INSTRUCTIONAL DESIGN FRAMEWORK AR4A Standard Analyze linear functions by investigating rates of change and intercepts.

Transcription

1 AR4A Analyze linear functions by investigating rates of change and intercepts. Linear functions Rates of change Intercepts Analyze Investigate Linear functions Intercepts Rates of change Slope 1. I will analyze linear functions by investigating rates of change. 2. I will analyze linear functions by investigating intercepts. Given the slope of a line and a point on the line OR given two points on the line, write the equation of the line and then graph the line. Use a graphing calculator to investigate how the line changes when the slope changes, when the y-intercept changes, and when both change. Given the equation of several lines, identify the x- and y-intercepts. Given equations of several lines, students will analyze the slopes to determine whether they are parallel, perpendicular, or neither. Given several pairs of lines, determine whether they are parallel, perpendicular, or neither, by graphing and by finding the slopes

2 AR2B Apply appropriate properties of exponents to simplify expressions and solve equations. Properties of exponents Expressions Equations Apply Simplify Solve Properties of exponents Expressions Equations 1. I will apply appropriate properties of exponents to simplify expressions. 2. I will apply appropriate properties of exponents to solve equations. Given any right triangle, students will apply the Pythagorean Theorem to solve for the missing length of a side of the triangle. Use the Pythagorean Theorem to discover the formulas for and triangles. Use the Pythagorean Theorem to determine whether a triangle is right, obtuse, or acute. Given a rectangular field and its dimensions, calculate the difference between the distance to walk around the field and the distance to walk across on the diagonal

3 GS2A Make conjectures and solve problems involving 2-dimensional objects represented with Cartesian coordinates. Conjectures 2-dimensional objectscartesian di t Make Solve Represent Conjectures Cartesian coordinates Coordinate plane Ordered pair 2-dimensional objects (circles, polygons, including special quadrilaterals) 1. I will make conjectures involving 2-dimensional objects represented with Cartesian coordinates. 2. I will solve problems involving 2-dimensional objects represented with Cartesian coordinates. Given any polygon, the student will represent it on a coordinate plane and then classify the polygon using slope and lengths of sides. Given pairs of points, use the distance formula to find the length of the segment between the points. Then, use the Pythagorean Theorem to find the length of the segments. Given pairs of points, count rise and then run to determine the slope. Then, use the slope formula to determine the slope of the line. a) Given the equation of a circle, graph the center and radius on a coordinate plane and then sketch the circle. b) Given a sketch of a circle, determine the radius and center and write the equation

4 AR1C Compare and contrast various forms of representations of patterns. Representations of patterns Compare Contrast Compare Contrast Representations of patterns I will compare and contrast various forms of representations of patterns. Create a poster describing and illustrating a pattern found in nature and compare it to a numerical pattern. Research a pattern found in nature, such as flower petals, honeycombs, and pine cones. Given a conjecture that appears to be true based on several examples, find a counterexample (if possible) that proves the conjecture is false. Given a pattern, identify the equation and graph that represent the pattern

5 NO1A Compare and order rational and irrational numbers, including finding their approximate locations on a number line. Rational numbers Irrational numbers Number line Compare Order Find Compare Irrational number Order Rational number Approximate Number line 1. I will compare and order rational numbers and locate them on a number line. 2. I will compare and order irrational numbers and locate them on a number line. Students will find the arithmetic and geometric means of several sets of data and plot the data and means on number lines. Identify perfect squares and use them to simplify given radicals. Discuss situations in which it is more appropriate to use simple radical form and when it is more appropriate to use an approximation. Given two numbers, calculate the geometric mean and the arithmetic mean and graph them on a number line. Compare the results and determine if any relationships exist

6 GS3A Use and apply constructions and the coordinate plane to represent translations, reflections, rotations and dilations of objects. Constructions Coordinate plane Translations Use Apply Represent Constructions Coordinate plane Transformations Translations Reflections Rotations Dilations Vectors Symmetry 1. I will use and apply constructions to represent translations, reflections, rotations and dilations of objects. 2. I will use and apply the coordinate plane to represent translations, reflections, rotations and dilations of objects. Create a wallpaper border using a frieze pattern influenced by the art, architecture, and culture of another country. The pattern must include a rotation, a reflection, and a translation. Use a sheet of patty paper to trace a given figure. Then use this to find the image after a translation, reflection, and/or rotation. Research the artwork, architecture, culture, customs, and other distinctive features of another country. Given the coordinates of a triangle and a scale factor, calculate the coordinates of the image of the triangle after a dilation and then draw both triangles on the coordinate plane

7 GS4B Draw or use visual models to represent and solve problems. Visual models Problems Craw Use Represent Solve Draw Represent Visual models Solve 1. I will draw or use visual models to represent problems. 2. I will draw or use visual models to solve problems. Students will build a structure out of popsicle sticks, wooden skewers, etc., consisting of pairs of congruent triangles. They will also draw a blueprint of the structure in which they classify the types of triangles by sides and angles. Given a diagram and given information, students will label the congruent pairs of corresponding parts. Given several triangles, students will classify them by sides (scalene, isosceles, equilateral) and by angles (acute, obtuse, right). Given several pairs of congruent triangles with a variety of given information, determine the method (SSS, SAS, ASA, AAS, HL) that could be used to prove the triangles congruent

8 DP1A Formulate and collect data about a characteristic. Data Characteristic Formulate Collect Formulate Data Collect Characteristic 1. I will formulate data about a characteristic. 2. I will collect data about a characteristic. Construct a circle graph based on music preferences of students in the class. Poll the class and make a list of their music preferences. Convert the data collected about music preferences into percents. Convert the percents from the music preference data into degrees and use a protractor to draw a circle graph

9 AR1B Generalize patterns using explicitly or recursively defined functions. Patterns Explicitly defined functions Recursively defined functions Generalize Using Generalize Explicitly defined functions Recursively defined functions 1. I will generalize patterns using explicitly defined functions. 2. I will generalize patterns using recursively defined functions. Students will determine the interior angle sum of any polygon. Given the first three geometric figures in a pattern, students will draw the next two figures. Given the first several terms of a numeric sequence, students will describe the pattern and predict the next two terms in the sequence. Students will divide several polygons into triangles and compare the number of triangles to the number of sides of the polygon to discover the interior angle sum formula

10 DP1C Select and use appropriate graphical representation of data and given one-variable quantitative data, display the distribution and describe its shape. Graphical representation One-variable quantitative data Select Use Display Describe Graphical representation Distribution One-variable quantitative data Shape 1. I will select and use appropriate graphical representation of data. 2. I will display the distribution and describe its shape, given one-variable quantitative data. Given a set of data, determine what type of graph best fits the information and then construct the graph. Find the measures of central angles in several given circles. Given a circle graph, line graph, bar graph or pictograph, display the information in the form of a table. Given sets of data, determine an appropriate scale and interval to use when drawing the axes

11 AR3A Identify quantitative relationships and determine the type(s) of functions that might model the situation to solve the problem. Quantitative relationships Functions Model the situation Solve the problem Identify Determine Solve Quantitative relationships Types of functions 1. I will identify quantitative relationships. 2. I will determine the type(s) of functions that might model the situation to solve the problem. Students will be given a story problem situation and will define the variables, set up, and then solve an appropriate equation. Given a set of data, determine whether it is linear, quadratic, or exponential by graphing or by finding a pattern. Use the vertical line test on several graphs to determine whether or not the graphs represent functions. Given several written phrases, students will write them using symbolic algebra

12 GS3A Identify types of symmetries of 2- and 3-dimensional figures. Symmetry 2-dimensional figures 3-dimensional figures Identify Symmetry Line symmetry Rotational symmetry 2-dimensional figures (including circle, polygons) 3-dimensional figures (including solids, polyhedra) 1. I will identify types of symmetry of 2-dimensional figures. 2. I will identify types of symmetry of 3-dimensional figures. Students will use patty paper to identify types of symmetries. Given several figures, students will trace each figure on patty paper and fold the paper on the proposed line of symmetry to see if the figure maps onto itself. Given several 2- and 3- dimensional figures, students will use patty paper to determine whether each has linear symmetry, rotational symmetry, or both. Given several figures, students will calculate the degrees of rotational symmetry each figure has

13 NO3D Judge the reasonableness of numerical computations and their results. Reasonableness of numerical computations Judge Judge Numerical computations Reasonableness Results 1. I will judge the reasonableness of numerical computations. 2. I will judge the reasonableness of the results of numerical computations. Students will use trigonometric ratios to solve for missing angle and side measurements of right triangles. Given a right triangle, students will identify the opposite leg, adjacent leg, and hypotenuse for the acute angles and then write the ratios for sine, cosine, and tangent to discover the relationships between these ratios. Given several angles, students will use their calculators to find the sine, cosine, and tangent in both degrees and radians. Given several right triangles, students will use SOH-CAH-TOA to determine which trigonometric ratio should be used to solve the problem

14 M2B Solve problems of angle measure, including those involving triangles or other polygons and of parallel lines cut by a transversal. Angle measure Solve Angle measure Triangle Polygons Parallel lines Transversal Corresponding angles Alternate interior/exterior angles Consecutive interior angles 1. I will solve problems of angle measure involving triangles. 2. I will solve problems of angle measure involving polygons. 3. I will solve problems of angle measure involving parallel lines cut by a transversal. Students will create a brochure on angle pairs created by parallel lines cut by a transversal. Students will draw a diagram of two parallel lines cut by a transversal and identify each of the different angle pairs formed. Using a protractor, students will discover the properties of corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles, given parallel lines cut by a transversal. Given several angle pairs with their measures written as algebraic expressions, students will set up and solve equations to find the measurements

15 NO3E Solve problems involving proportions. Proportions Solve Solve Problems Proportions I will solve problems involving proportions. Given the measurements of objects in an actual room, students will use proportions to find the scaled measurement of each object and then use a ruler to draw them in a floor plan. Given similar figures, students will write and solve proportions using the scale factor. Students will use a ruler to draw line segments of varying lengths in different units. Given the scale factor between similar solids, students will use proportions to find measurements

16 M2C Determine the surface area and volume of geometric figures, including cones, spheres, and cylinders. Surface area Volume Geometric figures Determine Surface area Volume Geometric figures Cones Spheres Cylinders Prisms Pyramids 1. I will determine the surface area of geometric figures, including cones, spheres and cylinders. 2. I will determine the volume of geometric figures, including cones, spheres and cylinders. Given any 3-dimensional shape, the students will use formulas and area of faces to find surface area and volume. Students will research to find the diameter and then calculate the surface area of each planet in the solar system and arrange them by size. Given a cylindrical sugar cone and a waffle cone, students will determine which cone will hold the most ice cream. Students will identify the faces, vertices and edges of 3-dimensional shapes and then test Euler s formula for each solid

17 GS1A Use inductive and deductive reasoning to establish the validity of geometric conjectures, prove theorems and critique arguments made by others. Inductive reasoning Deductive reasoning Use Establish Prove Critique Inductive reasoning Validity Theorems Deductive reasoning Conjectures Arguments 1. I will use inductive reasoning to establish the validity of geometric conjectures. 2. I will use deductive reasoning to establish the validity of geometric conjectures. 3. I will use inductive reasoning to prove theorems. 4. I will use deductive reasoning to prove theorems. 5. I will use inductive reasoning to critique arguments made by others. 6. I will use deductive reasoning to critique arguments made by others. Create a fictional product and write a slogan for it. Write the slogan in the form of an if-then conditional statement and determine the converse, inverse and contrapositive. Students will use symbolic notation to write the converse, inverse and contrapositive of several conditional statements. Students will identify the next object in a pattern using inductive reasoning Given several conditional statements, students will test the validity of the statement and its converse, inverse and contrapositive.

18 NO1B Use real numbers and various models, drawing, etc. to solve problems. Real numbers Models Drawing Use Solve Real numbers Models Drawing 1. I will use real numbers to solve problems. 2. I will use various models, such as drawings, to solve problems. Using a tape measure and clinometer, students will find the heights of objects in the commons area using right triangle trigonometry. Given several diagrams with a variety of sides and/or acute angles given, students will determine the appropriate trigonometric ratio to use when solving a right triangle. Students will draw and label the angle of elevation or angle of depression in a story problem. Given several right triangles, students will determine the opposite leg, adjacent leg and hypotenuse in relation to either acute angle

19 M2E Use unit analysis to solve problems. Unit analysis Use Solve Unit analysis Solve Problems I will use unit analysis to solve problems. Students will find the cost of remodeling a room, given dimensions and costs of carpet, paint and tile. Students will simplify several given ratios in which the numerator and denominator are in different units. Students will find the area and perimeter of several different types of polygons. Given several metric and customary units, students will convert them to another unit within that system

20 GS4A Draw and use vertex-edge graphs or networks to find optimal solutions and draw representations of 3-dimensional geometric objects from different perspectives. Vertex-edge graphs Networks 3-dimensional geometric objects Draw Use Find Vertex-edge graphs Networks Optimal solutions Representations 3-dimensional geometric objects Perspectives 1. I will draw and use vertex-edge graphs to find optimal solutions. 2. I will draw and use networks to find optimal solutions. 3. I will draw representations of 3-dimensional geometric objects from different perspectives. Given a map showing locations and mileage on a delivery route, students will determine the shortest route. Given several networks, students will use the traceability rules to determine whether or not each network is traceable. Given an isometric drawing of a 3-dimensional object, students will draw the front, side and bottom views on dot paper. Given several networks, students will count the vertices to determine the degree

21 NO2D Apply operations to real numbers, using mental computation or paper-and-pencil calculations for simple cases and technology for more complicated cases