Alignment to College and Career Ready Standards: Overall Summary

The instructional materials for the Meaningful Math series partially meet expectations for alignment to the CCSSM for high school. For focus and coherence, the series showed strength in making meaningful connections in a single course and throughout the series. For rigor and the mathematical practices, the series showed strengths in the following areas: supporting the intentional development of students' conceptual understanding, utilizing mathematical concepts and skills in engaging applications, displaying a balance among the three aspects of rigor, supporting the intentional development of reasoning and explaining, and supporting the intentional development of seeing structure and generalizing. Since the materials partially meet the expectations for Gateways 1 and 2, evidence for usability in Gateway 3 was not collected.

See Rating Scale
Understanding Gateways

Alignment

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Partially Meets Expectations

Gateway 1:

Focus & Coherence

0
9
14
18
10
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
9
14
16
12
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Usability

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Not Rated

Not Rated

Gateway 3:

Usability

0
21
30
36
0
30-36
Meets Expectations
22-29
Partially Meets Expectations
0-21
Does Not Meet Expectations

Gateway One

Focus & Coherence

Partially Meets Expectations

Criterion 1a - 1f

Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).
10/18
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Criterion Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. Overall, the instructional materials partially attend to the full intent of the mathematical content contained in the high school standards for all students and partially let students fully learn each non-plus standard. The instructional materials partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards, and they do not spend the majority of time on the CCSSM widely applicable as prerequisites (WAPs). The instructional materials make meaningful connections in a single course and throughout the series, and although the materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, they do not vary the types of numbers being used. The materials do not explicitly identify Grade 6-8 standards when addressed in the materials, but there is some evidence that the materials build on knowledge from Grades 6-8 standards to the high school standards.

Indicator 1a

The materials focus on the high school standards.*
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Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Some non-plus standards are not addressed by the instructional materials of the series. Additionally, there are instances where all aspects of some non-plus standards are not addressed across the series.

The following standards were not addressed across the courses of the series:

  • F-IF.9: Evidence was found for comparing two functions represented in like ways, but the review did not find instances where students would compare two functions represented in different ways.
  • G-CO.6: In Geometry, Shadows, Triangles Galore, page 47, students determine whether triangles are congruent; however, the definition of congruence in terms of rigid motions is not used to determine congruence. Then, Geometry, Geometry By Design, Do It Like the Ancients, pages 134-136 states: “Two figures are congruent if they can be placed one on top of the other and they match up perfectly.”
  • G-CO.7: In Geometry, Shadows, Triangles Galore, pages 45 and 47, students experiment with corresponding angles and sides in triangles to "prove" triangles are congruent or not congruent; however, the definition of congruence in terms of rigid motion is not used to determine congruence.
  • G-CO.8: In Geometry, Shadows, Triangles Galore, page 47, students are given the SSS Triangle Congruence Property without reference to transformations. In a group activity, Geometry by Design, Do It Like the Ancients, pages 125-126, students draw specified triangles and place triangles on top of each other to determine if the two triangles coincide. In another Do It Like the Ancients activity, page 130, students work with congruent triangles. In both activities, the definition of congruence in terms of rigid motion is not used to determine congruence.
  • G-SRT.2: In Geometry, Shadows, The Shape of It, page 30, students decide if two figures are similar but do not use the definition of similarity in terms of similarity transformations. The materials state, “Two polygons are called similar if their corresponding angles are equal and their corresponding sides are proportional,” but do not make connections to the combination of a rigid motion with a dilation. More such references are included in Shadows, Triangles Galore.
  • G-SRT.3: In Geometry, Shadows, Triangles Galore, page 42, students are introduced to the AA criterion to decide if figures are similar; however, the AA criterion is not established using the properties of similarity transformations.
  • G-GPE.7: Perimeters and areas of figures are computed; however, these computations are not determined on the coordinate plane.

Additionally, the following standards were identified as only being partially addressed. Details concerning the aspects of the standards that were not addressed are shown below.

  • A-REI.3: The review found many instances of solving linear equations and inequalities in one variable; however, opportunities to solve equations or inequalities which included coefficients represented by letters were not found.
  • G-CO.13: Students construct an equilateral triangle, square, and equilateral hexagon; however, these constructions are not inscribed in a circle. Students construct a hexagon inscribed in a circle in Geometry, Geometry by Design, Do It Like the Ancients, page 122.
  • G-C.3: In Geometry, Orchard Hideout, Supplemental Activities, pages 395-396, the materials describe the constructions for the inscribed and circumscribed circles of triangles, but the materials do not prove properties of angles for a quadrilateral inscribed in a circle.
  • G-GMD.4: There are opportunities for student exposure to cross sections of cylinders in Geometry, Orchard Hideout, Coordinates and Distance, page 338; however, cross sections of three other dimensional solids were not found. The review also found no evidence of three-dimensional objects generated by rotating two-dimensional figures.
  • S-ID.3: In Algebra 1, The Pit and the Pendulum, Statistics and the Pendulum, students interpret the center of two data sets on page 217 and the spread of four data sets on pages 221-222. Additionally, students explore what ignoring the highest and lowest values from a data set will do to the spread. There is no evidence for making connections between possible extreme values and the center of a data set.
  • S-IC.3: The materials provide opportunities for students to work with sample surveys, experiments, and observational studies; however, the materials do not provide opportunities for students to explain how randomization can be applied to each.
  • S-IC.5: Students engage in several experiments and simulations within Algebra 1, The Pit and the Pendulum; however, no opportunities were located where students use data from a randomized experiment to compare two treatments.
  • S-CP.2: The correlation document identifies this standard as being addressed in Algebra 2, The Game of Pig, In The Long Run, page 139. Students are exposed to independent events in the scenario of a one-and-one situation in basketball, and students determine independence through area models and tree diagrams but not multiplying the probability of each event. The product of probabilities for two events is not used to determine if the events are independent.
  • S-CP.4 : While students do construct and interpret two-way frequency tables and compute conditional probabilities, they do not use two-way tables to decide if events are independent. In Algebra 1, Overland Trail, Supplemental Activities, page 132, students compute conditional probabilities after constructing a two-way table. In Algebra 2, Is There Really a Difference?, A Tool for Measuring Differences, pages 465, 471-472 and Comparing Populations, page 494, students interpret two-way frequency tables but do not use the tables to decide if events are independent.

Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. Various aspects of the modeling process are present in isolation or combination, but the full intent of the modeling process is not used to address more than a few of the modeling standards by the instructional materials of the series.

Throughout the series, students perform Problems of the Week (POWs). This component of the materials routinely uses portions of the modeling process. While these problems are often open ended, causing students to consider the variables required along with the methods needed to solve the problems, they do not ensure that the entire modeling process will occur. Some problems provide significant scaffolding and guidance, which diminishes student engagement in one or more parts of the modeling process. Opportunities for making choices, assumptions, and approximations are not routinely experienced by students. Many POWs offer opportunities to formulate models, interpret results, and validate conclusions, and POWs also require significant reporting of student arguments, solutions, and findings. While the POWs are likely sites for aspects of the modeling process, other activities also contain aspects of modeling and were examined in reference to the modeling process with similar findings. While some POWs and other activities contain elements of the modelling process, there is no guidance provided to students or teachers to ensure that the complete modelling process would occur within a single mathematical task.

Some examples of where the modeling process is incomplete are:

  • Algebra 1, The Overland Trail, Reaching the Unknown, pages 79-80 scaffolds students through aspects of the modeling process for A-CED.1 and A-CED.2. In the problem, the variables are identified in the materials before students compute possible answers. Students write an equation and graph the function to find possible answers in order to validate their original guesses.
  • Geometry, Do Bees Build It Best?, The Corral Problem, page 264 provides an opportunity for students to engage with aspects of the modeling process for G-SRT.8. Students compute the area of a regular pentagon using the amount of fencing provided in the activity. Students do not interpret their findings within the context of the problem, validate their results, or report their results.
  • Algebra 2, Small World, Isn’t It?, POW 2, pages 27-28 scaffolds students through aspects of the modeling process for N-Q.2 and F-BF.1. In the problem, the variables are identified in the materials before students formulate a model based on computations when manipulating the variables in the problem. Students validate their work when they explain why their formulas make sense within the context of the problem.

Examples of tasks that utilize the full modeling process but do not address non-plus standards from the CCSSM include:

  • In Algebra 1, The Pit and Pendulum, Statistics and the Pendulum, pages 232-233, students use a pan balance to find the lightest of eight bags of gold weighing them as few times as possible. This POW does not align to any standards from the CCSSM.
  • In Algebra 2, Is There Really a Difference?, A Tool for Measuring Differences, pages 481-482, addressing S-IC.1 and S-IC.2, students work with a partner and create a hypothesis and null hypothesis for two populations, decide on a method for collecting sample data from the two populations, and then analyze their results using a chi-square analysis to determine if the two populations are statistically different. Students identify the problem of interest, formulate hypotheses, collect data to draw conclusions, interpret the survey results, compare their results to the original hypotheses (Is the accepted hypothesis true, false, or not proven? How could students modify their hypothesis or sample technique to reach a conclusion?), and report their results in a presentation to the class. However, this problem does not align to any standards from the CCSSM.

Indicator 1b

The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
0/0

Indicator 1b.i

The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.
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Indicator Rating Details

When used as designed, the instructional materials reviewed for Meaningful Math series partially meet expectations for spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs, and careers. The instructional materials for the series do not spend a majority of time on the WAPs, and some of the remaining materials address prerequisite or additional topics that are distracting.

The publisher-provided alignment document indicates the Algebra 1 course addresses the WAPs in less than half of the Algebra 1 activities, with Geometry and Algebra 2 addressing these standards less frequently than in Algebra 1. Similarly, in examining each activity of the course independently of the alignment document, reviewers verified the greatest focus on the WAPs is in Algebra 1, with less attention to these standards as the series progresses, and overall, the majority of the time across the series is not spent on the WAP standards. Examination of the publisher-provided pacing guide indicated similar findings.

While many of the topics below relate to content in the series, they are distracting topics from the WAPS as either being prerequisite, plus standards, or additional topics that are not a part of the CCSS for high school mathematics. Examples of this include:

  • In Algebra 1, All About Alice includes work with the properties of integer exponents (8.EE.1) and scientific notation (8.EE.4). Also, many activities in The Overland Trail focus more on the input-output relationship of functions (8.F.1) rather than the concept of function using domain and range and function notation (F-IF.1).
  • In Geometry, many activities in Shadows focus on proportional relationships (7.RP.2), and in Do Bees Build it Best? the materials address the area of triangles and rectangles using tools such as geoboards (6.G.1).
  • In Algebra 2, Small World, Isn’t It? addresses the concept of derivatives, and Is There Really a Difference? includes statistical analysis using the chi-square distribution. These are topics that do not align to any of the CCSSM.
  • In Algebra 2, High Dive addresses work with velocity that aligns to N-VM.3. World of Functions includes activities that address the following concepts: vectors that align to N-VM.4a and 5, matrices that align to N-VM.7, and composition of functions that align to F-BF.1c, 4b-d. These are plus standards.

Indicator 1b.ii

The materials, when used as designed, allow students to fully learn each standard.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for letting students fully learn each non-plus standard. The materials do not enable students to fully learn the following non-plus standards:

  • N-RN.1: In Algebra 1, All About Alice, Curiouser and Curiouser!, pages 160, 163, and 164, students extend the properties of integer exponents to rational exponents; however, there is minimal evidence where students can make connections between the notation for radicals in terms of rational exponents.
  • N-CN.2: In Algebra 2, The World of Functions, Supplemental Activities, page 425, students perform addition and multiplication with complex numbers but do not subtract with complex numbers.
  • A-SSE.3c: Students have some opportunities to rewrite exponential expressions using the properties of exponents in Algebra 1, All About Alice. In Algebra 2, Small World, Isn’t It?, The Best Base, page 63, students use properties of exponents to show how two exponential expressions are equivalent by transforming the base, but students do not transform expressions for exponential functions.
  • A-SSE.4: In Algebra 1, All About Alice, Supplemental Activities, pages 180-181, students develop the formula for the sum of a geometric series, but there are limited tasks that involve using the formula to solve problems: Algebra 1, All About Alice, Supplemental Activities, pages 180-181 and Algebra 2, Small World, Isn’t It?, Supplemental Activities, pages 93-94.
  • A-APR.1: In Algebra 2, The World of Functions, Supplemental Activities, pages 425-426, the materials show that polynomials form a system analogous to the integers, yet they do not show that polynomials are closed under the operations of addition, subtraction, and multiplication.
  • A-APR.3: In Algebra 2, The World of Functions, Supplemental Activities, page 431, students find the roots and x-intercepts of a polynomial written in factored form in Exercise 1, yet students do not construct a graph of the function using these x-intercepts. In Exercise 3, students use the given roots of a polynomial to write the polynomial in standard form, graph the function using a graphing calculator, and determine the x-intercept(s). Students do not construct a rough graph of a polynomial function using the given roots.
  • A-APR.4: Students are introduced to the Binomial Theorem in Algebra 2, The World of Functions, Supplemental Activities, page 429 and explore the difference of squares, cubes, fourths, and higher powers in Algebra 1, Fireworks, Supplemental Activities, page 443. These activities do not have students use these polynomial identities to describe numerical relationships.
  • A-APR.6: In Algebra 2, The World of Functions, page 426, Exercise 1, students use long division, but the expression is not presented as a rational function in the form a(x)/b(x).
  • A-REI.4b: In Algebra 1, Fireworks, Supplemental Activities, page 438, students use the quadratic formula to find x-intercepts of a quadratic equation and compare the number of x-intercepts to the discriminant, but no other problems were found where students recognize when the quadratic formula gives complex solutions and when it doesn’t. The quadratic formula is used to find complex solutions and write the solution in the form of a + bi for a few exercises in Algebra 2, High Dive, A Falling Start, page 270.
  • F-IF.7e: In Algebra 1, The Overland Trail, Supplemental Activities, page 128, and various activities in All About Alice, students graph exponential and logarithmic functions; however, there is little emphasis on intercepts and end behavior.
  • F-BF.2: In Algebra 1, The Overland Trail, Supplemental Activities, page 105, students write a recursive formula for the number of diagonals in a polygon in Part 1 and then translate to an explicit formula in Part II. In Algebra 2, Small World, Isn’t It?, Supplemental Activities, pages 91-94, students write arithmetic and geometric sequences with an explicit formula. Students do not translate from an explicit formula to a recursive formula in these activities.
  • F-TF.8: Students work with the derivation of the Pythagorean Identity in Algebra 2, High Diver, A Trigonometric Interlude, page 250. Students continue to work with the Pythagorean Identity in Algebra 2, High Dive, Supplemental Activities, page 307, but exercises are limited to the first quadrant of the coordinate plane.
  • G-CO.5: Students use tracing paper, compass, and ruler to draw geometric figures. Several instances are included in which students rotate, reflect, or translate figures in Geometry, Geometry by Design unit. In Geometry by Design, Put the Pieces Together, page 205, students determine a sequence of transformations to map a pre-image onto an image; however, the sequence is a series of reflections. Students do not specify a sequence of varying transformations that will carry a given figure onto another.
  • G-C.1: In Geometry, Geometry by Design, Put the Pieces Together, page 196, students determine whether the statement, "All circles are similar," is true or false. The statement that all circles are similar is made, but no proof is given in the materials or completed by the students.
  • G-GPE.6: In Geometry, Orchard Hideout, Coordinates and Distance, page 339 and Orchard Hideout, Supplemental Activities, page 383, students partition a segment to find the midpoint. However, students do not partition a segment into other ratios (thirds, fourths, etc.).
  • S-ID.2: The materials provide several opportunities for students to compare the spread of two or more data sets in Algebra 1, The Pit and the Pendulum, Statistics and the Pendulum, pages 221-223, 228. Students compare the center between two data sets in Algebra 1, The Pit and the Pendulum, Statistics and the Pendulum, pages 217-218. This activity provides minimal practice for students to compare the center of two or more data sets.
  • S-IC.4: In Algebra 2, The Game of Pig, Coins, and Dice, page 445, students informally estimate the margin of error of an event occurring, but students do not develop a margin of error through the use of simulation models for random sampling.
  • S-CP.3: In Algebra 2, The Game of Pig, students engage in problems related to conditional probability, but students do not interpret the independence of events by calculating conditional probabilities.
  • S-CP.6: In Algebra 2, The Game of Pig, Supplemental Activities, page 509, students calculate the conditional probability, "Given that someone has emphysema, what is the probability that the person is a smoker?" However, one problem does not provide sufficient practice for students to fully develop their understanding of calculating conditional probabilities of two events.
  • S-CP.7: In Algebra 2, The Game of Pig, Supplemental Activities, page 159, students apply the addition rule when determining the probability that Paula will get a pizza she likes. However, one problem does not provide sufficient practice for students to fully develop their understanding of the addition rule.

Indicator 1c

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, yet the materials do not vary the types of real numbers being used.

Multiple examples of applying the key takeaways from Grades 6-8 were found across the series. For example:

  • In Algebra 1, The Overland Trail, Traveling at a Constant Rate, pages 55-56, students consistently use functions to model linear relationships. Students apply knowledge of line of best fit from Grade 8 to make predictions regarding the amount of beans, sugar, and gunpowder travelers will need on the Overland Trail based on data from previous travelers along the trail.
  • Students apply their knowledge of ratios and proportional relationships from Grades 6-8 when learning about similar triangles. In Geometry, Shadows, The Shape of It, page 40 and Shadows, Triangles Galore, page 46, students use proportions to find missing side lengths in similar triangles.
  • Students apply their knowledge of proportional relationships of solving percent problems from Grade 7 (7.RP.3) when they solve contextual problems related to percentage growth and depreciation in Algebra 1, All About Alice, Supplemental Activity, page 177.
  • Students apply their skills related to geometric measurements (area, perimeter, and the Pythagorean Theorem) developed in Grades 6-8 in Geometry, Do Bees Build it Best?, The Corral Problem, pages 261-264, 266. Students calculate different geometric measurements to determine the size of the corral using various shapes (rectangle, equilateral triangle, regular pentagon) and given price constraints.

In the instructional materials, contexts are appropriate for high school students. Each unit contains a unit problem where students apply the mathematical topic of the unit. For example, in Algebra 1, Cookies, students develop skills in solving systems of linear equations within the context of a bakery trying to maximize its profits. Students learn about circles and coordinate geometry as they find out how long it will take for trees to grow in an orchard so that the center of the orchard cannot be seen from the outside world, within Geometry, Orchard Hideout. In Algebra 2, students study world population data trends and predict future populations as they learn about rate of change, derivatives, and exponential growth.

The instructional materials do not vary the types of real numbers being used. Across the series the majority of work is done with integers or simple rational numbers, such as 1/2, 3/2, 1/4, 1/10. A few examples of this include:

  • In Geometry, Shadows, Triangles Galore, page 46, students use proportions to find unknown lengths in a triangle where some side lengths are integers while others are decimals to the tenths.
  • In Geometry, students calculate area and volume problems with integers.
  • In Geometry, Orchard Hideout, Cable Complications, page 367, students complete the square to write the equation of a circle. In all four exercises, the coefficients of the quadratic terms are 1, and the coefficients of the linear terms are even integers with the exception of one portion of Exercise 3.
  • In Algebra 2, Small World, Isn’t It?, All in a Row, page 29, students find an equation for a line with a given slope and point and find the equation for a line given two points. Of the seven exercises, one incorporates a fraction (slope of 2/3 in Exercise 4).

Indicator 1d

The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Teacher notes delineating coherent connections within a course are present; however, regular reference to the CCSSM are not present. Coherence for each unit is built around a unit problem and activities are clearly related in an intentional sequence to support the mathematics of the unit. In some instances, course materials refer to previous units or activities, both within and across courses. Examples of coherence within and across Algebra 1 and 2 were identified more often than between Geometry and either Algebra course.

Overall, the instructional materials foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards. Examples of this include:

  • In Algebra 1, The Overland Trail, students construct linear equations and graph linear functions (F-IF.1, F-IF.2, F-IF.4, F-IF.7). Students build upon this knowledge in The Pit and the Pendulum, with students fitting a line to a set of data. In The Pit and the Pendulum, Graphs and Equations, pages 250-251, students create a scatterplot showing the relationship between prices and expected profit, sketch a line of best fit, and then use that line to make a prediction about the maximum profit (S-ID.6).
  • The theorem, “Every point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the segment,” (G-CO.9) is proved in Geometry, Geometry by Design, Construction and Deduction, pages 151-152, using SAS congruence of triangles. Students revisit this theorem in Geometry, Orchard Hideout, Orchards and Mini-Orchards, page 330, using the Pythagorean Theorem.
  • In Algebra 2, students explore slopes of linear functions and the connections to graphs and equations of linear functions (F-IF.7, F-BF.1, F-LE.1b) in Small World, Isn’t It?, All in a Row, pages 21-22. Students build upon these connections as they explicitly explore the relationship between tables of linear functions and the equations of linear functions (F-LE.1) in The World of Functions, Tables, page 333. In this unit students extend their thinking to other families of functions.
  • Students are introduced to the concept of exponential functions and logarithmic functions in Algebra 1, All About Alice. In this unit, students define the exponential function, work with properties of exponents (N-RN.A), and graph exponential and logarithmic functions (F-IF.7e, F-LE.1). Students continue their study of exponential and logarithmic functions in Algebra 2, Small World, Isn’t It?. In this unit students use exponential functions to model real-life scenarios, including compound interest problems (F-LE.1, F-LE.2).
  • In Algebra 1, The Overland Trail, Reaching the Unknown, page 81, students explore the connection between a possible solution set of an equation represented as numerical coordinates and the geometric points on a coordinate plane. In this activity, students suggest three possible solutions, plot those solutions on a coordinate plane, and connect the points to discover a linear relationship as an appropriate model. Using the line, students identify two more possible solutions, determine the equation of the line, and then algebraically check to verify those additional two solutions fit the linear model. This activity provides a visualization tool for how the solution set of an equation becomes a geometric curve (A-REI.10).
  • In Algebra 2, Small World, Isn’t It, All in a Row, page 31 and Small World, Isn’t It, Supplemental Activities, page 85, students engage in geometric proofs on the coordinate plane to connect geometric and algebraic concepts (G-SRT.2, G-SRT.5, G-GPE.4, G-GPE.5).

Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The series does not explicitly identify standards from Grades 6-8 addressed in the instructional materials for teachers or students. Limited references are included within the teacher materials indicating that students are building upon knowledge from Grades 6-8. For example:

  • In Algebra 1, All About Alice, the Teacher Overview states: “... Students will derive several rules for computing with exponents and extend their understanding of exponential expressions to include zero and negative integers,” without explicit reference to any standards from Grades 6-8.
  • In Algebra 1, The Pit and the Pendulum and the CCSSM document available to teachers, the publisher notes, “These specific content standards are addressed in The Pit and the Pendulum. Additional content is covered that reinforces standards from earlier grades and courses.” The standards from earlier grades are not explicitly identified.
  • In Algebra 1, the Teacher Overview for Cookies does not refer to systems of equations or inequalities in the opening statement of intent, although it is included as the central mathematical focus under the “Mathematics” heading. Within the Teacher Overview there is no indication that this unit focuses on standards that represent an extension of 8.EE.8 or any other standards from Grades 6-8.

Although the materials do not explicitly identify Grade 6-8 standards when addressed in the materials, evidence that the materials build on knowledge from Grades 6-8 Standards to the high school standards is shown below:

  • Students build upon their knowledge of solving systems of linear equations (8.EE.8) in Algebra 1, Cookies. Throughout this unit students solve a system of linear equations embedded in a real-world context. Students extend their knowledge as they solve linear programming problems in several activities and the unit problem. Students consider constraints and identify feasible regions as they maximize profits and minimize costs in activities (A-REI.D).
  • Students work with vertical angles, corresponding angles, alternate interior angles, angles in triangles, and angles in parallelograms in Geometry, Geometry by Design (7.G.5). Not only do students solve for unknown angle measures, they also prove theorems using theorems, axioms, and postulates about angle measures and relationships (G-CO.A).
  • Students display quantitative data sets using dot plots, histograms, and boxplots in several activities within Algebra 1, The Pit and the Pendulum, to reinforce 6.SP.4. Students build upon 6.SP.5 as they summarize data sets using the mean, median, range, and standard deviation. Additionally, students calculate r-values and interpret them (S-ID.C) in Algebra 1, The Pit and the Pendulum, Supplemental Activity, page 271, which builds upon 8.SP.2.

Indicator 1f

The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.
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Indicator Rating Details

The instructional materials reviewed for the Meaningful Math series use the plus standards to coherently support the non-plus standards. Generally, the plus standards are explicitly identified in the teacher materials but not in the student materials. When included, plus standards are integrated into the units in such a way that omitting the materials aligned to the plus standards would disrupt the coherence of the remainder of the mathematical content in the series. Activities that include the plus standards support the mathematics relevant to the unit problem to provide coherence with the content aligned to non-plus standards.

The following plus standards are fully addressed:

  • N-CN.3: This standard is not explicitly identified by the publisher, but evidence of this standard was found in Algebra 2, High Dive, Supplemental Activities, page 308, when students are introduced to complex conjugates and find the quotient of complex conjugates. In Supplemental Activities, page 309, students find the moduli of complex numbers.
  • N-CN.4: In Algebra 2, High Dive, Falling Start, page 271, students graph complex numbers in rectangular form. In Algebra 2, High Dive, Supplemental Activities, pages 310-311, students graph complex numbers in polar form. In Exercise 2, students graph an equation in both rectangular and polar forms to understand how the two forms can represent the same complex number.
  • N-CN.9: In Algebra 2, High Dive, Supplemental Activities, pages 314-315, the materials state the Fundamental Theorem of Algebra. In Exercises 2 and 3, students work with quadratic polynomials as they find roots and explain the meaning of a double root for a given quadratic equation.
  • N-VM.3: In Algebra 2, High Dive, Components of Velocity, pages 282-283, students solve problems involving velocity that can be represented by vectors.
  • N-VM.4a: In Algebra 2, High Dive, Falling Start, page 271 and High Dive, Components of Velocity, page 282, students add vectors.
  • N-VM.4b: In Algebra 2, High Dive, Components of Velocity, page 283, students find the magnitude and direction of the sum of two vectors given in magnitude and direction form.
  • N-VM.5: In Algebra 2, High Dive, Supplemental Activities, page 314, students multiply vectors by scalar values.
  • N-VM.7: In Algebra 2, High Dive, Supplemental Activities, page 322, students multiply 2 x 1 column matrices to produce new matrices.
  • A-APR.5: In Algebra 2, The World of Functions, Supplemental Activities, page 429, students multiply (x+y) by itself several times to generate a pattern and derive the Binomial Theorem in Exercise 1. In Exercise 2, students use Pascal’s Triangle to determine an equivalent expression for (x+y)^6.
  • F-BF.1c: In Algebra 2, The World of Functions, Composing Functions, pages 381-388, students compose functions.
  • F-BF.4b: In Algebra 2, The World of Functions, Composing Functions, page 394, students verify that one function is the inverse of another function using composition.
  • F-BF.4c: In Algebra 2, The World of Functions, Composing Functions, pages 390-391, students complete an In-Out table for a function and its inverse function in several exercises before reaching conclusions regarding the relationship between the table of values for a function and the table of values for its inverse function. Additionally, students graph a function and its inverse on the same x and y axes in several exercises before reaching conclusions regarding the relationship between the graph of a function and the graph of its inverse.
  • F-BF.5: In Algebra 2, Small World, Isn’t It?, A Model for Population Growth, pages 50-51, students explore the connection between exponential and logarithmic equations. In Algebra 2, Small World, Isn’t It?, Supplemental Activities, page 96, students connect the domain of an exponential function to the range of a logarithmic function and the range of an exponential function to the domain of a logarithmic function in Exercises 3 and 4.
  • G-SRT.9: In Geometry, Do Bees Build It Best?, Supplemental Activities, page 312, students derive the formula for the area of a triangle using sine and then use the formula to find the area of a given triangle.
  • G-C.4: In Geometry, Orchard Hideout, Supplemental Activities, page 343, students construct a tangent line from a point outside a given circle.
  • S-MD.5: In Algebra 2, The Game of Pig, In the Long Run, pages 133-135, 138, students find expected payoffs and expected values within the contexts of basketball games, the lottery, and insurance.
  • S-MD.6: In Algebra 2, The Game of Pig, In the Long Run, page 132, students use probability to determine whether a dart game is fair or not for the two players playing the game.
  • S-MD.7: In Algebra 2, The Game of Pig, students use probability concepts to analyze decisions and strategies in several activities throughout the unit. Ultimately, students apply probability concepts to answer the unit problem of the best strategies to use when playing The Game of Pig.

Parts of the following plus standards were addressed:

  • N-VM.1: In Algebra 2, High Dive, Supplemental Activities, page 322, students find the magnitude of vectors, but students do not use appropriate symbols to represent the magnitude.
  • N-VM.10: In Exercises 1b, 1c, 3b, and 3c, Algebra 2, The World of Functions, Composing Functions, pages 393-394, students explore how the zero and identity matrices play a role in matrix addition and multiplication. Students do not have the opportunity to understand that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
  • F-IF.7d: In Algebra 2, The World of Functions, Going to the Limit, pages 350-352, students graph rational functions, accounting for vertical asymptotes and values of x that make the denominator equal to zero. In The World of Functions, Going to the Limit, page 356, students explore the end behavior of several function families, including rational functions. No emphasis is placed on identifying the zeros of rational functions.
  • F-BF.4d: In Algebra 2, The World of Functions, Supplemental Activities, page 424, students investigate inverse trigonometric functions with restricted domains, but students do not produce an invertible function from a non-invertible function by restricting the domain.
  • F-TF.4: This standard is not explicitly identified by the publisher, but the review found evidence of this standard in Algebra 2, High Dive, Trigonometric Interlude, pages 252-255. Students use the unit circle to explore the periodicity of the sine and cosine functions. In Algebra 2, The World of Functions, Supplemental Activities, pages 414-415, students recognize the sine function as an odd function by showing sin(30) = sin(-30), but students do not directly explain symmetry using the unit circle.
  • F-TF.6: In Algebra 2, The World of Functions, Supplemental Activities, page 424, students recognize that not every function has an inverse unless the range is restricted, such as for the sine and cosine functions. While students recognize that restrictions take place, materials emphasize a restriction on the range and not a restriction on the domain of a function.
  • F-TF.7: In Algebra 2, High Dive, The Height and the Sine, page 220, students solve a trigonometric equation using trigonometric inverses with technology and interpret their solution in terms of the context provided. The modeling context is not present as the quantities are defined for the students and the trigonometric equation is provided.
  • G-SRT.10: In Geometry, Do Bees Build It Best?, Supplemental Activities, pages 308-309, students derive the Law of Cosines and Law of Sines. Students solve problems using the Law of Cosines in Do Bees Build it Best?, Supplemental Activities, page 308, and Geometry by Design, Supplemental Activities, page 216. Students do not use the Law of Sines to solve problems.
  • G-SRT.11: In Geometry, Geometry by Design, Supplemental Activities, page 216 and Do Bees Build It Best?, Supplemental Activities, page 308, students apply the Law of Cosines to find unknown measurements in right and non-right triangles. Students do not solve problems to find the unknown measurements in right and non-right triangles using the Law of Sines.
  • G-GPE.3: In Geometry, Orchard Hideout Supplemental Activities, pages 411-414, students derive the equation of an ellipse when the difference from the distance of the foci is 8 and then derive the equation of a hyperbola when the difference from the distance of the foci is 2. Students generalize their results for the equation of an ellipse in “standard position” with its center at the origin and its foci on the x-axis at (c, 0) and (-c, 0). Students do not generalize their results to derive the general equation of a hyperbola.
  • G-GMD.2: In Geometry, Orchard Hideout, Supplemental activities, pages 400-401, students solve a problem involving the volumes of a sphere and a cone in Exercise 2. However, there is no informal argument provided relating the formula for the volume of a sphere using Cavalieri’s principle.
  • S-MD.1: In Algebra 2, The Game of Pig, students conduct several simulations where they collect data. In Pictures of Probability, page 127, students draw a graph of their data and collate their data results to create a frequency bar graph for the entire set of class data, but the graph that is drawn is not of a probability distribution.
  • S-MD.2-4: In Algebra 2, The Game of Pig, In the Long Run, pages 131, 133-135, students calculate expected values but do not connect them to probability distributions.

There was no evidence found for the following plus standards:

  • N-CN.5
  • N-CN.6
  • N-CN.8
  • N-VM.2
  • N-VM.4c
  • N-VM.6
  • N-VM.8
  • N-VM.9
  • N-VM.11
  • N-VM.12
  • A-APR.7
  • A-REI.8
  • A-REI.9
  • F-TF.3
  • F-TF.9
  • S-CP.8
  • S-CP.9

Gateway Two

Rigor & Mathematical Practices

Partially Meets Expectations

Criterion 2a - 2d

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.
7/8
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Criterion Rating Details

The instructional materials reviewed for the Meaningful Math series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. Overall, conceptual understanding and application are thoroughly attended to, but students are provided limited opportunities to develop procedural skills and fluencies.

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.
2/2
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Indicator Rating Details

The instructional materials for the Meaningful Math series meet expectations that the materials develop 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 throughout the series.

Examples of the materials developing conceptual understanding and providing opportunities for students to independently demonstrate conceptual understanding are highlighted below:

  • A-APR.B: In Algebra 1, Fireworks, A Quadratic Rocket, page 385, students make the connection between the shape of a parabola and the number of x-intercepts. Students determine the vertex from quadratic equations written in vertex form, whether the graph is concave up or down, and the resulting number of x-intercepts. Also, in Algebra 1, the zero product rule is defined in the Glossary on page 459, and concept development of the property begins in Algebra 1, Fireworks, Intercepts and Factoring, page 415, where factoring is introduced. “Factored form is useful for finding x-intercepts. The x-intercepts are the values of x that make y = 0.” Students engage with this activity, using a polynomial’s factors to unveil the x-intercepts, however the term zero is not introduced. In subsequent activities, students factor expressions, graph parabolas, and solve a cattle pen application using the factored form and the x-intercepts. In Algebra 2, The World of Functions, Supplemental Activities, page 431, students use their knowledge of the Remainder Theorem on page 427 to show connections between roots, equations, and graphs of polynomial functions. The term roots is used rather than zeros.
  • A-REI.A: In Algebra 1, The Overland Trail, Reaching the Unknown, pages 90-91, students build conceptual understanding of solving equations within the context of a using a pan balance scale. Mystery bags filled with gold and lead weights are arranged on the pan balance, and students determine how much gold is in each bag. Through a series of subsequent activities on pages 92-95, students move from the pan balance analogy to solving one-step, two-step, and multi-step equations.
  • A-REI.10: In Algebra 1, The Overland Trail, The Graph Tells the Story, page 51, students make a table of values based on a provided in-out rule, and then plot the points until they gain “a good idea of what the whole graph looks like.” Later, in Algebra 1, All About Alice, Curiouser and Curiouser!, page 157, students use a similar method to gain understanding of the properties of the graph of y=2x and in Algebra 2, The World of Functions, Going to the Limit, pages 351-352, for the graphs of rational functions.
  • F-IF.A: In Algebra 1, students use in-out tables to investigate functions and develop an understanding of what a function is in The Overland Trail, The Importance of Patterns, page 12 and The Overland Trail, The Graph Tells the Story, pages 49-51. Additionally, sequences are used as a means to establish the meaning of functions in Algebra 1, The Overland Trail, Supplemental Activities, page 104, when students look for a pattern in a provided sequence and write a description of the pattern, a method for how to find the next few terms in the pattern, and an equation for the sequence.
  • G-SRT.2: In Geometry, Shadows, Triangles Galore, page 57, students experiment with congruent triangles to see if they can find a small triangle inside a larger triangle so that the two triangles are similar. In this investigation, students verbalize their conclusions and describe the line segments that can be used to create a small triangle that is similar to the larger triangle.
  • G-SRT.6: In the Geometry Shadows unit, students develop their conceptual understanding of trigonometric ratios by looking at right triangles and have multiple opportunities to independently demonstrate that understanding. The materials highlight past experience with using ratios in their work with triangles and similarity, “You’ve also seen that ideas of similarity involve ratios of sides of triangles. So it’s natural to think about ratios of sides within right triangles” (Shadows, The Sun Shadow, page 73), and the materials assign names to the trigonometric ratios on pages 74-75. Students engage in creating trigonometric tables on page 77 to examine the relationship between the sine and cosine of complementary angles on page 78 and apply trigonometric ratios to solve real-world problems on pages 79-81.
  • S-ID.7: In Algebra 1, The Overland Trail, Traveling at a Constant Rate, page 59, students graph a data set to make predictions about the water supply of travelers on the Overland Trail. In Exercise 4, students estimate how much water each family used per day (understanding what slope means in context) and how much water each family started with (understanding what y-intercept means in context).

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.
1/2
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Indicator Rating Details

The instructional materials for the Meaningful Math series partially meet expectations that the materials provide intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters.

Students’ independent demonstration of procedural skills is often limited to a few problems. The following are examples of how the instructional materials provide students with limited opportunities to independently demonstrate procedural skills throughout the series.

  • N-RN.2: In Algebra 1, All About Alice, Curiouser and Curiouser!, page 160, students use work from a previous problem to develop a general way to define 2pq for any fraction pq. In Exercise 3 on pages 163-164, students rewrite five expressions in simpler form using properties of exponents with rational exponents. There is minimal practice for students to rewrite radical expressions.
  • A-APR.2: In Algebra 2, The World of Functions, Supplemental Activities, pages 427-428, the materials state the Factor Theorem, specifically referring to the Remainder Theorem on page 428, and explain why the theorem is true. Yet students have limited opportunities to apply the remainder theorem when working with polynomials.
  • A-APR.6: In Algebra 2, The World of Functions, Supplemental Activities, page 426, the materials provide students with two examples of long division. Students then practice polynomial division on three problems.
  • A-REI.4b and N-CN.7: In Algebra 2, High Dive, A Falling Start, page 270, students solve quadratic equations with complex solutions. There are three problems where students develop and independently demonstrate solving quadratic equations with complex solutions. Overall, there are limited opportunities for students to recognize when the quadratic formula will result in complex solutions.
  • F-TF.2: In Algebra 2, High Dive, Supplemental Activities, pages 316-317, students explore radian measures and the unit circle. While students work with the unit circle in other activities in the High Dive unit, these activities do not use radian measures.
  • G-GPE.5: There is minimal evidence students develop procedural skills in using the criteria for perpendicular and parallel lines to solve problems. In Geometry, Geometry by Design, Isometric Transformations, page 179, students use the slope criteria for perpendicular lines to find the equation of the line that passes through a given point and is perpendicular to a given line in two exercises. In Algebra 2, Small World, Isn’t It?, All in a Row, page 23 and Small World, Isn’t It?, Supplemental Activities, page 85, students develop the concept that parallel lines have the same slope.

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.
2/2
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Indicator Rating Details

The instructional materials for Meaningful Math series meet expectations that 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 problem-based nature of the series lends itself to using application problems throughout all courses. In each unit, a series of activities are used to tie together mathematics content as students seek to solve the overarching unit problem. Typically, the unit concludes with students writing up their solutions to the unit problem, which provide a regular opportunity for students to independently demonstrate their use of mathematics in solving applications. The mathematical activities within the unit primarily consist of contextualized applications. The units within all three courses follow this structure.

Examples of engaging high school applications in real-world contexts include:

  • A-REI.12: Algebra 1, Cookies focuses on Systems of Equations and Linear Programming as students build a conceptual understanding in the first activity when they find combinations that satisfy a given criteria. The following activity modifies the criteria and increases the demand for combinations. Later during the unit, students are introduced to inequalities and apply this newly acquired concept to solving for combinations that satisfy a new set of criteria. Before the unit is complete, students have the opportunity to apply the skill to other contextual situations, for example, dog diets and music.
  • In Algebra 2, students apply knowledge of solving equations to solving quadratic contextual problem situations in Small World, Isn’t It?, Beyond Linearity, pages 35-36, as well as exponential problem situations in Small World, Isn’t It?, A Model for Population Growth, pages 53-55.
  • In Algebra 1, Fireworks, A Quadratic Rocket, page 376, students use quadratic equations or other representations of data to determine the population of rats after a period of time.
  • G-SRT.5: In Geometry, Shadows, The Lamp Shadow, students apply their experience with similar triangles to solve problems that involve indirect measurement (e.g., height of an object, pages 63 and 65). These opportunities transition to establishing and applying trigonometric ratios (G-SRT.8) (e.g., encountering angles of elevation and depression, page 80).
  • A-SSE.3: In Algebra 1, Fireworks, Putting Quadratics to Use, page 406, students convert an equation modeling the path of a rocket from standard form to vertex form. By converting the equation, students are able to identify the maximum height of the rocket and how long it took for the rocket to reach the maximum height.
  • G-SRT.8: In Geometry, Do Bees Build It Best?, Area, Geoboards, and Trigonometry, page 245, students use trigonometric ratios to find the missing side of a right triangle as they seek to determine if people stranded on a sailboat will make it to shore safely. In the second part of the activity, students use inverse trigonometric ratios to solve for a missing angle in a right triangle within the context of a tree’s shadow.
  • F-IF.B: In Algebra 2, Small World, Isn’t It?, Average Growth, page 13, students apply their knowledge of functions to create an in-out table, graph, and equation to represent the spread of an oil slick after an explosion of an oil tanker at sea in Part I. Students use their rule in Part II of the activity to see if the cleanup operation can eventually neutralize the oil spill.

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.
2/2
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Indicator Rating Details

The instructional materials for Meaningful Math series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All units emphasize applications. In general, tasks use a real-world context and units are organized around an overarching real-world problem. Conceptual understanding is developed through the applications by teaching through problem solving. Units often feature limited opportunities for practicing procedural skills, but when present, procedural skills are integrated into the problem-solving scenarios.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples of this include:

  • In Algebra 1, The Overland Trail, students plan provisions for the trip for several fictional families. As the families progress through the trip, students encounter graphs, lines of best fit, equations, and rate problems. The mathematical concepts are applied to the migration. The activities are often presented within a context. For example, The Overland Trail, Setting Out with Variables, page 35, includes the application of a procedural skill (Students calculate the price to cross a river.) and prompts “explain your reasoning,” where students use a given formula to explain the cost for different categories of customers. The In-Class and Take-Home Assessments use a related modern-day context, prompt for reasoning, and include the use of procedural skill, which balances the three aspects of rigor in The Overland Trail.
  • In Geometry, Orchard Hideout, students consider a circular piece of land where they will plant an orchard and make a projection about how long it will take before the orchard is a dense hideout. For example, in Orchards and Mini Orchards, pages 326-333, students use radius, midpoint, perpendicular bisector, circumcenter, tangents, area of a circle, and the distance formula to promote conceptual understanding and mathematical application. There is also practice with procedural skills.For example, on page 333, students decide whether points shown as ordered pairs are inside, outside, or on the boundary of the orchard.
  • In Algebra 2, The World of Functions, students reason about the relationship between speed and stopping distance using multiple representations as they are introduced to the unit problem on page 326. Students make connections between verbal descriptions and graphs on pages 328-331 and 334, and students use tables to explore patterns and properties of linear, quadratic, cubic, and exponential functions on pages 333, 335, 338-342, and 347. Students assign functions to tables in Who’s Who? on page 361. The unit concludes with students returning to the unit problem as they explain what function family they think best represents data given in a table.

There are some instances where procedural skills activities are not presented simultaneously with other aspects of rigor. Examples of this include:

  • In Algebra 1, Overland Trail, Reaching the Unknown, page 92, students solve one-step, two-step and multi-step equations containing variables on both sides of the equal sign.
  • In Algebra 1, Cookies, Points of Intersection, page 339, students solve linear equations and linear systems.
  • In Algebra 1, Fireworks, Intercepts and Factoring, page 416, students factor quadratic equations.
  • In Geometry, Shadows, The Shape of It, pages 37, 38, and 40, students create proportions based on similar figures and solve the proportions to find the lengths of missing sides.
  • In Algebra 2, Small World, All in a Row, page 29, students find the equation of a line given specific information.

The instructional materials embed conceptual understanding and application in contexts such that these two aspects of rigor are simultaneously being addressed. For example:

  • In Algebra 2, The World of Functions, Composing Functions, page 381, students develop their conceptual understanding of composition within the context of a student who is trying to save enough money to travel across the country. Students can either make a graph or a table to show student earnings as they apply one function to another function.
  • In Algebra 2, Small World, Isn’t It?, All in a Row, page 23, students make a connection between the slope of parallel lines and the graph of parallel lines within the context of teammates saving money to help buy new basketball uniforms. They develop formulas to describe the amount of money each of the friends has at any time and consider how these formulas relate to their respective slopes and graphs.

Criterion 2e - 2h

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
5/8
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Criterion Rating Details

The instructional materials for the Meaningful Math series partially meet expectations that the materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials develop each of the Mathematical Practices, except for MP5 and MP6. For MP5, students do not have opportunities to choose an appropriate tool to use to solve a problem because the materials include directions which specify which tool(s) to use, and for MP6, the materials do not always use precise mathematical vocabulary and definitions.

The instructional materials do not identify or support the development of the MPs in the units or activities for teachers or students. At the beginning of each unit, there is a document titled "(Unit Name) and the Common Core State Standards for Mathematics," and in each of these documents, there is the following general statement, "The eight Standards for Mathematical Practice are addressed exceptionally well throughout the Meaningful Math curriculum." A publisher-provided document, that is separate from the digital materials, entitled "Correlation of Interactive Mathematics Program (IMP), Years 1-4, Common Core Edition (2014) to Common Core State Standards (June 2010)", lists activities within courses for each MP that are representative of the MP, but other than a description of the activity, there is no identification of the MPs for those activities. The lack of identification of the MPs is reflected in the scoring of indicator 2e, and does not affect the scoring of indicators 2f, 2g, or 2h.

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.
0/2
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Indicator Rating Details

The instructional materials for Meaningful Math series do not 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. In addition to not developing MP6 to its full intent, the materials do not identify the MPs for teachers or students as evidenced in the EdReports.org Criterion Summary for the MPs.

Students often make sense of problems and persevere in solving them, and several tasks address a general problem-solving process and are not connected to the high school content standards. There is intentional development of MP1 across the series, but MP6 is not developed to its full intent as the materials do not always use precise mathematical vocabulary and definitions.

Problems of the Week (POW) provide opportunities to make sense of problems and persevere in solving them (MP1). Examples include:

  • In Algebra 1, The Pit and the Pendulum, Edgar Allan Poe - Master of Suspense, POW 7, pages 198-199, the materials present a modified chess board and explain how a knight moves. Students determine if it is possible to move each knight from one spot on the board to another spot on the modified board. In order to determine if the movements are possible, students make sense of how a knight moves, and they also make sense of how to record the movements of the knights. Students persevere in the task as they record multiple combinations of moves in order to determine if the knights can land in the desired spaces on the modified board.
  • In Geometry, Do Bees Build it Best?, From Two Dimensions to Three, POW 10, pages 270-271, students plan to create a patchwork quilt and find a piece of satin that can be used to make patches for the quilt. Students are given the dimensions of the piece of satin and have to determine how many 3 inch by 5 inch patches could be cut from the large piece.
  • In Algebra 2, The Game of Pig, Pictures of Probability, POW 5, pages 123-124, pairs of students play a game in which each can remove a limited number of objects from a group (e.g., remove one, two, or three objects from a group of ten). The winner is the player who takes out the last object. After playing several variations of the game, students describe their best strategies, make generalizations about the structure of the game, and give justification for their findings.

The materials do not develop MP6 to its full intent as they do not always use precise mathematical vocabulary and definitions. Examples of how the materials do not use precise mathematical vocabulary and definitions include:

  • Functions are introduced in Algebra 1, The Overland Trail, within the context of in-out tables and are defined in the Glossary on page 451 as “a process or rule for determining the numerical value of one variable in terms of another. A function is often represented as a set of number pairs in which the second number is determined by the first, according to the function rule.” The materials do not use the definition of a function as assigning each element of the domain exactly one element of the range (F-IF.1).
  • In Algebra 1, The Pit and the Pendulum, Supplemental Activities, page 292, the term domain is defined as “intervals on the x-axis” and used in relationship to piecewise functions, and the term range is not defined or used in relationship to functions in Algebra 1. In Algebra 2, Small World, Isn’t It?, Supplemental Activities, pages 95-96, the terms domain and range are examined in the context of the relationship between exponential and logarithmic functions. Also, in Algebra 2, The World of Functions, Supplemental Activities, pages 408-409, students determine the domains of rational and radical expressions. The terms domain and range are not used or defined for other types of functions, including polynomial functions, in the series.
  • In Geometry, Geometry by Design, Do It Like the Ancients, page 134, the definition of congruent is written as, “Two figures are congruent if they can be placed on one on top of the other and they match up perfectly.” The materials do not define congruence in terms of rigid motions.
  • In Algebra 2, High Dive, The Height and the Sine, page 213, students model the movement of a Ferris Wheel using a trigonometric function and examine how the amplitude, period, and frequency affect the graph and equation modeling the Ferris Wheel. The materials do not use the term frequency when referring to trigonometric functions, but in Exercise 1, students modify the frequency by changing the period of the trigonometric graph.
  • The term zeros is used in Algebra 2, High Dive, Supplemental Activities, page 306, but there is no other evidence for the use of this term. The term is also not used in any of the problems that are a part of the Supplemental Activity on page 306.
  • No evidence of the use of the term interquartile range (S-ID.2) was found.

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.
2/2
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Indicator Rating Details

The instructional materials for Meaningful Math series meet expectations that the materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. MP2 and MP3 are used to enrich the mathematical content throughout the series, and there is intentional development and full intent of MP2 and MP3.

Some examples of MP2 include:

  • In Algebra 1, The Overland Trail, Setting Out with Variables, page 31, students are given detailed information about the quantities of boots, shoes, and shoelace that travelers will need on the Overland Trail. Students decontextualize the quantities in order to determine the total amount of shoelace a family travelling on The Overland Trail will need, and they recontextualize their calculations and final amount to describe how they obtained the amount of shoelace in relationship to the members of the family.
  • In Algebra 1, Cookies, Picturing Cookies, page 311, students reason about the quantities provided as they define variables to represent different quantities and use those variables to write a system of inequalities that describe the constraints of the problem.
  • In Algebra 2, High Dive, Falling, Falling, Falling, pages 225-226, students are given a particular example of the distance travelled by a falling object and develop a general formula for the height of a falling object after a given number of seconds.

When engaging in group activities throughout the course, students construct arguments and critique the reasoning of others as they collaborate and discuss in groups. Some examples of MP3 include:

  • In Algebra 1, The Overland Trail, Who’s Who, page 7, students solve a problem and write up their solution. The final part of the write-up prompts students to show that their “answer fits the information and that it is the only answer that fits the information.” This allows students to provide an argument for why their answer is correct by disproving other possible answers.
  • In Algebra 1, All About Alice, Curiouser and Curiouser!, pages 161-162, students consider three problems which pose several students’ reasoning regarding the additive law of exponents, multiplying expressions with the same exponent, and raising exponential expressions to powers. Students critique the reasoning of each student to determine whether any of the student answers are correct and then justify why a particular student is correct.
  • In Geometry, Shadows, How to Shrink It?, page 28, three students share their strategy for shrinking the size of a house while keeping the shape exactly the same. Students critique the reasoning of others as they determine whether each strategy works and explain why the method does or does not work.
  • In Geometry, Geometry by Design, Dilation, page 191, a student seeks advice from five friends about how to enlarge a figure on a copier. Students critique each friend’s response as to whether it produces the desired enlargement, and if it doesn’t, students determine what size enlargement was actually made.

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.
1/2
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Indicator Rating Details

The instructional materials for Meaningful Math series 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. Students create and use models throughout the series and use tools to solve real-world problems; however, students are often told which tool to use.

Some examples of MP4 include:

  • In Algebra 1, Fireworks, The Form of It All, pages 399-340, students design a drinking trough for a farmer to use on his farm. Students are provided with the width and length of the metal sheet and must determine the height that would maximize the volume of the trough.
  • In Geometry, Do Bees Build It Best?, The Corral Problem, pages 260-264 and 266, students build a corral for a rancher. This corral can be the shape of any regular polygon, but the rancher has the budget for a particular amount of fencing. These activities provide students the opportunity to model with mathematics as they explore the relationship between perimeter and area of regular polygons.
  • In Algebra 2, Small World, Isn’t It?, Average Growth, page 13, students consider the growth of an oil spill at sea and determine a function to model the growth of the oil spill. Students create a different function that represents relief efforts in the clean-up process and determine whether the clean-up efforts will eventually counteract the growth of the oil spill.
  • In Algebra 2, The World of Functions, Back to the Beginning, page 403, students select activities that helped them to see connections between tables, graphs, equations, and situations. This activity supports student engagement with multiple representations of functions to represent real-world scenarios.

In the series, students often use tools, but students generally do not choose which tool to use. Some examples of not choosing a tool include:

  • In Algebra 1, The Overland Trail, Reaching the Unknown, pages 79-80, students write an equation representing a given context. The materials state: “Graph the function from Question 4 on your calculator,” and then, “Use the trace feature on your calculator to find three more pairs of possible” solutions. Students do not have the opportunity to make decisions about whether to construct a graph by hand or use a calculator, nor do they consider the advantages/limitations of finding possible solutions by guess and check or using the calculator.
  • In Geometry, Do Bees Build it Best?, Area, Geoboards, and Trigonometry, page 240, students derive the area formulas for a parallelogram and a trapezoid. The material specifies two approaches, one of which involves using geoboard paper, and the other involves students drawing figures on paper and cutting out the figures to see how the pieces fit together. By including these approaches, the materials take away the opportunity for students to determine what tool(s) would be helpful in deriving the area formulas for parallelograms and trapezoids and identifying strengths/limitations of the tool(s).
  • In Algebra 2, Small World, Isn’t It?, Beyond Linearity, page 38, students graph a function on the graphing calculator. The materials explicitly state to, “Graph this function, and adjust the viewing window so your graph includes the point (50, 400).” With this hint, students do not have an opportunity to consider how to use the tool appropriately so that they could see a useful view of the graph.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for Meaningful Math series meet expectations that the materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. In the instructional materials, students often look for structure in patterns and generalize the patterns in addition to generalizing findings from regularity in repeated reasoning.

Some examples of MP7 include:

  • In Algebra 1, All About Alice, Extending Exponentiation, page 152, students examine a list of powers of 2 (25=32, 24=16, 23=8, 22=4, 21=2, 20=1, 2-1=?, 2-2=?, 2-3=?, 2-4=?) and describe the pattern of values on the right side of the equality statements. Students use the pattern to determine the missing values for powers of 2 with negative exponents. Students use the structure of powers to similarly determine a list of negative powers for 1/2 on page 153.
  • In Algebra 1, Fireworks, The Form of It All, pages 390-391, students consider the multiplication of two two-digit numbers using an area model. This structure of the area model is built upon as students multiply algebraic expressions on pages 392-393. Factoring is informally introduced using the area model in Exercise 4 on page 393 when students are given the total area and seek to find the length and width to set the stage for factoring quadratic expressions using this model later in Fireworks, Intercepts and Factoring, page 415.
  • In Algebra 2, High Diver, A Falling Start, Page 269, students find the values of i3, i4 ,i5, and students use the structure to write an equivalent form of i3057 and a general procedure for finding the value of in.

Some examples of MP8 include:

  • In Geometry, Shadows, What Is a Shadow?, page 16, students find a formula to represent how many wood strips would be needed to build square windows of different sizes. Students are given a diagram showing how many wood strips would be needed for a 3 x 3 window. Students then draw windows of different sizes and make an in-out table of values for the different windows. Students use the table of values or the picture to obtain a formula for any n by n window.
  • In Geometry, Shadows, The Shape of It, page 22, students use protractors to discover the angle sums of triangles and quadrilaterals. Students build upon this knowledge in the following activity on page 23 as they consider other polygons. During this activity, students generalize their findings for a few specific polygons to find an expression for the sum of the angles in a polygon as a function of the number of sides in that polygon.
  • In Algebra 2, Small World, Isn’t It?, A Model for Population Growth, page 49, students consider a population that doubles every 12 hours. In Exercise 1, students figure out how many creatures there are at specific times then generalize and create a formula to find the size of the population for any number of days in Exercise 2.
  • In Algebra 2, The World of Functions, Tables, page 333, students consider f(x) = 4x + 7 and look for a pattern in the output values based on input values that have a constant difference between them. Students create their own linear functions, look for patterns in the output values based on their own functions, and develop a generalized statement regarding the pattern in constant differences of output values within a table for all linear functions. On page 338, students work with specific quadratic functions and express regularity in the repeated reasoning to develop a general conclusion regarding constant second differences in outputs with constant changes in x within a table for all quadratic functions.

Gateway Three

Usability

Not Rated

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
0/8

Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
0/2

Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
0/2

Indicator 3c

There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.
0/2

Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
0/2

Indicator 3e

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
0/0

Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
0/8

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
0/2

Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
0/2

Indicator 3h

Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.
0/2

Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
0/2

Indicator 3j

Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
0/0

Indicator 3k

Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
0/0

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
0/0

Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
0/10

Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
0/2

Indicator 3n

Materials provide support for teachers to identify and address common student errors and misconceptions.
0/2

Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
0/2

Indicator 3p

Materials offer ongoing assessments:
0/0

Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
0/2

Indicator 3p.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
0/2

Indicator 3q

Materials encourage students to monitor their own progress.
0/0

Criterion 3aa - 3z3ad

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
0/10

Indicator 3aa

0/

Indicator 3ab

0/

Indicator 3ac

0/

Indicator 3ac.i

0/

Indicator 3ac.ii

0/

Indicator 3ad

0/

Indicator 3r

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
0/2

Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
0/2

Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
0/2

Indicator 3u

Materials provide support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
0/2

Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.
0/2

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
0/0

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
0/0

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
0/0

Indicator 3z

0/

Indicator 3z3ad

0/

Criterion 3aa - 3z

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
0/0

Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
0/0

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
0/0

Indicator 3ac

Materials can be easily customized for individual learners.
0/0

Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
0/0

Indicator 3ac.ii

Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
0/0

Indicator 3ad

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
0/0

Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
0/0

Additional Publication Details

Report Published Date: Tue Apr 03 00:00:00 UTC 2018

Report Edition: 2014

Title ISBN Edition Publisher Year
Meaningful Math Algebra 1 Student Edition 978-1-60720-601-9 2014/2015 Activate Learning 2015
Meaningful Math Geometry Student Edition 978-1-60720-602-6 2014/2015 Activate Learning 2015
Meaningful Math Algebra 2 Student Edition 978-1-60720-784-9 2014/2015 Activate Learning 2015
Meaningful Math Algebra 1 ebook 978-1-68231-202-5 2014/2015 Activate Learning 2015
Meaningful Math Geometry ebook 978-1-68231-204-9 2014/2015 Activate Learning 2015
Meaningful Math Algebra 2 ebook 978-1-68231-206-3 2014/2015 Activate Learning 2015

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Once a review is complete, publishers have the opportunity to post a 1,500-word response to the educator report and a 1,500-word document that includes any background information or research on the instructional materials.

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All team members look at every grade and indicator, ensuring that the entire team considers the program in full. The team lead and calibrator also meet in cross-team PLCs to ensure that the tool is being applied consistently among review teams. Final reports are the result of multiple educators analyzing every page, calibrating all findings, and reaching a unified conclusion.

Math HS Rubric and Evidence Guides

The High School review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The High School Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

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