High School - Gateway 2
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Rigor & Mathematical Practices
| Score | |
|---|---|
Gateway 2 - Partially Meets Expectations | 81% |
Criterion 2.1: Rigor | 7 / 8 |
Criterion 2.2: Math Practices | 6 / 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 materials reviewed for Big Ideas Learning AGA meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Overall, conceptual understanding and procedural skills are developed and demonstrated by students, but students are provided limited opportunities to engage in non-routine application of mathematics.
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 materials reviewed for Big Ideas Learning AGA meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Many of the opportunities for conceptual understanding are in the Explore It, activities located at the beginning of every chapter. In the Teacher Edition, there are also additional opportunities for conceptual understanding given in Laurie’s Notes.
Examples across the series that develop conceptual understanding include:
A-APR.2: In Algebra 2, Chapter 4, Section 3, Explore It, students use technology to explore the graph of $${{x}^{3}}+2{{x}^{2}}-x-2$$ divided by the binomial x+a for given values of a. Students develop conceptual understanding through making observations about the graphs and drawing conclusions about their observations.
G-SRT.6: In Geometry, Chapter 9, Section 4, Explore It, students use technology to construct a right scalene triangle with several perpendicular segments drawn to the base of the triangle. Students use the definition of tangent to write the tangent ratios for the acute angles and then compare the values. Students generalize their finding by determining if the size of the right triangle affects the tangent values or whether the angle measure affects the tangent ratios. In Geometry, Chapter 9, Section 4, students demonstrate understanding of tangent ratios by explaining how the tangent of an acute angle in a right triangle changes as the angle measure increases.
S-ID.6a: In Algebra 1, Chapter 4, Section 4, Explore It, students are given data from a survey of 179 married couples. Each person in the survey gave their age and the data collected was shown in a graph. Students develop conceptual understanding of linear functions suggested by the data to make observations, notice patterns in the data, and explain how patterns represent data.
Examples across the series where students independently demonstrate conceptual understanding include:
N-RN.1: In Algebra 2, Chapter 5, Section 1, students are given the expressions $${{({{a}^{\frac {1} {n}}})}^{m}}$$, $${{({\sqrt[{n}] {a}})}^{m}}$$, $${{({\sqrt[{m}] {a}})}^{-n}}$$, and $${{a}^{\frac {m} {n}}}$$. Students must determine which expression does not belong and justify their decision.
A-REI.12: In Algebra 1, Chapter 5, Section 7, students are given a graph of two linear equations with five plotted points. Students must replace the equal sign in the two graphed linear equations with inequality symbols such that two of the plotted points are solutions.
G-CO.1: In Geometry, Chapter 1, Section 1, students develop conceptual understanding of parallelism by determining if parallel lines always, sometimes, or never intersect. Within the same sections, students also determine if two points always, sometimes, or never form a line.
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 materials reviewed for Big Ideas Learning AGA meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Opportunities for students to independently develop procedural skills are located within the Self-Assessment exercises, the Practice exercises, the Review and Refresh exercises, and the supplemental resources.
Examples of the materials developing procedural skills and students independently demonstrating procedural skills include:
A-SSE.2: In Algebra 2, Chapter 6, Section 1, students justify the steps needed to rewrite an exponential function. In Algebra 2, Chapter 6, Section 3, students rewrite functions in exponential and logarithmic form. In Algebra 2, Chapter 6, Section 5, students rewrite logarithmic expressions by condensing or expanding them.
A-APR.1: In Algebra 1, Chapter 7, Section 1, students independently calculate the sum and difference of polynomials. In Algebra 1, Chapter 7, Section 2, students independently calculate products of polynomials using the Distributive Property and a table. In Algebra 1, Chapter 7, Section 3, students independently calculate the products of polynomials.
F-BF.4a: In Algebra 2, Chapter 5, Section 7, students find inverses of linear, quadratic, cubic, square root, and cube root functions.
G-GPE.4: In Geometry, Chapter 5, Section 8, students write coordinate proofs to prove geometric theorems. In Geometry, Chapter 10, Section 7, students use coordinates to prove or disprove whether a coordinate lies on a circle.
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 materials reviewed for Big Ideas Learning AGA partially meets expectations that the materials support 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. Most sections within the series have “Modeling Real Life” exercises where students independently engage in routine application problems, but the materials have missed opportunities for students to engage in non-routine application of mathematics.
Examples that show missed opportunities for students to engage in non-routine application of mathematics throughout the series include, but are not limited to:
A-CED.3: In Algebra 1, Chapter 5, Section 6, students are given the weight of two large boxes and three small boxes as well as the weight of one large box and four small boxes. Students are asked to write and graph an inequality representing the amount of large and small boxes a 200 pound person can carry in an elevator with a 2000 pound weight limit. Students are instructed how to solve the problem using a similar mathematical procedure that is found within the section.
A-REI.4: In Algebra 2, Chapter 3, Section 3, students are given the equation y^2+10y+20x-15=0. Students are instructed to write the equation in standard form and identify the vertex, focus, and directrix. Students are not engaging in non-routine application, since they are being instructed what to do.
F-IF.6: In Algebra 1, Chapter 10, Section 2, students are instructed to write a cube root function given that it passes through the point (3, 4), and has an average rate of change of -1 over the interval of x= -5 to x=2. Within the same section students are given an example showing how to compare the average rate of change of a cubic function graphed and a cubic function. During the solution steps of the example, students are shown how to calculate the rate of change of a cubic function over a given interval.
F-IF.7a: In Algebra 2, Chapter 2, Section 2, the materials provide an example with a graph modeling the path of a golf ball and an equation modeling the path of a second golf ball. In the example, students determine which shot travelled higher and which shot travelled farther before hitting the ground. Within the same sections, students are given a verbal description of a kicker soccer ball and an equation modeling the path of a second kicked soccer ball. Students determine which ball travelled high and which ball traveled farther before hitting the ground.
G-SRT.8: In Geometry, Chapter 9, Section 5, the materials provide an example where students use trigonometric ratios to calculate the distance from a base of a mountain to the student. The example provides a visual of the mountain and the pertinent information needed to calculate the distance. Within the same section, students independently use the visual given in the example to calculate the distance from the base of the mountain to the student given a new angle of depression.
G-GMD.3: In Geometry, Chapter 12, Section 5, students research the amount of propane needed to heat their current residence in a year. Students are given a model of a propane tank with a height of 37.5 inches and a width of 9 feet 11 inches. Students must calculate the volume of the propane tank to determine how often the tank will need to be refilled. Although there is some research involved where students need to calculate the amount of propane needed to heat their residence, the calculation of the volume is routine.
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 materials reviewed for Big Ideas Learning AGA meet expectations 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. The Explore It sections contain multiple aspects of rigor to develop students’ mathematical understanding while the independent practice sections are mainly procedural with a balance between conceptual understanding and application throughout the section.
Application problems are presented throughout the materials in the Explore Its, Independent Practice, and Self-Assessments. Examples of procedural skills and conceptual understanding being presented independently throughout the materials include:
In Algebra 1, Chapter 7, Section 5, students develop procedural skills by factoring polynomials and solving polynomials by factoring (A-SSE.3a).
In Algebra 2, Chapter 4, Section 1, Explore It, students develop conceptual understanding by identifying functions that are polynomials, graphing the polynomials, describing end behaviors, and determining the effects of exponents and leading coefficients. Students compare two graphs to determine if the graphs are cubic or quartic. Conceptual understanding is also developed when students generalize characteristics of cubic functions and quartic functions (F-IF.7c).
Examples of multiple aspects of rigor being engaged simultaneously to develop students mathematical understanding of a single topic of study throughout the materials include:
In Algebra 1, Chapter 8, Section 6, Explore it, students develop conceptual understanding through an application problem involving three cars traveling at the same time at different speeds. Students develop conceptual understanding by determining which car has a constant speed and which car has the greatest acceleration (F-LE.3).
In Geometry, Chapter 9, Section 6, Explore it, students develop procedural skills by calculating measure of an angle when given a trigonometric ratio. Students demonstrate conceptual understanding by explaining how technology can verify angle measures and explaining how to calculate the measures of two acute angles when given side lengths. Within the same section, students use procedural knowledge of trigonometry to calculate the acute angles in a right triangle created by leaning a firefighter ladder against a building (G-SRT.8).
In Algebra 2, Chapter 8, Section 3, Explore it, students develop procedural skills by determining different probabilities involving six pieces of paper numbered 1 through 6. Students develop conceptual understanding by comparing and contrasting the probabilities found. Within the same section students are given two events. Event A is the probability of the first number being divisible by 3 and Event B is the probability of the second number being greater than 2. Students calculate P(B|A), P(A and B), and P(A). Students develop conceptual understanding by using the previous probabilities to generalize a formula for P(B|A) in terms of P(A and B)and P(A) (S-CP.6).
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The materials reviewed for Big Ideas Learning AGA partially meet expectations for Practice-Content Connections. The materials intentionally develop the following mathematical practices to their full intent: reason abstractly and quantitatively (MP2), construct viable arguments and critique the reasoning of others (MP3), use appropriate tools strategically (MP5), attend to precision (MP6), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8). The materials partially meet expectations for intentionally developing the following mathematical practices: make sense of problems and persevere in solving them (MP1) and model with mathematics (MP4).
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 materials reviewed for Big Ideas Learning AGA partially meet expectations that the materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. Although the materials use MP6 to enrich the mathematical content, there are many examples of misleading identifications of MP1.
Examples of misleading identification of MP1 include, but are not limited to:
In Algebra 1, Chapter 6, Section 4, students calculate the approximate monthly percent increase in a population and estimate the population after 4 years given an initial population of 25,000 with an annual percent increase of 5.5%. This problem solving task prompts you to look at Example 6 since both tasks are similar, therefore students will not have to make sense of the problem to determine that they can replicate the problem-solving method from Example 6.
In Algebra 1, Chapter 11, Section 4, students are given expressions to represent the amount of child and adult tickets sold for both the main floor and a balcony. Students are then asked to find what percent of tickets are adult tickets and what percent of child tickets are balcony tickets. Since this problem is in a section about making two-way tables, students will not have to make sense of the problem to determine that a two-way table can be used as an entry point to the solution.
In Geometry, Chapter 4, Section 6, students have to determine whether the composition of rotations and dilations preserve the commutative property. The students will not need to persevere as these properties of rotations and dilations have already been explored in 8th grade mathematics.
In Geometry, Chapter 13, Section 3, students are given a table to determine if a company should change the recipe of a snack, based on whether the consumer market will change very little or expand very rapidly. Similar examples are presented throughout the section where students have to find conditional probability and use conditional probability to make a decision. Students will not need to make sense of the problem to determine that the probability models from this section will allow them to solve this task.
In Algebra 2, Chapter 6, Section 1, students are given a formula that predicts the eggs a chicken can produce given its age in weeks. Similar problems are completed in this section so students do not need to make sense of the problem to look for entry points to the solution.
In Algebra 2, Chapter 7, Section 4, students determine the time it would take to finish a race given a table showing rates of swimming, biking, and running. This task does not allow students to persevere in the grade level math because students can substitute the given value from the question into the expressions given. This will allow them to find the total time it takes to complete the race without having to persevere to solve this problem.
Examples where MP6 is used to enrich the mathematical content includes:
In Algebra 1, Chapter 11, Section 3, students describe why the mean is used to describe the center and the standard deviation is used to describe variation for a symmetric data set. Students also describe why the median is used to describe the center and the 5 number summary is used to describe the variation for a skewed data set.
In Geometry, Teacher Edition, Chapter 1, Section 4, the materials include an example explaining how to calculate the perimeter of a triangle in a coordinate plane. The distance formula is used to find the lengths of two sides and lengths are expressed as radicals. After finding the perimeter, the final solution is a decimal. Laurie's notes extends the example by having students discuss the difference between leaving the answer as an exact answer using radicals versus an approximation using decimals.
In Algebra 2, Chapter 8, Section 1, Explore It, students describe possible outcomes of different experiments. Students must then understand mathematical terms by discussing the difference between an event being impossible and an event being certain.
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 materials reviewed for Big Ideas Learning AGA 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:
Algebra 1, Chapter 3, Section 5, Example 3 provides the equation 6x+10y=180 to model the number of people that can sit at a small table and a large table at a banquet for 180 people. The material include a Mathematical Practice box asking students what the terms 6x and 10y represent in this context. Students are attending to the meaning of quantities.
In Geometry, Chapter 4, Section 3, students determine the coordinates of two endpoints after rotation 630 degrees and 900 degrees about the origin. Students reason abstractly to determine the new coordinate for the endpoints.
In Algebra 2, Chapter 5, Section 7, students are given the ordered pairs (-2, 5), (0, 1), (3, -6), and (7, n) and are told a function passes through the points. Students find values of n so that the inverse is a function and must explain their reasoning.
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, Chapter 7, Section 7, Exercise 41, students describe two methods use to simplify $${{(2x-5)}^{2}}-{{(x-4)}^{2}}$$. Then, students choose which of the two methods they would use to simplify the expression and provide an explanation about their choice.
In Geometry, Teacher Edition, Chapter 7, Section 2, Laurie’s Notes, students write a proof for the Parallelogram Opposite Sides Theorem. After writing the proof, students compare and critique proofs made by other students.
In Algebra 2, Chapter 9, Preparing for Chapter 9, students are given a table of results describing 3 different surveys used to determine if people would like more funding to monitor volcanic activity. Students collaborate with a partner to explain confidence levels in the conclusions of the results. Finally, students must write a survey question and sample to use for a valid conclusion.
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 for Big Ideas Learning AGA partially meet expectations that the materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. Although the materials use MP5 to enrich the mathematical content, materials do not develop the full intent of MP4.
Examples where the full intent of MP4 is not developed include, but are not limited to:
In Algebra 1, Chapter 5, Section 1, students determine whether two people starting at different points of a hiking trail, who are walking at different speeds will meet after one hour of hiking and also determine how much longer would it take the second finisher to complete the hiking trail than the first. Students are not able to make a conjecture because the conjecture is made for them via the friend’s suggestion. Additionally, students are not prompted to build or improve upon a model which represents the problem.
In Algebra 1, Chapter 9, Section 5, students write an equation to model the height of a football after being kicked with a given initial height and vertical velocity. Students are instructed to use the function h=-16t^{2}+v_{0}t+s_{0} where h is the height, t is the time, v_{0} is the initial vertical velocity, and s_{0} is the initial height. Students need only substitute the given information into the given equation.
In Geometry, Chapter 1, Section 5, students are given an image of a bird sculpture and are instructed to find the measure of the bird’s total wingspan. In the model, students are given two angles and their corresponding measures. Students subtract the two measures to find another angle. The model is provided instead of allowing students to create a model to represent the scenario.
In Geometry, Chapter 10, Section 4, students are asked to determine the length of a line segment that shows a seam on a circular cutting board. A model of a right triangle is drawn on an image of the cutting board, so students do not need to create a model or determine a strategy to solve the task.
In Algebra 2, Chapter 3, Section 3, students calculate the maximum height of a birdie and the amount of time it takes for the birdie to hit the ground. Students are given the model h=-16t^{2}+32t+4 where h represents height and t represents seconds after the birdie is hit.
In Algebra 2, Chapter 6, Section 6, students calculate the diameter of a telescope lens revealing stars with an apparent magnitude of 12. Students are given M=5logD+2 where M represents magnitude and D represents diameter.
In Algebra 2, Chapter 10, Section 6, students are tasked with volunteering to clean a beach at a large body of water 3 days from now. Students are told that the best time to do this is low tide. Students have to find the accurate times and depths of low and high tides on a recent day at a location of their choice by researching. Then they are instructed to write a sinusoidal model using this data. Students do not need to determine a representation or strategy as it is given to them.
Examples where MP5 is used to enrich the mathematical content includes:
In Algebra 1, Chapter 7, Section 6, Explore It, students use algebra tiles to factor polynomials that do not have a leading coefficient of 1. Students are given an algebra tile model. Students then write the polynomial represented by the algebra tiles and write the polynomial in factored form. Students must also describe a strategy to factor a trinomial that does not use algebra tiles. Students are also asked, “Why might algebra tiles be an inefficient way to factor polynomials involving greater numbers, such as 2x^{2}+47x+23?” This reminds students about the limitation of using algebra tiles as a tool.
In Geometry, Chapter 6, Section 1, Explore It, students use technology to explore the distances between any point on a perpendicular bisector and the distances between any point on the angle bisector and the sides of the angle. Students choose which technology tools to utilize to explore the relationships.
In Algebra 2, Chapter 4, Section 1, students are given the following scenario: “Your friend uses technology to graph f(x)=(x-1)(x-2)(x+12) in the viewing window -10\leq x\leq10,-10\leq y\leq10, and says the graph is a parabola.” Students use technology to verify the friend’s assumption while detecting errors that can occur when utilizing technology.
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 materials reviewed for Big Ideas Learning AGA 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 Algebra 1, Chapter 11, Section 2, Explore It, students interpret two box-and-whisker plots. One box-and-whisker plot represents the body mass of a ninth-grade class and the other represents the height of roller coasters. Students are asked to determine what the box and length represent and what the whiskers and its length represent within the context of the data.
In Geometry, Chapter 7, Section 5, students use their understanding of the structure of trapezoids and mid-segments to write an equation and calculate the value of x.
In Algebra 2, Chapter 1, Section 1, students are given the functions f(x)=|x-4| and g(x)=|x|-4. Students attend to structure by determining if the two functions are the same.
Examples of MP8 being used to enrich the mathematical content and intentionally developed to reach the full intent of the MP include:
In Algebra 1, Chapter 6, Section 1, Explore It, students explore properties of exponent by choosing several values for variables to find a pattern for exponent properties. Students find patterns in order to generalize rules for the properties.
In Geometry, Chapter 13, Section 6, students complete a table calculating nPr and nCr when n=3and r=0,1,2,3. Students generalize their finding to write an inequality relating nPr and nCr for any value.
In Algebra 2, Chapter 6, Preparing for Chapter 6, students are given a table representing the percentage of carbon-14 remaining after the death of an organism. Students must analyze the table to determine if calculations are repeated. Then, students write an equation using y to represent the amount of carbon-14 and t to represent years after death based on patterns discovered within the table.