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Waltham Public Schools

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Integrated III

WPS Math - Integrated Math III

Core Connections Integrated III is the third course in a four-year sequence of rigorous college preparatory mathematics courses that starts with Core Connections Integrated I and continues through Calculus. It aims to apply and extend what students have learned in previous courses by focusing on finding connections between multiple representations of functions, transformations of different function families, finding zeros of polynomials and connecting them to graphs and equations of polynomials, modeling periodic phenomena with trigonometry, and understanding the role of randomness and the normal distribution in making statistical conclusions.

On a daily basis, students in Core Connections Integrated III use problem-solving strategies, questioning, investigating, analyzing critically, gathering and constructing evidence, and communicating rigorous arguments justifying their thinking. Under teacher guidance, students learn in collaboration with others while sharing information, expertise, and ideas.

The course is well balanced between procedural fluency (algorithms and basic skills), deep conceptual understanding, strategic competence (problem solving), and adaptive reasoning (extension and transference). The lessons in the course meet all of the content standards, including the “plus” standards, of Appendix A of the Common Core State Standards for Mathematics. The course imbeds the CCSS Standards for Mathematical Practice as an integral part of the lessons in the course.

Unit Timeframe Big Ideas (Statements or Essential Questions) Major Learning Experiences from Unit
Unit 1
Exploring Algebraic and Geometric Relationships		 8/31-10/1
  • Am I able to describe, classify, and name basic polygons based on their attributes such as parallel sides and symmetry?
  • Am I able to represent a pattern of growth using tables, graphs, equations, and make connections between expressions and geometric models?
  • Am I able to identify and prove relationships between pairs of angles formed by intersecting lines and among sides and angles of triangles?
  • Classify polygons.
  • Mark parts of polygons that are equal in length and measure.
  • Understand area and perimeter and how area and perimeter change as shapes change (similarity).
  • Write algebraic expressions and equations representing side lengths and area.
  • Understand factoring quadratic expressions.
  • Write equations describing graphs.
  • Review complementary, supplementary and congruent angle pairs.
  • Understand the attributes of angle pairs that are formed by parallel lines intersected by transversals…particularly corresponding, alternate interior, alternate exterior and same side interior angles.
  • Understand the Triangle Angle Sum Theorem.
Unit 2
Justification and Similarity		 10/4-11/12
  • Am I able to write proofs for triangle similarity and the qualities of their corresponding parts?
  • Am I able to use similarity to solve everyday problems?
  • Review and understand the conditions triangle congruence (ASA, AAS, SSS, SAS, and HL).
  • Use flowcharts to organize proofs of triangle congruence.
  • Prove by contradiction and use the converse of theorems.
  • Learn about dilations and how corresponding parts change.
  • Learn triangle similarity conditions (AA, ASA, SSS).
  • Use flowcharts to organize proofs for triangle similarity.
  • Apply triangle similarity to real world problems.
Unit 3
Probability and Trigonometry		 11/15-12/17
  • Am I able to model situations involving probability using tree diagrams and/or area models?
  • Am I able to compute probabilities of unions, intersections and complements of events?
  • Am I able to calculate expected values in games of chance?
  • Am I able to see the relationship between the slope of a line and the slope angle?
  • Am I able to use the concept of slope ratio to determine missing measurements of a right triangle and solve everyday problems?
  • Use a probability area model to represent a situation of chance.
  • Develop complex tree diagrams to model probabilities for events that are not equally likely.
  • Use tree diagrams and area models to represent and solve probability problems.
  • Learn how to calculate probabilities of unions, intersections and complements of events.
  • Learn how to calculate expected value.
  • Recognize that all slope triangles on a given line are similar to each other.
  • Connect that specific slope ratios are related to specific angle measures.
  • Use the connections between slope ratios and their related angles to calculate missing side lengths and angle measures in right triangles.
  • Use technology to generate slope ratios for new angles and find missing side lengths in right triangles.
  • Start to understand that the slope ratio is called tangent.
  • Be able to reorient a right triangle in order to see which leg is opposite and which is adjacent to a given acute angle.
  • Use the slope ratio to find indirect measurements (Proportions).
Unit 4
Factoring and More Trigonometry		 12/20-1/28
  • Am I able to change a quadratic expression written as a sum into factored form?
  • Am I able to see patterns involved in factoring quadratic expressions?
  • Am I able to use sine, cosine and their inverses to find missing side lengths and angle measure in right triangles?
  • Am I able to use trigonometric functions to model situations and solve problems?
  • Use tiles to build rectangles and identify patterns for determining the dimensions of a completed area model.
  • Discover that the products of the terms in each diagonal of an area model are equal.
  • Factor quadratic expressions with missing terms, quadratic expressions not in standard form and quadratic expressions with more than one factored form.
  • Recognize when a quadratic expression is a perfect square trinomial or a difference of squares and find ways to factor these special cases.
  • Learn about sine and cosine ratios and start to create a Triangle Graphic Organizer.
  • Identify the appropriate trigonometric ratio based on the relative position of the reference angle and the given sides involved.
  • Use inverse trigonometric ratios to determine the unknown angle measure in right triangles.
  • Use trigonometric ratios to solve application problems.
Unit 5
Quadratic Functions		 1/31-3/11
  • Am I able to create a quadratic functions web using graphs, tables, and equations?
  • Am I able to use the zero product property to determine the x-intercepts of a parabola?
  • Am I able to model everyday situations using quadratic functions?
  • Am I able to solve quadratic functions by completing the square?
  • Am I able to solve quadratic functions using the quadratic formula?
  • Am I able to determine the number of solutions to a quadratic equation and choose the best strategy to solve it based on the given equation?
  • Investigate graphs of quadratic functions in order to learn about their shape and key features.
  • Describe the graphs of quadratic functions using appropriate vocabulary.
  • Identify connections between different representations of quadratic functions; an equation, a table, a situation, and a graph.
  • Relate the intercepts and vertex of a parabola in context such as launch and landing point as well as the maximum height of an object following a parabolic trajectory.
  • Learn how to model a quadratic situation using a graph.
  • Determine x-intercepts by factoring and using the zero product property.
  • Write a quadratic function when given a table.
  • Write a quadratic function when given a graph of a parabola.
  • Solve quadratic equations that are in perfect square form.
  • Express solutions to quadratic equations as exact and approximate values.
  • Complete the square to form perfect square quadratic equations and solve.
  • Use the quadratic formula to solve quadratic equations.
  • Decide whether to use factoring (zero product property), completing the square, or the quadratic formula to solve quadratic equations.
  • Write quadratic functions to model everyday situations.
  • Perform operations on complex numbers and solve quadratic equations with non-real solutions.
Unit 6
More Right Triangles		 3/14-4/8
  • Am I able to apply the Pythagorean Theorem and properties of similar right triangles to discover patterns in special right triangles; 45-45-90 and 30-60-90 degree triangles and those containing Pythagorean Triples?
  • Recognize the similarity ratios in 30-60-90 and 45-45-90 degree triangles and begin to apply them as shortcuts to find missing side lengths.
  • Recognize 3:4:5 and 5:12:13 triangles and other Pythagorean Triples and use dilations of each in application problems.
  • Connect trigonometric ratios to special right triangles and identify exact values for trigonometric ratios of special angles.
  • Interpret fractional and integer exponents and rewrite radical expressions using such exponents and vice versa.
Unit 7
Polygons and Circles		 4/11-4/29
  • Am I able to find shortcuts and generalize the rules for finding perimeters and areas of polygons?
  • Continue working with quadrilaterals and triangles to focus on the angles, area and perimeter of polygons with any number of sides.
  • Build polygons out of congruent triangles and develop vocabulary to describe these shapes.
  • Use knowledge of triangle angle sum and other angle relationships to make discoveries about the interior and exterior angles of polygons.
  • Develop strategies to find the area of a regular polygon with any number of sides.
  • Examine the relationships between areas of similar figures and discover that the ratio of the areas between similar figures is equal to the zoom factor (similarity ratio).
  • Extend the zoom factor generalization to finding the area and perimeter of a regular polygon with an infinite number of sides in order to develop the area and circumference formulas for a circle.
Unit 8
Circles		 5/2-5/20
  • Am I able to write equations of circles?
  • Am I able to rewrite quadratic equations to write circle equations in different forms?
  • Am I able to understand the relationships of angles, arcs, chords and tangents in a circle?
  • Am I able to use geometric tools to learn more about the planet Earth?
  • Learn how to determine the equation of a circle graphed on the coordinate axes.
  • Use completing the square to write equations of circles in graphing form.
  • Learn about the relationships between central angles, inscribed angles and arcs they intercept.
  • Learn the difference between arc length and arc measure.
  • Learn that an angle inscribed in a semicircle always measures 90 degrees.
  • Learn and justify that opposite angles in an inscribed quadrilateral are supplementary.
  • Develop different methods to calculate the length of a chord.
  • Use similar triangles and proportionality to understand the relationships between the lengths created by intersecting chords.
  • Apply knowledge of chords, angles and arcs to solve problems involving circles.
Unit 9
Solids		 5/23-6/10
  • Am I able to measure the surface areas and volumes of three dimensional solids?
  • Calculate the surface area and volume of non-rectangular prisms and cylinders.
  • Learn that the volume of a slanted cylinder or prism remains constant as long as the height remains constant.
  • Learn how to sketch prisms and cylinders on paper.
  • Learn that the ratio of the volumes of similar 3-D figures is the cube of the scale factor.
  • Use the scale factor relationship between similar 3-D figures in application problems.
  • Describe the features of a pyramid and name the pyramid using the shape of its base.
  • Calculate the total surface area of a pyramid.
  • Calculate the volume of a pyramid and justify that it is one-third of the volume of a prism with the same base and height.
  • Understand that the volume of an oblique pyramid remains constant as long as the height remains constant.
  • Learn how to calculate the volume and surface area of a cone.
  • Solve application problems involving cones.
  • Calculate the surface area and volume of a sphere.