High School - Gateway 1

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.

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.

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.

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.

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).

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.