Research to Support Algebra Intervention By Sharon S. Whitehead, Ph.D. Secondary Math Specialist, Mesa, Arizona
High school students study algebra more than any other course in mathematics (Usiskin 1980). Perhaps because of the abstractness of the material, the ages of the students, the pedagogical styles of the teachers, or the complexity of the curriculum, algebra can be very difficult and stressful for many students. However, students will have difficulty in all mathematical courses subsequent to Algebra I if they do not master the material in this pivotal course. And, almost half of the states now require students to pass Algebra I before graduation. So, the question becomes, “How can we best help these students who are having difficulty with the concepts presented in Algebra I?” The first thing to do is to remind teachers why algebraic thinking is so important. Algebra helps students engage in problem solving and develop rich analytical skills that will serve them throughout their lifetimes. The type of thinking encouraged in an Algebra I course helps students seek solutions, not just memorize procedures; explore patterns, not just memorize formulas; and formulate conjectures, not just do exercises. The National Research Council (1989) encourages this type of thinking to be developed in mathematics courses. Active Algebra: Algebra I Crash Course models all of these practices. The National Council of Teachers of Mathematics (NCTM) (2000) challenges teachers to “imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction.” Algebra I Crash Course is a high-quality, engaging curriculum designed for grades 8 through 12. It can be used for standardized test preparation, credit recovery, Algebra I intervention, and summer school. This program is ideally suited for students having trouble with Algebra I. Not every student learns the same way, and the flexibility of this program allows teachers to engage students in different ways. The program is set up in such a way that it helps students grasp complex algebra concepts with a conceptual understanding that ensures deep procedural knowledge. Conceptual knowledge can be defined as knowledge that involves relations or connections (but not necessarily rich ones) (Star 2005). Star defines deep procedural knowledge as involving “comprehension, flexibility, and critical judgment [as] distinct from (but possibly related to) knowledge of concepts” (p. 116). Too often, mathematical knowledge is placed only into one category or the other. This eliminates the necessity of the dialogue between both conceptual and procedural knowledge. Crash Course moves students between these two types of knowledge fluidly.
Correlation to the NCTM Standards NCTM Standard Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers. Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency— mentally or with paper and pencil in simple cases and using technology in all cases.
Lesson Title and Page Number
Adding Integers (pp. 47–53) Multiplying and Dividing Integers (pp. 54–57)
Writing One-Variable Equations (pp. 58–63) Point-Slope Form (pp. 115–117) Graphing Inequalities (pp. 121–123) Solving Multistep Inequalities (pp. 128–131) Compound Inequalities (pp. 132–137) Absolute Value Inequalities (pp. 138–142) Graphing Two-Variable Inequalities (pp. 143–146) Comparing Systems (pp. 151–153) Substitution Method for Solving Systems (pp. 154–157) Addition Method for Solving Systems (pp. 158–161) Systems of Equations (p. 162) Rational Equations (pp. 224–229)
Develop an initial conceptual understanding of different uses of variables.
Collecting Like Terms (pp. 68–72)
Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.
Solving Linear Equations 1 (pp. 76–82)
Distributing and Collecting (pp. 73–75)
Solving Linear Equations 2 (pp. 84–87) Solving Equations (pp. 92–94) Equations as Relations (pp. 104–106) Factoring the Greatest Common Factor (pp. 184–187) Factoring Trinomials (pp. 188–194) Factoring Special Types of Problems (pp. 195–198)
Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
Diagnostic Test Directions: Solve each equation. 1. –5 – 6 = A. 11 B. –1
C. –11 D. 1
2. –4 + 7 = A. 12 B. 3
C. –12 D. –3
Directions: Simplify each given expression.
3. 7 – 15 = A. –8 B. 22
C. 8 D. –22
4. (–4)(3) = A. 12 B. 9
C. 7 D. –12
5. –24 = –4 A. 20 B. 6
C. –6 D. –28
6. (6)(–3)(–2) = A. 11 B. –36
C. 1 D. 36
Directions: Read each question and circle the correct answer. 7. Find the expression for the given problem. A chicken sandwich costs $1.50 more than a hamburger. A. h + $1.50 C. $1.50h
B. h – $1.50
8. Find an equation for the given problem. The width of a rectangle is 9 meters. The width is 3 less than twice the length. What is the equation for the length of the rectangle? A. 3 – 2x = 9 C. 2x – 3 = 9 B. 2x + 3 = 9 D. 3 + 2x = 9
$1.50 D. h
#11002 (i2567)—Algebra I Crash Course
9. 4x – 7 + 3x2 – x + 3 – x2 + 7x A. 4x2 + 8x + 4 C. –2x2 – 10x + 4 B. 2x2 –10x + 4 D. 2x2 + 10x – 4 10. 10 – m – m2 – 13 – 4m + 5 + 3m2 – m A. 4m2 – 6m + 2 C. 2m2 – 6m + 2 B. –2m2 + 6m – 2 D. –4m2 + 6m – 2 11. m + 10 – (6m + 4) – 3 A. 5m + 3 C. 5m + 11 B. –5m + 3 D. –5m – 11 12. –4x – 3(2x + 3) + 7x + 12 A. –3x + 3 C. 3x + 3 B. –3x + 21 D. 3x – 21 Directions: Solve for each variable. 13. 3x – 5 = 4x + 3
Data-Driven Instruction Standards-based instruction must begin with the goal that all students can reach the standards if given appropriate instruction, materials, and support. Therefore, it is the responsibility of teachers to use various assessment and data-collection strategies to help identify when a student has misconceptions that will affect further learning of mathematical skills and concepts. Frequent formal and informal assessments provide teachers with the data needed to make informed decisions about what to teach and how to teach it. This is the only way that teachers will know who is struggling with various concepts and how to address the difficulties that students are experiencing with the curriculum.
Formal Assessments The following formal assessments are included throughout the Algebra I Crash Course program:
• Diagnostic Test—This assessment is given at the beginning of the program and at the end of the program. When students are retested, they should have the same classroom conditions and directions. This assessment is meant to be given within the increments of one school year and not stretched out beyond a single school year of instruction. In this way, the increments between testing are not too stretched out to detect growth. The final post-test use of the assessment becomes the ultimate summative evaluation that measures whether students have mastered what they were taught throughout the whole program.
• Quizzes—The quizzes are integrated into the lessons throughout the program. They measure the learned objectives for a specific lesson or series of lessons. These assessments can be used by teachers to determine whether reteaching is necessary. There are two versions of each quiz. Form A is in the appendix and Form B is only on the Teacher Resource CD. Use both versions at the same time to discourage copying, or use one version as the initial assessment and the other version as the retest.
Informal Assessments The following informal assessments are included throughout the program. These allow teachers to frequently watch for markers of comprehension in student responses.
• Problems in the Lessons—Within each lesson, practice problems are provided for modeling. In the lessons, teachers first model the problem while the students watch. Then, the teachers model the problems while the students help. Next, the students model the problems while the teacher helps. Finally, the students model the problems while the teacher watches.
• Guided Practice Book—This student consumable contains activity sheets to be used in correlation with lessons. These activities should be used to practice skills and concepts, and it is recommended that these activities are only given a completion grade versus a formal grade.
The object of the game is to be the first student to have a straight line of answers covered on his or her MATHO card.
• Transparencies folder—Radicals MATHO Problems (trans19.pdf) • Appendix C: Games—Radicals MATHO Answer Sheet (page 267; radmatho.pdf) • Games folder on the CD—MATHO cards (mathocrd.pdf) • Games folder on the CD—MATHO chips (mathochp.pdf)
Procedure Step 1
Cut the Radicals MATHO Problems transparency into strips with one problem and corresponding letter on each strip. Give each student a MATHO card and a handful of small game markers/chips. Students also need copies of the Radicals MATHO Answer Sheet. Explain the directions to the student: • This game is played like Bingo. • Put one problem strip at a time on the overhead. Have the students solve each problem.
Note: The problem is printed on one end of the strip and the answer on the other for easy location of winning answers. When placing a problem on the overhead, make sure the answer is not visible. It should be hanging off the side.
• Require all students to play. They should write the problems and answers in an organized manner on notebook paper. If necessary, give students grades for this work.
• Once students have solved each problem, they should look on their MATHO cards for the answer. If a student’s card has the right answer, he or she covers it with a marker.
• Tell students to call out, “MATHO” when they get a straight line of covered answers. You can also play for diagonal lines or blackouts (the whole card).
• Have the winning student(s) call out their answers and check them on the transparency strips that have been played.
• Understands basic concepts, applications, and solution methods of radical equations. • Students will understand the Pythagorean theorem and learn how to solve radical equations.
Materials • PowerPoint folder on the CD—Solving Radical Equations (lesson41.ppt) (optional) • Guided Practice Book—Solving Radical Equations (page 134; page134.pdf)
Procedure Step 1
Present the Notes on the Pythagorean Theorem (pages 208–209). These notes are provided as part of the lesson’s PowerPoint slide show on the CD (lesson41.ppt). • Discuss the bolded questions with students. • You may want to share some of the history of the Pythagorean theorem with students.
Notes on the Pythagorean Theorem What is the Pythagorean theorem? a2 + b2 = c2
c2 a2 b2
What is the hypotenuse? • It is represented by the variable c. • It is the longest side. • It is always the side across from the right angle. What are the legs? • They are represented by the variables a and b. • When solving for one of the variables, the order of a and b can be switched because of the commutative property.
When is the Pythagorean theorem used? • It is used to find the length of the third side of a right triangle when the other two sides are known. • It is used to determine if a right triangle can be made from three given line segments. How could the Pythagorean theorem help carpenters? • Discuss with students how buildings are constructed with right angles. Examples Complete these examples with the students by working out the formula for each triangle. Round the answers to the nearest hundredth. a. 5
If available, you may want students to check their solutions on graphing calculators using the Home Screen method. Store a value for a variable, input the equation, and press enter. The calculator will use Boolean logic to determine if the value stored is true (1) or false (0).
Discuss the principle square root in the above example. • Tell students that the symbol represents the principle square root and that it is a positive root. • Ask students, “Can a principle square root equal a negative value?” No; therefore, the answer to the problem is ø. m. y + 3 = y + 1 ( y + 3)2 = (y + 1)2 y + 3 = y2 + 2y + 1 –y – 3
–y – 3
0 = (y + 2)(y – 1)
y + 2 = 0
+ 1 +1
y=1 • Remind students to substitute solutions into the original equation to find the correct answer.
Algebra I Review 8 GPB (pages 124–127) 1. 4 hours 2. 40 people 3. C 4. D 5. C 6. y = –3x + 2 7. m = slope = rise run b = y-intercept 8. 3,4,5; 5,12,13; 7,24,25; 8,15,17; 9,40,41 9. C 10. check graph 11. check graph 12. Perimeter—add all sides; 2l + 2w Area—length x width 13. D = set of all x’s or independent variable or 1st coordinate R = set of all y’s or dependent variables or 2nd coordinate 14. t = 2s r = 3b b=f+8 s= 1r 3 15. n = –2 m=8 16. use one variable 17. 1 variable: 4w + 8 = 38 or 2w + 2(w + 4) = 38; 2 variables: 2w + 2l = 38, l=w+4 18. D 19. A Standardized Test Practice 8 GPB (pages 128–133) 1. B 2. C 3. C 4. A 5. B 6. D 7. A
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A B D B D C A B A B B D D
Solving Radical Equations GPB (page 134) 1. x = 49 2 2. p = 100 3. y = 25 4. Ø 5. x = 256 3 6. x = 4 7. x = 16 8. a = 45 9. m = –2, m = –3 Identifying the Axis of Symmetry and the Vertex GPB (page 135) 1. opens downward 2. (2, 64) 3. The x-axis represents time in seconds, and the y-axis represents the position of the ball in feet. 4. It tells us that at the time of two seconds, the ball is at its maximum height of 64 feet. 5. x = 2 Simplifying Rational Expressions GPB (page 136) 1. x + 32 6x 2. x – 3 3. mn 2 –y2 4. 3x2