Resources by Strand
Strand: MATHEMATICAL PRACTICES (MP)
The Standards for Mathematical Practice in Secondary Mathematics II describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes
(Standards MP.1-8).
Standard SII.MP.1
Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, "Does this make sense?" Consider analogous problems, make connections between multiple representations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check answers to problems using a different method.
Standard SII.MP.2
Reason abstractly and quantitatively. Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebraically and manipulating symbols fluently as well as the ability to contextualize algebraic representations to make sense of the problem.
Standard SII.MP.3
Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying questions, and critiquing the reasoning of others.
Standard SII.MP.4
Model with mathematics. Apply mathematics to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Standard SII.MP.5
Use appropriate tools strategically. Consider the available tools and be sufficiently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts.
Standard SII.MP.6
Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
Standard SII.MP.7
Look for and make use of structure. Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Standard SII.MP.8
Look for and express regularity in repeated reasoning. Notice if reasoning is repeated, and look for both generalizations and shortcuts. Evaluate the reasonableness of intermediate results by maintaining oversight of the process while attending to the details.
Strand: NUMBER AND QUANTITY - The Real Number System (N.RN)
Extend the properties of exponents to rational exponents (Standards N.RN.1–2)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
Standard N.RN.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
Standard N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers (Standard N.RN. 3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- Calculating the square root of 2
This Illustrative Mathematics task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. - Coordinates of equilateral triangles
The purpose of this Illustrative Mathematics task is to examine angles of triangles whose vertices have specific integer coordinates. - Operations with Rational and Irrational Numbers
This Illustrative Mathematics task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality. - Sums of rational and irrational numbers
The goal of this Illustrative Mathematics task is to examine sums and products of rational and irrational numbers.
Standard N.RN.3
Explain why sums and products of rational numbers are rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational. Connect to physical situations (e.g., finding the perimeter of a square of area 2).
Strand: NUMBER AND QUANTITY - The Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standards N.CN.1–2).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
Standard N.CN.1
Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
Standard N.CN.2
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Limit to multiplications that involve i2 as the highest power of i.
Use complex numbers in polynomial identities and equations (Standards N.CN.7–9).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
Standard N.CN.7
Solve quadratic equations with real coefficients that have complex solutions.
Standard N.CN.8
Extend polynomial identities to the complex numbers. Limit to quadratics with real coefficients. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
Standard N.CN.9
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Strand: ALGEBRA - Seeing Structure in Expression (A.SSE)
Interpret the structure of expressions (Standards A.SSE.1–2)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Quadratic Functions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 1: Quadratic Functions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions.
Instructional Tasks
- 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.
- Adding and Subtracting Polynomials
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. - Adding and Subtracting Polynomials video
This video introduces and explains the topic. - Adding and Subtracting Rational Expressions
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. - Adding and Subtracting Rational Expressions video
This video introduces and explains the topic. - Multiplying Polynomials
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. - Multiplying Polynomials video
This video introduces and explains the concept. - Multiplying and Dividing Monomials
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. - Multiplying and Dividing Monomials video
This video introduces and explains the topic. - Multiplying and Dividing Rational Expressions
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. - Multiplying and Dividing Rational Expressions video
This video introduces and explains the topic. - Polynomials
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. - Polynomials video
The video introduces and explains the topic. - Simplifying Expressions with 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. - Simplifying Expressions with Exponents video
This is a video introduction and explanation of the topic. - Simplifying Radical Expressions
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. - Simplifying Radical Expressions video
This video introduces and explains the topic. - Solving Quadratic Equations by Completing the Square
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 Functions by Completing the Square video
This video introduces and explains the topic. - Special Products of Polynomials
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. - Special Products of Polynomials video
This video introduces and explains the concept.
Standard A.SSE.1
Interpret quadratic and exponential expressions that represent a quantity in terms of its context.★
- Interpret parts of an expression, such as terms, factors, and coefficients.
- Interpret increasingly more complex expressions by viewing one or more of their parts as a single entity. Exponents are extended from the integer exponents to rational exponents focusing on those that represent square or cube roots.
Standard A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems, balancing conceptual understanding and procedural fluency in work with equivalent expressions (Standard A.SSE.3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 2: Structures of Expressions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 2: Structures of Expressions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
- Factoring Special Products
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. - Factoring Special Products video
This video introduces and explains the concept. - Factoring Trinomials by Grouping 1
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. - Factoring Trinomials by Grouping 1 video
This is a video introducing and explaining the concept. - Factoring Trinomials by Grouping 2
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. - Factoring Trinomials by Grouping 2 video
This video introduces and explains this concept. - Non-linear Functions video
This video introduces non-linear functions. - Solve Quadratic Expressions by Factoring video
This video introduces and explains the concept.
Standard A.SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. For example, development of skill in factoring and completing the square goes hand in hand with understanding what different forms of a quadratic expression reveal.★
- Factor a quadratic expression to reveal the zeros of the function it defines.
- Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
- Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
- Adding and Subtracting Polynomials
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. - Adding and Subtracting Polynomials video
This video introduces and explains the topic. - IXL Game: Polynomials: Multiply polynomials
This game in the algebra domain will help students understand how multiply polynomials. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Multiplying Polynomials
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. - Multiplying Polynomials video
This video introduces and explains the concept. - Special Products of Polynomials
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. - Special Products of Polynomials video
This video introduces and explains the concept.
Standard A.APR.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Quadratic Functions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 1: Quadratic Functions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
- Applications of Quadratic Functions
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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - Equations of Circles 1
This lesson unit is intended to help educators assess how well students are able to use the Pythagorean theorem to derive the equation of a circle, as well as translate between the geometric features of circles and their equations. - Graphing Quadratic Functions
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. - Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems. - Non-linear Functions video
This video introduces non-linear functions. - Polynomials
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 Using the Quadratic Formula video
This video introduces and explains the topic. - Sorting Equations of Circles 2
This lesson unit is intended to help educators assess how well students are able to translate between the equations of circles and their geometric features, and sketch a circle from its equation.
Standard A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Standard A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Standard A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations; extend to formulas involving squared variables.For example, rearrange the formula for the volume of a cylinder V = π r2 h.
Strand: ALGEBRA - Reasoning With Equations and Inequalities (A.REI)
Solve equations and inequalities in one variable (Standard A.REI.4)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.''
- Applications of Quadratic Functions
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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - Graphing Quadratic Functions
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. - Graphing Quadratic Functions video
This video introduces and explains the topic. - Proof Without Words: Completing the Square
The act of "completing the square" involves taking half the coefficient of x in the quadratic x2 + ax and adding its square. But many students do not understand why this process works. - Solve Quadratic Expressions by Factoring video
This video introduces and explains the concept. - Solving Quadratic Equations Using the Quadratic Formula video
This video introduces and explains the topic. - Solving Quadratic Equations by Completing the Square
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 Functions by Completing the Square video
This video introduces and explains the topic. - Systems of Non-Linear Equations
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. - Systems of Non-Linear Equations video
This video introduces and explains the topic.
Standard A.REI.4
Solve quadratic equations in one variable.
- Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
- Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable.
Instructional Tasks
- 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.
Standard A.REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
Strand: FUNCTIONS - Interpret Functions (F.IF)
Interpret quadratic functions that arise in applications in terms of a context (Standards F.IF.4–6).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 4: More Functions, More Features - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 4, More Functions, More Features, shifts focus to consider piecewise functions, absolute value functions, and inverse functions. Understanding of piecewise functions is built from students understanding of graphs and the stories that they can tell. Students learn to write functions for contexts in which rates change, making a piecewise function. - Module 4: More Functions, More Features - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 4, More Functions, More Features, shifts focus to consider piecewise functions, absolute value functions, and inverse functions. Understanding of piecewise functions is built from students understanding of graphs and the stories that they can tell. Students learn to write functions for contexts in which rates change, making a piecewise function.
Instructional Tasks
- 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.
- Applications of Quadratic Functions
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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - GeoGebra
GeoGebra is dynamic online geometry software. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions. All of them can be changed dynamically afterwards. - Graphing Quadratic Functions
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. - Graphing Quadratic Functions video
This video introduces and explains the topic. - Intercepts of Linear Equations video
This video introduces the topic. - Non-linear Functions video
This video introduces non-linear functions. - Polynomials video
The video introduces and explains the topic. - Solving Quadratic Equations Using the Quadratic Formula video
This video introduces and explains the topic. - Solving Quadratic Functions by Completing the Square video
This video introduces and explains the topic. - Sorting Equations of Circles 2
This lesson unit is intended to help educators assess how well students are able to translate between the equations of circles and their geometric features, and sketch a circle from its equation.
Standard F.IF.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
Standard F.IF.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Focus on quadratic functions; compare with linear and exponential functions. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★
Standard F.IF.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★
Analyze functions using different representations (Standards F.IF.7–9).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 2: Structures of Expressions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 2: Structures of Expressions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 3: Solving Quadratics & Other Equations - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 3: Solving Quadratics & Other Equations - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 3, Solving Quadratics & Other Equations. The idea that different quadratic forms are useful in graphing is extended in Module 3, Solving Quadratic and Other Equations, when students use different forms of quadratic equations to find the roots or solve quadratic equations of a single variable. - Module 4: More Functions, More Features - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 4, More Functions, More Features, shifts focus to consider piecewise functions, absolute value functions, and inverse functions. Understanding of piecewise functions is built from students understanding of graphs and the stories that they can tell. Students learn to write functions for contexts in which rates change, making a piecewise function. - Module 4: More Functions, More Features - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 4, More Functions, More Features, shifts focus to consider piecewise functions, absolute value functions, and inverse functions. Understanding of piecewise functions is built from students understanding of graphs and the stories that they can tell. Students learn to write functions for contexts in which rates change, making a piecewise function.
Instructional Tasks
- 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.
- Applications of Quadratic Functions
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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - GeoGebra
GeoGebra is dynamic online geometry software. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions. All of them can be changed dynamically afterwards. - Graphing Quadratic Functions
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. - Intercepts of Linear Equations video
This video introduces the topic. - Non-linear Functions video
This video introduces non-linear functions. - Solve Quadratic Expressions by Factoring video
This video introduces and explains the concept. - Solving Quadratic Functions by Completing the Square video
This video introduces and explains the topic. - Sorting Equations of Circles 2
This lesson unit is intended to help educators assess how well students are able to translate between the equations of circles and their geometric features, and sketch a circle from its equation.
Standard F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
- Graph linear and quadratic functions and show intercepts, maxima, and minima.
- Graph piecewise-defined functions and absolute value functions. Compare and contrast absolute value and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.
Standard F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
- Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Standard F.IF.9
Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Strand: FUNCTIONS - Building Functions (F.BF)
Build a function that models a relationship between two quantities (Standard F.BF.1).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Quadratic Functions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 1: Quadratic Functions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 2: Structures of Expressions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 2: Structures of Expressions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations.
Instructional Tasks
Review Activitiy
- Applications of Quadratic Functions
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. - Applications of Quadratic Functions video
This video introduces and explains the topic. - Generalizing Patterns: Table Tiles
This lesson unit is intended to help educators assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops. - Medical Testing
This lesson unit is intended to help educators assess how well students are able to make sense of a real life situation and decide what math to apply to the problem, understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model, and represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams.
- Polynomials
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. - Representing Trigonometric Functions - Ferris Wheel
This lesson unit is intended to help educators assess how well students are able to model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions, and interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.
Standard F.BF.1
Write a quadratic or exponential function that describes a relationship between two quantities.★
- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Build new functions from existing functions (Standard F.BF.3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Quadratic Functions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 2: Structures of Expressions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations. - Module 2: Structures of Expressions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 2, Structures of Expressions is both a functions module and an algebra module. It is designed to extend students knowledge of quadratic functions and to reinforce two big ideas of functions, Functions can be transformed in the same, predictable way and Different algebraic forms have purpose in different situations.
Instructional Tasks
Review Activitiy
- Generalizing Patterns: Table Tiles
This lesson unit is intended to help educators assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types that are needed for a range of square tabletops. - Medical Testing
This lesson unit is intended to help educators assess how well students are able to make sense of a real life situation and decide what math to apply to the problem, understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model, and represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams.
- Representing Trigonometric Functions - Ferris Wheel
This lesson unit is intended to help educators assess how well students are able to model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions, and interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.
Standard F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Strand: FUNCTIONS - Linear, Quadratic, and Exponential Models (F.LE)
Construct and compare linear, quadratic, and exponential models and solve problems (Standard F.LE.3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 1: Quadratic Functions - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions. - Module 1: Quadratic Functions - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 1, Quadratic Functions, picks up where students left off with linear and exponential functions, using the same types of diagrams and representations to introduce quadratic functions. The entire module focuses on the features of quadratic functions, comparing quadratics to other functions, and representing quadratic functions.
Instructional Tasks
- 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.
- Modeling: Having Kittens
This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions and investigate an exponentially increasing sequence.
Standard F.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
- 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).
Standard F.TF.8
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 5: Geometric Figures - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 5, Geometric Figures. Formal proof is introduced in this module, beginning with students understanding the ways of knowing continuum based on authority, on experience with a few examples, on reasoning from a diagram and on statements accepted as true by the community of practice, including postulates, definitions and theorems. - Module 5: Geometric Figures - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 5, Geometric Figures. Formal proof is introduced in this module, beginning with students understanding the ways of knowing continuum based on authority, on experience with a few examples, on reasoning from a diagram and on statements accepted as true by the community of practice, including postulates, definitions and theorems. - Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
- 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.
- Angles
Students are introduced to all kinds of angles in this lesson plan, including acute, obtuse, right, vertical, adjacent, and corresponding among others. - Evaluating Statements About Length and Area
This lesson unit is intended to help educators assess how well students can understand the concepts of length and area, use the concept of area in proving why two areas are or are not equal, and construct their own examples and counterexamples to help justify or refute conjectures. - Proofs of the Pythagorean Theorem
This lesson unit is intended to help educators assess how well students are able to produce and evaluate geometrical proofs. - Solving Geometry Problems: Floodlights
This lesson unit is intended to help educators assess how well students are able to identify and use geometrical knowledge to solve a problem. - Student Task: Circles in Triangles
In this task, the students have to find the radius of circles inscribed in various sizes of right triangle. - Student Task: Floor Pattern
In this task, students will investigate the geometrical properties of a pattern of floor tiles - Tessellations: Geometry and Symmetry
Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson.
Standard G.CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Standard G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Standard G.CO.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Strand: GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT)
Understand similarity in terms of similarity transformations (Standards G.SRT.1–3).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
- 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.
- Evaluating Conditions for Congruency
This lesson unit is intended to help educators assess how well students are able to work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context. - IXL Game: Geometry: Similar Figures
This game will reinforce geometry dealing with similar figures by looking at side lengths and angle measures. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Introduction to Fractals: Infinity, Self-Similarity and Recursion
This lesson is designed to help students understand aspects of fractals, specifically self-similarity and recursion. - Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem.
Standard G.SRT.1
Verify experimentally the properties of dilations given by a center and a scale factor.
- A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
- The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Standard G.SRT.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Standard G.SRT.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity (Standards G.SRT.4–5)
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
- 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.
- Evaluating Conditions for Congruency
This lesson unit is intended to help educators assess how well students are able to work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context. - Proofs of the Pythagorean Theorem
This lesson unit is intended to help educators assess how well students are able to produce and evaluate geometrical proofs. - Solving Geometry Problems: Floodlights
This lesson unit is intended to help educators assess how well students are able to identify and use geometrical knowledge to solve a problem. - Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem. - Tessellations: Geometry and Symmetry
Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson.
Standard G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem (proved using triangle similarity).
Standard G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles (Standards G.SRT.6–8).
Review Activitiy
- Access Ramp - Student Task
This task has students design an access ramp, which complies with the Americans with Disabilities Act (ADA) requirements and include pricing based on local costs. - Applications of the Pythagorean Theorem
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. - Applications of the Pythagorean Theorem video
This video introduces and explains the topic. - Applying Radical Equations
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. - Applying Radical Equations video
This video introduces and explains the topic. - Calculating Volumes of Compound Objects
This lesson unit is intended to help educators assess how well students solve problems involving measurement. - Evaluating Conditions for Congruency
This lesson unit is intended to help educators assess how well students are able to work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context. - Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems. - Miniature Golf - student task
This task requires students to redesign a miniature golf course to make it more challenging. - Pythagorean Explorer
This applet challenges the student to find the length of the third side of a triangle when given the two sides and the right angle. - Solving Problems with Circles and Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems - Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem. - Student Task: Hopewell Geometry
The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles.
In this task, the student will look at some of the geometrical properties of a Hopewell earthwork.
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
- 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.
Standard G.SRT.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Standard G.SRT.7
Explain and use the relationship between the sine and cosine of complementary angles.
Standard G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Strand: GEOMETRY - Circles (G.C)
Understand and apply theorems about circles (Standard G.C.1–4).
Core Guide
OER Curriculum
The Math Vision Project- Module 7: Circles from a Geometric Perspective - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders. - Module 7: Circles from a Geometric Perspective - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders.
Instructional Tasks
- 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.
- Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems. - Student Task: Circles in Triangles
In this task, the students have to find the radius of circles inscribed in various sizes of right triangle. - Student Task: Temple Geometry
During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for students to investigate.
Standard G.C.1
Prove that all circles are similar.
Standard G.C.2
Identify and describe relationships among inscribed angles, radii, and chords. Relationships include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Standard G.C.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Standard G.C.4
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).
Review Activitiy
- Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems. - Student Task: Circles in Triangles
In this task, the students have to find the radius of circles inscribed in various sizes of right triangle. - Student Task: Temple Geometry
During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for students to investigate.
The Math Vision Project- Module 7: Circles from a Geometric Perspective - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders. - Module 7: Circles from a Geometric Perspective - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders.
Instructional Tasks
- 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).
Standard G.C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
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).
Core Guide
OER Curriculum
The Math Vision Project- Module 8: Circles & Other Conics - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 8, Circles and Other Conics, takes an algebraic approach to solving problems with circles, parabolas, and in the Honors course, ellipses and hyperbolas. The module includes several hands- on explorations to develop the equations for circles, parabolas, and ellipses. - Module 8: Circles & Other Conics - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 8, Circles and Other Conics, takes an algebraic approach to solving problems with circles, parabolas, and in the Honors course, ellipses and hyperbolas. The module includes several hands- on explorations to develop the equations for circles, parabolas, and ellipses.
Instructional Tasks
- 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.
Standard G.GPE.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically. Include simple proofs involving circles (Standard G.GPE.4).
Core Guide
Instructional Tasks
Review Activitiy
- Finding Equations of Parallel and Perpendicular Lines
This lesson unit is intended to help educators assess how well students are able to understand the relationship between the slopes of parallel and perpendicular lines. - Sorting Equations of Circles 2
This lesson unit is intended to help educators assess how well students are able to translate between the equations of circles and their geometric features, and sketch a circle from its equation.
Standard G.GPE.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Use coordinates to prove simple geometric theorems algebraically (Standard G.GPE.6).
Core Guide
OER Curriculum
The Math Vision Project- Module 6: Similarity & Right Triangle Trigonometry - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other. - Module 6: Similarity & Right Triangle Trigonometry - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 6, Similarity and Right Triangle Trigonometry, introduces the last of the transformations, dilation. A big idea of Module 6 is that two figures are similar if a sequence of rigid transformation and dilations exists that maps one figure onto the other.
Instructional Tasks
Review Activitiy
- Student Task: Square
In this task, students are given four points on a graph, and must prove that they are the corners of a square.
Standard G.GPE.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Explain volume formulas and use them to solve problems (Standards G.GMD.1, 3).
Review Activitiy
OER Curriculum
The Math Vision Project- Module 7: Circles from a Geometric Perspective - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders. - Module 7: Circles from a Geometric Perspective - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 7, Circles: A Geometric Perspective, is composed of four learning cycles. In the first learning cycle, students use rotations and perpendicular bisectors to find the center of a circle. The second learning cycle in Module 7 builds on the circle relationships that students have learned so far in the module to develop a formula for the perimeter and area of a regular polygon. The third learning cycle addresses relationships among central angles, radii, arcs, and sectors. Students calculate arc length and the area of a sector. The final learning cycle in Module 7 is an intuitive approach to volume of prisms, pyramids, and cylinders.
Instructional Tasks
- 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.
Standard G.GMD.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Informal arguments for area formulas can make use of the way in which area scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k2 times the area of the first. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Standard G.GMD.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Informal arguments for volume formulas can make use of the way in which volume scale under similarity transformations: when one figure results from another by applying a similarity transformation, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.★
Strand: STATISTICS - Interpreting Categorical and Quantitative Data (S.ID)
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).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 9: Probability - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 9, Probability, extends students work in representing and analyzing data to understand concepts in probability. In the module, students use representations such as tree diagrams, Venn diagrams, and two-way frequency tables to draw conclusions about the likelihood of an event. - Module 9: Probability - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 9, Probability, extends students work in representing and analyzing data to understand concepts in probability. In the module, students use representations such as tree diagrams, Venn diagrams, and two-way frequency tables to draw conclusions about the likelihood of an event.
Instructional Tasks
- 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.
- A Better Fire!!
A simulation of a forest fire and how it spreads helps students learn about probability in this lesson. - Conditional Probability and Probability of Simultaneous Events
This lesson plan is designed to help students understand and use the formula for probability of simultaneous independent events. - Crazy Choices Game
Students will understand probability by playing various games of chance in this lesson's applet. - Ideas that Lead to Probability
This lesson is designed so that students will understand the concept of probability
by working with random number generators. - Marbles
This activity helps the student understand randomness and probability by pulling marbles out of a bag. - Medical Testing
This lesson unit is intended to help educators assess how well students are able to make sense of a real life situation and decide what math to apply to the problem, understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model, and represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams.
- Modeling Conditional Probabilities 1: Lucky Dip
This lesson unit is intended to help educators assess how well students are able to understand conditional probability and represent events as a subset of a sample space using tables and tree diagrams.
- Racing Game with Two Dice
By using the applet embedded in this lesson plan, students can simulate a race and learn about probability.
Standard S.CP.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
Standard S.CP.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
Standard S.CP.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model (Standard S.CP.6).
Core Guide
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems. - Module 9: Probability - Student Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 9, Probability, extends students work in representing and analyzing data to understand concepts in probability. In the module, students use representations such as tree diagrams, Venn diagrams, and two-way frequency tables to draw conclusions about the likelihood of an event. - Module 9: Probability - Teacher Edition (Math 2)
The Mathematics Vision Project, Secondary Math Two Module 9, Probability, extends students work in representing and analyzing data to understand concepts in probability. In the module, students use representations such as tree diagrams, Venn diagrams, and two-way frequency tables to draw conclusions about the likelihood of an event.
Instructional Tasks
- 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.
- Medical Testing
This lesson unit is intended to help educators assess how well students are able to make sense of a real life situation and decide what math to apply to the problem, understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model, and represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams.
- Modeling Conditional Probabilities 1: Lucky Dip
This lesson unit is intended to help educators assess how well students are able to understand conditional probability and represent events as a subset of a sample space using tables and tree diagrams.
Standard S.CP.6
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
HONORS - Strand: NUMBER AND QUANTITY - Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standard N.CN.3).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
HONORS - Standard N.CN.3
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane (Standards N.CN.4–5).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
HONORS - Standard N.CN.4
Represent complex numbers on the complex plane in rectangular form, and explain why the rectangular form of a given complex number represents the same number.
HONORS - Standard N.CN.5
Represent addition, subtraction, and multiplication geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
HONORS - Strand: ALGEBRA - Reasoning With Equations and Inequalities (A.REI)
Solve systems of equations (Standards A.REI.8–9).
HONORS - Standard A.REI.8
Represent a system of linear equations as a single-matrix equation in a vector variable.
HONORS - Standard A.REI.9
Find the inverse of a matrix if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
HONORS - Strand: FUNCTIONS - Interpreting Functions (F.IF)
Analyze functions using different representations (Standards F.IF.10–11).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
HONORS - Standard F.IF.10
Use sigma notation to represent the sum of a finite arithmetic or geometric series.
HONORS - Standard F.IF.11
Represent series algebraically, graphically, and numerically.
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).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
HONORS - Standard G.GPE.2
Derive the equation of a parabola given a focus and directrix.
HONORS - Standard G.GPE.3
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
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).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
Review Activitiy
- Conditional Probability and Probability of Simultaneous Events
This lesson plan is designed to help students understand and use the formula for probability of simultaneous independent events. - Crazy Choices Game
Students will understand probability by playing various games of chance in this lesson's applet. - Events and Outcomes (Counting)
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. - Events and Outcomes (Counting) video
This video introduces and explains the topic. - From Probability to Combinatorics and Number Theory
This lesson is designed to help students understand probability theory and look at data structures. - Ideas that Lead to Probability
This lesson is designed so that students will understand the concept of probability
by working with random number generators. - Permutations and Combinations
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. - Permutations and Combinations video
This video introduces and explains the topic. - Probability of Independent Events
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. - Probability of Independent Events video
This video introduces and explains the concept. - Racing Game with Two Dice
By using the applet embedded in this lesson plan, students can simulate a race and learn about probability. - Tree Diagrams and Probability
This lesson focuses on how to figure probability by having the students create tree diagrams. - Two Colors Applet
The applet in this lesson helps the student understand dependent event probability.
HONORS - Standard S.CP.2
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
HONORS - Standard S.CP.3
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of B given A is the same as the probability of B.
Use the rules of probability to compute probabilities of compound events in a uniform probability model (Standards S.CP.7–8).
OER Curriculum
The Math Vision Project- Introduction to the Materials (Math 2)
Introduction to the Materials in the Mathematics Two of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
Review Activitiy
HONORS - Standard S.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
HONORS - Standard S.CP.8
Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
The Online Core Resource pages are a collaborative project between the Utah State Board of Education and the Utah Education Network. If you would like to recommend a high quality resource, contact Trish French (Elementary) or Lindsey Henderson (Secondary). If you find inaccuracies or broken links contact resources@uen.org.
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