High School - Gateway 2

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. There are instances in the materials where students are prompted to use multiple representations to further develop conceptual understanding. In addition, throughout the materials real-world context is used in order to give “concreteness” to abstract concepts, especially when introducing a new topic.

A few examples of the development of conceptual understanding related to specific standards are shown below:

• A-REI.A: In Algebra I Topic 8 students use algebra tiles to solve linear equations. Students check their work using tables and graphs. Algebra II Topic 13 uses graphing technology to introduce logarithmic equations as students examine graphs and tables of logarithmic functions in a real-world context before solving them analytically in the lesson Analytic Techniques.
• A-APR.B: Algebra I Topic 18 begins by using a garden to connect x-intercepts of a graph to zeros of a function. There are a series of questions that students work through in order to connect what is happening graphically with the factored form of a function. This is found again in Algebra II Topic 5 Higher Degree Polynomials. There are graphs and functions (standard and factored form) to show the relationship between zeros and factors of polynomials. Students construct polynomials given the zeros.
• N-RN.1: In Algebra I Topic 13 Laws of Exponents students make tables to see patterns in whole numbers raised to integer exponents, including zero and negative exponents. Students extend this idea to rational exponents by using positive integer exponents and radicals to understand rational exponents.
• F-LE.1: Algebra I Topic 14 begins by introducing students to linear and exponential growth. Students are introduced to different scenarios using fruit flies and fire ants. Materials use the contexts, along with graphs, tables, and functions, to develop students’ conceptual understanding around linear and exponential growth.
• G-SRT.2: Geometry Topic 12 contains applets throughout the lesson that allow students to manipulate triangles in order to further understand triangle similarity through student exploration and guided questioning in Advice for Instruction.

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for providing intentional opportunities for students to develop procedural skills. Within the lessons, students are provided with opportunities to develop procedural skills for solving problems. Guided Practice and More Practice sections are included within each lesson. These practice sections are often problems with no context and provide students the opportunity to practice procedural skills when called for by the standards.

Some highlights of development of procedural skills include the following:

• A-APR.1: In Algebra I Topic 16 students multiply binomials to determine the area of rectangles as well as simplify expressions. Students also simplify expressions using polynomial multiplication, addition, and subtraction in the More and Guided Practice sections. In Algebra II Topic 4 students multiply binomials to determine the volume of a rectangular prism as well as simplify expressions using addition, subtraction, multiplication, and division within the More Practice section.
• G-GPE.7: In Geometry Topic 14 students use the distance formula to compute the perimeter of a triangle as well as to determine if the diagonals of a rectangle bisect each other during More Practice. In Topic 21 students use the distance formula to compute the area of a rhombus as well as find the area and perimeter of a hexagon.
• F-BF.3: Students are given examples and applicable activities throughout both Algebra I (Topics 3, 5, 6, 12, 15, and 17) and Algebra II (Topics 3, 4, 9, 10, 12, 13, and 20). For example, Algebra II Topic 3 Making the Algebra-Geometry Connection presents several examples addressing this standard. Student Activity Sheet 4, as well as Practice and Assessment, provide opportunities for students to develop necessary skills.
• G-GPE.4: Geometry Topic 17 Polygons and Special Quadrilaterals explores this standard. Students are given ten examples to view. Example 1 provides definitions and an overview, and Examples 2-10 provide proofs for simple geometric theorems algebraically. Students are then given Student Activity Sheet 4 Practice Problems and Assessment to practice these skills.

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Students work with mathematical concepts within a real-world context. There are Topics for which the content of the Topic is framed by a real-world context through the Overview. The context used in the Overview is expanded upon throughout the lessons in the Topic.

Examples of students utilizing mathematical concepts and skills in engaging applications include:

• F-IF.B: The Algebra I Topic 4 Overview introduces students to a skateboarder and his motion. Students use the context of the skateboarder to match his motion to graphs and answer a variety of questions regarding the motion of the skateboarder throughout the lesson. The Topic also uses the idea of elevators and their movement to create graphs and understand various features of the graph. This Topic focuses on interpreting rates of change, and the entire Topic uses a variety of various contexts. In Topic 6 of Algebra I the Constructed Response Assessment problems are examples of applications where students are utilizing mathematical skills to answer various single- and multi-step problems. Students are asked to do things such as find the domain, find and interpret the y-intercept, find a parallel data set, and find the zero and interpret it in relation to the context of the problem. This standard is also found in Algebra II Topic 10 as students engage with the problems in the Topic using sets of data modeled by square root equations to further develop the mathematical skills for examining and identifying the features of graphs.
• G-SRT.8: Geometry Topic 15 Indirect Measurement begins with the student council of a school finding the height of the flagpole in the school courtyard. The lesson takes students through different strategies to find the height. The last question in the lesson uses an airplane to find the angle of depression. There are a number of application problems in the practice and assessment.
• S.ID.2: Algebra II Topic 19 has a number of application situations for students to use surveys and sampling. The Overview for the Topic uses real-world context by doing a survey to see if Americans believe life exists beyond earth. This application is used throughout in order to make sense of key terms. The questions in the Advice for Instruction provides the teacher with strategies to support students to work through the application in more meaningful ways. The entire Topic has a variety of application problems in Exploring, Practice, More Practice, Assessment, and Student Activity Sheets.

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for the three aspects of rigor being balanced with respect to the standards being addressed. The structure of the materials lends itself to balancing the three aspects of rigor.

Each Topic includes an Overview, Exploring, Practice, Assessment, and Student Activity Sheets.

• The Topic Overviews provide a focal point for students to begin thinking about the Topic. They allow for students to relate the topic to a real-world application and/or prior knowledge. This gives students an opportunity to develop conceptual understanding through applications and/or prior knowledge. For example, in Algebra II Topic 4 the teacher is provided with the pieces of a puzzle. In the opening to the lesson teachers state “suppose you buy 100 pencils for $25.” The teacher is then presented with several framing questions “What is the cost of each pencil? How do you know? What operation did you perform to find the answer? If you bought x pencils for $25, what expression would represent the cost of each pencil?”
• The Exploring section focuses on developing conceptual understanding, in context and/or by using applets. Students are given the tools to build their procedural skills throughout as algorithmic steps are connected to the concepts in this section.
• Practice has Guided Practice and More Practice for students. There are a variety of types of problems (multiple choice, multiple select, true or false, etc) with a focus on conceptual understanding and procedural skills. Students can get hints and immediate feedback if their answer is correct. If it is incorrect, students receive a statement/question to help direct their thinking.
• Assessment has two parts, Automatically Scored and Constructed Response. Automatically Scored includes Multiple Choice and Short Answer. This section has questions that require conceptual knowledge, procedural skills, and application of the Topic.
• Student Activity Sheets follow the online instruction but include additional procedural skill and application problems.

In addition to this, there are MARS tasks throughout that focus on conceptual understanding and application.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• Algebra I Topic 5 Moving Beyond Sloping Intercept (S.ID.7) has students study data from a table of a skateboarder and their distance traveled during a set of skateboard drills. They use the data table to match the graph of the motion detector data to the path created by moving the computerized skater on the app provided. Students discuss the two parameters necessary to match the graph and develop understanding around steepness (slope) of the line and the constant (intercept). Students use this knowledge to do more procedural skills around standard form and point-slope form. Throughout the Topic, students are given real-world context to explore all concepts in this Topic.
• Geometry Topic 17 Polygons and Special Quadrilaterals (G-GPE.4) includes examples that review key concepts from previous topics and subtopics. In the topic, the meaning of coordinate proof is given and then stepped through the idea of special quadrilaterals. Students have a series of questions posed to answer (developing conceptual understanding), examples of the process to do a coordinate proof (developing procedural skills), and more questions to check their understanding throughout the lesson on Coordinate Proofs. Students then work in small groups to write a coordinate proof showing that diagonals of a rhombus are perpendicular. After completing these things, students can do the Guided Practice and More Practice to master the skills just learned.
• Algebra II Topic 2 Understanding Inverse Relations (F-BF.4.a) begins with a real-world context to introduce the idea of inverse relations. Throughout the topic, students are given contextual situations to bridge the idea of functions and their inverses together. The use of tables, graphs, and equations are all used throughout in order for students to understand the idea of inverse. Students are also given the opportunity to compare a function and its inverse as well as practice the skill of finding an inverse from a function. Students are also given real-world context to answer questions which require a conceptual understanding of inverse relations.

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). The instructional materials develop both MP1 and MP6 to the full extent of the MPs. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. Teachers are given guidance on materials in the Advice for Instruction provided in each Topic. Topics that include a MARS Task list MPs used within the task in the Advice for Instruction. The Advice for Instruction also provides teachers with guidance to foster discussion throughout the materials. This discussion often stresses accurate vocabulary used to increase precision of mathematical language. Emphasis is placed on using units of measure and labeling axes throughout the series. Making sense of answers within the context of a problem is also emphasized. Students persevere in problem solving in lessons through the many real-world application scenarios.

• In Geometry Topic 14 Block 5 MARS task students make sense of problems and persevere in solving them (MP1) to determine angles and lengths of pieces of wood of a wooden garden chair using similarity and Pythagorean Theorem (G-SRT.8).
• In Algebra I Topic 1 Student Activity Sheet 1 the first question has students look at three linear graphs and asks them “How are the three graphs similar? How are the three graphs different?” The students analyze the graphs, make conjectures, and plan their strategy. As the teacher directs the lesson, the teacher is told to encourage students’ thinking and emphasize the importance of precise communication by helping students use precise language (MP6) such as slope, how steep, increasing, decreasing, and line versus segment (F-IF.6).
• Algebra I Topic 8 Student Activity Sheet 4 Question 31 introduces a babysitting situation through Maggie who charges $15 an hour for babysitting. The students look for the possible numbers of hours she can babysit to make enough money for the bag she wants to buy without having to pay taxes on her earnings. In this problem, students are asked to first understand the meaning of the problem, analyze the information, make a conjecture, and try to find an answer. Students need to check their answers and see if their answer makes sense (MP1) (A-CED.3).
• In Algebra II Topic 18 students are presented with an overarching scenario. In this scenario, Mr. Jones witnesses a crime and gives a witness description. In the scenario, students reason through multiple probability concepts for each portion of Mr. Jones’ account. “How likely is it that Rob is the bad guy? Mr. Jones indicated that the license plate had 6 non-repeating letters.” In this case, the students (playing the role of Rob’s attorney) will have to use the fundamental counting principle to determine how likely it is to have a license plate with non-repeating letters. Students then use basic probability concepts to determine probabilities of events occurring together and/or occurring given that another event occurred first. Lastly, students use the normal distribution to find the percentage of people in Arresta that have a height of 6 feet or more. Students persevere through multiple nuances of the problem to finally determine their solution to whether the jury comes back with a guilty or not guilty verdict (MP1) (S-ID.4).
• In Algebra II Topic 1 Exploring Arithmetic Sequences and Series students are first presented with an auditorium seating scenario. In this scenario, students are told that the first row has 20 seats and each subsequent row has 3 seats more than the row in front of it. Students are given a table to complete in Student Activity Sheet 2 question 3. From here, the students are asked to determine the constant difference of the linear function that models this situation and also determine the domain and range. The student then writes a precise arithmetic sequence that will help them predict the 21st term in the sequence (MP6) (A-SEE.4).

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to reason abstractly and quantitatively, and students are expected to construct viable arguments by justifying along with a few opportunities to critique the reasoning of others in each course. Support and guidance is provided for teachers in the Advice for Instruction for each lesson to assist teacher development of these MPs although the MPs are not explicitly listed in the Advice for Instruction with the exception of the MARS Tasks. MARS Tasks identify specific MPs used within each task in the Advice for Instruction.

• In Algebra I Topic 3 Functions, Exploring Modeling with Functions begins with the real-world application of a soccer team selling roses as a fundraiser. Students are asked at the beginning of the lesson “What a successful project looks like” and “How will the factors in the list influence the success of the project?” Students think about successful money-making ventures and discuss the factors that make the ventures successful. Students are then given amounts and pricings for two shops to determine which shop would be the best deal for them. Students are then asked to determine costs for ordering a specific numbers of roses from each shop and explaining their reasoning to the class. In this scenario students formulate their own reasoning and explain the evidence and/or prompts that led to their reasoning, illustrating both MP2 and MP3 (F-IF.1).
• In the Geometry Topic 15 MARS Task students reason abstractly (MP2) about whether a particular triangle is a right triangle by finding the side lengths of neighboring triangles and then using the Pythagorean Theorem to draw a conclusion. Students must also explain how they decided the triangle was a right triangle (MP3) (G-SRT.8, G-SRT.5).
• One example where students are given the opportunity to critique the reasoning of others (MP3) can be found in Algebra II Topic 21 Student Activity Sheet 5 Problem 8. Students are shown a graph of a cotangent function along with a student’s description of the function. Students are directed to critique the student’s description of the graph of the cotangent function.

The instructional materials reviewed for Agile Mind Traditional series meet the expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5).

The materials fully develop MPs 4 and 5 as students build upon prior knowledge to solve problems, create and use models within many lessons, and choose and use appropriate tools strategically across the courses. The materials pose problems connected to previous concepts and a variety of real-world contexts. Students are provided meaningful real-world problems in which to model with mathematics and use tools.

Examples where students model with mathematics include:

• In Algebra I, Topic 6, Exploring, students examine a graph of shoe size versus height of a person. Students create a function to model the information provided in the graph. Students were able to use previous knowledge to show a correlation between each of the figures and thus draw a trend line. The students formulated the problem (with some help from the book), then were asked to compute the trend line that would correlate that data and lastly students checked their work and reported out. This process is present for multiple scenarios within the guided practice and more practice sections. 
• In Algebra I, Topic 15, MARS Task: Functions, students “model each of two subsets of a set of points on a scatterplot. Students must go beyond simple visual inspection of a graph to sort the set into two subsets and justify their sorting by applying their knowledge of fundamental characteristics of different function families.” In this activity, students must write a linear function to represent the scatterplot and determine a non-linear model for the rest of the points in the scatterplot. Students must verify their solutions with their partner and report their findings.
• In Algebra II, Topic 21, Student Activity Sheet 4, students are given data representing the number of hours of daylight in Tallahassee, Florida for the year 1998. Students are asked to “Make a scatterplot of these data using your graphing calculator. What type of function do you think would model these data? Do you think these data are periodic?” Later, students are asked, “What trigonometric function would you use to model the data?” and then “What is the period of the graph?” “What is the amplitude of the sinusoidal graph?” “Is the graph shifted horizontally and/or vertically from the parent function y = sin x? If so, by how much is it shifted?” “Transform the parent function, y = sin x, to fit the data.” Finally, students are asked to “Use your model to find the days when Tallahassee had more than 12 hours of daylight.” Through this set of problems, students apply prior knowledge to new problems, identify important relationships and map relationships with tables, diagrams, graphs, rules, draw conclusions as they pertain to a situation, create, and use models.

Examples where students choose and use appropriate tools strategically include:

• In Geometry, Topic 8, Block 4, Advice for Instruction states, “Encourage students to use tools to help them make sense of the problem of dividing a triangle into sixths. Make Patty Paper, rulers, protractors, dynamic geometry software (optional) and scissors available to students. Give students enough time to really try to answer this question.”
• In Geometry, Topic 11, Block 1, Advice for Instruction, students follow a paper-and-pencil activity with a construction activity in which they are to “use tools of their choice.” Later in the same block, in Technology tip, pages 6-7, Exploring “Congruent segment and angle bisector constructions,” students choose between compass and straightedge or an online construction tool.
• In Algebra II, Topic 20, Student Activity Sheet 2, Question 10, students design and carry out a simulation. They choose from a variety of tools to carry out the simulation, including a coin, a random number table, a random number generator, or a statistical software package.

The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, support is present for the intentional development of seeing structure and generalizing.

• In Geometry Topic 13 Geometric Mean students define the geometric mean. Students receive instruction on the connection between the geometric mean and right triangles. They work a puzzle to apply what they have learned and determine the difference between the arithmetic mean and the geometric mean. They are asked to make a general statement about when the arithmetic mean and the geometric mean of two numbers are the same. This provides students with the opportunity to see structure and generalize to a larger idea (MP7).
• In Algebra II Topic 19 Conditional Probability and Independence students are asked to determine the conditional probability that a British ship is armed given that it appears armed. This question arises from British ships’ need to deter pirates. They then use their conclusions from Exploring Conditional Probability to help determine calculations from the French ship data to show that the event of selecting an armed French ship is not dependent on the event of selecting a French ship that appears to be armed. This allows the students to use requisite knowledge to make generalizations from the sample data (French Ships) to the population (British Ships) (MP7).
• In Geometry Topic 21 students make estimates about the area of the model and then refine that estimate by using different scales for estimating the area through different geometric figures through repeated reasoning (MP8).
• In the Algebra I Topic 4 MARS Task Differences students look for patterns and express regularity in repeated reasoning by completing a table (MP8). By completing the table, students are able to notice patterns in a sequence and determine the 7th and 8th term in the sequence. Students must also use their completed table, along with a table of expressions, to look for patterns in both tables to determine the coefficients of the expressions given in the second table.