Math - Secondary Math II

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Math 2
Instructional Tasks

Stand alone tasks are organized to support learning of content standards. These tasks can be used as initial instruction or to support students who are struggling with a particular topic.

 

Strand: MATHEMATICAL PRACTICES (MP)
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Strand: NUMBER AND QUANTITY - The Real Number System (N.RN)
Extend the properties of exponents to rational exponents (Standards N.RN.1–2)
  • Checking a calculation of a decimal exponent
    This Illustrative Mathematics task is primarily suited for instruction as written, though could be used for assessment in settings where varied answers can easily be checked, in particular for formative assessment. The purpose of the task is to connect properties of fractional exponents with ordering of real numbers. One can place the numbers on the number line to emphasize this.
  • Evaluating Exponential Expressions
    The goal of this Illustrative Mathematics task is to use properties of exponents for whole numbers in order to explain how expressions with fractional exponents are defined.
  • Evaluating a Special Exponential Expression
    The purpose of this Illustrative Mathematics task is to study the rules of exponents in the context of trying to make sense of a very interesting mathematical expression.
  • Extending the Definitions of Exponents, Variation 2
    The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2)
  • Fractional Exponents
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
  • Kepler's Third Law of Motion
    The purpose of this Illustrative Mathematics task is to solve some expressions requiring fractional exponents in an interesting modeling context.
  • Rational or Irrational?
    This Illustrative Mathematics task makes for a good follow-up task on rational irrational numbers after that the students have been acquainted with some of the more basic properties, asking students to reason about rational and irrational numbers (N-RN.3) in a variety of ways. In addition to eliciting several different types of reasoning, the task requires students to rewrite radical expressions in which the radicand is divisible by a perfect square (N-RN.2).
  • Rules of Exponents
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
  • Rules of Exponents video
    This video introduces and explains the topic.
Use properties of rational and irrational numbers (Standard N.RN. 3).
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Strand: NUMBER AND QUANTITY - The Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standards N.CN.1–2).
  • Complex Cube and Fourth Roots of 1
    This Illustrative Mathematics task has students further their understanding of the algebra of the complex numbers.
  • Complex Square Roots
    This Illustrative Mathematics task is intended as an introduction to the algebra of the complex numbers, and also builds student's comfort and intuition with these numbers.
  • Complex number patterns
    This Illustrative Mathematics task serves as a possible first student exploration after an initial introduction to the form and arithmetic of complex multiplication.
  • Computations with Complex Numbers
    This task asks students to perform computations involving complex numbers.
  • Powers of a complex number
    The goal of this Illustrative Mathematics task is to practice making calculations with complex numbers and to visualize what happens when two complex numbers are multiplied.
  • Vertex of a parabola with complex roots
    The goal of this Illustrative Mathematics task is to study the relationship between the roots of a quadratic function and its graph.
Use complex numbers in polynomial identities and equations (Standards N.CN.7–9).
  • Completing the square
    The goal of this task is to solve quadratic equations with complex roots by completing the square. Students could of course directly use the quadratic formula, but going through the process of completing the square helps reinforce the mathematics behind the quadratic formula.
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Strand: ALGEBRA - Seeing Structure in Expression (A.SSE)
Interpret the structure of expressions (Standards A.SSE.1–2)
  • A Cubic Identity
    This Illustrative Mathematics task presents a challenging exercise in both algebraic manipulations and seeing structure in algebraic expressions.
  • Animal Populations
    This Illustrative Mathematics task students have to interpret expressions involving two variables in the context of a real world situation.
  • Equivalent Expressions
    This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure.
  • Math Task: Proving Patterns
    Math Task Overview: Students will analyze quadratic patterns related to the difference of squares and use patterns with the number line. They will then prove the general rule with a sequence of calculations that model inductive reasoning.
  • Sectors of Circles
    This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic.
  • Seeing Dots
    The purpose of this Illustrative Mathematics task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context.
  • Solving Quadratic Equations by Factoring
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
  • Sum of Even and Odd
    An Illustrative Mathematics task that supports A.SSE.2.
Write expressions in equivalent forms to solve problems, balancing conceptual understanding and procedural fluency in work with equivalent expressions (Standard A.SSE.3).
  • Algebra Team: Overview of Teaching Styles
    This Teaching Channel video illustrates how two teachers' collaboration impacted their algebra program. This site provides a lesson plan and student handouts. (10 min.)
  • Comparing Investments
    This lesson unit is intended to help you assess how well students are able to interpret exponential and linear functions.
  • Forms of exponential expressions
    There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.
  • Horseshoes in Flight (pdf)
    Students analyze the structure of algebraic expressions and a graph to determine what information each expression readily contributes about the flight of a horseshoe. This task is particularly relevant to students who are studying (or have studied) various quadratic expressions (or functions). The task also illustrates a step in the mathematical modeling process that involves interpreting mathematical results in a real-world context.
  • Ice Cream
    This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard A-SSE.B.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.
  • Identifying Quadratic Functions (Standard Form)
    This task has students explore the relationship between the three parameters a, b, and c in the equation f(x)=ax2+bx+c and the resulting graph. There are many possible approaches to solving each part of this problem, especially the first part.
  • Profit of a company
    This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.
  • Representing Conditional Probabilities 2
    This lesson unit is intended to help educators assess how well students understand conditional probability.
  • Representing Quadratic Functions Graphically
    This lesson unit is intended to help educators assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation.
  • Rewriting a Quadratic Expression
    The goal of this task is to complete the square in a quadratic expression in order to find its minimum or maximum value.
  • Sorting Equations and Identities
    This lesson unit is intended to help educators assess how well students are able to recognize the differences between equations and identities, as well as substitute numbers into algebraic statements in order to test their validity in special cases.
  • Taxes and Sales
    This task is not about computing the final price of the shirt but about using the structure in the computation to make a general argument.
  • Vertex of a parabola with complex roots
    The goal of this Illustrative Mathematics task is to study the relationship between the roots of a quadratic function and its graph.
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Strand: ALGEBRA - Arithmetic With Polynomials and Rational Expressions (A.APR)
Perform arithmetic operations on polynomials. Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x (Standard A.APR.1).
  • Manipulating Polynomials
    This lesson unit is intended to help educators assess how well students are able to manipulate and calculate with polynomials.
  • Math Task: Proving Patterns
    Math Task Overview: Students will analyze quadratic patterns related to the difference of squares and use patterns with the number line. They will then prove the general rule with a sequence of calculations that model inductive reasoning.
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Strand: ALGEBRA - Creating Equations (A.CED)
Create equations that describe numbers or relationships. Extend work on linear and exponential equations to quadratic equations (Standards A.CED.1–2, 4).
  • Basketball
    This task provides a simple but interesting and realistic context in which students are led to set up a rational equation (and a rational inequality) in one variable, and then solve that equation/inequality for an unknown variable.
  • Optimization Problems: Boomerangs
    This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.
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Strand: ALGEBRA - Reasoning With Equations and Inequalities (A.REI)
Solve equations and inequalities in one variable (Standard A.REI.4)
  • Braking Distance
    The purpose of this task is to give an application arising from a real-world situation in which a quadratic equation arises, and where it is natural to use a graphical method to find an approximate solution and the quadratic formula to find an exact solution
  • Completing the square
    The goal of this task is to solve quadratic equations with complex roots by completing the square. Students could of course directly use the quadratic formula, but going through the process of completing the square helps reinforce the mathematics behind the quadratic formula.
  • Quadratic Sequence 1
    This task belongs to a series of three tasks that presents students with a sequence of tile figures with the property that the n-th figure in the sequence has f(n) tiles, for some quadratic function f.
  • Quadratic Sequence 2
    This task belongs to a series of three tasks that has students process a sequence of tile figures with the property that the n-th figure in the sequence has f(n) tiles, for some quadratic function f.
  • Solving Quadratic Equations by Factoring
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
  • Solving Quadratic Equations: Cutting Corners
    This lesson unit is intended to help educators assess how well students are able to solve quadratics in one variable.
  • Sorting Equations and Identities
    This lesson unit is intended to help educators assess how well students are able to recognize the differences between equations and identities, as well as substitute numbers into algebraic statements in order to test their validity in special cases.
  • Springboard Dive
    The problem presents a context where a quadratic function arises. Careful analysis, including graphing, of the function is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.
  • Two Squares are Equal
    This classroom task is meant to elicit a variety of different methods of solving a quadratic equation.
  • Vertex of a parabola with complex roots
    The goal of this Illustrative Mathematics task is to study the relationship between the roots of a quadratic function and its graph.
  • Visualizing Completing the Square
    The purpose of this task is to help provide geometric intuition for the algebraic process called ''completing the square.''
Solve systems of equations. Extend the work of systems to include solving systems consisting of one linear and one nonlinear equation (Standard A.REI.7).
  • A Linear and Quadratic System
    The purpose of this task is to give students the opportunity to make connections between equations and the geometry of their graphs. They must read information from the graph (such as the vertical intercept of the quadratic graph or the slope of the linear one), use that information to construct and solve an equation, then interpret their solution in terms of the graph.
  • Optimization Problems: Boomerangs
    This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, and interpret and evaluate the data generated and identify the optimum case, checking it for confirmation.
  • Pythagorean Triples
    This task has two separate goals: the first is to solve a pair of equations, one quadratic and the other linear. The second goal is to observe that a point, such as (35,45), that lies on C gives rise to a Pythagorean triple, in this case the triple (3,4,5).
  • The Circle and The Line
    This task is assessing a simple but important piece of conceptual understanding, namely the correspondence between intersection points of the two graphs and solutions of the system.
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Strand: FUNCTIONS - Interpret Functions (F.IF)
Interpret quadratic functions that arise in applications in terms of a context (Standards F.IF.4–6).
  • Solving Quadratic Equations by Factoring
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
Analyze functions using different representations (Standards F.IF.7–9).
  • Carbon 14 dating in practice I
    In the task ''Carbon 14 Dating'' the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died and, as this task shows, this is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.
  • Functions and Everyday Situations
    This lesson unit is intended to help educators assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, and interpret algebraic functions in terms of the contexts in which they arise. Students will also reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
  • Functions and the Vertical Line Test
    The vertical line test for functions is the focus of this lesson plan.
  • Representing Quadratic Functions Graphically
    This lesson unit is intended to help educators assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation.
  • Solving Quadratic Equations by Factoring
    This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
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Strand: FUNCTIONS - Building Functions (F.BF)
Build a function that models a relationship between two quantities (Standard F.BF.1).
Build new functions from existing functions (Standard F.BF.3).
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Strand: FUNCTIONS - Linear, Quadratic, and Exponential Models (F.LE)
Construct and compare linear, quadratic, and exponential models and solve problems (Standard F.LE.3).
  • Comparing Investments
    This lesson unit is intended to help you assess how well students are able to interpret exponential and linear functions.
  • Functions and Everyday Situations
    This lesson unit is intended to help educators assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, and interpret algebraic functions in terms of the contexts in which they arise. Students will also reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
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Strand: FUNCTIONS - Trigonometric Functions (F.TF)
Prove and apply trigonometric identities. Limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean Theorem and the distance formula (Standard F.TF.8).
  • Calculations with sine and cosine
    The goal of this task is to study some values of sinx and cosx looking for and then explaining patterns.
  • Finding Trig Values
    This is a very straight forward task that addresses the second part of F-TF.C.8 exactly. It could be used as an introductory example, practice or assessment.
  • Trigonometric Ratios and the Pythagorean Theorem
    The purpose of this task is to use the Pythagorean Theorem to establish the fundamental trigonometric identity sin2θ+cos2θ=1 for an acute angle θ. The reasoning behind this identity is then applied to calculate cosθ for a given obtuse angle. In order to successfully complete part (c) students must be familiar with the definitions of trigonometric functions for arbitrary angles using the unit circle (F-TF.2).
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Strand: GEOMETRY - Congruence (G.CO)
Prove geometric theorems. Encourage multiple ways of writing proofs, such as narrative paragraphs, flow diagrams, two-column format, and diagrams without words. Focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning (Standards G.CO.9–11).
  • Angles
    Students are introduced to all kinds of angles in this lesson plan, including acute, obtuse, right, vertical, adjacent, and corresponding among others.
  • Classifying Triangles
    The goal of this task is to help students synthesize their knowledge of triangles.
  • Congruence of parallelograms
    Triangle congruence criteria have been part of the geometry curriculum for centuries. For quadrilaterals, on the other hand, these nice tests seem to be lacking. This task addresses this issue for a specific class of quadrilaterals, namely parallelograms.
  • Congruent angles in isosceles triangles
    The goal of this task is to establish that base angles in an isosceles triangle are congruent.
  • Congruent angles made by parallel lines and a transverse
    The goal of this task is to prove congruence of vertical angles made by two intersecting lines and alternate interior angles made by two parallel lines cut by a transverse.
  • Finding the Area of an Equilateral Triangle
    This task examines how to calculate the area of an equilateral triangle using high school geometry.
  • Is this a parallelogram?
    This task develops an alternative characterization of a parallelogram in terms of congruence of opposite sides.
  • Midpoints of Triangle Sides
    The goal of this task is to use similarity transformations to relate two triangles.
  • Midpoints of the Sides of a Paralellogram
    This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles.
  • Parallelograms and Translations
    The purpose of this task is to apply the definition of a parallelogram in the context of a geometric construction.
  • Points equidistant from two points in the plane
    This task gives the important characterization of the perpendicular bisector of a line segment as the set of points equidistant from the endpoints of the segment.
  • Seven Circles I
    This task is intended to help model a concrete situation with geometry.
  • Sum of angles in a triangle
    The goal of this task is to provide an argument, appropriate for high school students, for why the sum of the angles in a triangle is 180 degrees.
  • Tangent Lines and the Radius of a Circle
    This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches.
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Strand: GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT)
Understand similarity in terms of similarity transformations (Standards G.SRT.1–3).
  • Congruent and Similar Triangles
    The goal of this task is to understand similarity as a natural extension of congruence.
  • Dilating a Line
    This task asks students to "Verify experimentally" that a dilation takes a line that does not pass through the center to a line parallel to the original line, and that a dilation of a line segment (whether it passes through the center or not) is longer or shorter by the scale factor.
  • Similar Quadrilaterals
    The goal of this task is to study if the analogue of the AA criterion for similarity of triangles holds for different types of quadrilaterals.
  • Similar Triangles
    This task works toward establishing the AA criterion for similarity of triangles by providing a detailed sequence of transformations that moves one of the given triangles to the other.
Prove theorems involving similarity (Standards G.SRT.4–5)
  • Bank Shot
    This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.
  • Congruence of parallelograms
    Triangle congruence criteria have been part of the geometry curriculum for centuries. For quadrilaterals, on the other hand, these nice tests seem to be lacking. This task addresses this issue for a specific class of quadrilaterals, namely parallelograms.
  • Extensions, Bisections and Dissections in a Rectangle
    This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.
  • Finding triangle coordinates
    The purpose of this task is to use similar triangles in order to study the coordinates of points which divide a line segment in a given ratio.
  • How far is the horizon?
    The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make reasonable assumptions and find information that is not provided in the task statement.
  • Joining two midpoints of sides of a triangle
    This task is closely related to very important material about similarity and ratios in geometry.
  • Pythagorean Theorem
    The purpose of this task is to prove the Pythagorean theorem using similar triangles.
  • Slope Criterion for Perpendicular Lines
    The goal of this task is to use similar triangles to establish the slope criterion for perpendicular lines.
  • Tangent Line to Two Circles
    The purpose of this task is to use similar triangles, setting up a proportion in order to calculate a side length.
  • Unit Squares and Triangles
    This problem provides an opportunity for a rich application of coordinate geometry.
Define trigonometric ratios and solve problems involving right triangles (Standards G.SRT.6–8).
  • Ask the Pilot
    This task can be used as a classroom activity. There is a lot of opportunity to discuss the process of mathematical modeling. It serves to illustrate MP 4 - Model with Mathematics, not just by engaging in the practice, but also by investigating what this practice entails.
  • Coins in a circular pattern
    This task is intended for instructional purposes as an interesting activity which could accompany the other ''Seven Circles'' tasks. If it precedes these tasks, then the focus should be on recording information and looking for patterns.
  • Constructing Special Angles
    The goal of this task is to estimate the measure of angles in triangles with integer side lengths.
  • Defining Trigonometric Ratios
    The purpose of this task is to use the notion of similarity to define the sine and cosine of an acute angle.
  • Finding the Area of an Equilateral Triangle
    This task examines how to calculate the area of an equilateral triangle using high school geometry.
  • Mt. Whitney to Death Valley
    The purpose of this task is to engage students in an open-ended modeling task that uses similarity of right triangles, and also requires the use of technology (e.g., printed or electronic maps), thereby illustrating SMP 5 - Use Appropriate Tools Strategically.
  • Neglecting the Curvature of the Earth
    This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation.
  • Setting Up Sprinklers
    This task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.
  • Seven Circles I
    This task is intended to help model a concrete situation with geometry.
  • Seven Circles III
    This task is intended for instructional purposes only. It provides an opportunity to model a concrete situation with mathematics.
  • Shortest line segment from a point P to a line L
    This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle, which is crucial for many further developments in the subject.
  • Sine and Cosine of Complementary Angles
    The goal of this task is to provide a geometric explanation for the relationship between the sine and cosine of acute angles.
  • Solving Quadratic Equations: Cutting Corners
    This lesson unit is intended to help educators assess how well students are able to solve quadratics in one variable.
  • Tangent of Acute Angles
    The purpose of this task is to focus on studying values of tanx for special angles and conjecturing from these values how the function tanx varies when 0x<90.
  • Trigonometric Function Values
    The goal of this task is to explore the relationship between sine and cosine of complementary angles for special benchmark angles.
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Strand: GEOMETRY - Circles (G.C)
Understand and apply theorems about circles (Standard G.C.1–4).
  • Circumcenter of a triangle
    This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.
  • Circumscribed Triangles
    The goal of this task is to study where a circumscribed triangle can meet a given circle.
  • Inscribing a circle in a triangle I
    This task shows how to inscribe a circle in a triangle using angle bisectors.
  • Inscribing a circle in a triangle II
    This task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.
  • Inscribing a triangle in a circle
    This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter.
  • Locating Warehouse
    This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.
  • Opposite Angles in a Cyclic Quadrilateral
    The goal of this task is to show that opposite angles in a cyclic quadrilateral are supplementary.
  • Placing a Fire Hydrant
    This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle.
  • Right triangles inscribed in circles I
    This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle with one side of the triangle a diameter: the fact that these triangles are always right triangles is often referred to as Thales' theorem
  • Right triangles inscribed in circles II
    This task is designed to address the standard "Identify and describe relationships among inscribed angles, radii, and chords."
  • Similar circles
    The goal of this task is to work on showing that all circles are similar using these two different methods, the first visual and the second algebraic.
  • Tangent Lines and the Radius of a Circle
    This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches.
  • Tangent to a circle from a point
    This task is designed to allow students to construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles. Use this as a basis for introducing the radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course (Standard G.C.5).
  • Mutually Tangent Circles
    This is a challenging task which requires students to carefully divide up the picture into different pieces for which the area is known.
  • Sectors of Circles
    This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic.
  • Two Wheels and a Belt
    This task combines two skills from domain G-C: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment (G-C.2), and computing lengths of circular arcs given the radii and central angles (G-C.5).
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Strand: GEOMETRY - Expressing Geometric Properties With Equations (G.GPE)
Translate between the geometric description and the equation for a conic section (Standard G.GPE.1).
  • Coordinates of Points on a Circle
    The purpose of this task it to use geometry and algebra in order to understand the behavior of the trigonometric function f(x)=sinx+cosx. The task has been stated in an open ended fashion as there are natural solutions using geometry, or using the trigonometric identity sin2x=2sinxcosx, or algebraically solving a system of equations.
  • Explaining the equation for a circle
    Starting with explicit cases, this task derives the formula for an arbitrary circle in the plane.
  • Slopes and Circles
    The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever AXB is a right angle.
Use coordinates to prove simple geometric theorems algebraically. Include simple proofs involving circles (Standard G.GPE.4).
Use coordinates to prove simple geometric theorems algebraically (Standard G.GPE.6).
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Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Explain volume formulas and use them to solve problems (Standards G.GMD.1, 3).
  • Area of a circle
    This purpose of this task is to develop an understanding of the formula for the area of the circle.
  • Centerpiece
    The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario.
  • Doctor's Appointment
    The purpose of the task is to analyze a plausible real-life scenario using a geometric model
  • The Great Egyptian Pyramids
    The task is aligned to G-GMD.3 at each part involves either finding the volume of a pyramid or using the volume to find the base or height.
  • Volume Estimation
    This task has the dual purpose of having students apply geometric volume formulas, and to have them reason about modeling with geometric figures.
  • Volume formulas for cylinders and prisms
    The goal of this task is to establish formulas for volumes of right prisms and cylinders.
  • Volume of a Special Pyramid
    The goal of this task is to calculate the volume of a particular pyramid with square base, which is easily reasoned by viewing it as one sixth of a cube.
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Strand: STATISTICS - Interpreting Categorical and Quantitative Data (S.ID)
Summarize, represent, and interpret data on two categorical or quantitative variables (Standard S.ID.5).
  • Musical Preferences
    The basic idea is for students to demonstrate that they know what it means for two variables to be associated.
  • Support for a Longer School Day?
    The purpose of this task is to provide students with an opportunity to calculate joint, marginal and relative frequencies using data in a two-way table.
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Strand: STATISTICS - Conditional Probability and the Rules of Probability (S.CP)
Understand independence and conditional probability and use them to interpret data (Standards S.CP.1, 4–5).
  • Breakfast Before School
    The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.
  • But mango is my favorite
    This task can be used as an instructional task to develop students understanding of independence and students ability to calculate the probability of intersection events using the multiplication rule for independent events.
  • Describing Events
    The purpose of this task is to illustrate standard S.CP.1, which provides a review of the definitions of sample space and events. Students are asked to describe events both verbally and as subsets of a sample space.
  • False Positive Test Results
    This purpose of this task is to examine in a concrete situation a common statistical fallacy where two conditional probabilities are confused.
  • How Do You Get to School?
    This task is designed as an assessment item. It requires students to use information in a two-way table to calculate a probability and a conditional probability.
  • Rain and Lightning
    This task uses the same situation to explore different concepts of probability theory.
  • Representing Conditional Probabilities 2
    This lesson unit is intended to help educators assess how well students understand conditional probability.
  • Return to Fred's Fun Factory (with 50 cents)
    The task is intended to address standards regarding sample space, independence, probability distributions and permutations/combinations.
  • The Titanic 1
    This task guides students by asking the series of specific questions and lets them explore the concepts of probability as a fraction of outcomes, and using two-way tables of data.
  • The Titanic 2
    This task lets students explore the concepts of probability as a fraction of outcomes, and using two-way tables of data.
  • The Titanic 3
    This is the last task in the series of three, which ask related questions, but use different levels of scaffolding. This task uses a more detailed version of the data table. This is a very open ended task. It poses the question, but the students have to formulate a plan to answer it, and use the two-way table of data to find all the necessary probabilities.
  • Two-Way Tables and Probability
    The purpose of this task is to provide practice using data in a two-way table to calculate probabilities, including conditional probabilities.
Use the rules of probability to compute probabilities of compound events in a uniform probability model (Standard S.CP.6).
  • How Do You Get to School?
    This task is designed as an assessment item. It requires students to use information in a two-way table to calculate a probability and a conditional probability.
  • Representing Conditional Probabilities 2
    This lesson unit is intended to help educators assess how well students understand conditional probability.
  • The Titanic 1
    This task guides students by asking the series of specific questions and lets them explore the concepts of probability as a fraction of outcomes, and using two-way tables of data.
  • The Titanic 2
    This task lets students explore the concepts of probability as a fraction of outcomes, and using two-way tables of data.
  • The Titanic 3
    This is the last task in the series of three, which ask related questions, but use different levels of scaffolding. This task uses a more detailed version of the data table. This is a very open ended task. It poses the question, but the students have to formulate a plan to answer it, and use the two-way table of data to find all the necessary probabilities.
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HONORS - Strand: NUMBER AND QUANTITY - Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standard N.CN.3).
Represent complex numbers and their operations on the complex plane (Standards N.CN.4–5).
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HONORS - Strand: ALGEBRA - Reasoning With Equations and Inequalities (A.REI)
Solve systems of equations (Standards A.REI.8–9).
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HONORS - Strand: FUNCTIONS - Interpreting Functions (F.IF)
Analyze functions using different representations (Standards F.IF.10–11).
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HONORS - Strand: GEOMETRY - Expressing Geometric Properties With Equations (G-GPE)
Translate between the geometric description and the equation for a conic section (Standards G.GPE.2–3).
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HONORS - Strand: STATISTICS AND PROBABILITY - Conditional Probability and the Rules of Probability (S.CP)
Understand independence and conditional probability and use them to interpret data (Standards S.CP.2–3).
Use the rules of probability to compute probabilities of compound events in a uniform probability model (Standards S.CP.7–8).
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