8th Grade - Gateway 2

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards of cluster headings.

Materials develop conceptual understanding throughout the grade level, providing opportunities for students to independently demonstrate their understanding through various program components. Each lesson incorporates an Explore! activity for students to discover new concepts using diverse methods, a Lesson Presentation with slides that supports reasoning and sense-making through examples and communication breaks, a Student Lesson featuring mathematical representations, and a selection of Teacher Gems designed to target conceptual understanding through engaging activities such as Always, Sometimes, Never, Categories, Four Corners, and Climb the Ladder. Examples include: 

• Unit 2, Lesson 2.1, Lesson Presentation, Slide 4 & 5, Explore!, students develop conceptual understanding of the Pythagorean Theorem and its converse as they use Pythagorean triples to determine if a right triangle is formed (8.G.6). An example provided is as follows: “The lengths of the legs and the hypotenuse of a right triangle have a special relationship. Step 1: Using a ruler, draw a right triangle. Make one of the legs 3 centimeters long and the other 4 centimeters long. Use the chart below for steps 2-5. Each measurement is written in centimeters.” A chart provides four headings, Short Leg, Long Leg, Hypotenuse, Square of Short Leg, Square of Long Leg, and Square of Hypotenuse. Under the first column there are four rows. Row 1: 3, 4, . Row 2: 5, 12, . Row 3: 6, 8, . Row 4: 8, 15, _. “Step 2: Measure the length of the hypotenuse of the triangle you drew in Step 1. Round the measurement to the nearest centimeter. Fill in the blank in the first column with this measurement. Step 3: Find the square of the hypotenuse. Fill in the last column as shown in the middle column. Step 4: On scratch paper, draw another right triangle with the two legs given in the next row of the table. Measure the hypotenuse to the nearest centimeter. Complete the row. Step 5: Repeat the process for the remaining two right triangles. Step 6: Look at your chart. Do you see any patterns or relationships between the squares of the two legs and hypotenuse? If so, explain. Step 7: Create a rule for the lengths of the legs of a right triangle in relation to the hypotenuse length. Use a and b to represent the two legs and c to represent the hypotenuse.”

• Unit 3, Lesson 3.1, Explore!, students develop conceptual understanding of a function as they investigate a real-world situation represented on a graph (8.F.1). An example is as follows: “To find the relative age of a dog, some people use the rule of thumb that every year in a dog’s life is equal to seven years in a human’s life.” In Steps 1-3, students write an equation, complete a table, and graph. “The relationship between human years and dog years is a function. A function has inputs (in this case, human years) and outputs (in this case, dog years). Functions have exactly one output value for each input value. Step 4: 'Tamara was worried about her six-year-old dog, Suze, so she took her to three different veterinarians. Each vet told her the approximate age of her dog in dog years based on Suze’s health. Tamara graphed her dog’s age in dog years and included the three vets’ approximations at age 6. Do you think this graph is a function? Why or why not?” A graph is given with the x-axis labeled "Human Years" and the y-axis labeled "Dog Years," with the first 6 points showing a linear relationship and the next 3 points having the same x value. “Step 5: Sketch another graph that is not a function. Explain how you know it is not a function.”

• Unit 3, Lesson 3.3, Explore!, students develop conceptual understanding by exploring non-proportional linear functions and learning to calculate slope using slope triangles to find the change in 𝑦 over the change in 𝑥. (8.EE.6). An example provided is as follows: “Step 1: Both graphs below are linear functions. One is a proportional relationship and the other is not. Which is the proportional relationship? Why is the other not a proportional relationship? Step 2: For the proportional relationship above, what is the unit rate (or rate of change)? Explain how you calculated it. Step 3: The line on Graph B goes through (–2, –4), (0, –1) and (2, 2). For proportional relationships, you calculate the unit rate by dividing 𝑦 by 𝑥. Does this work for Graph B? Explain. To find the rate of change, which is also called the slope, on a linear graph (whether it goes through the origin or not) you can always use a slope triangle. A slope triangle is formed by drawing a horizontal leg and a vertical leg of a right triangle to connect two points. The segment on the line connecting the two points becomes the hypotenuse. Step 4: Draw a slope triangle on the copy of Graph B at the right. The slope (or rate of change) is the ratio of the change in 𝑦-values to the change in 𝑥-values. Step 5: How can you use your slope triangle in Step 4 to find the slope? What is the slope of the line? Step 6: The line at the right has a negative slope. What does this mean? Step 7: Draw a slope triangle and find the slope of the line for the graph at the right.” 

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 3, Lesson 3.2, Teacher Gems, Four Corners, students demonstrate conceptual understanding of proportional relationships as they model a proportional relationship in different ways (8.EE.5). Three equations, three graphs, three tables, and three situations all represent proportional relationships. Students are also given a blank Four Corners sheet split into four sections with the labels 'Equation,' 'Graph,' 'Table,' and 'Situation.' The student chooses one of the relationships given, which is placed into the corresponding section of the Four Corners. Then, they complete the other corners with the corresponding representation. 

• Unit 5, Lesson 5.1, Student Lesson, Exercise 11, students demonstrate conceptual understanding of systems of equations as they analyze and solve pairs of simultaneous linear equations (8.EE.8). An example is as follows: “Line A and Line B intersect at the point (−3, 4). Line A has a slope of 3. Line B’s slope is the opposite reciprocal of Line A. a. What are the equations of Line A and Line B? b. Graph Line A and Line B. c. The two lines are perpendicular. Define perpendicular in your own words. d. If two lines have slopes that are opposite reciprocals, then the lines are perpendicular. Show one more example of this by writing and graphing a new system of equations where the slopes of the lines are opposite reciprocals.”

• Unit 7, Lesson 7.5, Student Lesson, Exercise 10, students demonstrate conceptual understanding of transformations by describing the sequence that exhibits congruence between them (8.G.2). An example is as follows: “For Exercises 9-10, show that each set of figures is either similar or congruent by writing a series of transformation rules that maps the blue figure onto the green figure.” A coordinate grid is given with a blue triangle with vertices at (3,0), (0,2), (-2,0) and a green triangle with vertices at (2,0), (6,0), (4,-3).

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

There are opportunities for students to develop procedural skills and fluency in each lesson. The materials support the development of these skills and fluencies through Starter Choice Boards, Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. Examples include:

• Unit 2, Lesson 2.3, Student Lesson, Extra Example 2, students develop procedural skills and fluency in the application of the Pythagorean Theorem when given two coordinate pairs to find the distance between them (8.G.8). An example provided is, “Find the distance between (4,3) and (10,11). Round to the nearest tenth.” Quadrant 1 of a coordinate plane is given with integers labeled from 0-12 on each axis.

• Unit 3, Lesson 3.3, Lesson Presentation, Slide 9, Example 1, students develop procedural skills and fluency as the teacher guides them through examples of finding the slope of a line using a slope triangle (8.F.4). In Example 1a, students: “Draw a slope triangle for the line (when possible) and identify the slope of the line.” A graph is provided with a line drawn through the points (1,3) and (3,2), and the rise and run between the two points are indicated on the graph. Under the graph is the formula: “Slope = \frac{rise}{run}=\frac{-1}{+2}=-\frac{1}{2}.”

• Unit 5, Lesson 5.2, Teacher Gem: Partner Math, students develop procedural skills and fluency as they work with a partner to solve systems of equations by graphing two equations on a coordinate plane (8.EE.8). An example provided is as follows: “A. y=\frac{1}{2}x+4; y=-2x+9; B. y=x+5; y=-\frac{2}{3}x”

There are opportunities for students to develop procedural skill and fluency independently throughout the grade level. Examples include:

• Unit 1, Lesson 1.2, Leveled Practice P, students independently demonstrate procedural skill and fluency as they solve linear equations in one variable (8.EE.7). An example provided is as follows: “Solve each equation and check your solution. 4. 3(x+2)=4(x+2). 11. 6x+3=2x+3”

• Unit 7, Lesson 7.4, Student Lesson, Exercise 17, students independently demonstrate procedural skill and fluency as they use equations to find angle measures when parallel lines are cut by a transversal (8.G.5). An example provided is as follows: “Solve for x, then find the measure of each identified angle.” The problem involves a pair of parallel lines cut by a transversal, with alternate exterior angles labeled as (5x+10)^{\circ} and (6x-7)^{\circ}.

• Unit 8, Lesson 8.1, Student Lesson, Exercises 1-4, students independently demonstrate procedural skill and fluency as they use exponent properties to generate equivalent expressions (8.EE.1). An example provided is as follows: “Simplify. 1. x^{3}x^{2}. 2. (y^{5})^{2}. 3. (pq)^{5}. 4. (5gh^{2})^{2}.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

There are opportunities for students to develop routine and non-routine applications of mathematics in each lesson. The materials develop application through the Student Lesson Exercises in the Apply to the World Around Me section, Teacher Gems, a Storyboard Launch/Finale, and Performance Tasks.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 2.2, Teacher Gems: Masterpieces, students solve a non-routine problem by applying the Pythagorean Theorem to determine side lengths in right triangles in real-world problems (8.G.7). The materials state, Card 1 states, “Two fire trucks arrive at the same ten-story building. Truck 49 has the longer ladder so the firefighters on this truck are responsible for rescuing all people above 60 feet off the ground. Truck 25 has a ladder that extends to 72 feet. 1. The base of the ladder from Truck 49, when set on the ground, is 18 feet from the side of the building. The person they are trying to rescue is 80 feet up. To the nearest foot, how far will the firefighters have to climb to rescue the person?” 

• Unit 4, Lesson 4.3, Student Lesson, Exercise 9, students solve a non-routine word problem by interpreting the equation of a line given two points (8.F.3). Exercise 9 states, “Evan says it is impossible to create an equation for two points that have the same x-values, like (1, 5) and (1, 8), because the slope is undefined. Do you agree or disagree? Explain your reasoning.”

• Unit 5, Lesson 5.1, Student Lesson, Exercise 8, students solve a routine word problem by applying their knowledge of simultaneous linear equations and how they appear on a graph (8.EE.8). Exercise 8 states, “Describe how you can tell if two lines intersect by looking at the linear equations in slope-intercept form.”

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 1, Lesson 1.3, Leveled Practice T, Problem 12, students independently solve a routine problem by applying their knowledge of solutions to linear equations with one solution, infinitely many solutions, or no solutions (8.EE.7.a). Problem 12 states, “Eric and Lisa solved the equation 3(x-5)=3x-15. Eric said the answer was x=0 and Lisa said there are infinitely many solutions. Who is correct? Support your answer with your work.” 

• Unit 2, Lesson 2.2, Exit Card, Exercise 2, students independently solve routine problems by applying the Pythagorean Theorem to determine an unknown side length in three dimensions (8.G.7). Exercise 2 states, “A shipping company sells a rectangular box with dimensions of 12 inches by 12 inches by 18 inches. Find the length of the longest diagonal in the box.”

• Unit 4, Materials, Tic-Tac-Toe Board, Distance and Speed, students solve a non-routine problem by applying their understanding of functional relationships to sketch a graph that exhibits features of a function that has been described verbally (8.F.5). The problem states, “A little boy traveled home after his first day of kindergarten. He traveled at least three different speeds on his way home. He stopped at least once during his trip. Write a story about the little boy’s trip home. Explain how he traveled from school to home. Be creative and make it as adventurous as possible. Draw a graph that shows his distance from home over time. Draw a second graph showing his speed over time. Label the axes and include ordered pairs for important changes in distance or speed on the graph.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade level. 

All three aspects of rigor are present independently throughout each grade level. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 4, Lesson 4.6, Student Lesson, Exercise 7, students apply their understanding of functional relationships as they sketch a graph that exhibits the qualitative features of a function described verbally (8.F.5). Exercise 7 states, “Tatyana walked to the store at a rate of 4 miles per hour. She walked for one hour to get to the store. She shopped at the store for one hour before returning home walking at a rate of 4 miles per hour. a. How many miles is the store from Tatyana’s home? b. Sketch a graph of Tatyana’s trip using a coordinate plane. Be sure to include all ordered pairs when line segments change direction and label your axes.”

• Unit 5, Lesson 5.3, Leveled Practice P, Exercises 2-4, students demonstrate procedural skill and fluency to determine the solution to a system of equations by using the substitution method (8.EE.8b). The materials state, “2. x+y=6, x = 3y - 2. 3. -6x + 5y = 16, x = 5 - 3y. 4. 4x - 3y = -15, x + y = 5.”

• Unit 6, Lesson 6.2, Student Lesson, Exercise 9, students deepen their conceptual understanding of corresponding, alternate interior, and same-side interior angles formed when parallel lines are cut by a transversal by reasoning about the relationships between these angles (8.G.5). Exercise 9 states, “Answer the following questions about angle relationships. Explain your reasoning for each. a. Are corresponding angles always, sometimes or never congruent? b. Are alternate interior angles always, sometimes or never supplementary? c. Are same-side interior angles between parallel lines always, sometimes or never congruent?”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic or unit of study. Examples include:

• Unit 2, Lesson 2.2, Student Lesson, Exercise 13, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they analyze the dimensions of rectangular boxes, calculate spatial relationships, and determine which box can accommodate a given object in a real-world context (8.G.7). Exercise 13 states, “Elena needs to ship a 61 cm baton to a customer. She has two rectangular boxes. One is 25 cm by 25 cm by 50 cm. The other box is 10 cm by 12 cm by 58 cm. In which box will the baton best fit?” 

• Unit 5, Lesson 5.2. Explore!, use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they write and graph linear equations, analyze tables, and compare real-world payment options to make informed decisions (8.EE.8). The materials state, “Larry’s Landscaping offers two payment options for summer employees. Option #1 offers $100 base pay per week plus $25 for every job completed. Option #2 offers $160 every week plus $15 for every job completed. Step 1: Write an equation to represent the weekly salary, 𝑦, that could be earned for 𝑥 jobs completed if an employee chooses Option #1. Step 2: Write an equation to represent the weekly salary, 𝑦, that could be earned for 𝑥 jobs completed if an employee chooses Option #2. Step 3: Fill in the tables below by calculating the weekly salary for an employee under each plan for 0 through 10 jobs.” A 2-column table is provided for each Payment Option. It is labeled "Jobs Completed x’"and "Weekly Salary y". Students calculate and fill in the salary for 0-10 weeks. “Step 4: Graph Payment Option #1 and Payment Option #2 on the same coordinate plane. For how many jobs per week would employees earn the same amount for either option? Explain how you can tell that from the graph. Step 5: How could you have used the tables in Step 3 to determine the answer to Step 4 prior to graphing? Step 6: If an employee thinks he can complete 50 jobs in one week, which payment option should he choose? Explain your answer.”

• Unit 8, Lesson 8.2, Student Lesson, Exercise 14, demonstrate conceptual understanding and procedural fluency as they create and simplify algebraic expressions equivalent to a given expression using the properties of exponents and multiplication (8.EE.1). Exercise 14 states, “Alan’s teacher asked every student to find three different expressions that simplify to 6x^{2}y. Alan wrote \frac{12x^{2}y^{2}}{2y}. Write two other expressions that simplify to 6x^{2}y .”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 6, Materials, Performance Task, students make sense of problems and persevere in solving by finding missing angle measures, determining values of variables, and explaining the relationships between triangles, particularly in terms of similarity. The Performance Task states, “Part 1: For his math class, Isaac created the angle puzzle below:” An angle puzzle is given with three parallel lines intersected by two transversals. Some of the resulting angles are represented by variable expressions or letters indicating their measures. “1. What do you notice about the diagram? 2. What are some questions related to the situation above that could be solved using mathematics? PART 2: After creating his angle puzzle, Isaac’s teacher asked him to create an answer key for the puzzle. The answer key needed to include the values of each variable and the measures of each lettered angle. The key also needed to name the special angle relationship that would be used to solve for each variable. 3. Help Isaac create an answer key for his angle puzzle. PART 3: Finally, Isaac’s teacher had each student find a partner to solve their angle puzzle and to make sure that the angle puzzle met the requirements of the assignment. Each angle puzzle needed to include congruent and similar triangles. 4. Isaac partnered up with Justice and told her that he included three congruent triangles in his diagram. Justice told him that it was not possible to determine if there were any congruent triangles in his diagram. Who is correct? Explain. 5. In the space below, explain how Issac can modify his puzzle so that it meets the requirements of the assignment. Then return to page 1 of the activity and fix Issac’s angle puzzle.” 

• Unit 7, Lesson 7.5, Lesson Presentation, Example 2, with the support of the teacher, students make sense of problems and persevere solving as they describe the effect of dilations on a figure using coordinates. Example 2 states, “Write a series of two transformations that shows \triangle MNP is similar to \triangle M'N'P'. The new triangle appears to be longer and moved to the right. This can be done using a dilation and a translation. Record the original coordinates. M(0,0), N(4,0), P(0,3)” A coordinate grid of quadrant I is shown. Two triangles are drawn on the grid; \triangle MNP and \triangle M'N'P', M’(2,0), N’(10,0), P’(2,6). “The image appears to be larger. Find the scale factor by looking at the ratio of \triangle M'N'P' and the corresponding side of \triangle MNP. \frac{M'N'}{MN}=\frac{8}{4}=2 Multiply each vertex of \triangle MNP by a scale factor of 2. (x,y) (2x,2y) M(0,0) (0,0) N(4,0) (8,0) P(0,3) (0,6) A new triangle is then shifted 2 units to the right. Add 2 units to each x-coordinate for a translation rule of (x,y) (x+2,y); (0,0) M’(2,0), (8,0), N’(10,0), (0,6), P’(2,6). Since these points match \triangle M'N'P'on the graph above, the series of transformations (dilation (x,y) (2x,2y) and translation (x,y) (x+2,y) maps \triangle MNP onto \triangle M'N'P'. These two figures are similar (not congruent) because a dilation was part of the transformation sequence that made the triangle larger.”

• Unit 9, Lesson 9.2, Student Lesson, Exercise 12, students make sense of problems and persevere solving as they volume formulas for cones so solve a real-world problem. Exercise 12 states, “Frank is setting up a conical teepee. The box says it is 10 feet tall and has a volume of approximately 150.72 cubic feet. He needs to find the diameter of the teepee to find a space large enough to set it up. If Frank has a square patch of grass in his yard that has a perimeter of 28 feet, will the teepee fit in it?” 

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.2, Lesson Presentation, Example 3, with the support of the teacher, students reason abstractly and quantitatively as they write and solve equations representing real-world situations. Example 2 states, “Harper went to a plant nursery to buy flowers for her garden. The nursery had a deal of $7 off the purchase of a two-pack of geraniums. She decided to buy four two-packs of geraniums. Harper spent $20 on all the geraniums. What was the regular price of each geranium? Write an equation to represent this situation.” The equation 4(2x-7)=20 is given. The work for solving the equation is shown next to the written description of the steps. The materials state, “Distribute. Add 28 to both sides of the equation. Divide both sides of the equation by 8.” 

• Unit 4, Lesson 4.2, Student Lesson, Exercise 12, students reason abstractly and quantitatively as they write equations from a graph, interpret the slope and y-intercept in a real-world context, and use the equation to solve problems. Exercise 12 states, “Javier owns a car rental company. He provides the graph seen at the right for his customers to see the price for renting a sedan based on the number of miles they drive. a. Find the equation (in slope-intercept form) that represents the amount Javier charges based on the number of miles driven. b. What do the slope and y-intercept represent in this situation? c. Determine the amount a customer will have to pay if she rents a sedan and drives it 120 miles. d. Janelle rented a sedan from Javier. When she returned it, her bill was $38.50. How many miles did she drive?” A graph of the first quadrant is labeled with miles on the x-axis and cost on the y-axis. Three ordered pairs are plotted: (0, 12), (40, 32), and (70, 47), with a line passing through the points.

• Unit 10, Lesson 10.2, Student Lesson, Exercise 12, students reason abstractly and quantitatively as they use a linear model to solve real-world problems, including making predictions and analyzing trends. Exercise 12 states, “Mrs. Gonzalez noticed that class sizes at her school have been slowly declining over the years. In her first years of teaching starting in 1990, there was an average of 29 students in her classes. Using a line of fit, she found that the number of students per class (y) can be modeled by the equation y=29-0.2x, where x represents the number of years since 1990. a. Use the equation to predict the number of students per class in the year 2025. b. In what year will the student population reach 18 students per class? c. Does this equation accurately predict the class sizes at this school in the year 2050?”

The materials reviewed for EdGems Math Grade 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practice throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.5, Student Lesson, Exercise 9, students construct a viable argument as they explain the value of an irrational number. Exercise 9 states, “Sabah said there are an infinite number of integer values for x in the expression \sqrt{x} that would make the value of this expression fall between 4 and 5. Do you agree or disagree? If you disagree, how many possible integer values are there?”

• Unit 3, Lesson 3.1, Teacher Gems: Partner Math, students construct viable arguments when they determine if a relation is a function given a set of ordered pairs or a graph. The materials state, ”Determine if each relation represents a function. Explain why or why not. E. (3, 5), (5, 5), (9, 5) and (10, 5) F. (0, 1), (0, 3), (0, 6) and (0, 8).” 

• Unit 6, Lesson 6.3, Student Lesson, Exercise 10, students construct viable arguments as they use the sum of angles in a triangle to reason about the angles. The materials state, “In an obtuse triangle one angle is obtuse. What type of angle are the other two angles? Explain how you know your answer is correct.”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 4, Lesson 4.4, Student Lesson, Exercise 6, students critique the reasoning of others as they analyze equivalent equations in one variable. Exercise 6 states, ”Triffy claims that y=2(x-3)+5 and y=2x+5 are equivalent equations since both equations have a slope of 2 and a y-intercept of 5. Do you agree with Triffy? Explain your reasoning.”

• Unit 5, Lesson 5.2, Student Lesson, Exercise 9, students critique the reasoning of others as they examine the solution for a system of equations solved by a student. Exercise 9 states, “Polly took a quiz on solving systems of linear equations. She was not sure how she did on two of the harder problems when she turned in the quiz. Later, her friend told her she should have checked her answers. Did Polly answer the questions correctly? Explain your reasoning.” Polly’s work for the two problems is given. “a. y = 4x − 1, 2x − y = −13. Polly’s solution (−2, −9) b. 3x + 6y = 15, −2x + 3y = −3. Polly’s solution (3, 1).”

• Unit 7, Planning & Assessment, Unit 7 Assessment Form A, Problem 9, students critique the reasoning of others as they identify and address errors in proposed solutions, evaluate the soundness of the underlying logic or calculations, and offer clear, mathematically grounded explanations or corrections. Problem 9 states, “Owen made a mistake when writing a transformation rule to translate points one unit up and three units to the left. Transformation Rule: (x, y) → (x + 1, y − 3). a. What is wrong with Owen’s transformation rule? b. What is the correct transformation rule?”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for supporting the intentional development of MP4: Model with mathematics and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.2, Student Lesson, Exercise 12, students model with mathematics as they use the Pythagorean Theorem to calculate the tarp’s diagonal and determine if it fits the metal rod. Exercise 12 states, “Talia needs to paint a 9.5 foot metal rod. She wants to place it on a tarp so the paint does not drip on the floor. She has a rectangular tarp that is 6 feet by 8 feet. Will the metal rod fit on the tarp or does she need to buy a new tarp for the project?”

• Unit 5, Lesson 5.5, Explore!, with the support of the teacher, students model with mathematics when they write equations then solve a system of equations representing a real-world situation. The materials state. “The Rodriguez family and the Jacobson family go to the movies together. The Rodriguez family bought 3 adult tickets and 2 youth tickets for a total of $29.00. The Jacobson family bought 2 adult tickets and 5 youth tickets for a total of $31.25. Let 𝑥 represent the cost of an adult ticket and 𝑦 represent the cost of a youth ticket. Step 1: Write an equation to represent the Rodriguez family’s movie ticket purchase. Step 2: Write an equation to represent the Jacobson family’s movie ticket purchase. Step 3: Which method do you think will be best for solving this system of equations? Explain. Step 4: Solve your system of linear equations.” 

• Unit 10, Planning and Assessment, Performance Assessment, Exercise 5, students model with mathematics as they interpret a scatter plot and investigate patterns of association in bivariate data. Exercise 5 states, “The scatter plot below shows the age and current value of twenty cars. Each car was originally sold for $20,000 when it was new.” A scatter plot is provided with the x-axis labeled "Age of Car (in years)" and the y-axis labeled "Value of Car (in thousands of dollars)." Students are asked to evaluate the following statements: “5. For each of the statements below: ❖ Write 'True' if the statement is true. ❖ If the statement is not true, write a correct statement.” The statements given are, “The scatter plot shows a negative association between a car’s age and its value. Older cars in the scatter plot always have a lower value than a newer car. Point A in the scatter plot represents a car that is 3 years old and has a value of $17,250. The range of car values in the scatter plot is $9,250. The point (5,16) is an outlier.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 3, Lesson 3.4, Explore!, students choose appropriate tools when determining the slope of a line. The materials state, “Ginger got a job in downtown Oklahoma City. She bought a parking pass at a garage not far from her place of work. The table shows her total parking expenses based on the number of weeks she has been parking at the garage.” A table labeled "Week x" with values 6, 10, 12, and 24, and '"Total Expenses y" with values $50, $74, $86, and $158 is provided. “Step 1: Calculate the rate of change (the change in y over the change in x) for the table. Step 2: Graph the ordered pairs on the coordinate plane at the right. Draw a line through the points. Step 3: Make a slope triangle and determine the slope of the line. Step 4: What do you notice about the slope of the line? Step 5: If you were given the table of values in the table at right, what would the rate of change (or slope) ratio look like?” A table labeled "x" with values x₁ and x₂ and "y" with values y₁ and y₂ is provided. Step 6: The ratio developed in Step 5 is called the ‘Slope Formula.’ The subscripts identify two different points. Try your formula on these points from the tale at the beginning of this Explore!: (6, 50) and (12, 86). Did you get the same slope as you did in Step 1 and Step 3? Step 7: You have learned two methods for finding slope: slope triangles and the slope formula. Which method do you like the best? Why?”

• Unit 7, Lesson 7.2, Student Lesson, Exercise 8, students choose appropriate tools, with the support of the teacher, as they translate figures on the coordinate plane. Exercise 8 states, “A pre-image has coordinates A(7, 1), B(1, −2) and C(5, −3). The image has coordinates A′(7, 6), B′(1, 3) and C′(5, 2). Describe the translation that occurred to map the pre-image onto the image.” Teacher Guide, Focus Math Practice, Math Practices: Teacher and Student Moves, Teacher moves states, “Provide tracing paper, markers, geometric shapes, grid paper and other tools. Instruct students to choose one or more tools to explain the reason they chose the tools and how they will use them prior to solving problems. Then, ask them to reflect on their use of the tools and how they helped in solving a given problem.”

• Unit 9, Planning & Assessment, Performance Assessment, Exercise 1, students choose appropriate tools to solve a real-world problem involving volume of spheres. Exercise 1 states, “CJ’s Truffle Company is designing a new line of hazelnut truffles. The truffle will have a spherical hazelnut in the center and a spherical chocolate shell on the outside. CJ has two possible molds to use. Mold A has a radius of 1.2 centimeters and Mold B has a radius of 1.5 centimeters. The hazelnuts have an average radius of 6 millimeters. Use 3.14 for 𝜋. 1. How many cubic centimeters of chocolate will be used to create one truffle in Mold A? Show work that supports your answer. Round your answer to the nearest tenth. 2. How many cubic centimeters of chocolate will be used to create one truffle in Mold B? Show work that supports your answer. Round your answer to the nearest tenth. 3. The chocolate used to make the truffles costs $0.02 per cubic centimeter and the hazelnuts cost $0.05 each. The truffles are sold in packages of 10 truffles. CJ charges $2.50 for a package of truffles made in Mold A. CJ has determined he can charge 50% more for a package of truffles made in Mold B compared to Mold A. Which size of truffle would you advise he make and sell if he wants to produce just one size of truffle? Explain your reasoning. 4. CJ is considering changing Mold A to either a cone or a cylinder. He wants to use the same amount of chocolate as before. He also wants the radius of the new mold to still be 1.2 cm. a. What would the height of the new mold be if it was a cylinder? Show work that supports your answer. Round your answer to the nearest tenth. b. What would the height of the new mold be if it was a cone? Show work that supports your answer. Round your answer to the nearest tenth.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.4, Student Lesson, Exercises 10-12, students attend to precision as they estimate square roots to the nearest tenth. The materials state, “Estimate each square root to the nearest tenth. 10. \sqrt{103} 11. \sqrt{20} 12. \sqrt{89}.” Teacher Guide, Math Practices, Teacher Moves “Have students compare their estimates for Student Lesson Exercises #10-12. Encourage them to communicate why they chose their estimates and decide which estimates are more precise. They can check with calculators to see what the most exact answer would be.”

• Unit 4, Lesson 4.5, Student Lesson, Exercise 7, students attend to the specialized language of mathematics as they use their knowledge of linear equations and tables to identify a function as linear or non-linear. Exercise 7 states, “Is the function y=3x^{2}+1 linear or non-linear? Use a table of values to prove your answer is correct.”

• Unit 6, Lesson 6.1, Explore! students attend to the precise language of mathematics as they investigate the measures and locations of special angles formed when lines are cut by a transversal. The materials state, “Step 1: Draw two parallel lines in the space below. Then draw another straight line that intersects both parallel lines you drew. This line is called a transversal. Ensure that your diagram is large enough to easily measure the angles with a protractor. Label your angles 1 through 8. Step 2: Use a protractor to measure the eight angles on the drawing from Step 1. Record their measures on your drawing. Step 3: Look at the diagrams in the table below. What do you think it means for a pair of angles to be alternate exterior angles? What do you think it means for a pair of angles to be alternate interior angles?” A table containing diagrams of a pair of lines cut by a transversal and a pair of parallel lines cut by a transversal is given. Each diagram identifies the alternate exterior and alternate interior angles. “Step 4: Which pairs of angles in your diagram from Step 1 are alternate exterior angles? What do you notice about the degree measure of each pair of alternate exterior angles? Step 5: Which pairs of angles in your diagram from Step 1 are alternate interior angles? What do you notice about the degree measure of each pair of alternate interior angles? Step 6: Draw two lines that are not parallel. Draw a transversal that intersects both lines. Measure the eight angles formed by the transversal. Step 7: Write a conclusion about the alternate exterior angles formed by parallel lines compared to those formed by non-parallel lines.”

• Unit 8, Lesson 8.3, Student Lesson, Exercise 11, students attend to precision as they write numbers in scientific notation. Exercise 11 states, “Rewrite each number in scientific notation. a. 49\times10^{8} b. 0.6\times10^{-2} c. 0.05\times10^{3} d. 785\times10^{-5}.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year, with each lesson’s Unit Overview and Teacher Guide clearly outlining the teacher and student actions for implementing these practices.

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Planning and Assessment, Performance Assessment, Exercise 1, students make use of structure as they solve a real-world problem using irrational numbers. Exercise 1 states, “Henry is designing three different sizes of cubic blocks. Block A has a surface area of 150 square centimeters. Block B has a volume of 300 cubic centimeters. Block C has a volume of 512 cubic centimeters. 1. One of the blocks has a side length that is an irrational number. a. Which block is it? Justify your answer using words and mathematics. b. Predict the value of the irrational number from above. Justify your answer using words and mathematics.”

• Unit 5, Lesson 5.4, Explore!, with support from the teacher, students make use of structure as they make sense of a system of equations and use elimination to solve. The materials state, “Step 1: Look at the two equations below. 5+6=11. 10-6=4. How are the equations similar? How are they different? Step 2: You can combine two equations using addition. For the equations below, add all the values in blue on the left side of the equation and then add all the values in red on the right side of the equation. 5+6=11. 10-6=4. What do you notice about the sums on the left and right sides of the equation? Why do you think this is? Step 3: Using the same equations, find the sums of each vertical set of purple, green and orange numbers. 5+6=11. 10-6=4. How does your answer compare to your answer from Step 2? Step 4: Combine the two equations below by adding the pairs of vertical terms. 2x + 4y = 8. -2x + 3y = 6. Step 5: Can you solve the equation that was formed by adding the equations? If so, what is the solution? Step 6: In the previous step, you found the value of one of the two variables. How could you find the value of the remaining variable? Try your idea below. Step 7: The method used in Steps 5-6 is called the elimination method. Why do you think that is?”

• Unit 7, Lesson 7.3, Student Lesson, Exercise 7, students, with support from the teacher, make use of structure as they rotate images. Exercise 7 states, “Xavier predicted that every image formed by a rotation will be similar but not necessarily congruent to its pre-image. Do you agree or disagree? Explain your reasoning.” Teacher Guide, Math Practices, Teacher Moves, “As students learn additional types of transformations, they may begin to confuse rotations with other transformations (i.e., translations, reflections). Continue to reinforce the newly learned vocabulary in the unit by asking students to verbally describe different transformations using the correct terminology.” 

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Lesson 3.1, Student Lesson, Exercise 12, students use repeated reasoning of input and output values to create input-output tables that satisfy or violate the definition of a function. Exercise 12 states, “A relationship has a domain of {1, 3, 4} and a range of {−3, −1, 0}. a. Create an input-output table for this relationship that is a function. b. Create an input-output table for this relationship that is not a function.” 

• Unit 5, Lesson 5.6, Student Lesson, Exercise 15, students look for and use repeated reasoning as they informally understand decimal expansions for rational numbers. Exercise 15 states, “When a single digit repeats after the decimal point (i.e., 0.\overline{1} , 0.\overline{2} , 0.\overline{3}, 0.\overline{4} , etc.), what do you notice about the denominators of the equivalent fractions?”

• Unit 8, Lesson 8.1, Teacher Gems, Ticket Time, students, with the support of the teacher, look for and express repeated reasoning when simplifying expressions with exponents. The materials state, “Simplify 7^{4}\bullet7^{5}. Simplify (6^{3})^{4}. Simplify (ab^{2})^{4}\bullet b^{2}. Simplify (3y^{5})^{2}\bullet 5y^{2} .” Teacher Guide, Math Practices, Teacher Moves, “When learning methods for evaluating products of powers and powers of powers, students might mix up the rules for adding or multiplying the exponents. Encourage students to write expressions in expanded form to solidify their understanding of these properties.”