Lesson Structure and Resources

Lesson Narrative

Each lesson begins with a narrative of the mathematical work that will unfold within the activities of that lesson. Lesson narratives explain:

• the mathematical content of the lesson and its place in the learning sequence
• the meaning of any new terms introduced in the lesson
• how the mathematical practices come into play, as appropriate

Learning Goals

Teacher-facing learning goals appear at the top of the lesson overview. They describe, for a teacher audience, the mathematical, pedagogical, and language goals of the lesson.

Student-facing learning goals appear in student materials at the beginning of each lesson and start with the word "Let's." They are intended to invite students into the work of that day without giving away too much and spoiling the problem-based instruction. They are suitable for writing on the board before class begins.

Learning Targets

These appear in student materials at the end of each unit. They describe, for a student audience, the mathematical goals of each lesson.

We do not recommend writing learning targets on the board before class begins, because doing so might spoil the problem-based instruction. (The student-facing learning goals (that start with "Let's") are more appropriate for this purpose.)

Teachers and students might use learning targets in a number of ways. Some examples include:

• targets for standards-based grading
• prompts for a written reflection as part of a lesson synthesis
• a study aid for self-assessment, review, or catching up after an absence from school

Texas Essential Knowledge and Skills (TEKS)

A table listing all of the TEKS addressed in the lesson is provided along with an explanation of the depth of coverage.

• Math process coverage - While all of the process standards should be used within the course of instruction for the lesson, specific ones have been highlighted since they will be the one(s) most prominently used.
• Foundational - This type of coverage means the content in the lesson does not address the required rigor of the grade level TEKS but is necessary in providing access to the required expectation levels.
• Partial - This type of coverage means one or more of the component break outs for the TEKS has been addressed, but not all parts.
• Full - This type of coverage means all parts of the TEKS have been covered at the required level of rigor.

There are a few activities within some lessons that have been identified as exceeding the expectations of the TEKS. Notification of which activities fall into this category has been placed in narrative for the activity. These activities may be used for extension and differentiation.

Lesson Activities

A list of all of the primary learning activities for students is listed. The first and last instructional activity will be the Warm Up and Cool Down, respectively. Each learning activity denoted between the Warm Up and Cool Down will include a Self Check and Additional Resources.

If an activity in the list lacks a Self Check or Additional Resources, it is typically an activity that exceeds the expectation of the TEKS and may be used as an extension or extension activity.

Required Materials and Preparation

This section provides notification if special materials are required for any of the activities in the lesson. Such materials might include card sorts, graphic organizers, manipulatives, technology, etc.

Lesson Vocabulary

A list of the vocabulary words students are expected to learn and be able to use during the lesson are provided. The terms are divided between vocabulary that is new to the students and terms they should have learned in previous instruction.

Sample word wall cards are provided for use during discussions and activities. Model referencing the terms during instruction and encourage students to do so also.

To support newcomers or students identified at the beginning level of language proficiency, links to digital vocabulary cards in both English and Spanish have been provided. It is suggested to use the Spanish versions to anchor student understanding before bridging to the English versions.

The links have not been provided to students so that teachers may provide the appropriate set according to an individual student’s linguistic needs.

Support for English Language Learners

The list provided on the Teacher Overview for each lesson is not exhaustive, but it contains the ELPS that are most applicable for the activities in the lesson. Every lesson has been designed to provide ELPS instruction in at least three of the domains: speaking, listening, reading, writing, and thinking. The list of the targeted domains and standards for the lesson are in this list.

Directions for how to meet these standards involve the use of specific Math Language Routines [MLR] and have been provided within the Teacher Guides of each activity.

Support for Building Character

Each unit of instruction provides instructional suggestions and resources teachers may use to develop students’ workplace tools and learner behaviors. The Building Character activities at the lesson level include targeted tips teachers may use with or provide to students.

For instance, the Building Charter focus in Unit 1 is Social Intelligence. This topic will feed into developing students’ awareness of how to interact in cooperative learning environments, a key learner behavior students need to learn early in the year.

Each instructional activity (Warm Up, Primary Learning Activities, Cool Down, Practice) has a learning guide with suggestions for how to implement the content. The activities typically have three phases: Launch, Activity Work Time, and Activity Synthesis.

Activity Narrative

The Activity Narratives begin each page and include:

• the mathematical purpose of the activity and its place in the learning sequence
• what students are doing during the activity
• what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis
• connections to the mathematical process standards, when appropriate

Launch

During the launch, the teacher makes sure that students understand the context (if there is one) and what the problem is asking them to do. This is not the same as making sure the students know how to do the problem—part of the work that students should be doing for themselves is figuring out how to solve the problem. The launch invites students into the lesson and helps them connect to contexts that may be unfamiliar.

The launch for an activity also includes suggestions for grouping students.

Student Activity and Answers

Student Activities may follow a brief teacher mini-lesson or be used as the instructional portion of the lesson. This is a time when students have an opportunity to work individually, with a partner, or in small groups depending on the purpose of the activity.

Activity Synthesis

During the activity synthesis, the teacher orchestrates some time for students to synthesize what they have learned. This time is used to ensure that all students have an opportunity to understand the mathematics of the activity and situate the new learning within students’ previous understanding.

Differentiation Supports

Every lesson contains suggestions for how to meet the various needs of learners. More detailed descriptions of the specific supports and design principles are provided for teachers in the Supporting All Learners section of the course. The available supports listed in the Lesson Teacher Guides include:

• Support for English Language Learners This support aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional routines and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons.

• Support for Students with Disabilities Each lesson support is carefully designed to maximize engagement and accessibility for all students. These supports empower students and capitalize on their existing strengths and abilities so they can participate meaningfully in rigorous mathematical content.

• Are You Ready For More? Most primary learning activities include an opportunity for differentiation for students ready for more of a challenge.

Every extension problem is made available to all students with the heading "Are You Ready for More?" These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just "the same thing again but with harder numbers."

Self Check Problems

Self Check items assess understanding of the specific TEKS breakout expectation that was addressed in the Activity and consist of only one question. Thus, they rarely assess the full rigor expectation of the TEKS, but fully assess the breakout objective of the TEKS from the primary lesson Activity. If students incorrectly answer the question, they are automatically moved into a set of Additional Resources that can be used to refresh student understanding. Thus, these questions are always multiple choice including the option for “I am not sure.”

Additional Resources

Access to the Additional Resources is provided automatically to students who incorrectly answer the Self Check question. Students who answer the question correctly may access the Additional Resources content by using the right-navigation Lesson Menu.

The content on the Additional Resources page consists of two sections. First, students have access to a set of explanations or examples that was the basis of the content in the activity. Then, in the Try It section, students have an opportunity to complete a question based on the examples they just read or similar to the Self Check question.

There are several options for how teachers can use the information on the Additional Resources page.

• The content may be used as a remediation review component for the activity. If students reach this page after incorrectly answering the Self Check question, they have the opportunity to retry the Self Check question or move onto the next learning activity when they have completed reading the Additional Resources page and completing the Try It question.
• Students may also be directed to this page if the teacher chooses to use it for direct instruction prior to the activity.
• And, finally, teachers may choose to use the Additional Resources page as an outline for building a mini-lesson that can be used prior to the activity or within a small group instructional setting.

Anticipated Misconceptions

Recommendations for potential areas of misunderstanding and how to address them are provided for lessons where appropriate.

Suggestions include prompts and guidance to support the teacher in modeling, explaining, and communicating the concept(s).

Videos

Videos that explain the thinking and procedures needed for specific problems within the content are provided for students to access as needed.

Response to Student Thinking

The Response to Student Thinking on the Teacher Guides for the Cool Down provides guidance on how teachers might make adjustments based on specific student responses to a Cool Down. Next day supports, such as providing students access to specific manipulatives or having students discuss their reasoning with a partner, are recommended for Cool Down responses that should be addressed while continuing on to the next lesson. Teachers are directed to appropriate prior grade level support for Cool Down responses that may need more attention.

Practice Problems

Each section in a unit includes an associated set of practice problems. Teachers may decide to assign practice problems for homework or for extra practice in class. It is up to teachers to decide which problems to assign (including assigning none at all).

Lesson Synthesis

The Teacher Guide for the Lesson Synthesis portion contains questions and activities teachers can use to solidify student understanding for the full lesson. Sample answers are provided for teachers.

Lesson activities pose questions to students so that one response leads to another. A lesson may begin by appealing to student interest or sparking their curiosity, then, the questions build in complexity and rigor as students receive feedback about their responses.

This Warm Up example from one of the first lessons in the course, illustrates this design.

The level of questioning grows from requiring students to explain the meaning of a one-variable equation to the meaning of a multivariable equation to creating their own equation and describing its meaning. In this way, students experience questioning at increasing levels of rigor.

The course continues to increase the levels of rigor as students move from one activity to another within the lessons.

In the activities that follow the Warm Up example provided earlier, students begin to apply more abstract, academic vocabulary to their learning and consider the reasonableness of their responses.

As the lesson continues, students move towards applying their ability to write equations in differing situations and scenarios.

This example from the Practice problems for this lesson show how expectations are beginning to approach the TEKS expectation to “write linear equations in two variables given a table of values, a graph, and a verbal description.” (Algebra 2(C))

The previous examples are from the first activities and lesson of the unit where student development has not yet achieved the full rigor of the TEKS. That lesson is the first step in the learning progression for the whole unit. Ultimately, students acquire the rigor of the TEKS after completion of all the lessons within the unit.

The next example, from the middle portion of the unit, shows students writing equations for more complex scenarios and also determining values for varying situations.

Notice that students are now meeting one portion of the TEKS expectation. They are now able to write linear equations in two variables given a verbal description.

In one of the lessons towards the end of the unit, students are asked to find information from tables, equations, and graphs. This information will then be used to write linear equations in two variables.

Not only does this example illustrate the increase in rigor aligned to the TEKS, but it reinforces another point. As mentioned in a previous section about the developing rigor within the activities, note how the questions begin by asking students scaffolded questions about slope and y-intercept, information needed to write a linear equation.

The unit concludes with a Unit Quiz and Project. Both of these assessments are designed to assess the full rigor of the TEKS. Recall that Algebra 2(C) expects students to be able to write linear equations in two variables given a table of values, a graph, and a verbal description.

This first sample from the Unit Quiz illustrates mastery of writing a linear equation in two variables given a verbal description.

In the second portion of the Unit 1 Project, students are provided with images of graphs. Then, with a partner, they determine key features of each graph such as slope and y-intercept, use the information to write an equation for the line, and finally create a real world scenario that could be represented by the graph.

Other questions in the quizzes and practice problems provide students with tables and ask them to write equations.

Over the course of the unit, students fulfill the rigor expectations of all parts of the TEKS A.2(C). They begin the learning progression by answering questions about a topic that might hold interest to them, ordering and planning a pizza party, and writing expressions and equations about the constraints in that scenario. Then, they apply the ability to write linear equations to other situations and more specifically to situations involving two variables when data is provided through graphs, tables of values, and verbal descriptions.

The first instance of each routine in a course includes more detailed guidance for how to successfully conduct the routine. Subsequent instances include more abbreviated guidance, so as not to unnecessarily inflate the word count of the teacher guide.

Activities using the routines are flagged for the teacher, which is helpful for lesson planning and for focusing the work of professional development.

Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team. The purpose of each MLR is described in the Supporting All Learners section of the course.

This course utilizes the following instructional routines:

• Analyze It
• Anticipate, Monitor, Select, Sequence, Connect
• Aspects of Mathematical Modeling
• Card Sort
• Construct It
• Draw It
• Estimation
• Extend It
• Fit It
• Graph It
• Math Talk
• MLR 1: Stronger and Clearer Each Time
• MLR 2: Collect and Display
• MLR 3: Clarify, Critique, Correct
• MLR 4: Information Gap Cards
• MLR 5: Co-Craft Questions
• MLR 6: Three Reads
• MLR 7: Compare and Connect
• MLR 8: Discussion Supports
• Notice and Wonder
• Poll the Class
• Take Turns
• Think Pair Share
• Which One Doesn’t Belong?

Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem. The teacher circulates as students are working and notes groups using different approaches. Groups or individuals are selected in a specific, recommended sequence to share their approach with the class, and finally the teacher leads a whole-class discussion to make connections and highlight important ideas.

The tasks selected for this course have been chosen based on their ability to develop students’ mathematical reasoning. The following list describes the purpose for the different types of tasks students are asked to complete in this course.

• Provide experience with a new context. Activities that give all students experience with a new context ensure that students are ready to make sense of the concrete before encountering the abstract.
• Introduce a new concept and associated language. Activities that introduce a new concept and associated language build on what students already know and ask them to notice or put words to something new.
• Introduce a new representation. Activities that introduce a new representation often present the new representation of a familiar idea first and ask students to interpret it. Where appropriate, new representations are connected to familiar representations or extended from familiar representations. Students are then given clear instructions on how to create such a representation as a tool for understanding or for solving problems. For subsequent activities and lessons, students are given opportunities to practice using these representations and to choose which representation to use for a particular problem.
• Formalize a definition of a term for an idea previously encountered informally. Activities that formalize a definition take a concept that students have already encountered through examples, and give it a more general definition.
• Identify and resolve common mistakes and misconceptions that people make. Activities that give students a chance to identify and resolve common mistakes and misconceptions usually present some incorrect work and ask students to identify it as such and explain what is incorrect about it. Students deepen their understanding of key mathematical concepts as they analyze and critique the reasoning of others.
• Practice using mathematical language. Activities that provide an opportunity to practice using mathematical language are focused on that as the primary goal rather than having a primarily mathematical learning goal. They are intended to give students a reason to use mathematical language to communicate. These frequently use the Info Gap instructional routine.
• Work toward mastery of a concept or procedure. Activities where students work toward mastery are included for topics where experience shows students often need some additional time to work with the ideas. Often these activities are marked as optional because no new mathematics is covered, so if a teacher were to skip them, no new topics would be missed.
• Provide an opportunity to apply mathematics to a modeling or other application problem. Activities that provide an opportunity to apply mathematics to a modeling or other application problem are most often found toward the end of a unit. Their purpose is to give students experience using mathematics to reason about a problem or situation that one might encounter naturally outside of a mathematics classroom.