High School - Gateway 2

The instructional materials reviewed for the Glencoe Traditional series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.The instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding and most of the opportunities for students to develop conceptual understanding are located in the Interactive Student Guides (ISG). 

Examples of the materials developing conceptual understanding and students independently demonstrating conceptual understanding include: 

• A-APR.1: In the Algebra 1 ISG, Adding and Subtracting Polynomials, students compare the behavior of polynomials to integers under addition and subtraction and write a conjecture describing closure under addition and subtraction of polynomials. In the Algebra 1 ISG, Multiplying a Polynomial by a Monomial, students determine if polynomials are closed under multiplication by proving the product of a monomial and a polynomial is a polynomial.
• A-REI.11: In the Algebra 2 ISG, Solving Polynomial Equations, students use a graphing calculator to sketch $$f(x)=x^4-2x-5$$ and $$g(x)=-x^6+3x^2+3$$ and estimate the points of intersection. The materials state, “Explain the algebraic significance of a point of intersection on the graphs of f(x) and g(x).” Within the same lesson, students critique the reasoning of a student who claims two functions have no solutions due to no intersections and explain the reasoning behind their critique.
• F-LE.1: In the Algebra 1 ISG, Analyzing Functions with Successive Differences, students complete a table of values for a linear function, a quadratic function, and an exponential function. Students compare charts with other students in order to make conjectures about each function. Students “look for connections between different function types and their successive differences and successive ratios.” Throughout the lesson, students explain connections between the type of function and the model of the scenario.
• G-SRT.6: In the Geometry ISG, Trigonometry, students explore ratios in similar triangles by using Geometer’s Sketchpad. Students explain what they notice about side lengths when changing the size of the triangle. Students also examine two triangles with different side lengths and congruent angles to “explain why $$\frac{BC}{AB}=\frac{EF}{DE}$$. What does this tell you about any right triangle with a 37° angle.” From the task, students determine trigonometric ratios for right triangles with an acute angle.
• S-CP.5: In the Geometry ISG, Conditional Probability, students are given a scenario where event A represents owning a house and event B represents owning a car. Students must determine if the events are independent or dependent as well as compare the P(A|B) to P(B|A). After making the comparison, students must explain their reasoning using precise vocabulary.

The instructional materials reviewed for the Glencoe Traditional series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills throughout the series and students independently demonstrate procedural skills across the courses. 

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

• A-SSE.2: In Algebra 1, Lesson 8-5, students rewrite quadratics into equivalent factored forms. Students use the distributive property and factoring by grouping in order to rewrite quadratic expressions and polynomials.
• A-APR.3: In Algebra 2, Lesson 4-9, students factor polynomials to calculate the number and type of roots and sketch graphs based on the zeros. Students also match an appropriate graph to a given zero.
• F-IF.7e: In Algebra 2, Lesson 6-1, students graph exponential functions and state the domain and range of the function. In Algebra 2, Lesson 6-4, students graph logarithmic functions independently. In Algebra 2, Lesson 9-6, students graph trigonometric functions after identifying amplitude, period, vertical shift, and the equation of the midline.
• G-GMD.3: In Geometry, Lesson 11-2, students calculate the volume of different prisms, cylinders, and composite solids. In Geometry, Lesson 11-2, Preparing for Assessment, students calculate the volume of various three dimensional shapes.

The instructional materials reviewed for the Glencoe Traditional series meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially when called for in specific content standards or clusters. Students engage in routine and non-routine applications of mathematics throughout the series and demonstrate the use of mathematics flexibly in a variety of contexts.

Examples of students engaging in routine and non-routine application of mathematics and demonstrating the use of mathematics flexibly in a variety of contexts include: 

• F-TF.5: In Algebra 2, Lesson 9-6, students solve a non-routine application by writing a trigonometric function to model the movement of carousel horses according to a list of provided constraints. Students also consider how to make changes to their function if the characteristics of the carousel change.
• G-SRT.8: In the Geometry ISG, Angles of Elevation and Depression, students use trigonometric ratios to write a formula to calculate the height of a rock formation knowing the height of a geologist is 1.8m. Students also calculate the width of a lake given a helicopter with an altitude of 450 meters and two angles of depression.
• G-MG.2: In Geometry, Lesson 11-7, students determine a solid with the greatest density when given certain conditions. The materials state, “A cylinder and a sphere have the same mass. The height of the cylinder is equal to its radius. The radius of the sphere is equal to the radius of a cylinder.” Students determine the greater density and explain their reasoning.
• S-ID.2: In the Algebra 1 ISG, Comparing Sets of Data, students plan a vacation based on weather history. Students examine two charts showing the rainy days over a 10-year period. Students use the data to “determine the shape of each distribution and use the appropriate statistics to find the center and spread for each set of data.” Students provide advice on the better place to visit for vacation and explain their reasoning.

The instructional materials reviewed for the Glencoe 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. The majority of the time MP2 and MP3 are used to enrich the mathematical content and are intentionally developed to reach the full intent of the MPs. 

Examples of MP2 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• In Algebra 1, Chapter 8 Performance Task, students write expressions to represent different perimeters and areas of swimming pools, hot tubs, and lap pools. Students reason abstractly and quantitatively while computing both perimeters and areas.
• In the Geometry ISG, Dilations, students use a figure representing an architect’s plan for a bedroom where one unit is equal to one foot. Students enlarge or reduce the size of the bedroom based on given perimeters. Students also “write an equation that can be used to find the scale factor (x) of a dilation given any perimeter (y).” Students reason and make connections between the drawing and the physical dilation that will occur.
• In Algebra 2, Lesson 6-1, students reason to write functions representing situations. Students determine if the function represents exponential growth or an exponential decay, identify the growth factor, and graph the function.

Examples of MP3 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• In Algebra 1, Lesson 2-5, students perform an error analysis on the work of fictional students. Students determine if the fictional students are correct and explain any error in their reasoning.
• In the Geometry ISG, Proving Triangles Congruent ASA, AAS, the materials state, “Raj says that he can draw two triangles that have two sides and a nonincluded angle congruent and that the two triangles are congruent.” Students agree or disagree with Raj justifying his claim with the SSA Congruence Theorem and provide a counterexample to disprove Raj’s claim.
• In the Algebra 2 ISG, Designing a Study, students analyze a study on cat food and weight gain. Students calculate the mean weight of the cats and determine if the difference in weights is significant or not. Students construct an argument to explain their reasoning.

The instructional materials reviewed for the Glencoe Traditional series meet expectations for supporting 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 and are intentionally developed to reach the full intent of the MPs.

Examples of MP7 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• In the Algebra 1 ISG, Solving Systems by Elimination, students explain two different pathways to solve a system of equations using elimination. Then students use one of the pathways described to calculate a solution to the system. In order to identify multiple solution pathways, students make use of structure to analyze the task.
• In the Geometry ISG, Dilations, students dilate a triangle using different scale factors. Students make use of structure to find patterns when dilating with a scale factor less than 1, equal to 1, or greater than 1.
• In the Algebra 2 ISG, Properties of Logarithms, students make use of structure to approximate solutions of $$log_{10}a=0.903$$ and $$log_{10}b=2.477$$ when given $$log_{10}2\approx0.301$$ and $$log_{10}3\approx0.477$$. Students approximate the solutions without the use of technology.

Examples of MP8 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:

• In the Geometry ISG, Angles and Arcs, students investigate the proportionality between arc length and the radius of a circle. Students calculate arc lengths for circles of different radii in order to generalize the relationship between arc length and radius.
• In the Geometry ISG, Similar Triangles, students solve equations for the scale factor in order to write equal ratios. Students “state a generalization based on your findings.”
• In the Algebra 2 ISG, Real and Complex Numbers, students use a calculator to find the values of (1+i)$$^2$$, (1+i)$$^4$$, (1+i)$$^6$$, (1+i)$$^8$$, (1+i)$$^10$$,(1+i)$$^12$$. Students also calculate the solution of (1+i)(1+i) without using a calculator. Students use repeated reasoning to generalize a solution pathway of (1+i)$$^4$$ without using a calculator.