High School - Gateway 2

The instructional materials for the Interactive Mathematics Program 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 Year 2, Fireworks, A Quadratic Rocket, page 17, the Group Activity connects the number of x-intercepts with the shape of the parabola. Quadratic functions are given in vertex form, and students determine where the vertex lies, whether the parabola is concave up or concave down, and use these facts to determine how many x-intercepts it has. In the Glossary on page 504 in Year 2, the zero product rule is defined. This is explored in Year 2, Fireworks, Intercepts and Factoring, page 47, Group Activity, where students factor to find zeros and make a statement that connects the idea of x intercepts to “the values of x that make y = 0.” There are subsequent activities in Year 2, Fireworks that students independently engage in factoring expressions, graphing parabolas, and solving a cattle pen application using the factored form and the x-intercepts. Year 3, The World of Functions, Supplemental Activities, page 423 connects the idea between roots, equations, and graphs of polynomial functions. Note the term roots is used rather than zeros.
• A-REI.A: In Year 1, The Overland Trail, pages 92-96, students build understanding around balancing equations and explain what they are doing, which is pertinent in the conceptual development for this cluster of standards. In Year 2, Fireworks, Supplemental Activities, pages 68-69, students are exposed to extraneous solutions. Students are asked “See if you can find a rule for determining when an extraneous solution will occur.”
• A-REI.10: In Year 1, The Overland Trail, The Graph Tells a Story, page 51, students are given in-out relationships to make a table, plot the ordered pairs, and sketch a graph. Students connect “in and out” to independent and dependent variables on a graph and then graph equations on a coordinate plane. In the same year, All About Alice, Curiouser and Curiouser!, page 443, students use a similar method to graph y=2x. In Year 3, The World of Functions, Going to the Limit, pages 343-344, students investigate rational function graphs.
• F-IF.A: In Year 1, The Overland Trail, The Importance of Patterns, page 12 and The Graph Tells a Story, pages 49-51, students use in and out tables and real-world context to develop the idea of functions. Students extend their idea of functions using sequences later on in The Overland Trail, Supplemental Activities, page 104 where students find next terms and write equations for the sequences.
• G-SRT.6: Year 1, Shadows, pages 305-309 provide students a task to draw a right triangle and investigate the idea of the ratios that lead to defining sine, cosine, and tangent. Students are asked if their classmates will get similar results for their ratios and have to “Explain in detail why or why not.”
• S-ID.7: Year 1, The Overland Trail, page 66 begins to develop the idea of slope by giving real-world scenarios, and students write linear equations and answer questions. Pages 74 and 75 extend the conceptual development of slope by providing opportunities for students to find equations of real-world scenarios and answer questions related to the rate and the starting point (y-intercept). It should be noted that rate of change, slope, and y-intercept are not used.

The instructional materials for Interactive Mathematics Program 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.

Students engage in and practice problem solving, solve non-routine problems, and apply math in contextual situations with increasing sophistication across the courses.

Examples of engaging high school applications in real-world contexts are shown below:

• N-Q.1: Students choose and interpret the scale and origin in graphs and data displays frequently, including choosing axes for various real-world situations in Year 1, The Overland Trail, The Graph Tells a Story, pages 46-47. Students also use an appropriate scale as they graph data on supplies settlers used while travelling by wagon train in Year 1, The Overland Trial, pages 55 and 59. In Year 1, Cookies, Cookies and the University, page 389, students make an appropriate graph to decide how many cookies to produce for maximum profit.
• A-SSE.3: In Year 2, students use equivalent forms of expressions to show properties of quantities represented by those expressions as they learn about relationships between equations and graphs of quadratic functions. In Fireworks, A Quadratic Rocket, pages 4-5, 11-12, and 14-17; The Form of It All, pages 21-27 and 29-30; Putting Quadratics to Use, pages 36-38 and 41; and Back to Bayside High, page 45, students also use completing the square to rewrite quadratic equations to solve problems involving rockets.
• G-SRT.5: Students apply triangle congruence and similarity to solve problems through indirect measurement in various contexts for Year 1, Shadows, The Lamp Shadow, pages 294-298 and 302 and then use this idea to develop and use trigonometric ratios for Year 1, Shadows, The Sun Shadow, pages 305-313.
• F-IF.4: Students sketch graphs showing key features of the relationship between two quantities for given stories in Year 3, The World of Functions, The What and Why of Functions, pages 320-321 and 326.
• F-IF.6: Students calculate and interpret the average rate of change in a number of contexts, including graphs showing distance travelled each day in Year 2, Small World, Isn’t It?, Average Growth, page 392 and 396; a graph showing population growth in Year 2, Small World, Isn’t It?, page 398; and a graph of an equation representing the height of a falling object in Small World, Isn’t It?, Beyond Linearity, page 415.

The instructional materials for Interactive Mathematics Program 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 Year 1, Overland Trail, Setting Out with Variables, pages 33-39, procedural skills and applications are integrated. Students find the value of algebraic expressions and solve algebraic equations in the context of planning a trip by wagon train.
• In Year 2, Fireworks, The Form of It All, pages 19-32, students build a conceptual understanding of completing the square by working with area models of multiplication, applying those area models to the distributive property of multiplication over addition and the multiplication of binomial expressions, and using those ideas to derive the process of completing the square. In Putting Quadratics to Use, pages 33-41, students use completing the square to solve maximization problems involving quadratic functions, with practice of this procedural skill distributed throughout the unit and included in Supplemental Activities, pages 72, 73, and 77.
• In Year 3, The World of Functions, page 318, students reason about the relationship between speed and stopping distance using multiple representations as they are introduced to the unit problem. Students continue to make connections between a verbal description and an appropriate graph on pages 320-323 and 326. Continuing in World of Functions, Tables, students use tables to explore patterns and properties of linear, quadratic, cubic, and exponential functions, pages 325, 327, 330-334, 339, then assign functions to tables in Who’s Who?, page 353. The unit concludes with students returning to the unit problem to 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 Year 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 Year 1, Shadows, The Shape of It, pages 269, 270, and 272, students create proportions based on similar figures and solve the proportions to find the lengths of missing sides.
• In Year 1, Cookies, Points of Intersection, page 387, students solve linear equations and linear systems.
• In Year 2, Fireworks, Intercepts and Factoring, page 48, students factor quadratic equations.
• In Year 2, Small World, Isn’t It?, All in a Row, page 410, 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 Year 2, Small World, Isn’t It?, All in a Row, page 404, 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.
• In Year 3, The World of Functions, Composing Functions, page 373, 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.

The instructional materials for Interactive Mathematics Program 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 of MP2 and MP3 that reaches the full intent of the MPs.

Students often reason abstractly and quantitatively (MP2). Examples include:

• In Year 1, The Overland Trail, The Graph Tells a Story, pages 49-50, students answer a series of questions that lead them to analyze graphs, create corresponding tables, and write rules based on the information.
• In Year 2, Geometry by Design, Isometric Transformations, page 148, students solve a problem about a right triangle with a given slope and then describe a general solution for any slope.
• In Year 3, High Dive, Falling, Falling, Falling, pages 217-218, 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.

Students often construct viable arguments and critique the reasoning of others (MP3). Examples include:

• In Year 1, All About Alice, Curiouser and Curiouser, pages 447-448, 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 Year 2, Geometry by Design, Dilation, page 161, 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 then students determine what size enlargement was made.
• In Year 3, Is There Really a Difference?, Data, Data, Data, page 431, a scenario is given where students are constructing a mathematical argument for a jury. The student then becomes a member of the jury to see if there is enough evidence. This scenario develops further by having students include an explanation of additional evidence that might be needed to win the case.

The instructional materials for Interactive Mathematics Program 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 find and use patterns and generalize findings from regularity in repeated reasoning.

Students often look for and make use of structure (MP7). Examples include:

• In Year 1, All About Alice, Extending Exponentiation, pages 152-153, students examine a list of integer powers of 2 from -4 to 5, describe the structure they see, and use their results to find other powers of 2 with negative exponents.
• In Year 2, Fireworks, The Form of It All, pages 22-23, students consider the multiplication of two two-digit numbers using an area model. The structure of an area model is built upon in pages 24-25 as students multiply algebraic expressions using the same format. Factoring is informally introduced using the area model in Exercise 4 on page 25 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 47.
• In Year 3, Pennant Fever, Baseball and Counting, students write formulas for combinations and permutations by looking for structure in their work with specific examples from previous activities.

Students often look for and express regularity in repeated reasoning (MP8). Examples include:

• In Year 1, Shadows, The Shape of It, page 254, students use protractors to discover the angle sums of triangles and quadrilaterals. Students build upon this knowledge in the following activity on page 255 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 its sides.
• In Year 2, Small World, Isn’t It?, Average Growth, pages 398-399, students find the rate of change of a function from several graphs of real-world situations and use these repeated examples to determine a general expression for rate of change of a linear function given any two ordered pairs.
• In Year 3, The World of Functions, Tables, page 325, students consider f(x) = 4x + 7 and look for a pattern using equally-spaced inputs. Students then consider other linear functions of their own choosing and reach a generalized statement regarding the pattern in constant differences in outputs within a table for all linear functions. On page 338, students work with concrete examples and then generalize to reach a conclusion regarding constant, second differences in outputs with constant changes in x within a table for all quadratic functions.