High School - Gateway 2

The instructional materials reviewed for HMH Into AGA meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The series has Build Understanding in each lesson, which often includes Turn and Talks, designed for students to demonstrate their conceptual understanding.

Examples of the materials developing conceptual understanding and students independently demonstrating it include:

• N-RN.1: In Algebra 2, Lesson 6.1, students understand how the Power of a Power Property is extended to a rational exponent by solving an equation where the variable is an exponent and justifying each step. Students translate between rational exponents and radical expressions. In Lesson 6.2, students investigate the properties of rational exponents given the rules for integer exponents in a table. Students predict and show that the rules for integer exponents extend to rational exponents.
• A-APR.B: In Algebra 2, Lesson 5.1, students analyze the graph of a polynomial function in factored form, find zeros, write the function in standard form, and determine how the zeros are related to the standard form of the function.
• A-REI.A: In Algebra 1, Lesson 2.2, Journal and Practice Workbook, students solve equations, justify their solution steps, and check the solution. In Problem 9, students critique Kevin's reasoning by explaining his error, correcting his error, and completing the correct work to solve the equation. 
• F-IF.A: In Algebra 1, Lesson 4.1, Turn and Talk, students explain if a relation in which every domain value corresponds to the same range value is a function. Students state the change that would need to be made in a table of values so that the relation becomes a function. Students determine if it is sensible to complete a horizontal line test for functions and explain their reasoning.
• G-CO.8: In Geometry, Lesson 8.2, students explain the reason triangles provided on a coordinate plane are congruent using a triangle congruence theorem and describe a sequence of transformations that maps one triangle to the other. In Lesson 8.3, students use the SSS triangle congruence theorem and distance formula to show that two triangles provided on a coordinate plane are congruent.
• G-SRT.2: In Geometry, Lesson 12.1, students compare coordinates in a dilation to determine if two figures are similar. Students answer questions to explain or justify their answers using transformations. Students also determine whether each pair of figures is similar using similarity transformations and explain their reasoning.

The instructional materials reviewed for HMH Into AGA meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters.The instructional materials develop procedural skills, and students independently demonstrate procedural skills throughout the series. 

Examples from On Your Own in the student materials where students independently demonstrate procedural skills include: 

• N-RN.2: In Algebra 2, Lesson 6.1, Problems 52-60, students rewrite expressions with rational exponents as radicals. In Problems 66-74, students rewrite radicals as expressions with rational exponents.
• A-SSE.2: In Algebra 1, Lesson 15.3, Problems 14-37, students find special products of binomials. In Algebra 2, Lesson 4.4, Problems 7-24, students factor polynomials, including factoring by grouping.
• A-APR.1: In Algebra 1, Lesson 15.1, Problems 3-8 and 23-32, Lesson 15.2, Problems 7-9 and 23-30, Lesson 15.3, Problems 14-37, and Lesson 16.1, Problems 21-44, students add, subtract and multiply polynomials. 
• A-APR.6: In Algebra 2, Lesson 11.2, Problems 34-37, students rewrite rational functions to find the quotient and the remainder.  
• G-SRT.5: In Geometry, Lesson 8.3, Problems 7-12, students determine if triangles are congruent using SSS, ASA, or SAS triangle congruence.
• G-GPE.4: In Geometry, Lesson 4.3, Problems 8-13, students determine whether two lines are congruent by using the distance formula.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially when called for in specific content standards or clusters. Application problems occur toward the end of each lesson, and the Journal and Practice Workbook follows the same pattern of having at least one application problem at the end of each lesson. 

Examples of students utilizing mathematical concepts and skills in engaging applications include:

• N-Q.A: In Algebra 1, Lesson 1.3, students choose a level of accuracy appropriate for the finances of a family budget of expenses that include rent, food, transportation, clothing, and entertainment. Students explain their reasoning of the level of precision that will be acceptable to assemble the budget.
• A-SSE.3: In Algebra 1, Lesson 19.3, students write a function in intercept form from a given function denoting the monthly profit of a party planning business after time, t, in months. Students provide intercepts and their meaning in a situation, write a function in vertex form, and provide an interpretation for the vertex in a situation.
• F-IF.B: In Algebra 2, Unit 2, Project Network Functions, students sketch a graph of a polynomial function that models the number of users connected to the network t hours after 6:00 AM. Students describe the characteristics of the graph and decide if they would classify the graph as an even or odd function and explain their reasoning. 
• F-BF.1: In Algebra 2, Lesson 11.1, students solve inverse functions using various applications, such as vibrating violin strings. Another problem provides a table of carpentry projects, and students determine the associated equation. Students also compare pressure on snow depending on the type of shoes worn and find the height of a tree from a given distance. 
• G-SRT.8: In Geometry, Lesson 13.1, students apply the tangent function to find the distance a person is standing from the Washington Monument given the height of the monument and the angle between the ground and the top of the monument from the place where the person is standing.
• G-GMD.3: In Geometry, Lesson 19.2, students find the amount of plastic needed to make a cylindrical-shaped, plastic cup. Students also calculate the cost per cubic inch of plastic, and the amount Dan would pay for a cup with the same radius if the height were 6 inches.
• S-ID.2: In Algebra 1, Lesson 22.1, students find measures of center and spread for each data set given about swim times for two swimmers. Students compare the two data sets and identify if either data set has an outlier, choose the better swimmer, and justify the statistics used to determine the answer.

The instructional materials reviewed for HMH Into AGA meet expectations for not always treating the three aspects of rigor together and not always treating them separately. All three aspects of rigor are present independently throughout the materials, and multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each module begins with Are You Ready that reviews prior concepts and skills needed for that module; these problems are procedural. The lessons are intentionally developed so that students have opportunities to practice each aspect of rigor throughout each lesson.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• In Algebra 1, Lesson 2.1, Spark Your Learning, students build conceptual understanding through application by comparing two stores offering different deals on the same-priced phone and determine the strategy to use to answer the problem. Students use procedural skills to write an expression to find the store that has the better deal.
  • In Algebra 1, Lesson 7.1, On Your Own, students use procedural skills to extend various types of patterns for arithmetic sequences to determine the common difference and write a recursive rule in function notation. Students extend this knowledge to write an explicit rule for arithmetic sequences. Students complete an application problem by writing a recursive rule and using that rule to determine the amount a caregiver earns in a home health job.
• In Geometry, Lesson 5.1, students use procedural skills to draw figures with given vertices and the images after a translation by a given value. Students use conceptual understanding as they reason a classroom map where students are standing at different spots, and students complete an application problem about a cyclist to determine what vector describes her position from the starting position to her final destination.
• In Geometry, Lesson 13.2, students build conceptual understanding of side ratios of a right triangle. In Turn and Talk, students predict how their answer would change if angle A is increased in a given triangle. Students use procedural skills to calculate Sine and Cosine ratios in the On Your Own problems. 
• In Algebra 2, Lesson 3.2, On Your Own, students use procedural skills to find key features of polynomial functions. Later, students engage in application problems using polynomial functions.
• In Algebra 2, Lesson 4.5, Journal and Practice Workbook, students use procedural skills when using synthetic substitution to find p(-3) for each polynomial, and determine whether the given polynomial is a factor of the polynomial p(x). Students apply their understanding of polynomial functions in the context of profit for a game by determining a lesser number of games a company could produce and still make the same profit. Students use conceptual understanding to determine if the set of polynomials is closed under subtraction and either justify or give a counterexample.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards.

Examples where the materials identify MP2 as MP Reason and students reason abstractly and quantitatively include:

• In Algebra 1, Lesson 2.4, students write an inequality and explain their reasoning of the amount of pillows that must be sold during the third month to have a profit for the quarter in a company with a fixed monthly expense of $3,200. 
• In Algebra 1, Lesson 3.1, students interpret graphs of a real-world scenario of purchasing T-shirts and sweatshirts with $100 and determine the solutions that make sense in the context of the situations.
• In Geometry, Lesson 4.3, students explain why x$$_1$$$$\not =$$x$$_2$$ and y$$_1$$$$\not =$$y$$_2$$ when using the Pythagorean theorem to prove that the distance between two points on the coordinate plane is given by the distance formula.
• In Geometry, Lesson 5.2, Workbook, students decide what regular polygon Cindy has and explain their reasoning. Students also decide the smallest angle of rotation for an image is, based on given information, and explain their reasoning.
• In Algebra 2, Lesson 4.4, students use spatial and algebraic reasoning to determine the dimensions of prisms, from given expressions for the volumes of four small rectangular prisms that form one large rectangular prism.
• In Algebra 2, Lesson 15.1, Workbook, students calculate the angle a larger gear turns if the smaller gear turns through a central angle of$$\frac{3\pi}{16}$$ radians.

Examples where the materials identify MP3 as MP Critique Reasoning or MP Construct Arguments and students construct viable arguments and critique the reasoning of others include:

• In Algebra 1, Lesson 17.1, students construct an argument about viewing windows where graphs cross the x-axis. 
• In Algebra 1, Lesson 16.2, students critique John’s answer to a problem of how a given expression represents a sequence and explain why or why not his given sequence is correct. 
• In Geometry Lesson 8.2, students determine if Zach has made a mistake and correct his error through an explanation of a given proof showing two triangles are congruent. 
• In Geometry Lesson 8.4, students construct an argument by explaining if the given claim that two right triangles share a hypotenuse, then the triangles must be congruent is correct.
• In Algebra 2, Lesson 4.4, students construct an argument if it is possible to factor any polynomial with four terms using factoring by grouping. 
• In Algebra 2 Lesson 13.3, students critique if John’s simplified formula for the sum of the first n terms of the geometric series is correct and explain why or why not.

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. The materials support the intentional development of MP4 and MP5.

Examples where MP4 is identified as MP Model with Mathematics and students engage in parts of model with mathematics include:

• In Algebra 1, Lesson 5.4, Problem 23, students write an equation that gives the university’s expected enrollment and write an equation for the inverse of that function. Students also predict the number of years it will take for enrollment to reach 10,000 students.
• In Algebra 1, Lesson 20.2, students choose possible dimensions and areas for a chicken coop given some restraints. There is one correct answer for each restraint, but students do revise their solutions based on the changing constraints.
• In Geometry, Lesson 4.3, students use the distance formula to write and solve an equation about a car traveling down a straight road starting at the origin. Students also find the midpoint of the path of the car.
• In Geometry, Lesson 17.3, students write an equation to solve a construction site problem with sectors from two different circles outlined. Students are provided a hint as to how to solve the problem if it is needed.
• In Algebra 2, Lesson 3.1, students write a polynomial function to model the mass of a cylindrical concrete pier used in house construction as shown in an image. Students graph the relationship between the mass and the radius of the cylinder to determine if a specified mass is a reasonable estimate for the radius and explain their reasoning.
• In Algebra 2, Lesson 10.2, students write an exponential equation to represent the cost of schooling given the average cost and percent increase. Students determine the number of years it will take for the cost of schooling to exceed a given amount.

Examples where MP 5 is identified as MP Use Tools and students choose appropriate tools strategically include:

• In Algebra 1, Lesson 4.3, Spark Your Learning, students develop a mathematical question about a high-speed train traveling between stations. Students are specifically asked what strategy and tool they would use to solve their question (Part C). 
• In Algebra 1, Lesson 17.1, Spark Your Learning, students develop a mathematical question about a pallet of rolled sod. Students are specifically asked what strategy and tool they would use to solve their question (Part C). 
• In Geometry, Module 1, Performance Task, students choose an appropriate tool for approximating the perimeter of the irregular region enclosed by the fire edge as well as the length of the controlled fire edge. Students might use the given coordinate grid, a ruler, and/or a piece of string.
• In Geometry, Lesson 15.1, Spark Your Learning, students develop a mathematical question about an outdoor amphitheater. Students are specifically asked what strategy and tool they would use to solve their question (Part B). 
• In Algebra 2, Module 8, Performance Task, students  choose appropriate research tools to determine a reasonable average annual growth or interest rate for an investment as well as the inflation rate for the period 1960–present in order to estimate how much an investment of $1,000,000 in 1960 would be worth today. 
• In Algebra 2, Lesson 13.2, Spark Your Learning, students develop a mathematical question about the Louvre Pyramid. Students are specifically asked what strategy and tool they would use to solve their question (Part C).

The instructional materials reviewed for HMH Into AGA meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. The materials support the intentional development of MP7 and MP8.

Examples where the materials identify MP7 as MP Use Structure and students look for and make use of structure include:

• In Algebra 1, Lesson 2.2, students see algebraic expressions as single objects being composed of several objects as they notice that 3(4x - 7) + 2(4x - 7) = 5 has the form 3[ ] + 2 [ ] = 5 where [ ] = 4x - 7. Students explain how to use this observation to solve an equation, then use the same method to solve 2(3x + 4) - 5(3x + 4) = 33.
• In Algebra 1, Lesson 5.1, students look for, identify, and generalize relationships and patterns in transformations of quadratic functions by matching equations relating a function written in transformational function notation to its corresponding pair of graphs. 
• In Geometry, Lesson 6.1, students look for patterns and make generalizations by writing a coordinate rule to dilate a figure with center (a, b) not at the origin. 
• In Geometry, Lesson 18.3, students use structure to break apart solids and explore familiar two-dimensional faces by decomposing solids and developing surface area formulas for a regular pyramid and a right cone.
• In Algebra 2, Lesson 3.2, students look for overall structure and patterns as it relates to the behavior of graphs by matching factored polynomial functions to their graphs.
• In Algebra 2, Lesson 4.1, students look for mathematical structures by evaluating two functions, f(x) and g(x), given x = -1, 2, and 5. Students use the values of f(x) and g(x) to evaluate f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x)/g(x) for each of the values of x. Students then combine the function expressions to form four new functions, and evaluate each new function when x = -1, 2, and 5 to verify the results are the same. 

Examples where the materials identify MP8 as MP Use Repeated Reasoning and students look for and express regularity in repeated reasoning include:

• In Algebra 1, Lesson 2.3, students use repeated reasoning to rewrite the given perimeter formulas for specific types of regular polygons so that each formula gives the polygon’s side length s in terms of its perimeter P. In Part B, students then express regularity in their reasoning by writing the general formula.
• In Algebra 1, Lesson 7.1, students use repeated reasoning by observing that each figure in a pattern contains 3 more dots than the previous figure. Students then express regularity in their reasoning by concluding that the numbers of dots form an arithmetic sequence and write an algebraic rule for the number of dots in the nth figure.
• In Geometry, Module 2, Performance Task, students reason repeatedly about the data points in the graph by observing that the y-coordinate of each point is about 10 times the x-coordinate, meaning that an object’s weight in newtons is about 10 times its mass in kilograms. Students can then express regularity in their reasoning by conjecturing that the equation y = 10x approximates an object’s weight y if its mass is x.
• In Geometry, Lesson 5.4, Task 4, students reflect several points across the y-axis and then generalize their results by observing that when a point (x, y) is reflected across the y-axis, its image is (–x, y).
• In Algebra 2, Lesson 3.2, Task 1, , students use repeated reasoning to determine features of the graphs of specific polynomial functions. Later in the task, students express regularity in their reasoning by drawing general conclusions about the graphs of polynomial functions.
• In Algebra 2, Lesson 13.3, Spark Your Learning, students examine the first five stages of a tree fractal, look for a pattern in the number of new branches at each stage, and then express regularity in their reasoning by answering the question, “What is a general rule for the number of new branches at each stage of the tree fractal?”