High School - Gateway 1

The instructional materials reviewed for the Interactive Mathematics Program series partially meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials include many instances where all or some aspects of the non-plus standards are not addressed across the courses of the series.

The following standards have at least one aspect of the standard that is not addressed across the series.

• A-REI.3: Students solve linear equations and inequalities in one variable in Year 1, The Overland Trail, Reaching the Unknown, pages 82, 92, 95-96 as well as in Year 1, Cookies, Cookies and Inequalities, pages 347 and 349; and Year 1, Cookies, Points of Unknown, page 382, but there are no opportunities to solve equations with coefficients represented by letters.
• G-CO.13: Students construct equilateral triangles, squares, and regular hexagons in Year 2, Geometry by Design, Do It Like the Ancients, page 91, Construction and Deduction, page 123, and Supplemental Activities, page 178, but they do not construct these shapes inscribed in a circle.
• G-C.3: The materials describe the constructions for the inscribed and circumscribed circles of triangles in Year 3, Orchard Hideout, Supplemental Activities, pages 167-168, but the materials do not include proofs of properties of angles for a quadrilateral inscribed in a circle.
• S-ID.3: Students interpret differences in the shape, center, and spread of data sets without outliers in several of The Pit and the Pendulum tasks but do not have an opportunity to do so for data sets with outliers.
• S-IC.5: Students conduct experiments and use simulations in several tasks of Year 1, The Pit and the Pendulum, but do not compare two treatments.
• S-CP.2: Students find the probability of independent and dependent events in contextual tasks in Year 2, The Game of Pig, Pictures of Probability, page 215 and Year 2, The Game of Pig, In the Long Run, page 219 using area models and tree diagrams. However, no formal definition of independence is included and the products of probabilities are not used to determine if an event is independent.
• S-CP.4: Students construct two-way tables for numerous contextual tasks in Year 3, A Difference Investigation, but there is no opportunity to determine if events are independent using two-way tables.

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

• F-IF.9: Tasks include a single function represented in different ways and two functions represented in the same way, but none compare properties of two functions represented in different ways.
• G-CO.6: There are no opportunities to use the definition of congruence in terms of rigid motions to decide if two figures are congruent.
• G-CO.7: There are no opportunities to use the definition of congruence in terms of rigid motions to show that two triangles are congruent.
• G-CO.8: There are no opportunities to explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions.
• G-SRT.2: Transformations are not used in relation to similarity.
• G-SRT.3: Although students use the AA criterion to identify the similarity of two triangles, transformations are not used in relation to the AA criterion.
• G-GPE.7: Tasks include finding the perimeters of polygons and areas of triangles and rectangles, but the figures are not plotted on the coordinate plane.

The instructional materials reviewed for the Interactive Mathematics Program series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. Throughout the series, aspects of the modeling process are present in isolation or combinations, but the full intent of the modeling process is not used to address more than a few of the modeling standards.

Throughout the series, students complete open-ended problems that include defining the variables and selecting methods for solving the problems, but the problems do not ensure that the entire modeling process will occur. Some problems provide significant scaffolding and guidance, which diminishes students’ opportunities to make choices, assumptions, and approximations. Many Problems of the Week (POWs) offer opportunities to formulate models, interpret results, validate conclusions, and report on conclusions, but there are few opportunities for students to improve their first model.

Some examples of standards for which the modeling process is incomplete are:

• N-Q.1: In Year 1, Overland Trail, The Graph Tells a Story, pages 45-50, students use graphs to describe the relationship between two quantities and supply reasonable units on the axes for such relationships, but the tasks include guiding questions that help students make sense of the relationships rather than having students make their own assumptions.
• A-SSE.3: In Year 1, Overland Trail, Reaching the Unknown, page 79, students are given the variables to use. Students write their own equation in two variables, solve the equation for one of the variables, graph the solved equation, and use the graph to determine multiple solution possibilities.
• A-SSE.4: In Year 2, Small World, Isn’t It?, Supplemental Activities, pages 474-475, students are given a definition for a geometric sequence but are not given the term or sum formulas. Students are told to use the variable n to write a term and sum expression following the steps given. Students are told to use and to apply the formulas to “check” with specific terms of the sequence.
• A-CED.1: In Year 2, Fireworks, A Quadratic Rocket, pages 6-7, students answer questions about population growth for rats using a process they devise. In addition to giving their answer and describing their solution process, students describe attempts that did not work and evaluate how confident they are in the correctness of their answer. Students do not, however, make and test their own assumptions or decide if the results are acceptable.
• A-REI.11: In Year 4, Meadows or Malls?, Equations, Points, Lines, and Planes, page 33, students write constraints for a system of linear equations based on a cookie-selling scenario, and they represent the given situation with inequalities. In Equations, Points, Lines, and Planes, page 36, students write and solve systems of linear equations for given word problems, but they are told to use substitution to solve.
• F-IF.4: In Year 3, World of Functions, The What and Why of Functions, page 320, students model and analyze situations involving profit and tickets sold, but they are told to formulate the problem as graphs. In Year 3, World of Functions, Tables, pages 325-334, students model linear, quadratic, cubic, and exponential functions, but are told to use tables.
• F-BF.1a: In Year 1, The Overland Trail, Reaching the Unknown, pages 81-82, students write functions to model weekly pay rates and time for shifts, choosing rates and checking their functions with the given criteria, but are led through steps for using graphs, words, and equations.
• F-LE.1c, 2: In Year 1, All About Alice, Who’s Alice?, pages 422-423, students explore exponential growth and decay, as well as give and explain a rule that models a situation from Alice’s Adventures in Wonderland, but the variables are defined for them.
• S-ID.6a: In Year 1, The Pit and the Pendulum, pages 142-145, 152, and 203, students gather data from an experiment using a pendulum and then graph, analyze, and find a function that fits the data, but the process is scaffolded for students throughout the activities by the information provided and questions asked within the materials.

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

• In Year 1, The Pit and Pendulum, Statistics and the Pendulum, pages 174-175, 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 Year 3, Pennant Fever, Play Ball, pages 6-8, students determine on which day of the week a person was born given the date the person was born. This POW does not align to any standards from the CCSSM.

The instructional materials reviewed for the Interactive Mathematics Program series partially meet expectations, when used as designed, 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 that each course of the series addresses these standards less frequently as the series progresses. Similarly, in examining each activity of the course independently of the alignment document, reviewers verified the greatest focus on the WAPs is in Years 1 and 2, with less attention to these standards as the series progresses. Overall, the majority of the time across the series is not spent on the WAP standards, and 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 Year 1, The Overland Trail focuses on understanding functions (8.F.1) rather than tasks for the related high school standard, F-IF.1 (function notation, domain, and range).
• In Year 1, the majority of Shadows addresses unit rates (7.RP.1) and proportional relationships (7.RP.2).
• In Year 2, Do Bees Build It Best?, Area, Geoboards, and Trigonometry, pages 291-299, students spend a majority of the time on tasks involving area, triangles, and geoboards (6.G.1).
• Year 2, The Game of Pig, Pictures of Probability, pages 209-217 addresses probability (7.SP.C) through area models.
• In Year 2, Small World, Isn’t it: Beyond Linearity, Speeds, Rates, and Derivatives, page 421, students solve derivatives of functions at given points. This is a topic that does not align to any of the CCSSM.
• Year 3, Is There Really a Difference?, A Tool for Measuring Differences, pages 460-475 focuses on statistical analyses including chi-square. This is a topic that does not align to any of the CCSSM.
• In Year 3, Pennant Fever, students use combinatorics (S-CP.9) to develop the binomial distribution (A-APR.5) and find the probability that the team leading in the pennant race will ultimately win the pennant, addressing topics that are not widely-applicable prerequisites for postsecondary work. These are plus standards.

The instructional materials reviewed for Interactive Mathematics Program series partially meet expectations for letting students fully learn each non-plus standard when used as designed. The following standards are addressed in a way that provides limited opportunities for students to fully learn these standards.

• N-RN.1: The reviewers found minimal evidence for denoting radicals in terms of rational exponents: Year 1, All About Alice, Curiouser and Curiouser!, pages 442, 443, 449, and 476.
• N-CN.2: One task (Year 3, High Dive, A Falling Start, page 262, Exercise 3) includes use of the relation i² = -1 to evaluate higher powers of i and one task that includes addition of complex numbers (Year 3, High Dive, Complex Components, Exercise 3) but no tasks that include subtraction of complex numbers.
• A-SSE.3c: The reviewers found no tasks related to using the properties of exponents to transform expressions for exponential functions.
• A-SSE.4: Students derive the formula for the sum of a finite geometric series (Year 1, All About Alice, Supplemental Activities, page 466-467), but reviewers found two tasks that involve using the formula to solve problems: Year 1, All About Alice, Supplemental Activities, pages 466-467 and Year 2, Small World, Isn’t It?, pages 474-475.
• A-APR.1: Properties of polynomials are described in Year 2, Fireworks, Intercepts and Factoring, page 54, but no tasks address understanding that polynomials form a system that is closed under addition, subtraction, and multiplication.
• A-APR.3: Students factor quadratics and find x-intercepts in Year 2, Fireworks, Supplemental Activities, pages 75 and 77, but the x-intercepts are not used to draw graphs.
• A-APR.4: The reviewers found limited opportunities for working with polynomial identities: Year 2, Fireworks, Supplemental Activities, page 74 and Year 3, The World of Functions, Supplemental Activities, pages 417, 421-422.
• A-APR.6: Students divide polynomial expressions in Year 3, The World of Functions, Supplemental Activities, pages 418-420, but the expression is not presented as a rational function in the form a(x)/b(x).
• A-REI.4b: In Year 2, Fireworks: Supplemental Activities, page 70, 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 them in the form of a + bi for a few exercises in Year 3, High Dive: A Falling Start, page 262.
• F-IF.7e: In Algebra 1, Supplemental Activity: The Growth of Westville (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: Students work with arithmetic and geometric sequences recursively and with explicit formulas (Year 1, All About Alice, Supplemental Activities, pages 466-467; Year 2, Small World, Isn’t It?, All in a Row, page 408 and Supplemental Activities, pages 472 and 474; Year 3, High Dive, A Trigonometric Interlude, pages 249-251) but do not translate between the two forms.
• F-TF.8: Year 3, High Dive, A Trigonometric Interlude, page 242 includes the derivation of the Pythagorean identity, but the reviewers found no tasks addressing use of the Pythagorean identity to calculate trigonometric ratios outside of the first quadrant.
• G-C.1: Year 2, Geometry by Design, Putting the Pieces together, pages 166-167 notes that all circles are similar but does not include a proof.
• G-GPE.6: Students find midpoints in Orchard Hideout, Coordinates and Distance, pages 111 but do not partition line segments in other ratios.
• S-ID.2: Students compare the spread of two data sets in Year 1, The Pit and the Pendulum, Statistics and the Pendulum, pages 163 and 170, but reviewers found limited practice comparing the center of two or more different data sets.
• S-CP.3: Although students engage in problems related to conditional probability in Year 2, The Game of Pig, In the Long Run, page 225, students do not interpret the independence of events by calculating conditional probabilities.

The instructional materials reviewed for Interactive Mathematics Program series partially meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. Tasks are set in relevant contexts and address numerous key takeaways from Grades 6-8, but the materials do not vary the types of real numbers being used.

Scenarios and situations presented are appropriate for high school students and address a variety of interests. For example, the series uses historical scenarios (Overland Trail), literature (Pit & Pendulum, Alice in Wonderland), games (Pig, Pennant Fever) and societal issues (Small World) to engage students.

Key takeaways from Grades 6-8 are addressed. For example:

• In Year 1, Shadows, Triangles Galore, pages 280-281, students use ratios and proportional relationships (6.RP.A, 7.RP.A, 8.EE.B) to convert recipes, calculate fuel mileage, and plan a dance.
• In Year 2, Supplemental Activities, Above and Below the Middle, pages 244-246, students build on knowledge of mean and median (6.SP.5c) to analyze results and determine outcomes of rolling a pair of dice until doubles appear.
• Year 3, The World Of Functions builds on functions (8.F) which is a key takeaway from middle school. Students build on in and out tables and linear functions having features such as “equal spacing” to quadratics, exponential, and cubic tables to examine their spacing and rates of change.

Most problems, however, include only integer values. Students have limited opportunities to work with fractions and decimals. For example:

• In Year 1, The Graph Tells a Story, page 53, students solve five one-variable equations in which all constants and coefficients are integers, and all solutions are integers.
• In Year 2, Fireworks, The Form of It All, page 27, completing the square problems is limited to integer values for constants and coefficients.
• In Year 3, Orchard Hideout, Coordinates and Distance, page 108, students solve problems involving circles on the coordinate plane, but the coordinates contain only integer values.

The instructional materials reviewed for Interactive Mathematics Program series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The instructional materials do not explicitly identify content from Grades 6-8, but the materials include and build on content from Grades 6-8.

Neither the teacher nor the student materials explicitly identify content aligned to standards from Grades 6-8. Examples of tasks that build on standards from Grades 6-8 to the high school standards that do not identify the standards from Grades 6-8 include:

• In Year 1, The Overland Trail, Reaching the Unknown, pages 91-93 and 95, students solve equations in one variable (8.EE.7) and explain their solutions (A-REI.1).
• Year 1, The Pit and the Pendulum, Edgar Allan Poe--Master of Suspense, page 149 connects the mean of a set of data (6.SP.5) with dot plots (S-ID.1).
• In Year 2, Fireworks, Putting Quadratics to Use, page 34, students use the Pythagorean Theorem (8.G.7) to write a quadratic equation (A-CED.2).
• In Year 2, Geometry by Design, Isometric Transformations, page 142, students rotate shapes onto themselves (G-CO.3) and state that the new image is the exact same as the original (8.G.2).
• Year 3, Orchard Hideout, Equidistant Points and Lines, page 113 connects finding the distance between two points in a coordinate system (8.G.8) and using coordinates to determine what type of quadrilateral a figure is (G-GPE.4).
• In Year 3, High Dive, Sand Castles, page 208, students graph and interpret trigonometric functions (F-IF.7e) and identify inputs and their corresponding outputs (8.F.1) to make sense of a context.