The purpose of this task is to build fluency in writing equivalent expressions for quadratic equations using factoring, completing the square, and the distributive property. Students will use the equations that they have constructed to analyze and graph quadratic functions.

The purpose of this task is to give students practice in writing proofs to show that the conjectures are true. The specific theorems being examined in this task are:
• Vertical angles are congruent • The measure of an exterior angle of a triangle is the sum of the two remote interior angles
• When a transversal crosses parallel lines, alternate interior angles are congruent, corresponding angles are congruent, and same-side interior angles are supplementary (add to 180°).

In addition to practice writing proofs, the idea of the converse of a statement should be brought up in the discussion of this task. After students prove that alternate interior angles are congruent and corresponding angles are congruent when a transversal crosses parallel lines, they need to examine the converse statements as theorems:
• If corresponding angles are congruent when a transversal crosses two or more lines, then the lines are parallel.
• If alternate interior angles are congruent when a transversal crosses two or more lines, then the lines are parallel.

Students must determine how much water might a dripping faucet waste in a year. This task asks students to select and apply mathematical content from across the grades, including the content standards.

In this task students will write precise definitions for the three rigid-motion transformations, based on the observations they have made in the previous tasks in this learning cycle. To prepare students for writing their own definitions, they will study the language used in the definitions given for circle, angle, and angle of rotation. They will also write a definition for the word degree based on the information given. In part 2 of this task students will use their definitions to justify that multiple images have been correctly drawn based on specified transformations. As part of this task students will also explore the idea that two consecutive reflections produce a rotation when the lines of reflection are not parallel.

This task provides an opportunity for formative assessment of what students already know about the three rigid-motion transformations: translations, reflections, and rotations. As students engage in the task they should recognize a need for precise definitions of each of these transformations so that the final image under each transformation is a unique figure, rather than an ill-defined sketch. The exploration and subsequent discussion described below should allow students to begin to identify the essential elements in a precise definition of the rigid-motion transformations, e.g., translations move points a specified distance along parallel lines; rotations move points along a circular arc with a specified center and angle, and reflections move points across a specified line of reflection so that the line of reflection is the perpendicular bisector of each line segment connecting corresponding pre-image and image points.

In this lesson, students review and practice applying the properties of exponents for integer exponents. Students model a real-world scenario involving exponential growth and decay.

In this lesson, students review place value and scientific notation. Students use scientific notation to compute with large numbers.

In this lesson, students calculate probabilities given a two-way table of data. Students construct a hypothetical 1000 two-way table given probability information. Students interpret probabilities in context.

In this lesson, students calculate quantities that involve positive and negative rational exponents.

In this lesson, students use a hypothetical 1,000 two-way table to calculate probabilities of events. Students calculate conditional probabilities given a two-way data table or using a hypothetical 1,000 two-way table. Students use two-way tables (data tables or hypothetical 1,000 two-way tables) to determine if two events are independent. Students interpret probabilities, including conditional probabilities, in context.

In this lesson, students rewrite expressions involving radicals and rational exponents using the properties of exponents.

In this lesson, students approximate the value of quantities that involve positive irrational exponents.Students extend the domain of the function ??(??) = ??^?? for positive real numbers ?? to all real numbers.

In this lesson, students use logarithms to determine how many characters are needed to generate unique identification numbers in different scenarios. Students understand that logarithms are useful when relating the number of digits in a number to the magnitude of the number and that base 10 logarithms are useful when measuring quantities that have a wide range of values such as the magnitude of earthquakes, volume of sound, and pH levels in chemistry.

This is a project that follows the PBL framework and was used to help students master the fundamentals of probability, specifically the laws of probability (NC.M2.S-CP.1 to 8). Note that the project was designed and delivered per the North Carolina Math 2 curriculum and it can be customized to meet your own specific curriculum needs and resources.

Students will extend their study of functions to include function notation and the concepts of domain and range by exploring examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

The purpose of this task is two-fold. The first purpose is for students to explore and generalize how the features of the equation can be used to graph the quadratic function. The second purpose is for students to deepen their understanding of quadratic functions as the product of two linear factors. In the task, students are asked to graph parabolas from equations in factored form. They are given several cases to provide an opportunity to notice how the x-intercepts, yintercept, and vertical stretch are readily visible in the equation. This also sets them up to notice the relationship between the ! -intercepts and the y-intercept. The task extends this thinking by asking students to start with any two linear functions, multiply them together and find the function that is created, which is quadratic. They graph both the initial lines and the parabola to find the relationship between x-intercepts and y-intercept and to highlight the idea that quadratic functions are the product of two linear factors.

A PBL project that allows high school physics and math students exercise voice & choice in the design of a new maker space.

Adapted from mathematicsvisionproject.com’s Material Overview:
The Mathematics Vision Project (MVP) was created as a resource for teachers to implement the Common Core State Standards (CCSS) using a task-based approach that leads to skill and efficiency in mathematics by first developing understanding. The MVP approach develops the Standards of Mathematical Practice through experiential learning. Students engage in mathematical problem solving, guided by skilled teachers, in order to achieve mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The MVP authors created a curriculum where students do not learn solely by either “internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own.”
The MVP classroom experience begins by confronting students with an engaging problem and allows them to grapple with solving it. As students’ ideas emerge, take form, and are shared, the teacher deliberately orchestrates the student discussions and explorations toward a focused math goal. Students justify their own thinking while clarifying, describing, comparing, and questioning the thinking of others leading to refined thinking and mathematical fluency. What begin as ideas become concepts that lead to formal, traditional math definitions and properties. Strategies become algorithms that lead to procedures supporting efficiency and consistency. Representations become tools of communication which are formalized as mathematical models. Students learn by doing mathematics.

This task gives students opportunities to practice applying the theorems of this and the previous module. The theorems students will draw upon include:
• Vertical angles are congruent.
• Measures of interior angles of a triangle sum to 180°.
• When transversals cross parallel lines, alternate interior angles are congruent and corresponding angles are congruent.
• A line parallel to one side of a triangle divides the other two sides proportionally.

Students will also apply the Pythagorean theorem to find the missing sides of right triangles, and conversely, to determine if a triangle is a right triangle. The last part of the task allows students to review their “ways of knowing” something is true through inductive and deductive reasoning. Students will collect data about the sums of the measures of the interior angles of quadrilaterals and pentagons. When combined with their knowledge of the sum of the measures of the interior angles of a triangle, students make a conjecture about the sum of the measures of the interior angles of polygons with any number of sides. Students are then asked to use deductive reasoning to prove their conjecture for n-sided polygons.