High School - Gateway 2

The instructional materials reviewed for Agile Mind Integrated series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 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 contexts provide “concreteness” for abstract concepts, especially when introducing a new topic.

Examples of the development of conceptual understanding related to specific standards are shown below:

• N-Q.1: In Mathematics I, Topic 2, Student Activity Sheet 1, Problem 5, students use a “different type of representation to solve the problem.” Joachim states, “For every 2 yards of flowerbed, you can plant 3 peonies.” Students “describe how [they] can use Joachim’s representation to solve the problem.” Students use multiple representations to demonstrate the meaning of the problem and make connections among the representations.
• F-BF.1a: In Mathematics I, Topic 13, Student Activity Sheets 1, Problem 6, students use geometric patterns in mosaics to make connections between geometric and algebraic representations. Students build conceptual understanding by connecting a real world situation, a pattern, and a sequence. Finally, students consider, “How many tiles will be in the tenth mosaic? How do you know?” Students draw out mosaics, use manipulatives, and apply the sequence to connect the different representations of the pattern.
• G-SRT.4: In Mathematics II, Topic 14, Student Activity Sheet 2, Problem 7, students prove the Triangle Proportionality Theorem. Students demonstrate their understanding of various parts of the proof and of the purpose of the proof in more than one way. In Problem 8, students connect their proof to “parallel lines that cut two transversals.” 
• S-ID.9: In Mathematics I, Topic 6, Student Activity Sheet 3, Problem 25, students examine a scatterplot showing a positive association between the number of ice cream cones sold and the number of shark attacks. Students consider, “Based on this scatterplot, could you make a reasonable prediction of the number of shark attacks if you knew the number of ice cream cones sold,” and, “Does this mean that eating ice cream causes sharks to attack?” Students differentiate between correlation and causation by considering the similarities and differences between the two questions.

The instructional materials reviewed for Agile Mind Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. 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 often contain problems with no context and provide students the opportunity to practice procedural skills when called for by the standards.

Examples of the development of procedural skills related to specific standards include:

• A-APR.1: In Mathematics II, Topic 3, students perform operations on polynomials. Students use area models and algebra tiles in the Exploring section and then practice polynomial multiplication in the Guided Practice, More Practice, and Student Activity Sheet sections. 
• A-REI.2: In Mathematics III, Topic 9, Exploring, Solution methods #1-11, students solve rational equations. Students independently practice in Guided Practice #11, More Practice #1-12, and Student Activity Sheet 2.
• A-REI.7: In Mathematics II, Topic 7, Student Activity Sheet 4 #11, students algebraically solve a system of equations with three variables. Students consider connections to the geometric representations of the system. 
• A-SSE.3: In Mathematics II, Topic 4, MARS Task, students develop procedural fluency by determining whether statements about quadratics are true or false. Students demonstrate algebraic skills in several ways: graphing, factoring, completing the square, and through transformations.
• F-IF.7b: In Mathematics I, Topic 9, Student Activity Sheet 1, students consider a scenario about a skater’s distance from a cone as he skates by at a constant speed.  Students sketch a graph showing the skater’s distance from the cone versus time and use the graph to determine its algebraic representation.
• G-SRT.6: In Mathematics III, Topic 18, Student Activity Sheet 1, students practice using trigonometry ratios to solve problems involving a waterslide.
• S-ID.7: In Mathematics I, Topic 6, Student Activity Sheet 2, students find the equation of the graph of a line. Later in the worksheet, students create equations to determine the equation for a trend line.

The instructional materials reviewed for Agile Mind Integrated series meet expectations for supporting 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. Within every topic, the Overview introduces the topic content with a real-world scenario and problems that are based on these real-world scenarios. Students begin each topic with a context to understand the mathematical idea.

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

• A-SSE.3: In Mathematics 2, Topic 5, Student Activity Sheet 2, students find different forms of a quadratic to answer questions about a dog kennel. Students interpret key features of a graph to determine the maximum area possible.
• A-REI.6: In Mathematics I, Topic 10, Student Activity Sheet 2, students use data to create a set of equations and then interpret the data based on whether the mowing season is almost over or has just begun.  Students consider, “It would be useful to know how many weeks it would take for the two options to result in the same total cost. How can Desmond figure that out?” Students also determine at what point advertising pays for itself.
• A-REI.10: In Mathematics I, Topic 3, More Practice, students independently construct a graph from a routine context. Students also engage in a non-routine problem to construct a general rule. 
• F-IF.B: In Mathematics I, Topic 4, students engage in different real-world scenarios involving rates of change: elevators, draining a pool, and selling baseball caps. Students make predictions and compare different rates in graphs and tables.  Students build understanding from different contexts and different representations. In Student Activity Sheet 2, students interpret a graph in the context of a descending and ascending elevator.
• G-CO.2: In Mathematics II, Topic 28, Student Activity Sheet 3, students compare the volumes of two triangular prisms with congruent bases but different heights.
• G-SRT.6: In Mathematics II, Topic 19, More Practice, students independently apply prior knowledge from isosceles triangles and the Pythagorean theorem to new applications.

The instructional materials reviewed for Agile Mind Integrated series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

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

• The Topic Overviews provide a focal point for students to begin thinking about the topic. Materials 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 Mathematics I, Topic 6, students interpret a scatter plot relating height and shoe size.  Students consider positive and negative correlation and a trend line. The teacher is then presented with several framing questions “Does the size of Tommy's shoes surprise you? Why or why not? How would you expect a shorter person's shoe size to compare to Tommy's shoe size? What would you expect to be true about a person with a shoe size that is larger than size 14? How could you confirm your predictions?”
• 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.
• The Practice section includes 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:

• A-REI.3: In Mathematics II, Topic 8, Exploring, students draw from prior conceptual understanding of linear functions to solve linear equations and systems of linear equations.  Students engage in interactive visual models to solve equations that supports the conceptual aspect of solving linear equations while gaining procedural fluency with algebra.
• F-IF.5: In Mathematics I, Topic 3, Student Activity Sheet 2, students use a function for the number of 1-square foot tiles in the border of a square pool to consider whether “the relationship between the length of the pool and the number of tiles in the border is a function?”  Students then interpret the meaning of f(10) in the context of the application to build conceptual understanding of a function. Later, in Student Activity Sheet 3, students determine the cost of roses ordered from several shops (procedural fluency) in order to recommend which shop a soccer team should use as their supplier (application). 
• F-IF.6: In Mathematics I, Topic 4, Exploring, Constant Rates, students develop a conceptual understanding of constant rates through real-world scenarios and interactive lessons by sketching graphs and making predictions. Later, students develop procedural fluency by writing and calculating rates.
• S-ID.6a: In Mathematics II, Topic 18, students start with regression and move into exponential functions while building on their prior knowledge about linear functions.  In Student Activity Sheet 3, students examine flu data to determine a rule and construct a scatterplot. Students complete a table of the flu infection rates (procedural fluency). Later, students extrapolate values for the future of the flu epidemic (application).

The instructional materials reviewed for Agile Mind Integrated series meet expectations that the materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards.

Examples where students make sense of problems and persevere in solving them include:

• In Mathematics II, Topic 6, Overview, students examine different rectangular dog kennels with equivalent perimeters. The teacher facilitates the task by having students draw one rectangular kennel that meets the requirements. The students then decide whether other solutions exist and persevere in finding them. The teacher aggregates the solutions so students can model all solutions with a quadratic expression. 
• In Mathematics I, Topic 13, Student Activity Sheet 1, students find the next three terms of a given sequence and explain their method. Students then determine whether the sequence is arithmetic or geometric. Students must persevere when the pattern is not obvious or their method for determining the type of sequence needs refinement.
• In Mathematics II, Topic 6, Exploring, students use a graph and table of revenues for two different smartphone manufacturers to consider “if these trends continue, how will the revenue of the two manufacturers compare in future years? Explain how you know.” Although students are provided with the data, they make sense of the problem by predicting future revenues and justifying that prediction. They persevere in solving the problem by exploring beyond just the given data.

Examples where students attend to precision include:

• In Mathematics II, Topic 2, MARS Task: Graphs, students comparing linear and quadratic functions. Students determine points of intersection by examining two graphs and then verify the coordinates algebraically. Students generate another graph and use algebra to determine its intersection point(s) with the original graphs. Students must identify precise coordinates, accurately set up equations, and maintain exact values while solving to test whether the two methods produce equivalent points of intersection.
• In Mathematics III, Topic 15, Student Activity Sheet 3, students solve the equation 2+3ln(x)=4-2ln(x) algebraically and verify the solution using a table and graph. Students compare an exact solution from algebra with an approximate solution from the graph. They grapple with precision in determining how different solutions can be numerically different while still being considered equivalent.
• In Mathematics II, Topic 13, Exploring, students draw a scalene triangle and dilate it by a factor of 3 around a given point. Students measure corresponding angles and sides of their triangles using rulers and protractors. Students must be precise in their transformation and measurements to verify that angle measures are preserved and lengths are three times longer.

The instructional materials reviewed for Agile Mind Integrated series meet expectations that the materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards.

Examples where students reason abstractly and quantitatively include:

• In Mathematics II, Topic 21, Overview, students are given a real-world problem of determining the maximum height for a truck to travel through a tunnel. Students are shown how a coordinate grid can be used to determine the maximum height of a vehicle. This scenario provides students an opportunity to reason abstractly as they use both the coordinate grid and the Pythagorean Theorem to solve the problem. In addition, students use the given information about the radius of the tunnel and the width of the truck to quantitatively determine the maximum height of a vehicle that can pass through the tunnel.
• In Mathematics I, Topic 10, Student Activity Sheet 1, students are provided constraints on Desmond’s mowing operations. Students must reason abstractly about the information provided to create a system of linear equations. Students must also reason abstractly and quantitatively about their solutions to determine which option Desmond should choose based on the amount of time left in the mowing season. 
• In Mathematics III, Topic 3, Student Activity Sheet 2, students are asked, “In a random survey of 100 adults, 50% responded ‘yes’ to the question ‘Should parents monitor teens’ cell phone usage?’” Then students are asked, “Describe a simulation to develop a margin of error for this study.” and “Execute the simulation. What is the margin of error? What is the interval in which we can be confident our true population value lies?” Students make sense of the data they collected and represent the data symbolically. Students are also expected to determine the confidence interval and provide justification. 

Examples where students construct viable arguments and critique the reasoning of others include:

• In Mathematics III, Topic 3, Overview, students are presented with a report stating that “Most Americans think there is intelligent life on other planets” and provides students with additional data from the report. Students are then asked if the conclusion drawn in the article is justified. This provides students with an opportunity to share and justify their own reasoning/argument and to support their reasoning based on the information given in the report. As students share their own justification, they are also provided with the opportunity to critique the reasoning of others. 
• In Mathematics I, Topic 14, Student Activity Sheet 1, students are given a picture of the golden rectangle with the Fibonacci spiral in the center as well as the Sierpinski's triangle. Students are asked, “Take a few minutes to explore these two patterns. Then, in your own words, explain how each pattern is generated.” Students interpret the images, describe the patterns, justify their conclusions, analyze the problem using definitions, and establish results in constructing their arguments.
• In Mathematics III, Topic 18, Student Activity Sheet 4, students are given information from two fictitious students, Albert and Sonya: “An isosceles triangle has an area of 4 square feet. The base is twice as long as each of the two legs. Find the three side lengths of the triangle.” They are then asked to “Explain what Sonya means by this” concerning a comment that there is “no such triangle.” Students examine Sonya’s comments and decide if they agree or disagree and then explain why or why not.

The instructional materials reviewed for Agile Mind Integrated series meet expectations that the materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. 

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 Mathematics 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 are able to use previous knowledge to show a correlation between each of the figures and thus draw a trend line. The students formulate the problem (with some help from the book), then are asked to compute the trend line that would correlate that data and lastly students check their work and report out. This process is present for multiple scenarios within the Guided Practice and More Practice sections. 
• In Mathematics III, Topic 16, 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? 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; map relationships with tables, diagrams, graphs, and rules; draw conclusions as they pertain to a situation; and create and use models.
• In Mathematics II, Topic 2, 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.

Examples where students choose and use appropriate tools strategically include:

• In Mathematics I, Topic 18, 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 Mathematics II, Topic 11, 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 Mathematics III, Topic 3, 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 Agile Mind Integrated series meet expectations that the materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. The majority of the time MP7 and MP8 are used to enrich the mathematical content. Across the series, there is intentional development of MP7 and MP8 that reaches the full intent of the math practices.

Examples where students look for and make use of structure include:

• In Mathematics III, Topic 7, Exploring- Comparing Direct and Inverse Variation and Exploring- Investigating Inverse Variation, students explore and compare the differences of direct and inverse variation using real-world examples such and the speed of a train, the time is takes to reach its destination, and the cost of tickets to Math Land based on how many students attend. Students explore and construct equations using tables and graphs of direct and inverse variation. Students have the opportunity to look for and make use of structure as they reason about the differences and similarities between the two equations, graphs, and scenarios and the different ways these situations can be expressed algebraically.
• In Mathematics III, Topic 14, Exploring- The Basics of Logarithms, students explore a population of fire ants doubling over time by modeling the number of fire ants as a function of the number of weeks that have passed. Students explore the structure of this function by creating a graph and writing a logarithmic function rule that models number of weeks as a function of the number of fire ants. 
• In Mathematics II, Topic 15, Student Activity Sheet 2, students are given a set of three problems that have been worked out. Students are asked to “Identify the mistake in each solution, and then solve correctly, showing your work.” In this process, students must look for, develop, and generalize relationships and patterns. Students must apply their knowledge of patterns and properties to this new situation. Students also must not only know the rules of exponents, but be able to apply them and combine them into complex problems.

Examples where students look for and express regularity in repeated reasoning include:

• In Mathematics I, Topic 8, Exploring- Solving Linear Equations, students use equations to model the rental of a dune buggy and solve the equation if the rental fee is $75. Students solve linear equations and systems of linear equations algebraically using a balance model and identify the algebraic property for each step in the process. Students view an animation of a balance scale and use algebra tiles to model the steps in solving an equation or systems of equations. These experiences provide students with an understanding of repeated reasoning used when solving equations and systems of equations.
• In Mathematics I, Topic 3, Student Activity Sheet, students are given a scenario about the online retailers “We-stock-it.com” and “Discountstore.com” and the costs for shipping and handling. Each question asks the student to take a step in working with the function. One question asks students “If you made a graph of each retailers’ shipping and handling costs, which variable would you graph along the x-axis? Which variable would you graph along the y-axis? Why?” Another question asks students to graph the data. Students look for shortcuts using patterns and repeated calculations. Students think about what the data means, how to use it, what they notice, what is happening in this situation, and what would happen if they switched the values on the x-axis and y-axis. 
• In Mathematics III, Topic 12, Student Activity Sheet 4, students are given a picture of a set of six exponential functions. Students are asked “What do you notice about these graphs? Show that $$y=\frac{1}{2}^x$$ is the same as $$y=2^{-x}$$.” Students look at general methods and shortcuts in the function graphs (the applications of transformation rules) and attend to the details of the calculations and movements. Students must consider the reasonableness of their results.