High School - Gateway 2

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Across the series, conceptual understanding is developed in the instructional portion of the lessons as students are guided through the mathematical concepts by the teacher and/or instructional videos. The instructional videos explain all aspects of the concepts of the lessons, so the instructional materials do not provide opportunities for students to independently demonstrate conceptual understanding throughout the series. 

Examples showing the development of conceptual understanding in the instructional portion of lessons and not allowing students to independently demonstrate their conceptual understanding include:  

• In Algebra II, Unit 4, Lesson 2 instruction, students, with teacher and/or instructional video guidance, observe how the expression $$16^{\frac{1}{2}}$$ is equivalent to $$\sqrt{16}$$. Students then test this observation by evaluating $$25^{\frac{1}{2}}$$, $$81^{\frac{1}{2}}$$, and $$100^{\frac{1}{2}}$$. The materials state the general relationship $$b^{\frac{1}{n}} =\sqrt[n]b$$, and students rewrite expressions with fractional exponents into expressions with radicals. The lesson concludes with materials stating the general relationship $$b^{\frac{m}{n}} =\sqrt[n]{b^m}$$, and students rewrite three expressions using this equivalence statement. In Lesson 2 Homework, students independently develop their fluency in rewriting and evaluating expressions involving rational exponents without having to explain how the definition of the meaning of rational exponents extends from the properties of integer exponents (N-RN.1).
• In Algebra I, Unit 8, Lesson 6 instruction, students, with teacher and/or instructional video guidance, identify the zeros of a graphed quadratic function, $$y=x^2-2x-3$$, and factor the quadratic expression, $$x^2-2x-3$$. The materials state the Zero Product Law, and the subsequent exercises use the Zero Product Law to find solutions of quadratic functions. While students encounter the relationship between zeros and functions of a quadratic, students do not investigate or explain the relationship on their own in this lesson. In Algebra I, Unit 8, Lesson 7 instruction, students, with teacher and/or instructional video guidance, find the zeros of a quadratic function algebraically and sketch a graph of the quadratic function. In Lesson 7 Homework, students consider a quadratic and cubic function and (i) find the zeros algebraically and (ii) sketch a graph of the function using their calculator. Students do not independently construct a graph using the zeros they calculated algebraically (A-APR.3).
• In Algebra II, Unit 9, Lesson 3 instruction, students encounter quadratic equations with complex solutions. Students, with teacher and/or instructional video guidance, consider the quadratic equation, $$y=x^2-6x+13$$, algebraically find the x-intercepts, and write the answers in equivalent complex forms. Students sketch a graph using their calculator before making an observation between the number of x-intercepts on the graphed parabola and the complex roots found algebraically. This is followed by an exercise in which students calculate the discriminant of several quadratic expressions to determine whether the given quadratic equations have x-intercepts. In Lesson 3 Homework, students use the discriminant to determine whether given quadratic equations have real or imaginary zeros (with no connection to graphing). In this lesson, students do not independently develop their conceptual understanding of recognizing when quadratic equations have complex solutions (A-REI.4b).
• In Algebra I, Unit 4, Lesson 10 instruction, the materials state “Any coordinate pair (x, y) that makes the equation or inequality true lies on the graph. Any coordinate pair (x, y) that makes the equation or inequality false does not lie on the graph.” In Lesson 10 Homework, students do not independently develop their understanding of this standard as they substitute (x, y) values to determine if a point lies on the graph of an equation (A-REI.10).

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially 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. The instructional materials do not routinely engage students in non-routine applications of mathematics throughout the series. The majority of the application problems are extensions of content requiring students to use skills introduced within the same lesson and not spanning unit or course topics. 

Throughout the materials, students have limited opportunities to answer questions using numerous methods. The materials often instruct students to solve problems using a specific mathematical procedure, which makes the problems routine. Examples that show missed opportunities to engage in non-routine problems throughout the series include:

• In Algebra I, Unit 8, Lesson 6, students determine how long before a baking soda rocket fired upward hits the ground. Students use the given quadratic function $$h(t)=-16t^2+ 80t$$ and calculate the time algebraically by factoring. The scenario follows the lesson on factoring to calculate the zeros of a function, and the materials instruct students how to calculate solutions in one way. In Algebra II, Unit 8, Lesson 6, students use the given equation $$h=-9.8t^2+32.2t+6.5$$ that represents the height of a missile above the ground. Students do not choose a method for calculating the solution, instead the materials instruct students to use the quadratic formula to calculate when the rocket will hit the ground (A-REI.4). 
• In Geometry, Unit 8, Lesson 1, students estimate the height of a ladder leaning against a wall. Students are given a model to represent the scenario and do not draw their own models to represent the scenario. The problem follows the lesson on similar, right triangles, and students use similar, right triangles to create a ratio to solve the problem (G-SRT.6).
• In Algebra II, Unit 6, Lesson 9, the materials provide a function, $$h(t)=-16t^2+80t+30$$, for the height of a tennis ball t seconds after it is thrown upwards. Students are instructed by the materials to determine the time of the tennis ball’s greatest height algebraically. Students are also directed to use a graphing calculator to sketch a graph of the ball’s height where $$t\geq0$$ and $$h\geq0$$. The materials then state for students to use the zero command on the calculator to calculate the time the ball stays in the air (F-IF.4). Students do not solve the scenario using different techniques since the materials provided instructions for calculating the solution.
• In Algebra II, Unit 3, Lesson 4, the materials provide a scenario of a rocket with 225 gallons of fuel taking off and using fuel at a constant rate of 12.5 gallons per minute. Students follow directions to create a linear model in the form, y = ax + b, where y is the amount of fuel and x is the number of minutes. The materials provide students with a general graph representing the scenario and instruct students to use a calculator to determine the intercepts of the model. The materials direct students to calculate the maximum number of minutes by graphing the horizontal line y = 50 to show the point of intersection (S-ID.C). The materials provide instruction for solving the problem which allows for one way of solving the scenario.

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. The materials intentionally develop MP2 to the full intent across the series, but the materials do not intentionally develop MP3 to the full intent across the series. 

Examples that demonstrate the intentional development of MP2 across the series include: 

• In Algebra I, Unit 10, Lesson 6, the materials provide a data set comparing the price people pay for their most expensive car and the current value of their house and a linear regression equation modeling the scenario. Students use the regression equation to quantitatively make predictions. Additionally, students reason abstractly about a possible causal relationship between the two variables. If students believe a causal relationship exists, then students identify which variable causes the other; whereas, if students do not believe a causal relationship exists, then students identify a third confounding variable.
• In Geometry, Unit 10, Lesson 10, the materials provide a model of a pill shaped like a cylinder with a hemisphere at each end. Students calculate the volume of the pill given the height of the cylinder as 12 mm and the height of the pill as 18 mm. Students determine the amount of Vitamin C a pill contains per cubic millimeter if the pill contains 100 milligrams of Vitamin C. Students compare the amount of Vitamin C in a second spherical pill with a diameter of 10 mm containing 1.5 milligrams of Vitamin C. In this problem, students attend to the meaning of quantities as they perform calculations with different units.
• In Algebra II, Unit 5, Lesson 2, students work with the Koch Snowflake and examine the perimeters of the first six iterations. The materials provide the perimeter of the first snowflake, and students quantitatively determine the perimeter of the second and sixth snowflakes. Students reason abstractly to explain why the perimeter would become infinitely large if the iteration process continued forever.

Throughout the series, the materials, and not students, make conjectures that students must examine or prove. Students do not recognize counter-examples when breaking down situations. While students justify or explain their reasoning throughout the series, students do not communicate their thinking to others or respond to arguments from other students. Students have limited opportunities to read the arguments of others and decide if they make sense, and students do not ask questions to clarify or improve the arguments.

Examples where the materials do not develop MP3 to the full intent of the practice include:

• In Algebra I, Unit 1, Lesson 6, students examine provided student work that represents a pattern. The materials tell students that the work is wrong, and students show why the work is wrong. Students find the correct pattern and explain their reasoning. The students do not independently determine if the student work is correct or incorrect.
• In Geometry, Unit 7, Lesson 8, students are given a model involving a triangle and parallel lines. Students explain how Francine incorrectly solved for a missing side length by using the Side Splitter Theorem, and students calculate the correct length. Students do not independently determine if the student work is correct or incorrect.
• In Algebra II, Unit 6, Lesson 5, the materials state, “Be careful when you use factoring by grouping. Don’t force the method when it does not apply. This can lead to errors.” After this, students explain the error in the factored expression of $$2x^3+10x^2+7x+21$$. Students do not independently determine if the work is correct or incorrect.

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. The materials provide opportunities for students to model with mathematics (MP4), but MP5 is not developed to the full intent of the practice.

Examples of MP4 being used to enrich the mathematical content to the full intent of the practice include: 

• In Algebra I, Unit 11, Lesson 4, students determine a linear function to model the volume of water in a tank over time. Students also determine an exponential function to model volume as a function of time. Lastly, students determine whether the exponential function or linear function is a better model for the problem. 
• In Geometry, Unit 10, Lesson 5, students determine the weight of three propellor blades by calculating the area of each sector. While students are provided a diagram, radii, and angle measures, students create their own equations to solve the problem.
• In Algebra II, Unit 4, Lesson 6, students determine, “In how many years, to the nearest year, will Red Hook have a greater population than Rhinebeck?” Students create and graph exponential equations to determine their point of intersection.

The instructional materials often specify tools needed to solve problems, so students have limited opportunities to choose the appropriate tools to use for a scenario. Examples of the materials not developing MP5 to the full intent of the practice include:

• In Algebra I, Unit 8, Lesson 7, students find the zeroes of the polynomial equation, $$y=x^3+2x-8x$$. The materials instruct students to use a graphing calculator and also provide a graph indicating the maximum and minimum values of the x- and y-axes. Students do not choose the tool needed, and students do not choose how they use the tool to sketch the graph.
• In Algebra I, Unit 9, Lesson 7, students use the Quadratic Formula to determine which functions have real zeroes and which do not have real zeroes. The materials instruct students to use a graphing calculator using the standard viewing window to verify their solution. Students do not choose a tool to verify their solutions.
• In Geometry, the materials provide icons at the beginning of each lesson to indicate the tool needed to complete the lesson. For example, a ruler symbol would indicate a ruler is used in the lesson. In Geometry, Unit 1, Lesson 5, students create constructions to explain or verify the characteristics of circles. The materials include the compass, ruler, and protractor icons at the beginning of the lesson. Within the same lesson, the materials explicitly instruct students when to use a compass and a straight edge. Students do not choose a tool appropriate to the lesson.
• In Algebra II, Unit 1, Lesson 6, students solve expressions given an x-value. Students are instructed to use the STORE feature on the calculator to evaluate each expression. Additionally, the materials state, “The STORE feature is particularly helpful in checking to see if a value is a solution to an equation.” After the statement, students are instructed to solve the equation 6x-3=4x+9. Students use the STORE feature to determine if their solution is correct. Students do not choose the tool needed to verify their solution.