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					                                 Algebra I
                       Suggested Teaching Strategies



The curriculum guide is a set of suggested teaching strategies designed to be
only a starting point for innovative teaching. The teaching strategies are optional,
not mandatory. A teaching strategy in this guide could be a task, activity, or
suggested method that is part of an instructional unit. It should not be
considered sufficient to teach the competency and the associated objective(s);
the teaching strategy could be one small component of the unit. There may not
be enough instructional time to utilize every strategy in the curriculum guide.

The 2007 Mississippi Mathematics Framework Revised includes the Depth-
of-Knowledge (DOK) level for each objective. As closely as possible, each
strategy addresses the DOK level specified for that objective or a higher
level. Suggestions or techniques for increasing the level of thinking may be
included in the strategy(ies). In addition, the process strands (problem
solving, communication, connections, reasoning and proof, and
representation) are included in the strategies.

The purpose of the suggested teaching strategies is to assist school districts
and teachers in the development of possible methods of organizing the
competencies and objectives to be taught. Since the competencies and
objectives require multiple assessment methods, some assessment ideas
may be included in the strategy.




October 2007
                                            2007 Mississippi Mathematics Framework Revised Strategies




Comp.     Obj.                        Suggested Teaching Strategies
                                                                              2
   1           a   Give students an algebraic expression such as 3x + 2y . Ask them to create
   2           g   as many different algebraic expressions that can be simplified to the given
                                                                                       2
                   expression. Some possible expressions might be: 1) 2x + 2(x + y ) – x;
                                                               2     2
                   2) 4(x – ay) + 2y(2a + y) – x; or 3) 3(x + y ) – y . Have students share their
                   expressions as others check to be sure that it can be simplified to the given
                   one. Any monomial, binomial or polynomial expression can be given to
                   students to express in different algebraic expressions that can be simplified
                   to the given expression.

   1           a   Using Algebra tiles™, ask students to find five representations of 3x + 2.
   2           g   They should consider the zero pairs (one positive unit and one negative unit,
   4           c   for example) to create those representations. Ask them to write their
                   representations as algebraic expressions. For example, if they use 4 green
                   longs, 3 yellow units, 1 red long, and 1 red unit, the expressions would be
                   4x + 3 + (–x) + (–1). How are the representations similar to simplifying
                   algebraic expressions. [Note, Algebra tiles™ are one manipulative that
                   could be used as an area model.]

   1           b   Give students a problem such as: A fast food company sells three specials.
                   The number of hamburgers, French fry orders, and drinks in each special is
                   summarized below. The day's receipt indicates that there were 300
                   hamburgers, 500 French fries, and 400 drinks sold. How many of each type
                   of special were sold? (Adapted from: http://www.mathme.com/henderson/
                   mrhenderson/algebra2/worksheets/matrixwordproblems.htm.)

                                               Special 1        Special 2         Special 3
                             Hamburgers        1                0                 1
                             French fries      2                1                 0
                             Drinks            1                2                 0

                   Have students work together in small groups to solve and then ask groups to
                   share their solutions and solution methods. During the discussion, ask
                   students to decide how a table like this one is similar to a matrix.

   2           a   Show an equation, such as: 5x – 9 = 15 + 2x. Assign each small group of
   2           b   students a method to use to solve the equation. These methods might
                   include: 1) algebraic manipulation, 2) graphing, 3) a table, 4) diagram or
                   Algebra tiles™, and 5) guess-and-test. Ask each group to share their
                   solution method. Have students compare and contrast the methods. How
                   are they alike? How are they different? How can you use the solution to the
                   equation to find the solution to: 5x – 9 ≥ 15 + 2x?

   2           c   Give students a table, such as the example below:

                         x           –10              –5            0              5          10
                         y           –32             –17           –2             13          28

                   Ask students to find an equation that expresses the relationship between x
                   and y in the table. Is this a function? Why or why not? What is the shape of
                   the graph that represents this relationship? Have students compare and
                   contrast it with this relationship:




October 2007                                                                                       2
                                          2007 Mississippi Mathematics Framework Revised Strategies



Comp.     Obj.                       Suggested Teaching Strategies

                         x            2             –2            0           1.5           –4
                         y            4             4             0          2.25           16

                   Students should note that the second table represents the squaring function
                   and will not be represented with a linear graph. [Note, when teaching this
                   particular objective, it is important to emphasize relationships that may not
                   have all real numbers as the domain and/or range.]

                   To generalize relations and functions, ask students to find an example of a
                   relation that is not a function. Have students explain why it is not a function.
                   Have students find an example of a function and explain why it is a function.
                   [Open-ended tasks of this type can be used to increase the DOK level of an
                   objective or task. They allow all students to access the mathematics at their
                   level and their responses give the teacher much information about the
                   understanding levels.]

                   Provide an equation such as y = 2x + 3. Ask students to find at least four
   2           d   solutions. As they find the solutions, have students place them in a table
                   such as:

                                                x                     y
                                               –1                     1
                                                0                     3
                                                1                     5
                                                2                     7

                   Ask students what they notice. They may note any of the following: 1) An
                   increase in x produces an increase in y; 2) a decrease in x results in a
                   decrease in y; 3) when x increases by 1, y increases by 2; and so on.

                   Provide students with a condition for a line, such as a y-intercept of 3. Ask
   2           e   them to draw three lines each meeting this condition. Have the students
                   compare and contrast the lines. How are they alike? How are they different?
                   They may notice some relationships relative to slope, which could lead to
                   generalizations about the position on the Cartesian coordinate plane when
                   the slope is positive, negative, zero or undefined.

                   When students are presented with a system of equations, encourage them to
   2           f   solve it in multiple ways (graphing, tables, diagrams, multiplication with
                   addition or subtraction, addition or subtraction, and substitution). Compare
                   and contrast the methods. Why might one method be more appropriate than
                   another for a specific system of equations? How are the methods alike? How
                   are they different?

                   Ask students to find a binomial that can be factored as the product of a
   2           h   binomial and a monomial. Have students share their binomials and ask other
                   groups to check its factorization. What do they notice about the binomials
                   that all the groups found? Is it possible to find a binomial that can be factored
                   as the product of two monomials? [Note, this type of strategy increases the
                   DOK level so that students must use higher-order thinking skills to solve the
                   problem. It also pushes generalizations so that students see the bigger idea
                   and concept for the skill.]




October 2007                                                                                      3
                                        2007 Mississippi Mathematics Framework Revised Strategies



Comp.     Obj.                       Suggested Teaching Strategies
   2           i   Ask students to find a quadratic equation that has only one solution. Have
                   students share their equations. How are the equations alike? How do they
                   differ? What are common characteristics of these equations? Find a
                   quadratic equation that has two solutions. As students share their solutions,
                   ask them to compare and contrast these equations to those that have only
                   one solution. Is it possible for a quadratic equation to have more than two
                   solutions? Why or why not? Ask students to support their answer and
                   reasoning. [Note, this type of strategy increases the DOK level so that
                   students must use higher-order thinking skills to solve the problem. It also
                   pushes generalizations so that students see the concept for the skill.]

   2           j   A prime polynomial is one that has only two factors, 1 and itself. This is
                   comparable to a prime number. Ask students to find a prime polynomial that
                   has 2 terms, a polynomial that has 3 terms, and a polynomial that has 4
                   terms. How are these polynomials alike? How do they differ? Discuss the
                   characteristics of these polynomials.
                                                                                             2
   2           k   Using graphing calculators, have students graph the equation y = x + 1.
                   What would be the effect on the graph if the coefficient of the x-squared term
                   was 2 instead of 1? –5 instead of 1? ½ instead of 1? What if the constant
                   was 5 instead of 1? –3 instead of 1? Have students predict the effect and
                   then graph to check. The same technique can be used with absolute value
                   equations. For example, graph x – 2 = 5. What would be the effect on the
                   graph if the coefficient of x was 3 instead of 1? –2 instead of 1? What if
                   instead of subtracting 2, x had been increased by 2?

   2           l   Give students the graph of an inequality and ask them to find the inequality.
                   You can increase the level of thinking by using tasks or questions like these
                   that ‘reverse’ the problem. That is, you give the students the answer or
                   solution and ask them to create the problem.

                   Ask students to find an inequality whose graph is only in quadrants I and II.
                   You can change the conditions of the inequality to provide variety.

   3           a   Consider presenting parallel and perpendicular line concepts and skills by
                   linking it to a system of equations with respect to consistent and inconsistent
                   systems. Parallel lines represent inconsistent systems while perpendicular
                   lines represent a consistent system. While students will find that parallel lines
                   have the same slope (and thus do not intersect), they may also generalize
                   that perpendicular lines also have common traits related to the opposite of
                   the reciprocal, one to the other.

   3           b   Give students a length of rope and a tape measure. Tell them they are going
   5           a   to tie nine (9) knots in the rope. Ask them to predict what a graph might look
                   like if the number of knots is graphed on the x-axis and the length of the rope
                   is graphed on the y-axis. Give them time to tie the knots and measure. They
                   can graph their ordered pairs (number of knots, length of the rope). What do
                   they notice about the ordered pairs? They will comment on the linearity of
                   the points and their placement in only Quadrant I as well as other
                   observations. However, one generalization or observation they may note is
                   that the slope of the line is related to the amount of rope in each knot.




October 2007                                                                                     4
                                        2007 Mississippi Mathematics Framework Revised Strategies



Comp.     Obj.                       Suggested Teaching Strategies
   4           a   Present problems similar to the following: Professional photographers are
                   aware that three measurements from the lens are critical and must be kept in
                   balance. When the camera is in focus, 1/f = 1/u + 1/v where f is the focal
                   length of the camera lens. The value of f depends on the curvature of the
                   lens. The u represents the distance between the lens and the subject. The v
                   is the distance between the lens and the film. In one photo shoot, the
                   distance from the lens to the film is 0.5 inch and the subject is 60 inches from
                   the lens. What should the focal length be so that the camera is in focus?
                   Have students solve the problem and share their solution methods. Other
                   problems of this type can be used during the instructional sequence related
                   to this objective. Note that measurement formulae can be embedded in
                   different contexts.

   4           b   Consider open-ended problems that have multiple solutions to meet the DOK
                   level for the objective as well as to raise the level of thinking. For example,
                   present a problem like the following: The length of a segment formed by two
                   points in a coordinate plane is 5 units. If one of the points is found by the
                   ordered pair (5, 10), what are four possible coordinates for the other point?
                   Explain how you found the possible coordinates. Allow students time to solve
                   the problem in pairs or small groups. Randomly select a pair or group to
                   present their solution and solution method to the class. A discussion rubric
                   that is created with the class can be used to assess the accuracy,
                   completeness, and appropriateness of the solution and the solution method.
                   Are other points possible? Why or why not?

   5           a   Students should have multiple experiences collecting data and graphing it.
                   The Knotty Problem (objective 3b) is one type of an experience that can be
                   used in instruction to address this objective. Other examples of relationships
                   that can be explored include 1) height and shoe size; 2) drops of water and
                   the diameter of the circle formed; 3) constant perimeter and the dimensions
                   of a rectangle; 4) distance traveled and time; 5) diameter of a circle object
                   and its circumference, and so on. For lab or explorations, a rubric can be
                   created to evaluate student exploration responses.

                   If CBR or CBL hardware is available, it could enhance student explorations.
                   The use of graphing calculators is appropriate as students complete
                   explorations related to this objective.

   5           b   Linear regression allows students to describe relationships between a
                   dependent and independent variable. Any of the data collection and graphing
                   explorations described in 5a can be used with linear regression instructional
                   sequences. For example, students can measure the diameters of circular
                   objects and the circumferences (perimeters) of the same objects. They
                   record them in a table and graph the ordered pairs (diameter,
                   circumference). The points will approximate a line. To find the equation of
                   the line that can represent the trend of the data, two points can be selected
                   to determine the slope of the line, and the y-intercept can be determined
                   from the graph. With that information, an equation can be found for the line.




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