High School - Gateway 1

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials attend to the full intent of the mathematical content contained in the high school standards for all students. Overall, the majority of non-plus standards are addressed throughout the series, though some aspects of the non-plus standards were not completely addressed or were omitted from the instructional materials of the series.

The following are some examples of standards that were fully addressed in the materials:

• F-IF.4: Students are presented with various functions throughout the instructional materials and provided the opportunity to “interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.”
  • In Math I Concept 9.1 Practice students analyze a function and determine what price of a sweatshirt would produce the greatest income.
  • Math II Concept 5.1 Investigation 2 provides students the opportunity to explore how key features of equations help when graphing functions.
  • Math III Concept 6.2 Investigation 2 requires students to identify key features of tables and graphs in context with temperatures.
• G-CO.10: Throughout the Math II materials, students are developing proofs of triangle theorems.
  • In Math II Concept 4 Investigation 1 students develop the Triangle Sum Theorem and Exterior Angle Theorem proofs. In Investigation 3 students prove base angles of isosceles triangles are congruent and equilateral triangle angles are congruent.

The following standards were partially addressed in the materials:

• G-CO.2: Math I Concept 3.1 provides students opportunities to “represent transformations in the plane” using software and to “describe transformations as functions that take points in the plane as inputs and give other points as outputs.” No evidence was found where students were expected to “compare transformations that preserve distance and angle to those that do not.”
• A-SSE.3b: While students solve equations by completing the square in Math II Concept 4.1 Investigation 4, no instruction was found wherein students complete the square for the purposes of revealing the maximum or minimum value of the function it defines.

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that the materials attend to the full intent of the modeling process when applied to the modeling standards. Opportunities to implement the modeling process are integrated throughout all the instructional materials. The materials provide students with intentional development of the modeling process throughout the courses in the series. Overall the modeling process is used to reach the full depth of the modeling standards.

In the beginning of the series, the materials are presented in a way that intentionally develops the modeling process. A few examples include:

• Math I Concept 1.1 Apply 3, Which Cell Phone Plan Should You Choose?, addresses standards N-Q.1 and 2 as well as A-SSE.1a and provides students exposure to the beginning stages of the modeling process, scaffolding by providing initial variables while allowing students to compute and validate their results.
• Math I Concept 1.2 Apply 1, How Long Will The Kayak Trip Take?, addresses standards A-SSE.1a and contains more aspects of the modeling process. In the Teacher’s Notes, students are encouraged to discuss why it is important that the kayakers’ speed in still water be greater than the speed of the current. Teachers are also encouraged to ask students how they might validate whether their answer for the time needed to complete the trip is reasonable.

Students have opportunities to experience the entire modeling process with increasing complexity. Here are some examples:

• Math I Concept 6.2 Apply 3, How Much Solar Power Should Your Community Purchase?, incorporates the modeling process in its entirety. Students define variables, formulate a system of inequalities to model all constraints, write and compute a function to model greenhouse gas emissions, determine how many megawatts of power to purchase, justify decisions, and report their rationale.
• In Math I Concept 4.3 Apply 2, How Can You Make The Most of $1,500?, students research and compare different financial institutions and their various types of accounts for investing. Students are expected to compare their investment options, validate accuracy using technology, interpret their analysis to make an informed decision and then report out by presenting their findings to the class.
• In Math II Concept 7.1 Apply 2, How Can You Design A Trail That Challenges Hikers With Different Skill Levels?, students research the type of information needed to build a hiking trail and design one with minimal prescribed constraints, computing at least two angles of elevation and two angles of depression. Students interpret those angles in terms of the context.
• In Math III Concept 8.2 Apply 1, Is it Normal?, students “(c)reate an event that is binomial, such as the coin flipping example or rolling a pair of number cubes and getting a 3 or not.” After students choose their event, they define and execute their experiment, including measures of central tendency. Results of the experiment are validated by comparing the actual results with the predicted result. Students interpret whether their data is normal and then report their findings by explaining their experiment and stating their conclusions.

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that the materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs, and careers.

The majority of concepts throughout each course focus on the WAPs. Examples of students engaging with the WAPs include:

• The majority of the concepts in Math I focus on the WAPs.
  • Standards within Quantities are addressed throughout Unit 1, and Concepts 2.2 and 6.2.
  • Standards within Algebra are addressed throughout Units 1 and 6, and Concepts 2.1, 2.2, 5.1, and 8.2.
  • Standards within Functions are addressed throughout Units 4, 5, 8, and 9.
  • G-CO.1 is addressed in Concept 2.3, and G-CO.9 is addressed is in Concept 3.3.
  • S.ID.2 is addressed in Concept 7.1 and S.ID.7 is addressed in Concept 7.2.
• The majority of the concepts in Math II focus on the WAPs.
  • Standards within The Real Number System are addressed throughout Concepts 3.1, 3.2, and Unit 4.
  • Standards within Algebra are addressed throughout Units 3, 4, and 11, and Concept 5.1.
  • Standards within Functions are addressed throughout Units 4, 6, and 11, and Concepts 5.1, and 7.3.
  • G-CO.1 is addressed in Concepts 1.1 and 1.2. G-CO.9 is addressed in Concepts 1.1 and 1.3. G-CO.10 is addressed in Concepts 2.2 and 9.1.
  • G-SRT.B standards are addressed in Concepts 1.4 and 2.3, and G-SRT.C standards are addressed in Concepts 7.1 and 7.2.
• The majority of the concepts in Math III focus on the WAPs.
  • Standards within The Real Number System are addressed throughout Concept 2.2.
  • Standards within Quantities are addressed throughout Concept 7.2.
  • Standards within Algebra are addressed throughout Units 1, 3, 5, 9, 10, and 11, and Concept 2.2.
  • Standards within Functions are addressed throughout Units 2, 3, 4, 6, and 11, and Concepts 7.1, 9.3, and 12.4.
  • S-IC.1 is addressed in Concept 8.2.

While there are additional and prerequisite topics found throughout the series, they do not distract from the work of the WAPs and other standards.

• In Math II Concept 3.1 problem 4 within Coach goes beyond the intent of the standard (N-RN.1) with fractions that have terms with rational exponents in the numerator and in the denominator with nested exponents and the expectation that students will “rationalize” the denominator. The unit test requires students to “order the steps” to rationalize a denominator.
• In Math II Concept 9.1 Investigations 1-4 coherently connect the structure and key features of polynomials with quadratics including factoring powers of four and cases where higher powers factor out as monomials. Investigation 4 then leads students through a visualization for the formulas of differences and sums of cubes which is beyond the stated intent of the standard (A-SSE.2). Investigation 5 contains explorations of Pascal’s triangle and binomial expansion.

The instructional materials reviewed for the High School Discovery Education Integrated series meet the expectation that materials require students to engage in mathematics at a level of sophistication appropriate to high school. Investigations and Apply problems provide students with tasks that are age-appropriate and of interest. The sophistication of numbers is lacking in the Investigations and Practice problems but appropriate in the Apply problems.

Throughout Math I, II, and III, students are presented problems with contexts that are age-appropriate with a level of complexity expected for high school.

• In Math I Concept 9.1 Investigation 2 students use the quadratic parent function to model the path of a basketball.
• In Math II Concept 5.1 Introduction presents students with the concept of world population size and growth rate.
• In Math III Concept 2.2 Investigation 2 students analyze radical functions through the medical context of BSA (Body Surface Area).

Throughout the materials, students are expected to apply key takeaways from middle school.

• Math 1 Unit 1 Assessment 1 extends and connects 7th and 8th grade understanding of integer expressions, distribution, and area (7.EE.1, 7.G.6, 8.EE.7.b) by having students associate the product of a binomial and a monomial with the area of a floor mat. Students interpret a part of a term with respect to its physical value. These extensions are strategically built from the Introduction of Math I Concept 1.1 wherein students make a conjecture for an expression, reflect on it, and then describe it in terms of constant, coefficient, variable, factor, and term.
• In the Math I Concept 2.2 Introduction students recall and extend their understanding of solving equations from 8.EE.7b by exploring numerical relationships in variable contexts. In Investigation 1, students explain the process of solving equations to extend the process to more complicated cases with multiple distributions with fractions.
• Math II Concept 8.3 Play problem 14 extends the middle grades focus on ratios and proportions when asking students to determine how many fish a tank can support given the following: “No more than 10% of the volume of the tank can be reserved for eels, rays and leopard sharks. The ratios for each of these marine animals are as follows:
  • 1 eel per 150 gallons
  • 1 ray per 180 gallons
  • 1 leopard shark per 400 gallons.”

Within Apply problems, and some Investigations problems, students work with more complex, age-appropriate numbers while Play problems are typically presented with simple numbers and do not vary the types of real numbers being used.

• In Math I Concept 7.1 Investigation 3 students analyze levels of protein in various breakfast cereals. The initial grams of protein consist of all whole numbers ranging from 1 to 6. As students begin to find the mean, deviation from the mean, and square of deviation, the students are now interpreting and analyzing more complex numbers ranging from -1.4 and 12.96.
• In Math II Concept 11.1 Apply 2 students create and analyze piecewise functions to determine “How Many T-Shirts Should You Purchase for a Fundraiser?”. Students are provided with a table from two different companies displaying the cost per shirt when purchasing various range of quantities. Best Tees company offers prices that are more simplistic while T-Shirts D-Luxe prices- such as $3.72, $4.08, and $4.64- are more complex.
• In Math III Concept 3.1 Apply 3 students explore how math can help them learn to play the guitar. Students are first asked, “What are the common ratios for the sequences for the string length and frequency?” The string length common ratio ends up being 0.9438, and the frequency common ratio is 1.0594. Students then use this information to predict the string length and frequency of the 13th fret.
• Math I Concept 2.1 provides 15 play problems, and all but two problems result in answers involving integers. One of these two problems results in a fraction, and the other one results in a decimal.

The instructional materials reviewed for the High School Discovery Integrated series meet the expectations that the materials explicitly identify and build on knowledge from Grades 6-8 to the High School Standards. Connections between Grades 6-8 and High School concepts are identified for teachers in the Model Lesson tab as well as the During Previous Instruction section of the Progressions and Standards tab.

In each unit, the Progressions and Standards tab in the teachers materials provides teachers with lists of standards linking previous grade standards as Reach Back, within the current unit, as Standards Covered and future targets as Reach Ahead.

• In Math III Concept 1.1 a graphic is used to link Reach Back Standards to the Standards Covered and Reach Ahead Standards. These are listed as “7.EE.B.3, 8.EE.C.7a, 8.EE.C.7b, 7.EE.B.4a, and 7.EE.B.4b” leading to “HSA-REI.B.3 and HSA-CED.A.3” and then to “HSA-REI.D.11 and HSF-IF.C.7b.”

Session Notes to Teachers call out prior knowledge to be used when appropriate.

• In Math I Concept 1.1 Session 1 the Introduction notes state “Students apply prior knowledge and understanding of variables and algebraic equations to model an observed behavior.”
• In Math I Concept 1.3 Session 2 Investigation notes state “Students are bringing prior experience with interest from Grade 7, but the video that introduces the investigation and the sequence of activities in the investigation provide solid support for students who may not have understood these concepts in prior grades.”