Unit Sequence and Resources

A unit-level teacher guide begins each unit of instruction. On this page, teachers find the learning targets for the unit along with a short list of the prerequisite skills tested in the Overview and Readiness assessment. Also on this page, a short introductory video provides a big picture look at the content for the unit.

Next, teachers have access to information for each specific section of the unit. This includes an overview video of the lessons within the section and a list of the Texas Essential Knowledge and Skills [TEKS] taught in each lesson. Note the list includes both process and content TEKS standards.

At the bottom of this page, Unit Resources give teachers an idea of where the unit fits within a student’s overall mathematical learning journey. A link to the Texas Education Agency’s TEKS Vertical Alignment Chart Grades 5 through Algebra 1 and Algebra 2 is provided along with a video that highlights the TEKS of the unit within the Vertical Alignment Chart. This section gives teachers guidance on additional prerequisite needs their students may face. To support teachers with addressing student prerequisite challenges, specific lessons from the OpenStax PreAlgebra textbook have been curated and provided. These PreAlgebra sections specifically address the TEKS from grades 5 - 8 identified by the Vertical Alignment Chart and can be shared directly with students or used in small group settings.

The last set of resources listed includes the assessments that check student mastery of the unit level learning targets: Unit Project, STAAR Review and blueprint, Unit Quiz, and Unit Wrap Up that contains the student self-assessment tool.

Using the right navigation menu, teachers can access additional information including a descriptive narrative of the learning progression for the unit, Family Support Materials, the English Language Proficiency Standards [ELPS] contained within the unit, and suggestions for building students’ workplace tools in a section titled Building Character.

More detailed information about each section’s lessons and learning progression is detailed on the Section Overview page. This page also provides teachers with another access to the section overview video. Typically sections contain 4 - 6 lessons.

All process standards are addressed within each lesson and unit, however, specific process standards have been included in the TEKS table to identify the most prominent ways students might learn the mathematical concepts and display their understanding. While the process standards contained within the list of TEKS is not a complete list of the ways in which students acquire and demonstrate mathematical understanding in the unit or lesson, guidance is provided within the Teacher Guides so teachers may use all of the process standards during instruction for each lesson.

The culminating Unit Project provides an opportunity for students to demonstrate their mastery of these process standards. These real-world tasks are designed for students to engage in the problem solving process where they select the needed tools and communicate their thinking and justifications using multiple representations. In the Teacher Guides for each project, teachers are prompted to ask students to reflect on the assumptions made before and during the problem solving process as well as the rationale for choosing a solution method. Opportunities for students to engage in gallery walks or make presentations permit a chance for them to exhibit mastery of precise mathematical language.

The ELPS section of the Unit Teacher Guide provides a core sampling of teaching strategies that align to the English Language Proficiency Standards. The alignments are not intended to be all inclusive. Rather, they demonstrate a sampling of ways to support English language learners within the mathematics classroom. Each lesson offers ELPS aligned to at least three different domains and across the entire unit, all five ELPS domains are represented and aligned. You can pull these strategies for learning (ELPS 1), listening (ELPS 2), speaking (ELPS 3), reading (ELPS 4), and writing (ELPS 5) for any of the lessons. To see examples of teaching routines for each standard, please view the Supporting All Learners section of this course.

This strand of mathematical proficiency describes a student’s ability to connect ideas across topics and use the connections to make sense of the mathematics. “A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes.” (p.119)

We see conceptual understanding reflected in TEKS such as:

• A.1(F) analyze mathematical relationships to connect and communicate mathematical ideas.
• A.7(B) describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions.

In this course, students have multiple opportunities to develop or demonstrate conceptual understanding.

In the provided image, notice how the activity starts by drawing students’ attention to the intercepts for the graph of the function. This lesson goes on to develop students’ understanding of the connections between the intercepts, the linear factors of the quadratic, and the different forms in which the function can be expressed.

The connections students make between these different representations help them determine when it is most efficient to use each equation, why different algebraic representations are preferred in specific situations, and how to graph quadratic functions efficiently (fluency).

The focus of this mathematical proficiency strand extends beyond basic computation and estimation to include mental strategies and applying procedures “flexibly, accurately, and efficiently.” (p. 121)

Procedural fluency should not be in competition with conceptual understanding, but the strands should reinforce each other.

Referring to the example displayed in the conceptual understanding section, when students understand the connection between the intercepts, graph, and algebraic expressions, they understand how to quickly graph a parabola no matter what form of the equation they are given. Their thinking can flexibly move between standard form and factored form as they use each to harvest different key points needed for graphing.

This strand of mathematical proficiency encapsulates the essence of mathematics in the real world. Many times students are repeatedly given the same types of problems that can be solved using the same method, but real-world problems are not like this. To gain strategic competence, students should encounter unique problem situations that are new or unfamiliar to them. The goal is to force them to analyze the problem and then try different problem solving strategies, refine the approach, and try again.

Examine the example of the Unit 5 project overview provided to students. Note how the task will require students to gather additional real world data and carefully determine which model to use for their predictions.

This project scenario not a typical regression-style problem. In the course of this project, students will encounter huge, “messy” numbers and data that does not indicate a specifically “correct” approach.

Upon completion of the project, students are expected to justify their model as they explain the assumptions they made, which data they chose to select, and what the implications for their prediction models might hold.

“In mathematics, adaptive reasoning is the glue that holds everything together.” (p. 129) Students possessing adaptive reasoning skills are able to reason for themselves if their answers or thinking is valid and correct. This component of mathematical proficiency means students have an ability to organize facts, skills, procedures, concepts, strategies, and methods into one logical whole way of thinking.

Opportunities for students to build on their understanding of representations is one way to develop adaptive reasoning and logical thinking. This course often asks students to display their thinking or products. Then, with guidance from the teacher, the use of annotations can help them build the connections (conceptual understanding) that form a logical set of verification strategies.

Another way to develop adaptive reasoning is the consistent use of tasks that require explaining and justifying their solution methods.

This example not only asks students to find the answer, but to also explain their reasoning, a common expectation in this course.

Every mathematics teacher strives to inspire productive disposition and the ability for their students to “see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.” (p. 131)

In this course there are a number of ways that productive disposition is addressed and developed. The use of goal setting tools and Character Building suggestions help teachers develop students’ self efficacy and feeling of success. The students’ experiences with these tools invite them to develop a mathematical identity and see themselves as mathematicians.

Another way to develop students’ positive attitude towards mathematics and help them see themselves as mathematicians is the way the problems are organized within the units. The first lessons in all the units seek to provide students with opportunities for success that gradually build to complex projects at the end of the unit.

Examine this example from the first lesson of unit 1 where students are asked to plan a pizza party for their classmates. The questions start by addressing a situation familiar to the students and seeks to appeal to their interests.

As the questions build through the activity and lesson, they begin to transition into thinking about the party plan in more abstract, mathematical terms. The next two samples illustrate the development in student thinking from words and ideas towards mathematical expressions.

Activities and questions like this not only develop students’ productive disposition, but they also support and provide a foundation for the other strands of mathematical proficiency.

Notice the questions teachers are prompted to ask students in the Lesson Synthesis for this activity.

All five strands of mathematical proficiency are evident in the lesson’s closing questions. The ability to gather information and consider alternative approaches for gathering the data exemplifies strategic competence. Considering the impact of different data gathering methods indicates adaptive reasoning. Conceptual understanding and procedural fluency are demonstrated as students make their estimations, consider how the estimation might change with different data, and apply the idea of constraints to other situations. Being able to see the value of expressions, equations, and inequalities and their usefulness pulls all the pieces together and clearly depicts productive disposition.

The culminating project for the unit takes all of the building blocks from the first lesson and develops student thinking so they are ready to demonstrate more complex mathematical models that involve expressions, equations, and constraints to include linear situations with two variables.

In the example of the Unit 1 project, notice that students are again asked to consider alternative thinking strategies when discussing the problems with their partner. They must also demonstrate fluency in their ability to gather data from various graphs and find the matching equations. And, finally, they write real-world scenarios for one of the graphs and explain how characteristics of the graph relate to their equation and the real-world context.

This culminating demonstration of mastery had its beginnings in a set of questions that asked students about their favorite pizza.

As mentioned in the research foundations of this course, achieving goals helps develop students’ self efficacy and assessments are one way to measure progress towards the goals.

Teachers should share with students the learning targets and learning progression for the unit before instruction begins. Then, to help students develop individual learning goals, use the Unit Overview and Readiness pre-assessment to provide a benchmark of student understanding of prior knowledge.

Guide students to develop a goal for each section or lesson so they have smaller, proximal goals that will help them achieve the longer-term unit goal. To assist with this process, access the Student Self-Assessment tool in the Unit Wrap Up.

Notice how the tool provides students with the opportunity to select “not yet” and “almost.” These choices reinforce that the learning is ongoing and a true progression towards mastery.

As students participate in the learning activities, encourage them to read the feedback provided for each question. In the primary learning activities, feedback may reinforce a student’s thinking process or provide an alternative method they had not considered.

The Practice questions are another place where students are encouraged to persist and focus on the learning target rather than a grade. Each of these questions provides students with multiple attempts. When an incorrect answer is submitted, the course provides feedback that directs students to the prior activity they can return to and review. Teachers should not only reinforce and support students through such a revision process, but help them to understand how it moves their understanding along the unit’s learning progression.

For more information on goal setting and growth mindset in this course, please access the Motivation in the Classroom section of Research in Practice.

Another feature of the materials in this course includes suggestions on Building Character. Throughout each unit, teachers are provided with resources to promote students’ character development as it relates to mathematics identity and self-efficacy. These resources come from Character Lab and their Playbooks.

The image from Unit 5 provides an example of the type of content provided to teachers in these sections.

The Building Character topics are provided for teachers in the Unit Level Guide with additional supporting activities embedded within the Lesson Teacher Guides. The list of topics in each unit include:

• Unit 1 - Social Intelligence
• Unit 2 - Growth Mindset
• Unit 3 - Purpose
• Unit 4 - Creativity
• Unit 5 - Grit
• Unit 6 - Judgment
• Unit 7 - Proactivity
• Unit 8 - Curiosity
• Unit 9 - Intellectual Humility

Topics such as social intelligence and judgment are crucial for helping students understand how to work in cooperative groups and strengthen their mathematical voice in classroom discussions, while creativity and curiosity are the foundation of problem solving. And research has shown that having a growth mindset and grit not only supports student self efficacy but it also plays a role in developing a student’s mathematical identity.

Students will also see embedded information sprinkled throughout the course on their primary activity pages.

Creating a team of support for students is another way to develop their self-efficacy. Including parents and guardians in a students’ learning journey ensures the student can find support at school and at home. Through the right navigation menu of the Unit Teacher Guide, teachers can access the Family Support Materials for this course.

The Family Support Materials are aligned to each unit of study and divided into three sections. The first section summarizes all of the content in the unit. Then, within this section, a real-world example is often provided. Whenever possible, the information includes tables and graphs to fully illustrate all the different ways that students will encounter the information.

In the next part, the Apply section, parents are asked to complete a task with their child. Their task is usually embedded in another real-world scenario and includes questions with the answers. Some of the answers include explanations, as needed.

Finally, in the last section, parents are offered an opportunity to review what they have learned or continue learning with their students through videos aligned to specific lessons in the unit. If the content is shared with parents digitally via emails or websites, the videos are linked through YouTube and can support the selection of closed captioning in a preferred language. In addition, the documents are available in both English and Spanish.

Teachers may also wish to display or project the provided content for Family Math Nights, Parent Open House, etc.

Creating a team of support for students is another way to develop their self-efficacy.

These resources are located in each unit’s Section Overviews for the teachers and in the Overview and Readiness sections for students.

Applicable information is dispersed throughout the student pages at the lesson level so they are intermittently reminded of how the Algebra 1 content plays a part in our lives.

The following sample appears at the end of Activity 2.9.2: Graphing Inequalities. Notice how it ties the algebra content from this specific lesson to the overall “Why Should I Care?” topic for the unit.