Algebra 1 Teaching Goals and Strategies
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Algebra 1 Teaching Goals and Strategies
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1. SUPPORT FOR STUDENT LEARNING Symbolic reasoning and calculations with symbols are “central” in Algebra 1. Through the study and use of algebra, the learner develops an understanding of the symbolic language of mathematics and the sciences. Algebra 1 develops the skills and concepts to help solve a wide variety of problems. Goals:
Below, are “Algebra 1 major concepts” aligned with
the California Algebra 1 Standards.
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Strategies: A. Emphasize conceptual understanding, that is stress the “big ideas.” These big ideas help all students understand how the current topic being studied connects to material previously learned. B. Teaching the “big ideas” helps link the Concepts and topics to the California Algebra Standards. C. Teach concepts or big ideas before introducing the Vocabulary of Algebra (See U. under Objectives) D. Use etymologies for “new” words. |
Goals, Standards, Concepts
“A.” Concept: Algebra Solving Strategies: (Std. 3) Students solve equations and inequalities; (Std. 4) Students simplify expressionsbefore solving linear equations and inequalities in one variable,such as 3(2x-5)+4(x-2) = 12; (Std. 5) Solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. *** |
Strategies, Approach, Plan to meet Goals,
Standards, Concepts A. Algebra Solving
Strategies Guess
& Check, Tables, Graphing, Equations, “Big Idea” Examples: 4x + 2 = 10, x2 + 3x = 10 4x = 10 – 2 or 4x = 8 or 4x/4 = 8/4 or x = 2 ½x½ = 4 ® x= 2 or x = -2
Emphasize multiple solving
strategies: Using (1.) words, (2.) tables of data, (3.) graphs, and (4.) algebraic symbols (ie equations). Multiple solving strategies is a “BIG IDEA” To illustrate multiple solving strategies – “A turtle walks at five feet per minute, and a snail crawls at three feet per minute. The turtle and the snail start from an oak tree and head toward an elm stree that is located thirty feet from the oak tree.” In this simple setting, students can investigate Patterns in words, tables, symbols, and graphs. Choose content rich problems combining Multiple Strands (eg pre-algebra, algebra, and geometry). See “Big Square” below. |
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“B.”
Concept: Writing Equations (Std. 15.) Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Goals,
Standards, Concepts |
Writing Equations
(See B. Concept Under Goals) Use the first 5 or 10 minutes of class for a warmup Problem or activity to: (1.) offer the student opportunity to be “ready for work.” (2.) Prepare the learner for today’s lesson Strategies, Approach, Plan to meet Goals, Standards, Concepts |
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“C.”
Concept: Manipulating Equations (Std. 4.0) Students simplify expressions before solving linear equations and inequalities in one variable, such as3(2x-5)+4(x-2) = 12; (Std. 6.0) Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by a linear inequality (e.g., the region defined by 2x+6y<4); (Std. 9.0) Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Also, they are able to solve a system of two linear inequalities in two variables and to sketch the solution sets; (Std. 10.0) Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques; (Std. 12.0) Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to their lowest terms; |
Manipulating Equations: (See C. Under Goals) 3(2x-5)+4(x-2) = 12 or distribute: 6x-15 + 4x-8 = 12 combine like term:10x – 23 = 12 add 23 to both sides: 10x = 35 divide both sides by 10: x = 3.5 Check: 3(2*3.5 –5) + 4(3.5-2) = ? 12 Or 3(7-5) + 4(1.5) =? 12 Or 3(2) + 6 =? 12 Or 6 + 6 =? 12 yes
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Goals, Standards, Concepts “D.”
Concept: Absolute Value (Std. 3.0) Students solve equations and inequalities involving absolute values); *** |
Strategies, Approach, Plan to meet Goals, Standards, Concepts Absolute Value
(See D. Under Goals) Meaning of: Absolute value of an expression, means the distance between the expression and the origin. Example: ½3½ = 3 and also = -3 because Both 3 and –3 measure 3 units from the origin. Example: Graph abs 3 on the number line
Remember: “Always simplify expression inside The “absolute value” symbols. |
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“E.”
Concept: Inequalities and Systems of Equations (Std. 3.0) Students solve inequalities involving absolute values; (Std. 4.0) Students simplify expressions before solving inequalities in one variable,such as 3(2x-5)+4(x-2) < 12; (Std. 6.0) Students are able to sketch the region defined by a linear inequality (e.g., the region defined by 2x+6y<4); (Std. 9.0) Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. *** |
Inequalities and Systems of Equations (See. E. Under Objectives) ½x-3½+ 7 £ 15 To isolate the absolute value, we subtract 7 from both sides ½x-3½ £ 8 this means that -8 £ x-3 £ 8 Add 3 to both sides -5 £x £ 11 x Î [-5, 11] |
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Goals, Standards, Concepts “F.”
Concept: Variables and Patterns of Change (Std. 15.0) Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems; (Std. 16.0) Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions; *** |
Strategies, Approach, Plan to
meet Goals, Standards, Concepts Variables and Patterns of
Change –Linear (See F., Under Objectives) |
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“G.” Concept: Variables and Patterns of Change (Std. 23.0) Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. *** |
Variables and Patterns of Change – 2nd
degree |
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“H.”
Concept: Linear Functions and Linear Equations
(Std. 4.0) Students simplify expressionsbefore solving linear equations in onevariable, such as 3(2x-5)+4(x-2) = 12; (Std. 5.0) Solve multi-step problems, including word problems, involving linear equations in one variable and provide justification for each step; (Std. 6.0) Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4); (Std. 7.0) Students verify that a point lies on a line, when given the equation for the line. Students are able to derive linear equations by using the Goals, Standards, Concepts “H.” Concept: Linear
Functions and Linear Equations
(continued) point-slope formula,
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Linear Functions and Equations – (see H. under Objectives) Explain the differencebetween a “function” and an“equation”: e.g., 3x + 6 = 18 is an “equation” while 3x + 6 = y is a “function.” That is to say, a linear function has MANY answers (often represented by a straight line graphed in the coordinate plane) while a linear equation has one answer for x represented by a single point on the straight line. Recognize further that the slope of the line is 3/1 (i.e. = rise/run) while the 6 in the function represents the y-intercept. Define a function, discuss domain and range of the function -- all “x” values make up the domain— and all the corresponding “y” values determine the range. Strategies, Approach, Plan to meet Goals, Standards, Concepts Linear Functions and Equations – (see H. under Goals) (continued) Develop linear functions from real life situations. Ask students to model real life events to form a linear equation. |
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“I.” Concept: Quadratic
Functions- Solving of (Std. 14.0) Students solve a quadratic equation by factoring or completing the square; (Std. 19.0) Students know the quadratic formula and are familiar with its proof by completing the square; (Std. 20.0) Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations; (Std. 21.0) Students graph quadratic functions and know that their roots are the x-intercepts; (Std. 22.0) Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points; *** |
Quadratic Functions – (see I. Standards under Goals) Contrast quadratic equations with linear equations, and describe a quadratic equation in standard form, factored form, the zero product property, and the difference of two squares. Relate that the graphical solution of a quadratic function is a parabola opening upwards or downwards. Introduce the discriminant as that part of the quadratic formula, which predicts how many roots the quadratic function, will have. Use memory aid techniques, such as jingles to aid in the learning the Quadratic Formula. |
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Goals, Standards, Concepts
“J.” Concept: Quadratic
Functions- the Solving of (Std. 23.0) Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity *** |
Strategies, Approach, Plan to meet Goals,
Standards, Concepts Quadratic Functions – (see J. under Goals) Contrast quadratic equations withlinear equations, and describe a zero product property, and the difference of two squares. Relate that the graphical solution of a quadratic function is a parabola opening upwards or downwards. Introduce the discriminant as that part of the quadratic formula, which predicts how many roots the quadratic function will have. Use memory aid techniques, such as jingles as an aid in learning the Quadratic Formula. Remember: The Quadratic Formula solves a Quadratic equation. |
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“K.”
Concept: Quadratic Formula (Std. 19.0) Students know the quadratic formula and are familiar with its proof by completing the square; (Std. 20.0) Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations; (Std. 22.0) Students use the quadratic formula or
factoring techniques or both to determine whether the graph of a quadratic
function will intersect the x-axis in zero, one, or two points; *** |
Quadratic Formula
– (See K. Under Objectives) The quadratic formula, derived by completing the square, looks like this.
It is used to solve any 2nd degree quadratic equation when it is written in the general form as:
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Goals, Standards,
Concepts
“L.” Concept: Exponents and Radicals
(Std. 2.0) Students know and use the rules of exponents, taking a root, raising to a fractional power. *** |
Strategies, Approach, Plan to meet Goals,
Standards, Concepts Exponents
and Radicals (see
L. under Goals) Explain and demonstrate the properties of exponents and radicals. Use memory aid techniques, such as jingles as an aid in learning the Quadratic Formula. Show the exponent rules and Their applications to positive numbers. Exponents and Radicals is =, sum +, difference -, product x, quotient ¸ |
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“M.”
Concept: Systems of Equations (Std. 9.0) Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically; *** |
Systems of Equations – (See M. Under Goals) Consider the case of two linear equations in two unknowns. Teach the substitution and elimination methods to find single solutions to systems of Linear and non-linear equations. Discuss consistent, inconsistent, and dependent systems of equations. |
Goals, Standards,
Concepts
“N.” Concept: Word Problems
(Std. 5.0) Solve multi-step problems, including word problems, involving linear equations in one variable and provide justification for each step; (Std. 9.0) Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically; (Std. 15.0) Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems; (Std. 23.0) Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity *** |
Strategies, Approach, Plan to meet Goals, Standards, Concepts Word Problems – See (N. under Goals) Choose content rich problems that combine Several concepts (pre-algebra, algebra, and geometry) Below is such a problem geared especially for The “visual learner” with emphasis on fractions and Area. Here is a problem that deals with proportion, Fraction, shapes, and area.
Some of the possible Questions are: Find Area of each part when the Total Area = 1 Which area is the largest, the smallest? Which parts have the same area? Many, many other questions could be asked for this content rich problem. |
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Goals,
Standards, Concepts “O.” Concept: Understanding
Polynomials (Std. 10.0) Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques;
(Std. 11.0) Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials; (Std. 12.0) Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to their lowest terms; *** |
Strategies, Approach, Plan to meet Goals, Standards, ConceptsUnderstanding Polynomials (See O. and
R.(Factoring) under Goals) For the visual learner, define a polynomial using shapes -- e.g. a big square Is x2 , “x by x”, a medium sized rectangle is x, “1 by x”, and a very small “1 by 1” square represents “1.” Explain what factoring “means,” demonstrate “collecting like terms,” and explain that a polynomial should be “simplified” before it can be factored. Simplifying means to count all the big shapes—the x2’s, all the medium shapes—the x’s, and all the small shapes—1’s, and determining or counting how many shapes there are of each type. Simplifying a polynomial means making sure all the shapes (the big square, the medium sized rectangle, and the small square) are of the same type— combining them – and that a polynomial must be simplified before it can be factored. Use visuals as an aid in factoring polynomials. |
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Goals,
Standards, Concepts “P.” Concept: Establish a “safe” classroom environment. |
Strategies, Approach, Plan to
meet Goals, Standards, Concepts “safe” classroom environment is a classroom where--- · everyone in class, including the teacher, is part of the community of learners; · takings risks by asking questions, volunteering answers, and so on, is encouraged and respected; · Lessons are molded, whenever possible, around the interests of individual students; · different solution methods and different ways of thinking about problems and mathematics are valued and give the teacher opportunities; · collaboration with classmates is valued. |
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“Q.”
Concept: Key Formulas *** |
Key Formulas (See “Q” Under Goals) |
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“R.”
Concept: Test Preparation for CAHSEE *** |
Preparing for the CAHSEE ( See R. Under Goals) Four Step Kaplan Approach –
is that which is being asked;
and disregard that which is unimportant;
etc or a combination);
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“S.” Concept: Distributing
and Factoring techniques in Algebra (Std. 11.0) Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials *** |
Distributing and Factoring techniques in Algebra (See. S. Under Goals) Explain what factoring “means” , demonstrate “collecting like terms” and explain that a polynomial must be “simplified” before it can be factored. Simplifying a polynomial means making sure all the images are of the same type—combining them and that a polynomial must be simplified before it can be factored. |
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Goals,
Standards, Concepts “T.”
Concept: Combining Like Terms *** |
Strategies, Approach, Plan to
meet Goals, Standards, Concepts Combining Like Terms – (See T. Under Goals)
Algebraic approach: collect “like terms” and we have Visual approach: we have 2 big x by x squares, 4 medium rectangles each is 1 by x, 3 little 1 by 1 squares, and 1 more big x by x square. Draw this picture and see what you have. Put them together. |
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“U.”
Concept: Vocabulary/Language of Algebra Graphing: origin, function, domain, slope, Range, variable, abscissa, ordinate, co-ordinate pairs, x-axis, y-axis, the coordinate plane Numbers: whole, real, integers, rational, fractions, irrational Equations: expression, terms, constant, variable, linear equation, quadratic equation, 2nd degree equation, coefficient, exponent, slope of a linear equation, y-intercept of a linear equation. Quadratics: quadratic equation, quadratic function, radicals, square root, ±, discriminant, general form of a quadratic equation, roots of a quadratic equation, parabola is the graph opening up or down, Factoring a quadratic equation, graphical Representation of a quadratic equation. *** |
Vocabulary/Language
of Algebra (see U. under Objectives) Use Examples to teach: algebraic expressions, variable, product, sum, term, coefficient, factor, common factor, constant, like terms, simplify (what it means), equation, expression. Be aware that language is often a challenge for students. Try not to overlook language and vocabulary issues. Build concepts first, then attach algebra vocabulary to the established ideas. Since students think and learn in many ways, Use a spectrum of approaches, including oral, written, visual, and kinesthetic modes. Rephrase math ideas as needed. Listen to students and then rephrase and clarify when necessary. Use choral response or rhythmic strategies. Such as, “a logarithm is an exponent,” or “a negative exponent means take the reciprocal.” |
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Goals, Standards, Concepts “V.”
Concept: Zero Product Concept If a*b = 0, then we have three possible cases: (1.) a = 0 (2.) b = 0 (3.) a and b both = 0 *** |
Strategies, Approach, Plan to meet Goals,
Standards, Concepts Zero Product Concept – (See V. Under Goals) This useful concept is especially useful when the product of two binomials is zero. For example: Given the quadratic equation (x+3)(x-7) = 0. We have x+3 = 0 or x = -3 And we also have x-7 = 0
or x = 7 |
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“W.” Concept: Multiple
Solving Strategies Utilize –
Goals,
Standards, Concepts “W.” Concept: Multiple Solving Strategies *** |
Multiple Solving Strategies – (See W. Under Goals) Two Oral Strategies: A. Students work in small groups B. Choral response approach Teacher: “What is a logarithm?” Class: “A logarithm is an exponent” Teacher: “What does an equation always have?” Class: “An equation always has an equal sign.” Teacher: “What’s an exponent?” Class: “It’s the little number to the right of the variable.” Use writing strategies – have the students do Problems like these, perhaps in their journals:
Now I know that a function is ___________.
with what it means in mathematics
in algebra, in studying functions, and in everyday English. What ideas do all the meanings share?
between the square of a number and the square root of a number? Strategies, Approach, Plan to meet Goals,
Standards, Concepts Multiple Solving Strategies –
(See W. Under Goals) Visual strategies – math graffiti A few examples below: para½½ e ½ for parallel lines
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