High School - Gateway 2

The instructional materials reviewed for the High School Discovery Education Integrated 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. This series is constructed in such a way that each Concept contains Discover, Practice, and Apply tabs. The Introduction, Investigations, Summary, and Extension tabs within the Discover tab are predominantly where students’ conceptual understanding of key mathematical concepts is developed.

Each Concept within each Unit contains a list of Conceptual Understandings to be found in that section. This list is located within the Progressions and Standards tab in the Teacher edition. For example, Math III Concept 2.2 includes the following list of Conceptual Understandings:

• "Explain why the inverses of quadratic and cubic functions are square root and cube root functions and may require restricted domains."
• "Recognize how the effects of changing parameters within the radical equation will affect the translation of the function’s graph."
• "Compare and contrast methods for solving radical equations of square roots and cube roots while realizing that some of these equations have extraneous solutions."

Included throughout teacher materials within the Session tabs are “Questions to promote development of conceptual understanding.” Within Math I Concept 5.1 the materials provide questions in the Session 1 and 2 Instructional Notes. “Questions to promote development of conceptual understanding” include the following:

• “Which attributes were more challenging to assign to the figures? What made them more challenging?"
• "Which attribute assignments did you and your partner disagree on? How did you resolve the dissenting analysis?"
• "How can you use the responses you made for these problems to compare and contrast linear and exponential functions?”

Examples of select cluster(s) or standard(s) that specifically relate to conceptual understanding include, but are not limited to, the following:

• F-LE.1: In Math I Concept 8.2 Investigation 1 students use equations and graphs to model how the amounts in three different accounts grow over time. Students use technology to analyze and verify how changes in parameters affect the graphs of the equations. Students identify which options involve adding a fixed quantity to the previous year’s amount and which options involve multiplying a fixed quantity by the previous year’s amount.
• G-SRT.6: In Math II Concept 7.1 Investigation 2 students use an online interactive to explore the relationship between the angle measures and the ratios of the sides of a right triangle to develop conceptual understanding of trigonometric ratios. Students look for any relationships that might help the team determine how far they are from the checkpoint. Students are asked to articulate their observations in writing.
• S-ID.7: In Math I Concept 7.2 Investigation 3 students collect measurements of their forearms and right feet to determine if there is a relationship between forearm length and foot length. After students examine results and write an equation of a line of best fit they are asked: “What does the slope of the line of best fit represent in this context? What does the y-intercept of the line of best fit represent in this context? Explain why the y-intercept is or is not realistic in this context.”

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials provide intentional opportunities for students to develop procedural skills and fluency, especially where called for in specific content standards or clusters. The instructional materials build fluency practice into multiple sections of concepts.

Each concept contains a Fluency box within the Progressions and Standards tab of the teacher materials where specific skills are identified for focused fluency development. For example, in Math III Concept 6.1 the Fluency box include the following:

• “Identify the radian measure for angles within a unit circle.”
• “Measure angles using radians by finding the measure of an angle as the length of the arc on the unit circle subtended by the angle.”
• “Convert between radians and degrees.”

Students are provided opportunities to develop procedural skills and fluencies in the Intro, Investigations, Play, and Check for Understanding tabs.

• The Play tab, which consists of a Coach and Play section, is designed to address procedural skill and fluency. The Play tab includes practice problems placed in a progression of learning that provides students the opportunity to build procedural fluency from conceptual development.
• Every Investigation contains a Check for Understanding where students determine their “current level of understanding”. Problems within the Check for Understanding are often focused on procedural skills and fluency. For example, Math III Concept 1.1 Check for Understanding includes:
  • "Hurricane strength is classified using the Saffir-Simpson scale. Category 3 hurricanes have wind speed greater than 110 miles per hour but at most 130 miles per hour. Which of these show possible wind speeds, w, in this category? Select all that apply."
  • "Which compound inequality is shown by the number line graphed?"

Examples of select cluster(s) or standard(s) that specifically relate to procedural skill and fluency include, but are not limited to:

• G-GPE.4: In Math I Concept 2.3 students are asked to use the Pythagorean Theorem to find locations of a statue in the Investigations as well as Check your Understanding. In Investigation I of Math II Concept 7.3 students are asked to write an equation that represents all the points on a circle when given the center and a point on the circle.
• A-SSE.1b: In Math I Concept 1.2 Investigation 3 students explore card tricks and write and simplify algebraic expressions to express the number of cards in a pile. In Trick 2 students ultimately write equations, based on expressions they wrote, to describe the results of the card trick. Investigation 3 also offers four questions providing students the opportunity to interpret expressions. In the Play tab students are given multiple scenarios and asked to write an expression/equation that represents the scenario or given options and asked to choose all that apply. In the Play tab of Math II Concept 3.2 students are again asked to simplify or write equivalent expressions and/or polynomials. In the Play tab of Math III Concept 9.1 students are given the opportunity to expand binomials, factor expressions, provide missing factors for expressions, and identify “Which of the following is the factored form of 16x^4+32x^3y+24x^2y^2+8xy^3+y^4?” and “Which of the following expressions show the difference of two perfect squares. Select all that apply.”

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials support 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. The third tab in each Unit is titled Apply. The Apply tab consists of one or more open-ended problems aimed at the development of students’ ability to utilize mathematical concepts and skills in engaging applications.

Each Concept has an Apply section where students are able to apply what they have learned in the investigation.

• In Math I Concept 5.1 Apply Problem 1, How Can You Use Functions to Make an Image?, students are expected to draw a recognizable figure using graphs of linear and/or exponential functions, vertical lines, and at least one pair of non-vertical parallel or perpendicular lines. This application task requires students to identify the domains of the functions they use, explain the relationship between the slopes of parallel or perpendicular lines, and identify and describe transformations of linear and exponential functions. The materials explains that architects, fashion designers, and engineers draw using technology. (F.IF.4)
• In Math II Concept 5.1 Apply Problem 1, How Can Math Help You Create a Successful Business?, students analyze the relationship of a demand function and a revenue function. They interpret the key features of the graphs and determine the price of the item that will maximize revenue. (F-IF.7)
• In Math III Concept 3.1 Apply Problem 1, How Many Digits Are Needed in a Telephone Number?, students watch a video about the history of telephone numbers and are posed with the question of whether or not “10 digits will be enough to ensure that everyone has a unique phone number.” After some guided questions to get students thinking, they are told “the world will continue to increase by 25 million every month.” Then they are asked, “How many digits are needed right now to accommodate all phones? When would more digits be needed for phone numbers so each user has one unique number?” (F-LE.2)

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed. As described in indicators 2a, 2b, and 2c, there is evidence that all three aspects of rigor are present in the materials. Overall, the series provides a balance of the three aspects of rigor throughout the materials.

Each Concept includes Discover, Practice, and Apply sections:

• Discover includes Introduction, Investigation, Summary, and Extension sections that give students the opportunity to build conceptual understanding of the mathematics and practice procedural skills, typically in the context of a real-world example;
• Practice focuses on procedural skills with a Coach section that provides student support to develop fluency- for example, leading students through solving an algorithmic problem and giving immediate feedback- as well as a Play section where students demonstrate procedural fluency without support; and
• Apply includes extended tasks based on real-world applications.

In the Model Lesson section of the teacher materials, Progressions and Standards includes a diagram that identifies for teachers the balance of conceptual understandings, procedural fluencies, and applications that should emerge from each Concept in a Unit.

Some examples that demonstrate this balance include:

• In Math 1 Concept 4.1 students "engage with real-world phenomena that involves a dependent relationship, constraints in data, and different mathematical representations of the data." The Concept progresses to students using "function notation to represent, interpret, and evaluate functions." (F-LE.5, F-IF.1, 2, and 9)
• In Math II Concept 5.2 students explain complex numbers and extend the commutative, associative, and distributive properties to operations with complex numbers. They explain the relationship between the discriminant and complex quadratic solutions and show procedural skills by comparing quadratics having both real and complex roots. Students then "apply the real-world implications of complex quadratics solutions to word problems." (N-CN.1, 2, and 7)
• Math III Concept 6.1 balances conceptual understandings, procedural fluencies, and applications as students explore angle measures and define radian measures for angles throughout the unit circle. In the Intro, students utilize Geometry software to discover patterns between central angle measure, radius length, and arc length. Throughout the Investigations, students explore angles within the coordinate plane, derive radian measure, and begin converting between degree and radian measures. The use of interactive software provides students the opportunity to discover relationships and develop conceptual understanding of radian measure. Each Investigation contains a Check for Understanding where students practice more procedural skills such as identifying “Which of the following angle measures given in radians is greater than 28.5°?” and completing a table of equivalent radian and degree measures. Students then apply their new understandings of radian measure when posed with two application problems where they are asked “How Would you Design Your Own Ferris Wheel?” and “How Can You Create a New Protractor?”. (F-TF.1)

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials support the intentional development of overarching mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the Standards for Mathematical Practice. Within each Unit Concept, the Standards for Mathematical Practice are identified under the Progressions and Standards tab. Additionally, the materials provide information regarding the Standards for Mathematical Practice within the Teacher Model Lesson under the Standards tab as well as within the Teacher Notes.

Examples of MP1 are as follows:

• In the Math I Concept 8.3 Intro students analyze geometric patterns, making sense of the algebraic relationship between the quantities. In Investigation 2, students are asked to confirm their solutions through graphic analysis. In Investigation 4 MP1 is evident as students analyze relationships between an initial investment and changes to an account. (F-IF.7e, 9, A-REI.11, F-LE.5, F-BF.2)
• In the Math II Concept 9.1 Intro students analyze givens, constraints, relationships, and goals as they explore and discuss different meanings for the center of a triangle. In Investigation 1 students start each exploration by explaining to themselves the meaning of the problem, analyzing goals, and looking for entry points to its solution. (G-C.3, C-CO.10)
• In Math III Concept 3.2 Apply 1 students collect data and select an appropriate time period when they would most likely see a wolf when visiting Yellowstone National park and provide strong support from graphical and algebraic analysis. Students then make predictions from their models. (A-CED.1, A-REI.11, F-BF.1b, F-IF.8,8b, F-LE.1c, 2, 5)
• In Math III Concept 5.1 Apply 2 students use data tables showing the number of transistors in various processors over the years. Students make predictions based on data of when processors will hit 10 billion transistors. Students must research their conclusions to confirm accuracy. (A-REI.7, 11)
• In Math III Concept 4.1 Investigation 4 MP1 is explicitly noted in the Instructional Notes for teachers to “encourage students to consider different methods of solving, including both algebraic and graphic approaches.”

Examples of MP6 are as follows:

• In the Math I Concept 1.1. Intro students are led to discuss precise vocabulary for parts of expressions. The rubric for Apply I requires students to identify all variables, explain what the expression represents, and include units. (N-Q.1, 2, A-SSE.1a)
• In the Math II Concept 3.2 Intro, the Teacher Notes state that students “will communicate clear definitions and state the meaning of the symbols they use (MP.6).” Later in the intro exercise, students “Discuss with a partner: What distinguishes a polynomial from an expression that is not a polynomial?” The Teacher Notes suggest referring students back to the initial class list of definitions for a polynomial. Students are expected to refine the definition based on the examples and nonexamples of polynomials. (A-APR.1, A-SSE.1a, 1b, N-RN.2)
• In Math III Concept 9.1 Investigation 2 students use MP6 as they attend to precision when describing characteristics of polynomial equations and their relationship to the graphs of the related functions. (A-APR.1, 2, 3, 4, 5(+), A-SSE.1b, 2, N-CN.2, 4(+))

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the Standards for Mathematical Practice. Within each Unit Concept, Standards for Mathematical Practice are identified under the Progressions and Standards tab. Additionally, the materials provide information regarding the Standards for Mathematical Practice within the Teacher Model Lesson under the Standards tab as well as within Teacher Notes.

Examples of MP2 are as follows:

• In the Math I Concept 5.1 Investigation 1, Teacher Notes state “MP.2 will also be evident as students decontextualize information to create a graph and analyze it to draw conclusions regarding the domain and range. They will use the graph to compare the density of different substances, such as water.” (A-REI.10, G-GPE.5, F-BF.1a, 1b, 1c(+), F-IF.1, 2, 4, 5, 6, 7a, 7e)
• In Math 1 Concept 8.3 Investigation 2 students formulate a process for solving exponential equations by analysis of quantitative relationships, identifying common structure to the base values. (F-IF.7e, 9, A-REI.11, F-LE.5, F-BF.2)
• In Math II Concept 10.1 Investigation 1 students contextualize and de-contextualize different scenarios to construct mathematical models and interpret the likelihood of winning. (S-CP.1 - 5)
• In Math II Concept 8.2 Investigation 1 students analyze and evaluate the distribution of cookie weight data. They analyze the variability in the data and explain what the variability means in the context provided. (S-ID.4, S-MD.6(+), 7(+))

Examples of MP3 are as follows:

• In the Math I Concept 2.3 Intro students relate dimensions and units to a coordinate plane. In Investigation I a variety of dimensions are presented that may or may not satisfy given constraints. Students are asked to justify their conclusions. (G-CO.1, G-GPE.4, 6, 7)
• The Math I Concept 5.1 Introduction to the Investigations states the following: “MP.3 will be evident as students compare characteristics among the figures, justifying their reasoning for similarities and differences. Allow students to discuss with a partner before reconvening the class to share their reflections. Questions to promote discussion: How did you know whether each figure on the image was exponential or linear? Under what conditions could an interval of an exponential function behave like a linear function?” (A-REI.10, G-GPE.5, F-BF.1a, 1b, 1c(+), F-IF.1, 2, 4, 5, 6, 7a, 7e)
• In the Math I Concept 6.2 Intro students debate options for buying economy versus quality brushes. (A-CED.3, N-Q.3)
• In Math II Concept 1.3 Investigation 2 students analyze givens and diagrams and develop proofs with classmates and on their own. Students choose and arrange the statements and reasons for the Corresponding Angles Theorem and Alternate Interior and Exterior Angles Theorems (paragraph, two-column, and flowchart proofs). Though the student and teacher view of the Techbook does not specifically invite students to critique the reasoning of others, the Instruction Notes in the Model Lesson tab suggests teachers “invite pairs of students to share different proof formats… During critiquing, make sure students can ascertain the validity of each proof.” Investigation 3 Instruction Notes suggest that teachers “invite students to present their proof of the Converse of the Consecutive Angles Theorem and have students critique selected proofs in pairs or small groups.” (G-CO.9, G-GPE.4, 5)
• In Math II Concept 8.1 Investigation 2 students use properties of dilation to justify the area of a circle. Within Investigation 3 students explore the concept of sampling variability and the margin of error. Students justify their conclusions and communicate them to others. (G-C.5, G-GMD.1)
• In Math III Concept 11.2 Apply 1 students extend their ability to analyze dimensions by finding ways of reducing the amount of aluminum used for soup cans, presenting advantages and disadvantages for sizing options. (A-CED.1, 2, A-REI.11, F-BF.4a, F-IF.5)

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the Standards for Mathematical Practice. Within each Unit Concept, Standards for Mathematical Practice are identified under the Progressions and Standards tab. Additionally, the materials provide information regarding the Standards for Mathematical Practice within the Teacher Model Lesson under the Standards tab as well as within Teacher Notes. Students have access to a wide range of tools throughout the series, and although the tools are prescribed in some cases, students are given the opportunity to choose appropriate tools strategically in many of the Apply problems.

Examples of MP4 are as follows:

• In Math I Concept 2.1 Investigation 4 students model and analyze the use of tickets at an amusement park. In Investigation 5 students extend their understanding of equations and inequalities to model a real-world scenario, analyzing the relationship between two options to determine when the options are equivalent or when one is more reasonable. (A-CED 1, 3, A-REI.1, 3, 11)
• In Math 1 Concept 7.1 Apply 1 students create a graphical display to represent data. Students justify their choice in data display. (S-ID.1-3)
• In the Math II Concept 5.1 Intro students model with mathematics by representing the data with different equations and graphs. In Investigation 1 students model a real-world situation by using both the equation and graph of a quadratic function and analyze this function quantitatively. (A-CED.1, F-IF.4, 5, 7a, 7c, 8a, 9, F-BF.1, 3, F-LE.3)
• In Math III Concept 3.2 Apply 1 students collect data and select an appropriate time period with strong support from graphical and algebraic analysis. Students then make predictions from their models. (A-CED.1, A-REI.11, F-BF.1b, F-IF.8, 8b, F-LE.1c, 2, 5)

When working through Investigations, students are often prescribed the appropriate tool(s) to utilize, providing students with guided practice using each tool. As students reach the Apply problems, the choice of tool becomes the decision of the student. Examples of MP5 are as follows:

• Math I Concept 3.1 Apply 2 students “determine the shortest path the [Mars] rover can take to visit all three rocks and then travel to its ending point. Your job is to determine this shortest path, and then model it by using transformations.” The materials provide students with a Mars map as well as an image of the Mars Rover and a rock on Mars. Students must utilize these provided tools and determine what additional tools they would need to accomplish this task. (G-CO.2, 3, 4, 5)
• In Math II Concept 2.1 Apply 1 students consider the question “How Would You Set Up a Projector to Show an Outdoor Movie?” Students experiment with their classroom projector and the concept of aspect ratio to determine the location of an outdoor movie projector for optimal size and clarity of the movie. Selection of appropriate tools for this task is left up to the student. (S-SRT.1, 1a, 1b, 2)
• In Math III Concept 3.2 Apply 3 students determine how much each person in the country would have to contribute to pay off the national debt within four years. Students conduct their own research and use technology (specific technology is not provided) to make calculations and display their data. The Teacher’s Note in the materials states, “Encourage students to use 21st-century publishing tools to present their responses. You may wish to have students use the Board Builder or a particular software tool or leave the choice of tools open.” (A-CED.1, A-REI.11, F-BF.1b, F-IF.8, 8b, F-LE.1c, 2, 5)

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the Standards for Mathematical Practice. Within each Unit Concept, Standards for Mathematical Practice are identified under the Progressions and Standards tab. Additionally, the series materials provide information regarding the Standards for Mathematical Practice within the Teacher’s Model Lesson under the Standards tab as well as within Teacher Notes.

Examples of MP7 are as follows:

• In Math I Concept 2.1 Apply 3 students consider a sales pattern and write a system of linear inequalities that would assist in identifying all possible solutions. (A-CED.1, 3, A-REI.1, 3, 11)
• In Math 1 Concept 9.2 Investigation 1 students use the structure of functions to determine which function will eventually have the greatest value. In Investigation 2 students make use of the structure of linear, exponential, and quadratic functions in order to compare them. (F-IF.9, F-LE.3)
• In Math II Concept 8.1 Investigation 1 students look for and make use of structure to determine the trend in the data and develop the understanding that the area of the circle will approach π as the grid size decreases. (G-C.5, G-GMD.1)
• In Math III Concept 12.1 Investigation 1 students look for patterns in determining the radius of the circle, based on repeated applications. In Investigation 3 students look for patterns in determining the foci and major and minor axes of the ellipse. (G-GPE.1, 2, 3(+))

Examples of MP8 are as follows:

• In Math I Concept 6.1 Investigation 1 students use repeated reasoning to create single equations by applying the substitution method. (A-CED.2, 3, 4, A-REI.5, 6, 11, 12)
• In Math II Concept 8.1 Investigation 5 students look for and express regularity in repeated reasoning as they examine the ratio of the arc length to the circumference for arcs of different measures and the ratio of the arc length to the radius for arc with the same measure. (G-C.5, G-GMD.1)
• In Math III Concept 4.1 Investigation 3 students analyze patterns of inputs and outputs of logarithmic and exponential function compositions and then use these patterns to discern structure relative to the properties and justify the inverse relationship algebraically. (F-BF.3, 4a, 4c(+), 5(+), F-IF.4, 7e)