### 3: Analytic Geometry

#### 3.1: Overall Expectations

3.1.1: demonstrate an understanding of the characteristics of a linear relation;

3.1.2: determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;

3.1.3: determine, through investigation, the properties of the slope and y-intercept of a linear relation;

3.1.4: solve problems involving linear relations.

#### 3.2: Understanding Characteristics of Linear Relations

3.2.1: design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g.,making tables, drawing graphs) (Sample problem: Design and perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);

3.2.2: construct equations to represent linear relations derived from descriptions of realistic situations, and connect the equations to tables of values and graphs, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil) (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs \$25, plus \$0.10 per minute of airtime.);

3.2.3: determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot).

#### 3.3: Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph

3.3.1: determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);

3.3.2: identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;

3.3.3: express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.

#### 3.4: Investigating the Properties of Slope

3.4.1: determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m = rise/run, m = the change in y/the change in x or m = delta y/delta x, m = (y2 - y1)/(x2 - x1), and use the formulas to determine the slope of a line segment or a line;

3.4.2: identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;

3.4.3: determine, through investigation, connections between slope and other representations of a constant rate of change of linear relation (e.g., if the cost of producing a book of photographs is \$50, plus \$5 per book, then the slope of the line that represents the cost versus the number of books produced has a value of 5, which is also the rate of change; the value of the slope is the value of the coefficient of the independent variable in the equation of the line,C = 50 + 5p, and the value of the first difference in a table of values);

3.4.4: identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate.

#### 3.5: Using the Properties of Linear Relations to Solve Problems

3.5.1: graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x - 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);

3.5.2: determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points) (Sample problem: Compare the equations of the lines parallel to and perpendicular to y = 2x - 4, and with the same x-intercept as 3x - 4y = 12.Verify using dynamic geometry software.);

3.5.3: describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is \$40 per evening, plus \$2 per person for equipment rental; the y-intercept, 40, represents the \$40 cost of renting the gym; the value of the slope, 2, represents the \$2 cost per person);

3.5.4: identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n,C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to \$50 to \$2550).

### 4: Measurement and Geometry

#### 4.1: Overall Expectations

4.1.1: solve problems involving the surface areas and volumes of three-dimensional figures;

4.1.2: verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.

#### 4.2: Solving Problems Involving Surface Area and Volume

4.2.1: solve problems involving the volumes of composite figures composed of prisms, pyramids, cylinders, cones, and spheres;

4.2.2: determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a squarebased pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);

4.2.3: solve problems involving the surface areas of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.);

4.2.4: identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];

4.2.5: explain the significance of optimal surface area or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);

4.2.6: pose and solve problems involving maximization and minimization of measurements of geometric figures (i.e., squarebased prisms or cylinders) (Sample problem: Determine the dimensions of a square-based, open-topped prism with a volume of 24 cm_ and with the minimum surface area.).

#### 4.3: Investigating and Applying Geometric Relationships

4.3.1: determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);

Correlation last revised: 8/18/2015

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