Students determine the sample space for a chance experiment. Given a description of a chance experiment and an event, students identify the subset of outcomes from the sample space corresponding to the complement of an event. Given a description of a chance experiment and two events, students identify the subset of outcomes from the sample space corresponding to the union or intersection of two events. Students calculate the probability of events defined in terms of unions, intersections, and complements for a simple chance experiment with equally likely outcomes.

The purpose of this task is for students to interpret information provided that allows them to make sense of and organize data in a tree diagram, a two-way table, and a Venn diagram. Students will solidify their understanding of conditional probability by writing statements supported by data collected to justify the flavor of ice cream preferred by most. In this task, students will:
● Organize data into a tree diagram, two-way table, and a Venn diagram
● Calculate probabilities and conditional probabilities 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 ● Highlight the different representations and become more familiar with what each representation highlights and conceals.
● Continue to become more familiar with probability notation.
● Make decisions about meaning of data.

The purpose of this and the following task is to give students practice in analyzing diagrams to identify conjectures, and then writing proofs to show that the conjectures are true. The specific theorems being examined in this and the following 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°).

This resource is meant to supplement lessons on completing the square (quadratic equations) in order to find the vertex of a quadratic function and write the function in vertex form. The practice problems provide scaffolding to allow students to slowly build up to fully using the process of completing the square.

Students will understand transformations by first exploring the notion of linearity in an algebraic context. This quickly leads to a return to the study of complex numbers and a study of linear transformations in the complex plane.

Students will study transformations and the role transformations play in defining congruence. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.

This student interactive, from Illuminations, allows students to explore the conditions that guarantee uniqueness of a triangle, quadrilateral, or pentagon regardless of location or orientation. Each set of conditions results in a new congruence theorem.

The purpose of this task is to establish ASA, SAS and SSS as sufficient criteria for claiming that two triangles are congruent and to show how the rigid-motion transformations, along with the given congruence criteria about the two triangles, allows us to prove that the two triangles are congruent. As students work on such proofs they often overlook or reveal misconceptions about how to use the given congruence criteria in their work. Consequently, they might create a sequence of transformations that they claim carries one triangle onto the other, similar to the work they did in the previous task Can You Get There From Here, but in doing so they often assume the triangles are congruent, rather than proving them to be congruent. Therefore, the purpose of this task is less about students creating their own arguments, and more about considering the details of how such arguments can be made. Students begin by analyzing a couple of different arguments about ASA criteria for congruent triangles—one that harbors some misconceptions and one that is more explicit about the details. Then they explore other criteria for congruent triangles, such as SSS and SAS, and begin to formulate their own arguments about how they might justify such criteria using transformations.

The purpose of this task is to provide students with practice in identifying the criteria they might use—ASA, SAS or SSS—to determine if two triangles embedded in another geometric figure are congruent, and then to use those congruent triangles to make other observations about the geometric figures based on the concept that corresponding parts of congruent triangles are congruent. A secondary purpose of this task is to allow students to continue to examine what it means to make an argument based on the definitions of transformations, as well as based on properties of congruent triangles. The focus should be on using congruent triangles and transformations to identify other things that can be said about a geometric figure, rather than on the specific properties of triangles or quadrilaterals that are being observed. These observations will be more formally proved in Secondary II. The observations in this task also provide support for the geometric constructions that are explored in the next task.

Students find a rule that agrees with a giventable. The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.

In this task students use their techniques for changing the forms of quadratic expressions (i.e., factoring, completing the square to put the quadratic in vertex form, or using the quadratic formula to find the x-intercepts) as strategies for solving quadratic equations.

Students learn about basic recursion by exploring patterns in the data they generate from two simple probability-based experiments.

The purpose of this task is extend the types of proportionality statements that can be written when two sides of a triangle are crossed by a line that is parallel to the third side. In previous tasks students have written proportionality statements based on the corresponding sides of the smaller and larger triangle. In this task, they observe that corresponding segments formed on the sides of the triangle are proportional, even though those segments are not sides of the triangles. This is sometimes known as “the side splitter theorem.”

Students will develop a set of tools for understanding and interpreting variability in data, and begin to make more informed decisions from data. They work with data distributions of various shapes, centers, and spreads. Students build on their experience with bivariate quantitative data from Grade 8. This module sets the stage for more extensive work with sampling and inference in later grades.

Students determine how far a brother and sister travel before they catch up with one another. The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.

The purpose of this task is to continue developing the ideas of formal proof, particularly moving from reasoning with a diagram to reasoning based on a logical sequence of statements that start with given assumptions and lead to a valid conclusion. Reasoning with a diagram is an important geometric thinking skill, and in this task students explore the logic behind the construction of diagrams—what features of a diagram must precede the addition of other features. Students also examine the diagram for possible conclusions that can be made—what else must be true. The last part of the task focuses on writing symbolic statements to match the verbal descriptions we are making about the diagram, and then to sequence those statements into a logical flow of ideas. The format of this work is suggestive of a two-column proof.

This lesson introduces Venn diagrams to represent the sample space and various events and sets the stage for the two lessons that follow, which introduce students to probability formulas. The purpose is to provide a bridge between using the two-way table approach and using formulas to calculate probabilities. Venn diagrams also provide an opportunity to visually represent the population needed to understand what is requested in the exercises.

Students will be able to have a better understanding of the slope of a line, the y intercept, and parallel lines.

The purpose of this task is for students to solidify their understanding about the various ways to solve for unknown values of right triangles. Students will build on their prior knowledge of using the Pythagorean theorem to find unknown side lengths as well as their knowledge of setting up trigonometric ratios to find unknown side lengths. In this task, students will also learn how to use inverse trigonometric relationships to find unknown angle measures. Students will also further their knowledge of solving application problems using trigonometry by discussing the angle of elevation and the angle of depression.