High School - Gateway 2
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Rigor & Mathematical Practices
| Score | |
|---|---|
Gateway 2 - Partially Meets Expectations | 62% |
Criterion 2.1: Rigor | 6 / 8 |
Criterion 2.2: Math Practices | 4 / 8 |
Criterion 2.1: Rigor
Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet the expectations for Rigor and Balance. The materials meet expectations for providing students opportunities in developing procedural skills and balancing the three aspects of Rigor, and the materials partially meet expectations for providing students opportunities in developing conceptual understanding and utilizing mathematical concepts and skills in applications.
Indicator 2a
Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.
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).
Indicator 2b
Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meet expectations for providing opportunities for students to independently develop procedural skills, especially where called for in specific content standards or clusters. Across the series, students independently demonstrate procedural skills throughout the materials, specifically in the homework activities located after every lesson.
Examples showing independent development of procedural skills include:
- In Algebra II, Unit 8, Lesson 3, students transform exponential expressions into equivalent expressions using properties of exponents. In Algebra II, Unit 8, Lesson 4, students transform radical expressions to equivalent expressions with rational exponents and vice versa. In Algebra II, Unit 8, Lesson 5, students transform rational expressions into equivalent expressions (A-SSE.3c).
- In Algebra I, Unit 7, Lesson 1, students add and subtract polynomials. In Algebra I, Unit 7, Lesson 2, students multiply polynomials (A-APR.1).
- In Algebra I, Unit 8, Lesson 3, students transform parabolas to identify the effects on the corresponding quadratic function. In Algebra I, Unit 9, Lesson 3, students transform square root functions to identify the effects on the corresponding function. In Algebra I, Unit 11, Lesson 1, students identify characteristics of function transformations and transform functions. In Algebra I, Unit 11, Lesson 2, students perform horizontal and vertical stretches on functions. In Algebra II, Unit 7, Lesson 1, students transform functions through vertical and horizontal shifts. In Algebra II, Unit 7, Lesson 3, students transform functions through vertical stretching. In Algebra II, Unit 7, Lesson 4, students identify characteristics of horizontal stretching of a function (F-BF.3).
- In Geometry, Unit 5, Lesson 1, students use slope criteria to show lines are parallel. In Geometry, Unit 5, Lesson 2, students use slope criteria to show lines are perpendicular. In Geometry, Unit 5, Lesson 3, students draw a line parallel or perpendicular to a given line passing through a given point. Students then write the equation of the line drawn in slope-intercept form. In Geometry, Unit 5, Lesson 4, students apply the point-slope form of a line to create equations parallel or perpendicular to a given line (G-GPE.5).
- In Geometry, Unit 3, Lesson 4, students use congruence criteria for triangles to prove relationships in triangles. In Geometry, Unit 7, Lesson 6, students use similarity criteria for triangles to prove triangles similar. In Geometry, Unit 7, Lesson 7, students use similarity criteria for triangles to prove relationships in triangles (G-SRT.5).
Indicator 2c
Attention to Applications: The 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 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.
Indicator 2d
Balance: 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.
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Procedural skills and fluency are present independently throughout the materials, and the materials develop conceptual understanding independently. Application problems are often connected with developing procedural skills and fluency throughout the materials.
Examples where all three aspects of rigor are present independently throughout the materials include:
- In Algebra I, Unit 7, Lesson 5 Homework, students build procedural skills by factoring trinomials. The materials separate the trinomials into three groups for students to build procedural skills gradually. In Algebra I, Unit 7, Lesson 6, students continue to build procedural skill by factoring a greatest common factor before factoring the trinomial completely.
- In Algebra I, Unit 8, Lesson 2 instruction, students are instructed to use a calculator to sketch $$y=2x^2$$, $$y=3x^2$$, $$y=4x^2$$, $$y=-2x^2$$, $$y=-2x^2$$, and $$y=-4x^2$$. After sketching the graphs, the materials provide statements about the direction a parabola opens that students must complete based on the quadratic equations graphed. Understanding of the relationship between parts of the equation of a parabola and how the parabola opens is developed.
- In Algebra II, Unit 8, Lesson 2 Homework, students build procedural skill in solving square root equations with one root.
Examples where multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic of study throughout the materials include:
- In Algebra I, Unit 5, Lesson 5, students solve a problem involving a local theater’s ticket prices for children and adults. Students write systems of equations to represent total tickets sold and the amount of profit. Students develop skill in solving systems of equations by calculating the number of child tickets sold and the number of adult tickets sold.
- In Geometry, Unit 9, Lesson 1, the materials provide terminology for students to conceptually understand relationships between parts of a circle. Within the Homework, students develop understanding by identifying parts of a circle and solving problems using characteristics of circles.
- In Algebra II, Unit 13, Lesson 1, the materials provide definitions of types of variability that can occur when collecting data. Students build conceptual understanding by providing real-world examples to represent each type of variability. Within the Homework, students apply understanding of the definitions to identify the type of variability present for several different real-world scenarios.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for Practice-Content Connections. The materials do not identify the mathematical practices for teachers or students which results in the deduction of a point in indicator 2e. The materials intentionally develop the following mathematical practices to their full intent: make sense of problems and persevere in solving them (MP1), reason abstractly and quantitatively (MP2), model with mathematics (MP4), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8).
Indicator 2e
The 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 mathematical practice standards.
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series do not meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. Throughout the series, the materials intentionally develop MP1 to the full intent of the practice standard. However, the materials do not intentionally develop MP6 to the full intent of the practice standard.
Examples that demonstrate the intentional development of MP1 to the full intent of the practice include:
- In Algebra I, Unit 2, Lesson 1, students complete a chart representing the percent of a body affected by a disease over a set amount of days. Students make sense of the data in the chart to provide an explanation to a possible patient about the disease and treatments.
- In Geometry, Unit 2, Lesson 3, students reflect a segment on a coordinate plane across the y-axis followed by a rotation of 90 degrees counterclockwise about the origin. Students determine the coordinates of the endpoints of the transformed segment. Students then use a ruler or tracing paper to determine whether the preimage and image are the same length. Finally, students consider whether their solution makes sense.
- In Algebra II, Unit 6, Lesson 3, students consider the trinomial $$6x^2-35x-6$$ and are given four possible factorizations. Students identify two factorizations that do not make sense and explain why they are unreasonable. Students then consider the two factorizations that are reasonable and determine which of the two are the correct factorizations.
Throughout the series, students are given few opportunities to choose variables when context is provided. The majority of the graphs in the series provide labels on the graph or provide instructions on how the graph is labeled. The materials frequently specify how students should round when calculating solutions, so students do not make sense of a degree of precision that is appropriate for any given context. Examples where the materials do not develop MP6 to the full intent of the practice include:
- In Algebra I, Unit 9, Lesson 6, the materials state, “Many times in applied problems it makes much greater sense to express answers, even if irrational, as approximated decimals.” This is followed by an exercise in which students are instructed to round their answer to the nearest tenth of a second. However, the materials do not provide guidance as to when it’s appropriate to round an answer.
- In Algebra II, Unit 3, Lesson 4, students write a linear model in point-slope and slope-intercept forms to represent the distance driven as a function of time. Materials identify the variables, D for distance and h for the number of hours, rather than students choose variables to represent the situation.
- In Algebra II, Unit 11, Lesson 9, the materials provide students the sinusoidal equation $$O(t)=11cos(\frac{\pi}{12}t)+71$$ representing the temperature of a summer day in New York. Students are instructed to graph the function given. Instead of allowing students to label the graph, the materials provide the graph with the y-axis representing the degrees and the x-axis representing the time in hours. The materials also provide the scale of the graph.
- In Geometry, Unit 10, Review, students find the combined area of fan blades to the nearest square centimeter in part (a) of the problem, calculate the volume of the blades to the nearest tenth of a cubic centimeter in part (b), and determine the density of the fan blades to the nearest hundredth of grams per cubic centimeter in part (c). The materials provide no rationale for why students express their answers to each of the different levels of precision for different parts within the same problem.
Indicator 2f
The 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 mathematical practice standards.
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.
Indicator 2g
The 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 mathematical practice standards.
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.
Indicator 2h
The 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 mathematical practice standards.
The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meets expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards.
Examples of MP7 being used to enrich the mathematical content to the full intent of the practice include:
- In Algebra I, Unit 2, Lesson 2, students consider the structure of equations in order to perform inverse operations. Within the same lesson, the materials instruct students to solve -2(x-4)+8=2 by reversing operations and using the distributive property first before solving equations. Students create equations from verbal expressions before solving.
- In Geometry, Unit 10, Assessment Form B, students consider a model for a plastic rocket that is composed of a cone and cylinder with a hemisphere hollowed out of the bottom of the cylinder. Students use structure to focus on the individual objects, rather than the entire model, in order to calculate the volume of the plastic that remains in the cylinder after it has been hollowed out.
- In Algebra II, Unit 10, Lesson 6, students consider the equivalent relationship between $$\frac{x^2- 4}{2x - 4}$$ and $$\frac{x + 2}2$$. Students rewrite the first expression to show the two expressions are equivalent. Then, students use a calculator to complete a table where they evaluate each expression for given values of x ranging from 0 to 4. Finally, students explain why the two expressions are not equivalent for x = 2.
- In Algebra II, Unit 11, Lesson 3, students explore patterns using the Pythagorean Theorem. The materials provide right triangles where students use the Pythagorean theorem to solve for missing values in radical form. Students use their discoveries to determine ordered pairs for different angles on the unit circle.
Examples of MP8 being used to enrich the mathematical content to the full intent of the practice include:
- In Algebra I, Unit 1, Lesson 7, the materials provide the steps for rewriting the expression, $$(2x^3)^3$$. Students write justifications to explain each of the steps, and students use repeated reasoning to write the extended products of $$(2^2)^4$$ and $$(x^3)^4$$ and write the extended products in the form of $$2^n$$ or $$x^n$$.
- In Geometry, Unit 3, Lesson 3, students prove triangles congruent using side-angle-side, angle-side-angle, and side-side-side criteria. Students attend to details to determine which congruent criteria will prove the triangles congruent. Within the same lesson, students choose a congruence theorem to justify why two triangles are congruent and explain their choice to evaluate the reasonableness of their solution.
- In Algebra II, Unit 5, Lesson 3, students find the sum for a variety of arithmetic sequences. Students summarize the results by determining the types of numbers used within the sequence in order to establish a rule for the sum of the arithmetic series.
- In Algebra II, Unit 10, Lesson 14, the materials state, “Squaring both sides of an equation is irreversible. Is cubing both sides of an equation reversible?”. Students answer the questions and provide numerical evidence to support their claim in order to evaluate the reasonableness of their solution.