maaa 037053
DESCRIPTION
Chicago Public Schools, Advanced Algebra Lesson plans, Days 37-53TRANSCRIPT
134
STRUCTURED CURRICULUM LESSON PLAN
Day: 037 Subject: Advanced Algebra Grade Level: 11
Correlations (SG,CAS,CFS): 6A1
TAP:Analyze and interpret data presented in charts,
graphs, tables and other displays
ISAT:Identify, analyze, and solve problems using
equations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Matrices and Systems of Equations
Instructional Focus/Foci
Using Matrices to Store and Represent Data and Adding or Subtracting Matrices
Materials
Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Give the following definitions:
Matrix: A rectangular arrangement of elements in rows and columns enclosed by brackets.
Example: M = 1 4 3 2
3 2 0 5
−−LNM
OQP
Matrix notation: An entry is named as mab, where a = row number and b = column number
In the example above, m13= 3; m24=5Dimension: the number of rows x the number of columns gives the dimension of amatrix. The dimension of the matrix M above is 2 x 4.
Have students place elements in their proper position, then have them enter the followingmatrices into their calculators:
135
To enter a matrix into the graphing calculator:1. press MATRX2. use the right arrow to EDIT, then ENTER3. enter number of rows, ENTER4. enter number of columns, ENTER5. enter each element, going across. After each element, press ENTER. If you
make a mistake, scroll to the location you want to change.
To perform operations on matrices, press MATRX, leave NAMES highlighted and scroll downto the matrix you want to use; press ENTER.
Students will find: A + B; B – A; A – A (call this C); C + B; –1 x B
Students will report findings to the whole class.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment: In a recent survey, 90% of high school students were opposed to a systemwideSlice Girls concert suggested by elementary school students.1. Work with a partner to conduct a survey in your school regarding the questions below.
A. Should the Slice Girls concert occur?B. Should high school students be mandated to attend?C. Should elementary school students be mandated to attend?D. What percent have no opinion?
Use the following matrix to report your data in percentage form.
Question Yes No Indifferent
A B=−
− −
L
NMMM
O
QPPP
=− −
−−
L
NMMM
O
QPPP
3 2 1 0
6 2 4 1
0 3 2 1
7 3 2 1
4 0 3 2
1 7 3 0
ABCD
136
Fine Arts:
Home: Parents will sign homework record weekly.
Remediation:
Technology:
Assessment
Have students answer the following questions:
a. What must be true about the dimensions of two matrices that are equal? b. Given matrices A, B, and A + B, why must they have the same dimension? c. What is the relationship between matrices A and –1A? d. What is a good name for matrix C?
Homework
Assign appropriate problems from your text.
Teacher Notes
Solutions to Assessment:
a. They must have the same number of rows and columns.b. The entries from the first matrix should correspond to the entries from the second matrix.c. Matrix -1 x A is the opposite of matrix A.d. Zero matrix
Solutions to in-class assignment:
A + B =
−−−−−
11101
17210
11110
B − A =
−−−−−−−
1541
3122
1354
A − A =
0000
0000
0000
C + B =
−−−−−
0371
2304
1237
-1 x B
−−−−
−−
0371
2304
1237
137
STRUCTURED CURRICULUM LESSON PLAN
Day: 038 - 039 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 6A1
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Performing Matrix Multiplication
Materials
Classroom set of graphing calculatorsOverhead graphing calculatorsClassroom set of graphing calculatorsOverhead projector
Educational Strategies/Instructional Procedures
Explain to students: To multiply two matrices Amxn and Bpxq, n must equal p (the inner
dimensions). The resulting matrix AB will have dimension m x q (the outer dimensions). If n≠p, the two matrices cannot be multiplied.
Example 1: A B=−
− −LNM
OQP =
−
L
N
MMMM
O
Q
PPPP3 2 0 1
4 3 3 2
0
3
2
1
138
dim[A] = 2x4, dim[B] = 4x1, so these two matrices can be multiplied (since 4=4), and theresulting matrix will have dimension 2x1.
Take row 1 of A and column 1 of B, multiply successive elements, add these products, and theresult is the corresponding element in the product matrix. Using the example above:
3•0 + 2•3 + 0•–2 + –1•1 = 5; 4•0 +0•3 + –3•–2 + -2•1 = 4. So the product matrix is 5
4
LNMOQP
Give students several matrices of different dimensions and ask if they can be multiplied.
Example 2:
a.
40
36
− 25
68b.
−−
82
43
96
−41
72c.
−036
894
−
−282
413
Solutions: a. yes b. yes c. no
Have students form small groups; use the graphing calculator to determine whether theassociative, commutative, or distributive properties hold for matrix multiplication.
Report findings to class. Use the matrices: A =
−06
14 B =
63
12 C =
−
−20
31
a. A*B = B * Ab . A(B*C) = (A * B ) * Cc . A(B+C) = AB + AC
Example 3:
Jane and Andrew enjoy shopping for music on a monthly basis. They each purchase thefollowing items.
CDs Tapes LPsJane 10 6 1
Andrew 8 2 3
If CDs cost $15, tapes cost $8, and LPs cost $10, use matrices to find the total cost that Jane andAndrew each spent on music.
139
Solution:
328
1610
10
8
15
=
166.
208
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment: Use graph paper to draw squares with areas of 2, 5, and 10 units.
Fine Arts:
Home: Parents will sign homework record weekly.
Remediation:
Technology: Have students review how their calculator can perform various operations withmatrices.
Assessment
Teacher observation
Homework
Assign appropriate problems from your text.
Teacher Notes
This is a two-day lesson. The second day (Day 39) should be used to review homework, forremediation exercises, or for extra practice, as necessary.
Solutions to in-class assignment:
a. A*B = B *A (not true) b. A(B*C) = (A *B) * C (true)
−612
25
−−
=348
214
−−
2412
195=
−−
2412
195
c. A(B+C) = AB + AC (not true)
invalid dimensions =
246
121
140
STRUCTURED CURRICULUM LESSON PLAN
Day: 040 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Identifying Systems of Two Linear Equations
Materials
Handout 4.3Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Students will: Form groups of 2. Complete Handout 4.3 (exploration on intersections of lines). Report findings to the class.
Present the following definitions:
Consistent system: lines intersect in one or more pointsIndependent system: lines intersect in one unique point (one solution)Dependent system: lines intersect in an infinite number of points (the lines coincide; infinitelymany solutions)Inconsistent system: lines do not intersect (parallel lines; no solutions)
141
Give examples of systems of equations and graphs of solutions of equations.
Have students match each system with its graph.
a. y = x + 1 b. y = -2x + 3 c. y = .5x - 2 y = x y = 3x + 1 y = -x + 4
1. 2. 3..
Solutions: a ⇔⇔⇔⇔ 3 b ⇔⇔⇔⇔ 1 c ⇔⇔⇔⇔ 2
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment: Have students discuss how to solve a system of three linear equations in threevariables.
Fine Arts:
Home: Parents will sign homework record weekly.
Remediation:
Technology: Students will review the various ways their calculator can graph and find theintersections of linear functions.
Assessment
Describe the three types of systems of linear equations in terms of intersecting lines.
Homework
Assign appropriate problems from your text.
142
Teacher Notes
Solutions to Handout 4.3:
Type 1: a. Intersects at (1, 2) b. Intersects at c. Intersects at (-7, 1)
Type 2: a. Always intersect b. Always intersect c. Always intersect
Type 3: a. Parallel b. Parallel c. Parallel
− −FHIK
5
8
1
2,
143
Handout 4.3
This exploration involves different types of systems of equations and their graphs. For each type,graph each pair of equations and the same coordinate plane. Then answer the questions.
Type 1:
1.x 2y 5
3x 2y 72.
4x 3y 1
4x y 33.
x 2y 5
x 3y 4
+ =
+ =
+ =
− − =
− − =
− + =RST
UVWRST
UVWRST
UVWWhich pairs of lines intersect?How is each pair of lines alike or different?
Type 2:
1.x y 4
3x 3y 122.
x 2y 6
4x 8y 243.
4x y 1
12x 3y 3
+ =
+ =
− =
− + =
+ = −
− − =RST
UVWRST
UVWRST
UVWWhich pairs of lines intersect?How is each pair of lines alike or different?
Type 3:
1.x 3y 15
x 3y 62.
2x y 13
4x 2y 143.
x 4y 1
x 4y 2
− = −
− = −
− + = −
− + =
+ =
− = −RST
UVWRST
UVWRST
UVWWhich pairs of lines intersect?How is each pair of lines alike or different?
Describe the three types of systems of linear equations in terms of intersecting lines.
144
STRUCTURED CURRICULUM LESSON PLAN
Day: 041 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Solving a System of Two Linear Equations Algebraically
Materials
Classroom set of graphing calculators
Educational Strategies/Instructional Procedures
Present the following situational problem: Teacher will tell students: I am thinking of twonumbers whose sum is 30 and whose difference is 10. What are the two numbers? (20, 10) Havestudent volunteers make up additional problems like this one for class to solve.
Ask students how to write a pair of equations to represent the problem.
x + y = 30x – y = 10
Ask: What happens if we add the two equations? [The y’s eliminate and we get2x = 40 x = 20.] How can we then find y? [Replace into one of the equations: y = 10.]
This is an example of solving a system of equations using elimination by addition.
145
Example 1:3x + 2y = 8 _ 3x + 2y = 8 2(2) + y = 52x + y = 5 –4x – 2y = –10 4 + y = 5
–1x = –2 y = 1x = 2 so, x = 2, y = 1
Example 2: 5x – y = 6 -y = -5x+6 y = 5x-6 y = 5x-6
0 =0
Example 3: x +y = 7 2x + 2y =14 2x + 2y =4 2x + 2y = 4
0 ≠ 10∴ no solution
.1. Different slope Lines intersect One solution Independent system2. Same slope andintercepts
Lines coincide Infinite number ofsolutions
Dependent system
3. Same slope anddifferent intercepts
Lines parallel No solution Inconsistent system
Cooperative Learning: Separate the class into pairs of students. Assign a system of equations andhave one student of the pair solve the system using the elimination method while the other solvesthe system using a graphing calculator and the TRACE feature. Have the pairs compare theiranswers and then share them with the class. Have pairs exchange roles and repeat.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment:
Fine Arts:
Home: Parent will sign homework record weekly.
Remediation:
Technology:
146
Assessment
Assign students a system of equations and have them describe the steps for solving them.
Homework
Assign appropriate problems from your text.
Teacher Notes
The cooperative learning activities must be monitored very closely.Prepare copies of Solving Systems Algebraically.
147
STRUCTURED CURRICULUM LESSON PLAN
Day: 042 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percents,explicitly stated or within context
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Matrices and Systems of Equations
Instructional Focus/Foci
Solving a System of Two Linear Equations Algebraically
Materials
Copies of Solving Systems Algebraically
Educational Strategies/Instructional Procedures
Substitution is the second algebraic method used to solve a system of equations. One equation issolved for one variable in terms of the other. This expression for the variable is substituted in theother equation.
Example 1: 3x + 2y = 9 y = x + 2
3x +2(x +2) = 9 y= x+2 3x+2x+4 = 9 y = 1+2
5x+4 = 9 y = 35x= 5x = 1
148
Example 2: x − 3y = -122x + 11y = -6
x = 3y –12 2(3y – 12) + 11y = -6 x−3 = -12 6y –24 +11y = -6 x = -9
18y – 24 = -618y = 18 y = 1
Have the students complete Solving Systems Algebraically.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
SC: Understand uses of scientific units and instrumentsApply scientific method to solve problems
Connection(s)
Enrichment: Have the students solve the following systems using substitution.1. R+S +K =-10 2. 2L + 3M+ N = 36 4 S + K = 6 5L – N = -3
3K = -21 2N = 8
Fine Arts:
Home:
Remediation: Work with students individually on their deficiencies.
Technology: Have students write a program for the graphing calculator to solve a system of twolinear equations.
Assessment
Evaluate Solving Systems Algebraically using the Structured Curriculum Scoring Rubric.
Homework
Assign appropriate problems from your text.
149
Teacher Notes
Emphasize that substitution is used only when one variable can be solved for easily.
Solutions to Solving Systems Algebraically:
1. Independent system - lines intersect
2. Independent system - lines intersect
3. Independent system - lines intersect
4. Dependent system - lines coincide
5. x = 2 , y = 0
6. x = 3
7
3
29 =y
7. x = 3
7
3
29 =y
8. x = 5 y = -7
9. x = 5
6
5
16 =y
10. x = 1 y = -1
150
Solving Systems Algebraically
Describe the graph of each system of equations.
1. 2x + 3y = 6 2. 3x + 2y = 9 2x – 3 y = 1 x – y = -2
3. 4x – 3y = -7 4. 5x – y = 4 x+ 2y = 1 -5x + = -4
Solve each system by substitution.
5. 3x + y = 6 6. x – 2y = 5 -6x – 9y = -12 x + y = 12
Solve each system by elimination.
7. 2x + 3y = 10 8. 4x + 2y = 6 3x + 2y = 12 x + y = -2
Solve each system by any method.
9. 3x + 5y = 26 10. 8x – 3y = 11 x – 3y = 4 2x+ y = 1
151
STRUCTURED CURRICULUM LESSON PLAN
Day: 043 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Use variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Identify, analyze, and solve problems using
equations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Exploring the First Quarter
Instructional Focus/Foci
Reviewing the First Quarter
Materials
Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Review previous day’s homework and answer students’ questions.
Review Units One, Two, and Three using a format from the appendices.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment:
Fine Arts:
Home:
Remediation:
152
Technology:
Assessment
Teacher observation
Homework
Assign appropriate review problems from the first quarter.
Teacher Notes
Prepare copies of the First Quarter Exam.
153
STRUCTURED CURRICULUM LESSON PLAN
Day: 044 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 6A1; 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Unit Focus/Foci
Exploring the First Quarter
Instructional Focus/Foci
Assessing the First Quarter
Materials
Copies of the First Quarter ExamClassroom set of graphing calculatorsOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Administer the First Quarter Exam.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
Connection(s)
Enrichment:
Fine Arts:
Home:
154
Remediation:
Technology:
Assessment
Evaluate the First Quarter Exam using the Structured Curriculum Scoring Rubric.
Homework
Review class notes and think of possible Science Fair projects.
Teacher Notes
Evaluate the First Quarter Exam.
Prepare copies of the Science Fair Guidelines.
Solutions to First Quarter Exam:
1. c 2. c
3. c 4. b
5. d 6. b
7. a 8. d
9. c 10. a
11. Inconsistent 12. Axis of symmetry
13. Matrix 13. Parabola
15. Domain 16. Function
17. 1-1 17. Irrational
19. Line of best-fit 20. Negative
155
21. a.
b. y1
8x 800= −
c. correlation coefficient is 1
8
22. a. f xx− =
+1 2
3( )
b. domain: {x|x is all real numbers} range: {y|y ≥ 0 q
23.
a. f(x) g(x)x 2
43 x 2 or
x 2 12 x 8
4+ = + −
+ −
b. f(x)*g(x) =3 x 2 x
4
3 2−
c. f(x)-g(x)= 3 x 2x
4
2
− − or − + −x x2 12 8
4
d. f(x)
g(x)
3 x 2
x
4
12 x 8
x2 2= =
− −
24.
a. f(g(x))= 3 x2
4− 2 or
3 x2 8
4
−
b. g(f(x)) = 3 x 2
4
9 x 12 x 4
4
2 2− − +=
b g
25.a.
b.
c.
156
First Quarter Exam
Select the best answer.
1. Find the zero of the function f(x) = 6x + 5.
2. Find the slope of the line passingthrough (-3, 8) and (-2, 4).
a. − 6
5c. − 5
6
b. 5
6d.
6
5
a. − 1
4c. -4
b. 4 d. 1
4
3. Name all values of x that are in the
Domain of h(x) = x
x2 9−.
4. Find the value of 4 2
1 3− −LNM
OQP .
a. {x |x = ± 3} c. {x |x ≠ ± 3}b. {x |x < 3} d. {x |x > 3}
a. -14 c. 11b. -10 d. 14
5. Given f(x) = x2 + 5x −4, find f(-3). 6. Solve the system 2 3
2
x y
x y
+ =− = −
.
a. -28 c. 2b. 20 d. -10 a.
1
3
7
3, − c.
5
3
11
3, −
b. 1
3
7
3, d. − 5
3
11
3,
7. If f(x) = 4x + 2 and g(x) = 3
1x −, find
(f+g)(x).
8. Find the slope of the lineperpendicular to 3y − 5x = -2.
a. 4 2 1
1
2x x
x
− +−
c. 4 4 3
1
2x x
x
− +−
b. 4 2 1
1
2x x
x
+ −−
d. 4 4 3
1
2x x
x
− −−
a. −2
3 c.
2
5
b. 5
3 d. − 3
5
9. Find the distance between (2, -1) and (6, 9).
10. For which value of k are the pointsA(-3, 1), B(0, 2), and C(k, 5) ?
a. 4 5 c. 2 29
b. 11 d. 8 2a. 9 c. − 5
3
b. 5
3 d. 21
157
Fill in the blank.
11. A system of equations that has infinitely many solutions is identified as _______________.
12. A line that goes through the vertex of a parabola and divides it in half is identified as_______________________.
13. A ______________ is rectangular array of elements in rows and columns enclosed bybrackets.
14. The graph of a quadratic equation is a _______________________.
15. The set of first coordinates of a relation is called the _______________ of the function.
16. The vertical line test is used to determine if a relation is a __________________.
17. Every function has a ___________________ correspondence.
18. π is an example of a ______________________ number.
19. The ______________________ approximates the linear relationship for a set of data.
20. A line that slants downward to the right has ________________ slope.
21. Graph the data below:
College Students Faculty
DePaul University 24,800 2,180
Loyola University 18,100 1,600
University of Illinois 36,000 3,700
Northwestern University 12,000 700
University of Chicago 23,900 1,450
a. Find the equation for the line of best-fit.b. What correlation best describes the data?c. What is the correlation coefficient?
158
For problems 22-24, f(x) = 3x − 2 and g(x) = x2
4
22. a. Find f x−1b g .
b. What is the domain and the range of g(x)?
23. a. f x g x( ) ( )+ b f x g x( )* ( )
c. f x g x( ) ( )− d. f(x)
g(x)
24. a. f* g(x) b. g* f(x)
25. Graph ∆ ABC with vertices A(-2, 3), B(2, 2) and C(2, 5).a. Reflect ∆ ABC over the x-axis.b. Reflect ∆ ABC over the y-axis.c. Reflect ∆ ABC through the origin.
159
STRUCTURED CURRICULUM LESSON PLAN
Day: 045 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): All
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUnderstand geometric properties and
relationships; apply geometric concepts andformulas
Apply a variety of estimation strategies:standard rounding, order of magnitude, front-ending, compatible numbers, andcompensation
Use variables, number sentences, and equationsto represent solutions and solve problems
Analyze and interpret data presented in charts,graphs tables, and other displays
Understand and apply principles of probability,central tendency and variability
Demonstrate understanding of measurementconcepts and apply measurement skills
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Understand and apply geometric concepts andrelationships
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs
Understand and use methods of data collectionand analysis, including tables, charts, andcomparisons
Demonstrate an understanding of measurementconcepts and skills
Unit Focus/Foci
Exploring Mathematics and Science
Instructional Focus/Foci
Applying Mathematics and Science
Materials
Copies of the Science Fair Guidelines
160
Educational Strategies/Instructional Procedures
Discuss the Science Fair Guidelines with the class. Take the students to the school library toresearch possible topics or research an approved topic. Remind students that all topics must beapproved by their mathematics or science teacher.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in contextDraw conclusions, inferring meanings from the text
SC: Analyze and interpret dataApply scientific method to solve problems
SS: Read and interpret maps, charts, graphs, and cartoons
Connection(s)
Enrichment:
Fine Arts:
Home: Discuss possible Science Fair projects with parent and ask for suggestions.
Remediation:
Technology: Students may use the Internet to search for a topic or to research an approvedtopic.
Assessment
Homework
Research Science Fair projects.
Teacher Notes
Discuss the mathematics department’s participation in the Science Fair with the sciencedepartment.
161
Science Fair Guidelines
Project Planning and Selection
1. Decide on the nature of the investigation.
! Consult with the science and/or mathematics teachers about the selection of a topic. Topicsmust be approved by your mathematics or science teacher.
! Be specific. Investigate and explore an interest--a fascination--an idea that raises a questionthat would be stimulating to answer. Improve upon a previous science fair project from adifferent point of view (new question).
2. Proceed scientifically.
! Decide exactly what the question or problem is and state it clearly.
! Be sure that the problem chosen is within abilities and resources available.
! Do research. The most important part of a project is finding out as much as possible aboutthe problem. Spend some time in the library at school, the regional library, and the mainChicago Public Library. If possible, visit the libraries of local colleges. For informationabout library sources, contact the local science fair coordinator and school librarian. Somelibraries may be closed to the general public, but many will allow you to use their collectionon the premises.
! Formulate a hypothesis for testing. Design experiments to test the hypothesis.
! Find ways to measure, observe, and record what happens in each aspect of the project.Remember that every experiment must have a control.
! Do not abandon negative results. Use them to modify the hypothesis; then test again.
! Set time schedules. Remember that constant evaluation of the research takes place from thefirst step to the final conclusion.
Remember, good projects are the result of careful work, careful planning, and constantrevision as data accumulate.
162
Exhibit Design and Evaluation
The exhibit must not exceed dimensions of 76 cm deep and 122 cm wide. Build theexhibit no higher than 122 cm. No overhang is allowed . If the scientific apparatus exceedsthe height limit, use photographs to show what has been done. No part of the project maybe placed on the floor.
! Construct your own exhibit; teachers and parents are to provide only the necessaryguidance, encouragement, and constructive criticism.
! Have a purpose and hypothesis for your project; it must apply to a definite scientificquestion. The reason for doing a research project is to make a significantcontribution to the body of scientific knowledge or solve a problem.
! Keep the title of your project brief, captivating, and prominently visible on theexhibit. It may contain no more than 45 characters and spaces. Titles in excess of45 characters will be shortened to fit into available space on the entry form.
! Make lettering neat and uncluttered. Make sure all words are spelled correctly.
! Determine the best way to present the research. The presentation must includegraphs, charts, posters, 35-mm slides, videotape, transparencies, demonstration ofapparatus, or other components.
! Be well versed in as many aspects of the project as possible.
! Be prepared to present the project.
! Prepare not only for direct questions pertinent to the research but also for relatedquestions.
163
STRUCTURED CURRICULUM LESSON PLAN
Day: 046 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Use variables, number sentences, and
equations to represent solutions and solveproblems
ISAT:Identify, analyze, and solve problems using
equations, inequalities, functions, andtheir graphs
Unit Focus/Foci
Matrices and Systems of Equations
Instructional Focus/Foci
Using Matrix Row Operations
Materials
Classroom set of graphing calculatorOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Write the following system on the overhead and instruct students to solve this system using theelimination method.
–4x + 3y = 36 2x + 5y = 34
The augmented matrix representing this system is: −LNM
OQP
4 3 36
2 3 34
|
|
An augmented matrix can be used to solve a system of linear equations, especially when thesystem has three or more equations and three or more unknowns.The elementary row operations used are:
1. Interchanging two rows.2. Multiplying all elements of a row by a non-zero constant.3. Adding a constant multiple of the elements of one row to the corresponding elements
of another row.By using these row operations, one of the rows can be reduced into Gaussian Reduction (oneequation with one unknown). Then we can “back substitute” to find the remaining unknowns.The teacher should carefully work through the following example, using the system above.
164
Example 1: −LNM
OQP
4 3 36
2 3 34
|
|
Multiply R2* by 2 ⇒−LNM
OQP
4 3 36
4 10 68
|
|
Add R1 to R2 and replace R2 ⇒−LNM
OQP
4 3 36
0 13 104
|
|
Divide R2 by 13 ⇒−LNM
OQP
4 3 36
0 1 8
|
|
*R2 represents Row 2
From R2 we see that y = 8. “Back substitute” into the equation –4x + 3y = 36, and we getx = –3. So the solution to this system is (–3, 8).
Example 2: Solve the system x – 2y + z = 7
3x + y – z = 2 2x + 3y+ 2z = 7
The augmented matrix is
1 2 1 7
3 1 1 2
2 3 2 7
−−
L
NMMM
O
QPPP
|
|
|
Multiply R1 by –3 ⇒− − −
−L
NMMM
O
QPPP
3 6 3 21
3 1 1 2
2 3 2 7
|
|
|
Add R1 to R2 and replace R2 ⇒−
− −L
NMMM
O
QPPP
1 2 1 7
0 7 4 19
2 3 2 7
|
|
|
Multiply R1 by –2 ⇒− − −
− −L
NMMM
O
QPPP
2 4 2 14
0 7 4 19
2 3 2 7
|
|
|
165
Add R1 to R3 and replace R3 ⇒−
− −−
L
NMMM
O
QPPP
1 2 1 7
0 7 4 19
0 7 0 7
|
|
|
Multiply R2 by – 1 ⇒−−
−
L
NMMM
O
QPPP
1 2 1 7
0 7 4 19
0 7 0 7
|
|
|
Add R2 to R3 and replace R3 ⇒−−
L
NMMM
O
QPPP
1 2 1 7
0 7 4 19
0 0 4 12
|
|
|
Multiply R3 by 1
4
1 2 1 7
0 7 4 19
0 0 1 3
⇒−−
L
NMMM
O
QPPP
|
|
|
Substitute z = 3 into the equation 7y – 4z = -19 and solve for y; y = -1.Substitute y = -1 and z = 3 into equation x – 2y + z = 7 and solve x; x = 2.Solution : (2, -1, 3)
Integration with Core Subject(s)LA: Understand explicit, factual information
Understand the meaning of words in context
Connection(s)
Enrichment:
Fine Arts:
Home:
Remediation:
Technology:
166
Assessment
Teacher observation
Homework
Assign appropriate problems from your text.
Teacher Notes
167
STRUCTURED CURRICULUM LESSON PLAN
Day: 047 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, within concepts
ISAT:Identify, analyze, and solve problems using
equations
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Finding the Inverse of a Matrix
Materials
Classroom set of graphing calculatorsOverhead graphing calculatorsOverhead projector
Educational Strategies/Instructional Procedures
Have students find the multiplicative inverse of each number.
− − − −33
83 5
9
5.
Review definitions: multiplicative inverse, multiplicative identity of a matrix. Introduce finding the inverse of a matrix using a graphing calculator.
Enter the following matrix on the graphing calculator:
13
02
Then go to NAMES (in MATRX). Enter [A], then press x–1.
Given a 2 x 2 matrix, have the students find its inverse. Check that A x A–1 = A–1 x A.
168
Give students the matrix equation Ax = B where A= 2 3
1 5
LNM
OQP , x =
x
y
LNMOQP , and B =
8
11
LNMOQP
Ask students if they think they can solve the equation for x by multiplying both sides of the
equation by A–1. After discussing students’ ideas, have them try multiplying and then look atthe resulting matrix.
Then ask students: What system of equations is represented by these matrices?[Answer: 2x + 3y = 8, x + 5y = 11]
Ask if the solution they found to the matrix equation Ax = B is also the solution to the system ofequations.[Answer: Yes, the solution is x = 1, y = 2.]
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
SC: Apply scientific method to solve problems
Connection(s)
Enrichment: Given a right triangle, an obtuse triangle, and an equiangular triangle, write asystem of equations that you would use to find the measure of each angle of the triangle. Writethe coefficient matrix.
Fine Arts:
Home: Parents will sign a weekly homework assignment sheet.
Remediation: Have students write four 2 x 2 matrices. Ask students to exchange matrices anduse a graphic calculator to find the inverse of each matrix. Then have each student exchangework with another student and check the answers by multiplying each matrix and its inverse.Ask students to keep a list of any matrices they find that do not have inverses. Discuss theresults.
Technology: A graphing calculator may be used to find the inverse of some 3 x 3 matrices.
Assessment
Give students a 2 x 2 matrix. Have them find the inverse and have them multiply the inversetimes the original matrix to check that it is the inverse. What is the result?
169
Homework
Assign appropriate problems from your text.
Teacher Notes
170
STRUCTURED CURRICULUM LESSON PLAN
Day: 048 - 049 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 9A1
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Using Matrix Algebra to Solve a System of Equations
Materials
Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector
Educational Strategies/Instructional Procedures
Review representing a system of equations as a matrix equation.Rewrite matrix equations using names of matrices.
Compare solving ax = b to solving AX = B
1/a (ax) = 1/a (b) (A–1) (AX) =(A–1) (B)
x = b/a X = (A–1)B
To find the solution (A–1)B, enter matrices A and B on a graphic calculator. Then find theproduct (A^–1)B (note the notation used on a calculator).
171
Students will solve the following systems of linear equations using algebra:1 2x + 3y - 2z = 4 2. x + 2y = -6 3x - 3y + 2z = 16 y + 2z = 11 6x - 2y + 8z = 10 2x + z = 16
Solutions: 1. x = 4, y = -3, and z = -5
2 2. x = 4, y = -5, and z = 8
Ask: What are the coefficients of the missing terms? (0)
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
SC: Analyze and interpret dataApply scientific method to solve problems
Connection(s)
Enrichment: If you pick a letter at random from the alphabet, what is the probability that it is inthe word Algebra? Express your solution as a common fraction.
Fine Arts:
Home: Parents will sign homework record weekly.
Remediation:
Technology: Use a graphing calculator to check solutions to in-class assignment.
Assessment
Have students write a paragraph explaining why it is important to check the solutions in eachoriginal equation.
Homework
Assign appropriate problems from your text.
172
Teacher Notes
This is a two-day lesson. The second day (Day 49) should be used to review homework, forremediation exercises, or for extra practice, as necessary. Prepare copies of Matrix Explorationsworksheet.
173
STRUCTURED CURRICULUM LESSON PLAN
Day: 050 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percentsexplicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Using matrices to represent and transform objects
Materials
Copies of Matrix ExplorationGraph paperColored pencilsCartoons from newspapers or magazines
Educational Strategies/Instructional Procedures
Divide students into groups of four. Have each group complete Matrix Exploration. Randomlyselect three groups to report their findings to the class.
Integration with Core Subject(s)
LA: Understand explicit, factual informationUnderstand the meaning of words in context
SC: Apply scientific method to solve problemsSS: Read and interpret maps, charts, graphs, and cartoons
174
Connection(s)
Enrichment: What percent of the whole numbers from 1 to 1000 inclusive are perfect squares?Solution: 3.1
Fine Arts: After completing the lesson, have students choose a cartoon from a newspaper ormagazine and trace it onto graph paper. Then have them explain how to enlarge the cartoon to1.5 times its original size by using a size-change matrix. Check students’ responses. The matrix
should be 15 0
0 15
.
.
LNM
OQP times the cartoon.
Home: Have parents sign homework record weekly.
Remediation:
Technology:
Assessment
Teacher observation
Homework
Assign appropriate problems from your text.
Teacher Notes
Solutions to Matrix Exploration:
1.
175
2. a. AB = 2 6 6 2
2 2 6 6− − − −LNM
OQP
c. a reflection about the x-axis
b.
3. a. AB = − − − −− − − −LNM
OQP
2 6 6 2
2 2 6 6
c. a reflection about the origin
b.
4. a. AB = − − − −LNM
OQP
2 6 6 2
2 2 6 6
c. a reflection about the y-axis
b.
5. a. AB = 4 12 12 4
4 4 12 12
LNM
OQP
c. enlarged by a multiple of 2
b.
176
6. a. 1 3 3 1
1 1 3 3
LNM
OQP
c. shrunk by 1
3
b.
177
Matrix Exploration
1. Graph the points F(2, 2) ,O(6, 2), R(6,6) and M (2,6) and sketch the quadrilateral.
Enter the vertices in B =2 6 6 2
2 2 6 6
LNM
OQP
For 2-6 below:a) Find A-Rb) Graph the imagesc) Describe the graph
2. A = 1 0
0 1−LNMOQP
3. A = −
−LNM
OQP
1 0
0 1
4. A = 0 1
1 0
−LNM
OQP
178
5. A = 2 0
0 2
LNMOQP
6. A=
1
20
01
2
L
N
MMMM
O
Q
PPPP
179
STRUCTURED CURRICULUM LESSON PLAN
Day: 051 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percents,explicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal-world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes, and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations
Unit Focus/Foci
Exploring Matrices and Systems of Equations
Instructional Focus/Foci
Reviewing Unit Four
Materials
Educational Strategies/Instructional Procedures
Have students check their work from the previous day’s homework.
Review Unit Four using a format from Appendix C.
Integration with Core Subject(s)
LA: Understand the meaning of key words and phrases in text
Connection(s)
Enrichment:
Fine Arts:
180
Home:
Remediation:
Technology:Assessment
Teacher observation
Homework
Study for the Unit Four Assessment.
Teacher Notes
Prepare copies of the Unit Four Assessment.
181
STRUCTURED CURRICULUM LESSON PLAN
Day: 052 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percents,explicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal-world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes, and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations
Unit Focus/Foci
Matrices and Systems of Equations
Instructional Focus/Foci
Assessing Unit Four
Materials
Copies of Unit Four Assessment
Educational Strategies/Instructional Procedures
Administer the Unit Four Assessment.
Integration with Core Subject(s)
LA: Understand the meaning of key words and phrases in text
Connection(s)
Enrichment:
Fine Arts:
Home:
182
Remediation:
Technology:
Assessment
Evaluate the Unit Four Assessment using the Structured Curriculum Scoring Rubric.
Homework
Have students select three samples of their work to place in their portfolio, then write anexplanation as to why each piece was selected.
Have students continue work on Science Fair projects.
183
Teacher Notes
Solutions to Unit 4 Assessment:
1.
2.
a. 2M = 10 4 16
0 12 6
−LNM
OQP b.
1
2M =
5
21 4
0 33
2
−L
N
MMMM
O
Q
PPPPc. -2M =
− −− −
LNM
OQP
10 4 16
0 12 6
3. a)
b)
c)
4. XY = − −
−LNM
OQP
12 24 12
11 40 8
5. YZ = 14
3
LNMOQP
6. ZY = no solution7. No solution for problem #6 due to a dimension mismatch.8. x = 3.6, y = 1.69. x = 3, y = 210. x = 774, y = 240,z = -188
184
Unit 4 Assessment
1. A triangle is described by the matrix M = 5 2 8
0 6 3
−LNM
OQP
The vertices of the triangle are (5,0), (-2, 6), and (8,3). Plot the vertices and sketch the triangle.
2. Calculate the value of the matrix given the following transformations.a. 2M
b. 1
2M
c. -2M
3. Plot and sketch the matrices from problem #2. Give a brief description of what happened to each matrix.
a. 2M b. 1
2M c. –2M
Let X=−LNM
OQP
6 0
4 3Y =
− −LNM
OQP
2 4 2
1 8 0 Z =
−
−
L
NMMM
O
QPPP
3
0
4
4. XY
5. `YZ
6. ZY
7. If problems 4-6 do not have solutions, explain why.
185
Given the following systems of equations, use matrices to solve for the variables.
8. 4x + 6y = 24 x + 9y = 18
9. –4x + 8y = 4
3x + 1
2y = 10
10.
1
3
2
32 42
2 3 12 12
4 6 9 36
x y
x y
x y z
+ + =
+ + =− + − =
186
STRUCTURED CURRICULUM LESSON PLAN
Day: 053 Subject: Advanced Algebra Grade Level: High School
Correlations (SG,CAS,CFS): 8D2
TAP:Perform arithmetic operations involving
integers, fractions, decimals and percents,explicitly stated or within context
Choose and apply appropriate operationalprocedures and problem-solving strategies toreal-world situations
Understand number systemsUse variables, number sentences, and equations
to represent solutions and solve problems
ISAT:Solve problems requiring computations with
whole numbers, fractions, decimals, ratios,percents, and proportions
Use mathematical skills to estimate,approximate, and predict outcomes, and tojudge reasonableness of results
Identify, analyze, and solve problems usingequations
Unit Focus/Foci
Matrices and Systems of Equations
Instructional Focus/Foci
Assessing Matrices and Systems of Equations
Materials
Unit Four Assessment (students’ copies)
Educational Strategies/Instructional Procedures
Return the Unit Four Assessment and review the solutions with the students.
Have the students write an essay in which they reflect upon their performance in Unit Four.Invite the students to use the following questions as guidelines for writing their essays:
1. Did I do my best?2. What gave me the most difficulty?3. Where do I need the most help?4. What methods or recourse did I use to improve myself?5. What would I do differently next time?
Remind students that an essay has a minimum of three paragraphs: introduction, body, andconclusion.
187
Integration with Core Subject(s)
LA: Understand the meaning of key words and phrases in textSC: Analyze and interpret data
Connection(s)
Enrichment:
Fine Arts
Home:
Remediation:
Technology:Assessment
Teacher observation
Homework
Have students select three samples of their work to place in their portfolio and write anexplanation as to why each piece was selected.
Have students continue to work on Science Fair projects.
Teacher Notes