 2004, #37(A)
TAKS(Obj. 5)
Grade 10
2009, #14(J); 2006, #42(G)
2004, #40(H)
Exit
2008, #3 (B) 2006, #18 (H)
TAKS
(Obj. 10)
Grade 9
2009, #6 (G)
Grade 10
2009, #29 (D)
2008, #3 (D)
2006 #33 (A)
Exit
2009, #55 (B) 2006, #30 (J)
TAKS(Obj. 1)
Grade 9
2009, #7 (D) 2008,: #3 (B)
2006, #37 (C)
Grade 10
2009, #23 (D) 2006, #49 (A)
Exit
2009, #31 (B) 2006, #59 (C)
TAKS(Obj. 2)
Grade 9
2009, #26 (H) 2008, #3 (D)
2006, #16 (G)
Grade 10 2006, #16 (F)
#46 (H)
Exit
2006, #48 (H)
Formative Assessment 5.1
Students will create either an original function dance or function song and display their products in Photobucket. / Prerequisites/Background Knowledge for Students
Students concretely demonstrated the Pythagorean Theorem in Middle School and used it to solve application problems (TAKS Obj. 7).
Students have solved quadratics using factoring and graphing in Learning Focus 5.1 (TAKS Obj. 5).
Students have studied linear models informally in middle school and formally in Algebra I during Learning Focus 3.1 (TAKS Obj. 3).
Connections to Future Objectives/ Assessments
Students will use graphs, tables, and algebraic methods to solve linear, quadratic, exponential, and many other types of equations in Algebra I, Geometry, Algebra II, and Precalculus.
Completing the square will be used with conic sections in Algebra II and Precalculus.
Essential Understandings/Guiding Questions
Multiple methods for determining solutions of a quadratic function include factoring, graphing, and the quadratic formula.
1. How is factoring used to solve quadratic equations?
2. How is graphing used in solving quadratic equations?
3. How can tables be used to solve quadratic equations?
4. How is the quadratic formula used in solving quadratic equations?
5. When would be an optimum situation to use any of these methods?
6. How does the value of the discriminant affect the solutions?
Multiple representations of quadratics functions may model real-world applications.
1. What is the interpretation of the intercepts in the real-world models? (see activities in the Resources column)
2. How are the maximum or minimum values related to the real-world model?
3. Looking at the graph of the parabola, what is happening at any particular point on the curve in relation to the situation?
Polynomials may represent area and perimeter measurements.
1. How may products of binomials represent area?
2. How can the maximum area be found for a two- dimensional figure, given a set perimeter?
Background Knowledge for Teachers
Critical Content
· Solve problems involving the Pythagorean theorem;
· Solve problems using the quadratic formula;
· Find graphical and tabular correlations; and
· Use quadratic models.
Look at additional methods of solving quadratic equations and correlate attributes of quadratic functions from the solutions to the graph to the table of values (Power Objective ALGI.10B).
Extend student vocabulary about imaginary numbers, discriminants, and square roots of negative numbers. The teacher may want only to relate how a negative discriminant relates to “imaginary number,” therefore, there will be no solution and no x-intercepts in the graph. Emphasize how the discriminant tells the number of solutions, and relate those solutions to their graphical representations.
In the Instructional Strategies column, watch for cue words that refer to the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate.
A full analysis of quadratic functions will include attributes of the graph such as maximum and minimums, intercepts, direction of the opening, how these attributes connect to the graph, and how they connect to the interpretation of the real-world model (Power Objectives ALGI.9D, ALG10A, ALGI.10B, and ALGI (8.15A).
Vocabulary
Academic / Content-specific
Multiple
representation
Advance
Organizers / Pythagorean Theorem
Leg of Right Triangle
Hypotenuse
Factor
Quadratic Formula
Graph
Tabular
Representation
Discriminant
Roots
Zeros
Maximum/Minimum
Intercepts
TAKS Tips
Teachers should conduct a review of Pythagorean Theorem, which will connect to the use of radicals in the quadratic formula, as well as to review for TAKS Objective 7. / On different days, use engagement activities such as Using the Pythagorean Theorem, The Spider, and TAKS Review for Objective 7 as topics to review for TAKS Objective 7 (Pythagorean Theorem). Use an application problem to start a discussion about right triangles.
Summarizing and Note Taking
- KWL
³ Prior to studying the quadratic formula, have students explore how they would solve equations with squares (see “Use Square Roots to Solve Quadratic Equations” in the Algebra I textbook). Give each pair of students one problem from this section in the book. Remind students how they solved first-degree equations by using inverse operations. Have the students report what they found and check their work graphically on the calculator. This strategy will assist in internalizing new basic and academic language by reusing the language in a meaningful way that builds concept and language attainment. By making parallel operations for solving quadratics to simple equations, students connect that solving is a series of reverse operations. 1(E)
Nonlinguistic Representations
Concretely introduce the process of completing the square by using algebra tiles. Have students find the necessary tiles to make a square area. For instance, in order for x2 + 4x to become a square area, 4 unit tiles must be added. The length of each side of the new square area
x2 + 4x+ 4 is (x + 2).
³ Depending on students’ readiness, explain the derivation of the quadratic formula. You may show that some elements from solving a squared equation have similarities to “proving” the quadratic formula. Use the contextual support to enhance and confirm understanding of the complexity of proving the quadratic formula. 2(E)
Identifying Similarities and Differences
Connect solutions obtained by using the quadratic formula to finding solutions from a graph. Students compare, contrast, and analyze the two processes and point out where the students can find parallel information.
Generating and Testing Hypothesis
Ask students to formulate a hypothesis concerning when the discriminant is greater than zero, less than zero, or equal to zero and how this relates to the graph of the equation.
Setting Objectives and Providing Feedback
³ Place student in cooperative groups for Reading a Graph, Projectile Motion, Braking Distance, and Quadratic Scavenger Hunt. If time is a factor, give a different activity to each group. Each group will analyze the problem situation of each activity. Students should use the scaffolding questions provided with each activity and answer in complete sentences. Allow students to use various methods or multiple representations to analyze and support their conclusions. Each student group should present their activity to the other groups.
Allow them to post their work on chart paper and allow the other students to critique in a gallery walk. Use post-it notes or a checklist to evaluate. Elements that students evaluate include: meaning of maximum or minimum point, the meaning of the intercepts, and the interpretation of the curve of the parabola at any given point. Students at this point should be able to elaborate in writing or verbally on these elements of a quadratic function. 1(B), 4(G), 5(G)
Homework and Practice)
In the LTF activity, “Quadratic Optimization,” students will write quadratic equations for real-world situations and find the maximum value.
For an additional assessment, use Formative Assessment 5.1 to either create a new product or expand on concepts discussed in the activities above. / Clarifying Activities: These activities vary from practice opportunities to real-world applications.
· Using the Pythagorean Theorem
· The Spider
· TAKS Review for Objective 7
· Building a Sandbox
Algebra I, McDougal-Littell, 2006:
· 11.4 Apply the Pythagorean Theorem and its Converse pp. 737-742
· Graphing Calculator Activity: Minimum and Maximum Values and Zeros pp. 649-650
· 10.4 Use Square Roots to Solve Quadratic Equations pp. 652-658
· 10.6 Solve Quadratic Equations by Quadratic Formula pp. 671-676
· 10.7 Interpret Discriminant pp. 678-683
Clarifying Activities: These activities vary from practice opportunities to real world applications.
· Reading a Graph
· Projectile Motion
· Braking Distance
· Quadratic Scavenger Hunt
Connecting Algebra I to Advance Placement Mathematics (LTF):
· Quadratic Optimizations pp. 264-266
· Graphing Quadratic Functions pp. 274-278
/ What is it we want all students to learn? / Objective Code Key:
EL – Content Area.Grade.Strand.Objective
HS – Course.Strand.Objective
è- Power Objective / T - Assessed on TAKS at grade level
(Obj) = TAKS objective tested
*(x) - Assessed on TAKS at specified grade level /
- Literacy Strategy / © Houston ISD – Curriculum
June 2008
Page 2 of 6
/ What is it we want all students to learn? / Objective Code Key:
Elementary – Content Area.GradeLevel.Objective
Secondary – Course.Objective / T - TAKS
(Obj) = TAKS objective tested
*(x) - Grade level Assessed on TAKS / ³ - ELPS (English Language Proficiency Standards)
· - Literacy Leads the Way Best Practices
è - Power Objective / © Houston ISD – Curriculum
DRAFT 2010 – 2011
Page 1 of 11
Algebra I