High School - Gateway 2

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).

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.

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.

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.

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.