High School - Gateway 2

The materials reviewed for Mathspace High School Traditional Series meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. Conceptual understanding is mostly developed within the Engage task, Lesson Exploration Tasks, and Assessments across the series.  

Examples of the materials providing opportunities to demonstrate conceptual understanding include:

• In Algebra 1, Subtopic 1.03, Practice, Worksheet, Question 5, students are given the following scenario, “Dylan purchased 3 pens, 4 pencils and a single note pad. The total cost of all these items was 3x+4y+5 dollars.” Students are tasked to interpret the variables in terms of the context and students must also explain what 5 represents in the scenario. Finally students, “Explain how this expression could be revised to represent purchasing an unknown number of notepads.” (A-SSE.1)

• In Geometry, Subtopic 10.06, Lesson, Teacher guide, Exploration, students use the definitions of focus and directrix to write an equation for a parabola in standard and vertex form. Students are given a graph of a parabola, with the focus labeled and the directix shown. “Consider the parabola: 1. Verify the point (8,7) lies on the parabola. 2. Write an equation that would allow you to find any point, (x,y), on the parabola. 3. Solve your equation for y. 4. Compare and contrast the equation from question 2 to the equation from question 3.” (G-GPE.2) 

• In Geometry, Subtopic 11.03, Lesson, Teacher guide, Exploration, students use an interactive applet to investigate the relationship between chords in a circle and their corresponding arcs. The materials state, “Using the applet below, move points C and D to change the lengths of the chords. Move point E to change the location of \overline{EF} around the circle. Move point B to change the size of the circle.” After exploring the applet, students are asked, “1. What can you conclude about the arc lengths of the arcs EF and CD? 2. How could we prove the arc lengths of EF and arc CD are congruent?”(G-C.2)

• In Algebra 2, Subtopic 6.01, Lesson, Teacher guide, Exploration, students use an interactive applet to explore the concept of a radian angle measure. “Drag the sliders to explore the applet. The different colors show the different lengths of each radian measure around the arc of the circle.” After exploring the applet students are to answer the following questions, “1. Is the number of arc lengths around the circle consistent for different radius measures? 2. How many arcs fit on the circumference of the circle? 3. Express the circumference as a multiple of the radius. 4. Determine the circumference of the circle in terms of \pi and the radius r. 5. How many radians are there in the circumference of a circle with a radius of 1?”(F-TF.1)

• In Algebra 2, Subtopic 8.04, Engage, students work in groups to build multiple models to represent population growth data and predict the population in the year 2030 based on each model. “Create two different models of Florida’s population using the growth shown in the table. Use each model to predict the population of the state in the year 2030. Which model is a better predictor of the population in 2030?” (A-CED.2, F-IF.4, F-IF.9 and F-BF.1)

The materials reviewed for Mathspace High School Traditional Series meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters. Procedural skills are mostly developed through the adaptive and worksheet practice questions located within each subtopic. 

Examples of the materials developing procedural skills and students independently demonstrating procedural skills throughout the series include: 

• In Algebra 1, Subtopic 1.04, Practice, Worksheet, students translate fluently between radical and exponential forms using properties of exponents and an understanding of how rational exponents are defined. Students also simplify radical expressions and write the expressions in reduced radical form. (N-RN.2)

• In Algebra 1, Subtopic 9.04, Practice, Worksheet, students rewrite quadratics into equivalent expressions. Students explore how to rewrite expressions by using a variety of factoring methods such as factoring out the Greatest Common Factors, and factoring by grouping, with polynomials of degree of 2 or higher. (A-SSE.2)

• In Geometry, Subtopic 1.03, Practice, Worksheet, students construct the copy of a line segment and bisect a segment. In Subtopic 1.04, Practice, Worksheet, students construct a copy of an angle and bisect an angle. In Subtopic 2.02, Practice, Worksheet, students construct parallel lines. In Subtopic 2.03, Practice, Worksheet, students construct perpendicular lines and a rectangle. In Subtopic 3.05, Practice, Worksheet, students explain how to construct the altitudes, and orthocenter of a triangle using a compass and straightedge. In Subtopic 3.06, Practice, Worksheet, students construct the midsegment of a triangle in at least two different ways. (G-CO.12)

• In Geometry, Subtopic 10.03, Engage, students explore the applet to create parallel and perpendicular lines. Students prove the slope criteria for parallel and perpendicular lines using algebraic calculations and geometric properties. In the Practice, Worksheet, students develop procedural skills by finding the equation of a line parallel or perpendicular to a given line passing through a given point. (G-GPE.5)

• In Algebra 2, Subtopic 1.01, Practice, Worksheet, students identify key characteristics of functions such as where the functions are positive, or negative, relative maximums and minimums, and end behavior for multiple function families. In Subtopic 1.03, students identify key characteristics of functions that have been transformed using reflections, translations, and other forms of transformations. (F-IF.4)

The materials reviewed for Mathspace High School Traditional 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. 

Examples of lessons that include multiple opportunities for students to engage in routine and non-routine mathematics applications include: 

• In Algebra 1, Subtopic 4.05, Engage, students build a parade float given constraints on size and budget. Students work in groups to define reasonable quantities for building the float, write a system of inequalities related to the constraints, and solve the system to explore reasonable solutions that fit the student council goals (maximum number of floats, least amount of empty space, longest parade, and least amount spent). (A-CED.3, A-REI.12, F-IF.B and F-BF.1)

• In Algebra 1, Subtopic 7.04, students compare two or more data distributions using appropriate measures for shape, center, and spread. In the Practice, Adaptive, students interpret the differences in shape, center, and spread to answer questions about data in a variety of contexts including but not limited to the following: comparing the beak size of two groups of birds to determine if they are the same species, examining the age distributions of employees working at two competing fast food restaurants on a Saturday night given the mean, median and range of each restaurant, or comparing the strength of two different types of bricks based on two blox plots. (S-ID.2 and S-ID.3)

• In Geometry, Subtopic 8.08, students apply trigonometry to solve right triangles in real-world contexts. In the Practice, Worksheet, Question 1, students are given the shortest straight-line distance to be 344 km, the angle of depression to be 32\degree, and the helicopter speed of 260 km per hr. “The pilot plans to fly until she is directly above the landing pad, then land vertically. Determine how long until she can begin her descent.” (G-SRT.8)

• In Geometry, Subtopic 9.01, Practice, Worksheet, Question 14, “The Tasmanian devil is native to the island of Tasmani in Australia. The populations of Tasmanian devils in the past 25 years had declined significantly due to cancer. In 1995, the population of devils was 140000 individuals. In 2020, the population was 20000 individuals.” Students are given a map of Tasmania broken up by shapes. “a. Use the breakdown of the map to estimate the area of Tasmania. b. State the population density of the Tasmanian devil in 2020, rounded to one decimal place. c. State the population density of the Tasmanian devil in 1995. d. Describe how the population density in 2020 compares to the population density in 1995.” (G-MG.2)

• In Algebra 2, Subtopic 5.05, Practice, Worksheet, Question 11, “Patricia goes for a run through Blue Spring State Park and the GPS function on her fitness wristband tracks her speed as she runs. The following function, R(x), models her speed in kilometers per hour when she is x kilometers from where she started.” Students must calculate R(0), find Patricia’s speed when she is 20 kilometers from where she started, and calculate Patricia’s distance from her starting point when her speed is 6 kilometers per hour. (A-CED.4, A-REI.2 and F-IF.8)

The materials reviewed for Mathspace High School Traditional Series meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. The three aspects are balanced with respect to the standards being addressed. Examples of lessons that engage students in the aspects of rigor include: 

• In Algebra 1, Subtopic 4.01, Engage, students use systems of equations to determine the number of servings a recipe can make within the defined constraints of a budget. Students working in pairs must choose one of four provided recipe options. “Determine the number of servings of your chosen recipe that can be made with a $15 budget. Create a visual mathematical model that supports your solution.” Students are provided a chart of various meats, eggs, dairy, grains, starches, and pantry items with various prices and weights. Students build procedural skills by creating graphs and equations representing the real-world prices, and develop conceptual understanding by determining the number of servings using their model. (A-CED.3, A-REI.6 and A-REI.11) 

• In Geometry, Subtopic 2.01, Engage, students investigate angle relationships in the application of creating weaving designs that includes parallel lines, a transversal and measurements of the angles they create. In the Lesson, students use applets to explore angle relationships of parallel lines cut by a transversal. In the Practice, students build procedural skills by identifying, solving, and proving angle relationships. (G-CO.9) 

• In Geometry, Topic 10, Assessment, Question 1, students develop procedural skills by finding the area and perimeter of a quadrilateral on a coordinate grid. Question 2, students develop conceptual understanding by stating whether the coordinate is possible for vertex R. “\triangle PQR has vertices P(2,2) and Q(6,8). The area of \triangle PQR is 28 square units. State whether each of the following are possible coordinates for R: a (10, 4) b (12, 3) c (14, 1) d (0, 13)” Performance task, Question 15, “Yasuko is investigating quadrilaterals in the coordinate plane when she comes up with a conjecture. She says that if you draw any quadrilateral and connect the midpoints of its sides to create a new quadrilateral, the new quadrilateral will always be a parallelogram. Is Yasuko’s conjecture valid? If so, use coordinate geometry to show why. If not, provide a counterexample.” Students develop conceptual understanding in an application problem by providing an explanation and model to prove Yasuko’s statement is valid. (G-GPE.4, G-GPE.5, G-GPE.6 and G-GPE.7)

• In Algebra 2, Topic 1, Topic Overview, Assessment, Question 1, students develop conceptual understanding by determining the domain and range of various graphed functions, using interval notation. Question 5, students develop procedural skills and conceptual understanding by creating a scatter plot representing data based on U.S. franchise applications for years since1900, determining the function family described by the table of values and explaining their answer. (F-IF.4, F-IF.5 and F-IF.6)

• In Algebra 2, Subtopic 7.02, Practice, Worksheet, Question 17, students develop procedural skills by completing a table of values based on a radical equation representing the length of a blue whale calf. Students illustrate conceptual understanding by graphing the length function and determining the constraints in terms of months, and writing an inequality for the possible values of k based on a new equation and the information that an orca whale calf is born shorter than a blue whale calf. (F-IF.4, F-IF.7b and F-BF.3)

The materials reviewed for Mathspace High School Traditional Series meet expectations that the materials support intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards. 

Each course has a Correlations and Alignment document which is found within the Materials Guide, Textbook Guides section. This document includes a list of the Standards of Mathematical Practices for each lesson in the course. Within the Subtopic overview, the “Lesson narrative” identifies and provides guidance on how the students engage with specific Mathematical Practices. Additionally, the Subtopic overview, “Standards” section list the Mathematical Practices that are addressed in the subtopic and offers suggestions for how the students can engage with them. 

Example of where and how the materials use MPs1 and/or 6 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include: 

• In Algebra 1, Subtopic 3.03, Practice, Worksheet, Question 7, students are given a graph relating distance from home and time along with the following, “The graph shows Yoichi’s distance from home over a 10-minute interval. Is the distance’s average rate of change between 3 and 8 minutes positive, negative, zero or undefined?” Students will need to determine a reasonable scale for distance, which is not given. (MP6)

• In Algebra 1, Subtopic 4.02, Engage, students analyze the similarities and differences between the graphs of two systems of linear equations. Students are then given the following, “Felipe’s band, Diamonds Under Pressures, are renting a venue for their first show. Diamonds Under Pressure must decide the venue and how many presale and door tickets they must sell for their first show.” Students must choose a venue based on cost, total hours of rental, maximum occupancy and age. After choosing a venue the group must decide how many presale and door tickets they must sell for their first show based on the following requirements: “Presale tickets will be 25% cheaper than door tickets. The band must sell enough presale tickets to cover a deposit worth 20% of the total venue cost. The band wants to make some profit, and cannot lose money.” Students analyze and make sense of the problem by defining the constraints, determining if their answers make sense by trying different options, and using both visual and mathematical descriptions. (MP1 and MP6)

• In Geometry, Subtopic 9.02, Engage, students are given pictures of four different cakes and asked, “Which one doesn’t belong?”. Students are then given pictures of four different cake pan shapes and instructed, “Pricscilla wants to bake an eight-layer rainbow cake where each layer is a different color. Before they start to create the cake, they must decide what shape of baking pan to use to bake the layers. Think about what a cake made with each pan may look like.” Groups are formed and students are asked to, “Work together to illustrate what the assembled cake might look like. Illustrate what the inside face of the cake will look like when Pricscilla cuts the cake in half to reveal the layers. Discuss and describe the similarities and differences between the shape of the cake layers and the shape revealed when the cake is cut in half.” Students must analyze the givens, constraints, and goals. Students make sense of the problems by using conjectures, drawing diagrams, and searching for trends. (MP1 and MP6)

• In Geometry, Subtopic 11.01, Engage, students are given a diagram of pizza sizes ranging from Mini 6’’ to Jumbo 18’’.“A new regulation for all food businesses requires the nutritional information for one serving to be provided. However, the variation in the size slice makes it difficult for the pizza place to accurately determine the nutritional information for a serving size of one slice.” Students work with a partner to determine a new way to slice the pizzas so that each slice represents one serving regardless of the size of the pizza. Then the pairs must, “Create a new sign for the pizza place announcing their new slices. Be sure to include: How many slices are included with each pizza size. The area of every slice. A mathematical justification for your slice choices.” Students make sense of determining the slices regardless of the pizza size and attend to precision as they create a sign for the pizza restaurant describing how many slices are included with each pizza size. Students must include a mathematical justification of the slicing method and area of the pizza. (MP1 and MP6)

• In Algebra 2, Subtopic 6.01, Overview, the materials states students will, “evaluate the solution notation for accuracy , paying particular attention to solutions written in radians. Encourage the use of clear and precise mathematical language when describing the solutions.” In the Practice, Worksheet, Question 6, students are given a series of true/false statements related to angle and radian measure. The students are to, “Determine whether the following statements are true or false. If it is false, correct the statement.” (MP6)

• In Algebra 2, Subtopic 7.02, Engage, students are given the following, “A tsunami wave moves according to the function: S=11.27\sqrt{d} where S is the speed in kilometers per hour and d is the depth of the water in kilometers. Your task is to determine how long it will take the waves of a tsunmai to reach different shores.” Students from groups and must make sense of the radical equation to find the ocean depth at an origin point of there choosing, the speed of a tidal wave starting from their origin point and determine the length of time for the tidal wave to reach three different locations on the map, including at least one island. Students are provided with a map of a section of the Pacific ocean and an elevation map key. (MP1 and MP6)

Although the materials use MP1 and MP6 to enrich mathematical content, there are instances where materials do not use accurate or precise mathematical language as it relates to MP6. Examples of where and how the materials do not attend to precision include, but are not limited to:

• In Algebra 1, Subtopic 1.03, Lesson, in the example for coefficients, the materials show the coefficient for the terms 3x+y and 1 is 3. The coefficients for the expression are 3 and 1. An exponent is defined as, “A number indicating how many times to use its based in a multiplication.” The definition does not consider variables. 

• In Geometry, Subtopic 5.03, Engage, the triangle congruent game generates triangles that can not exist based on the side lengths and angles.

• In Algebra 2, Subtopic 7.04, Lesson, Euler’s constant, the materials refers to Euler’s number, e. Later, the materials state, “Euler’s constant has many applications in natural growth and compound interest. We can evaluate powers of e using the button on our calculator.” Euler’s constant is \gamma, which is different from Euler’s number.

The materials reviewed for Mathspace High School Traditional Series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards. 

Examples of where and how the materials use MPs 2 and/or 3 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include: 

• In Algebra 1, Subtopic 6.03, Engage,  students work in groups of 3 to “create a sequence of shrinking shape images as divisions within a square and then analyze the patterns found within each shape. Students write a sequence representing the area of the blue region in each stage.” Students reason quantitatively by writing, “a sequence representing the perimeter of the purple region in each stage. Use your sequences to predict the blue area and purple perimeter of the 10th stage.” The materials encourage teachers to ask the class to analyze the similarities and differences between the patterns as well as to critique the efficiency of their classmates' strategy. (MP2 and MP3)

• In Algebra 1, Subtopic 10.03, Overview, the materials states, “Encourage students to describe how various parameters in a context relate to a function’s equation, graph and table.” In the Engage, students manipulate an applet to simulate a kicking contest. They make connections between the placement of the ball, the maximum height of the kick and the final location on the ground. By trial and error, students attempt to make money for charity. Students use the applet to record the equations for at least three successful kicking challenges. The materials provide teachers the following purposeful questions to check for understanding and encourage critical thinking: “What is the goal of this kicking challenge?, What is your plan to adjust the equation so that the kick hits the goal or is closer to hitting the goal? What makes you say that?, What observations do you notice about the graph and equation?, How do the parameters affect the graph?” (MP2)

• In Geometry, Subtopic 2.01, Practice, Worksheet, Question 8, students critique Henry’s claim that two angles are congruent within a diagram containing parallel lines and a set of congruent angles. “In the given diagram, a\Vert b and \angle{2}\cong\angle{3}. Henry claims that \angle1\cong\angle{8}. Select the statement that proves or disproves his claim.” (MP2 and MP3)

• In Geometry, Subtopic 2.02, students must justify their thinking when creating and analyzing proofs related to using angle measures to prove lines are parallel. In the Teacher guide, Example 3, teachers are guided to carry out the lesson with the following directions, “Observe students working and notice any common errors that may arise, or places where students are getting stuck. Present work to the class anonymously that includes this error and have students engage in error analysis. They should identify the error, explain why it is an error, and explain the correct steps that should have been taken.” (MP3)

• In Algebra 2, Subtopic 1.02, students identify the family a function belongs to based on its key features. In Practice, Worksheet, Question 14, students are given the following, “The graph shows the number of earthquakes that a particular country experiences in each year: a Find the average rate of change. b Interpret the average rate of change in the given context. c State which function family the graph belongs.”  Students reason abstractly and quantitatively and find solutions by connecting key features to the quantities they represent. (MP2)

• In Algebra 2, Subtopic 2.03, Lesson, Teacher guide, Exploration, students are asked to calculate the powers of i, from i^2 until i^8 explain any noticed patterns and use the pattern to find i^{23}, i^{30}, i^{33}, and i^{40}. In the Teacher guide, teachers are guided to ask purposeful questions including, “Do you notice any values repeating? What is the significance of i^{4}=1 for all higher powers of i?” (MP3)

The materials reviewed for Mathspace High School Traditional Series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards. 

Examples where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include: 

• In Algebra 1, Subtopic 5.01, Practice, Worksheet, Question 9 students are instructed to, “Draw the graphs of the function y(\frac{1}{2})^x, y=(\frac{1}{3})^x and y=(\frac{1}{5})^x by hand or using technology, then answer the following questions: a State whether the following statements are true for all of the functions: i All of the curves have a maximum value. ii All of the curves pass through the point (1,2). iii All of the curves have the same y-intercept. iv None of the curves cross the x-axis. b State the y-intercept of each curve. c Describe what happens to the values of y as x gets increasingly larger.”(MP5)

• In Algebra 1, Subtopic 6.07, Practice, Worksheet, Question 1, students construct models based on the data from a trial for anti-bacterial medication. “A trial for anti-bacterial medications used to contain an outbreak of an infection resulted in the slowing of the growth of bacteria. The samples used for the trial initially contained a total of 500 microbes. Medication A led to the growth of 1425 microbes after 4 hours while at 10 hours, the microbe growth was at 3160. The final recorded number of microbes in the sample was 3926 at 13 hours. Medication B showed growth at a rate of anywhere from 14.3% to 17.2% over the same time period. a. Construct a model and use it to compare the growth of the microbes with either medication. b. Scientists state that the threshold for a bacterial infection before a person’s safety is at severe risk is 7000 microbes. Use this information and the model constructed in part (a) to make a recommendation about the use of each treatment.” (MP4)

• In Geometry, Subtopic 4.04, Lesson, students use an applet that contains a manipulable line of reflection and multiple settings for the line of reflection. Students change the line of reflection and move it around to help them investigate the relationship between reflection and symmetry. Next, students use an applet in which they observe the rotation of different figures by changing the slider for the angle of rotation. The point of rotation for all the shapes is their center, which students can consider when observing when the image of the rotation overlaps with the pre-image. By the end of the lesson, students should be able to use a variety of tools to identify symmetry and map figures onto themselves using line and rotational symmetry. (MP5)

• In Geometry, Subtopic 12.01, Practice, Worksheet, Question 19, students are instructed to interpret a 3-part Venn diagram showing the languages available for students to study. “Five new students join the school in this year's group. One of these students studies German, one studies Italian and German, one studies Spanish and Italian, and two of them don’t study any languages. Draw a new Venn diagram representing the updated information and explain your reasoning.” (MP4)

• In Algebra 2, Subtopic 2.01, Practice, Worksheet, Question 17, students construct and solve a model based on Ryan’s savings, and state and explain whether statements are correct based on the scenario constraints. “Ryan wants to save up enough money so that he can buy a new sports equipment set, which costs $40.00. Ryan has $22.10 that he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows for $2 per window. Construct and solve a model for Ryan’s savings. b State whether the following statements are correct. Explain your thinking. i Ryan must wash more than 9 windows to be able to afford the equipment. ii Ryan must wash at least 8 windows to be able to afford the equipment. iii If Ryan washed 8 windows, and 95% of another window, he could afford the equipment. iv The number of windows Ryan must wash to be able to afford the equipment must be greater than or equal to 9.” (MP4)

• In Algebra 2, Subtopic 6.05, Lesson, students use various tools (technology, coordinate grid, tables and various equation forms) to understand transformations of sine and cosine functions. The materials encourage teachers to have students discuss the relative merits of using these different tools, throughout the lesson. (MP5)

The materials reviewed for Mathspace High School Traditional Series meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards. 

Examples where and how the materials use MPs 7 and/or 8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include: 

• In Algebra 1, Subtopic 6.01, Lesson, Example 2, students are instructed to calculate the first 6 terms in the sequence T_{n+2}=2(T_{n+1}-T_{n}) with T_1=-2 and T_2=0. In the Teacher Guide, Example 2, teachers are guided to, “Ask students to consider how they would describe the pattern in words.” In Example 4, students are given a pattern in a graph and asked to write the recursive rule, T_{n+1}, and the initial term T_0. In the Teacher guide, Example 4, teachers are guided to, “Ask students to consider and explain why the terms in the sequence alternate between negative and positive values.” Students use repeated reasoning to explore and understand sequence patterns and structures. (MP7 and MP8).

• In Algebra 1, Subtopic 7.02, Lesson, Example 2, students compare and analyze the mean and median of multiple data sets.” In the Teacher Guide, Example 2, teachers are guided to, “Ask students to note that for both classes the mean is slightly lower than the median. Ask students to look at the dot plots and discuss why that might be the case. This gets students to start thinking about the shape of the distribution and how it might affect the mean, median, and mode, and this idea will be developed in more detail in a future lesson.” (MP7)

• In Geometry, Subtopic 2.03, Lesson, Exploration, students manipulate the vertices on an isosceles triangle with a bisector using the applet to observe which properties of the triangle remain consistent throughout the changes. “1. What relationships in the diagram are always true? Can you explain why?” Students use repeated reasoning to determine that the bisector is a perpendicular bisector. (MP8)

• In Geometry, Subtopic 5.01, students make conjectures about the types of transformations that can be used to show congruency. Students look for patterns when exploring transformations and use them to make generalizations. In Lesson, Example 2, students determine whether \triangle DEF is congruent to \triangle JKL by using a sequence of transformations. Teachers are then guided to, “Ask students to provide another sequence of transformation that will map \triangle DEF to \triangle JKL.” (MP7 and MP8)

• In Algebra 2, Subtopic 6.07, students use the structural knowledge of reciprocal functions, basic trigonometric functions, and the unit circle to graph cosecant, secant, and cotangent functions. (MP7)

• In Algebra 2, Subtopic 7.01, Practice, Worksheet, Question 5, students are tasked with determining if an inverse function exists without any domain restrictions for a list of provided functions. Students must change the structure of the function to determine if an inverse function exists, and analyze patterns to make generalizations about which types of relations are one-to-one and which types of relations will require domain restrictions in order to have an inverse function. (MP7 and MP8)