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.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development.

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Key instructional support through resources designed to enhance teacher effectiveness. The Unit Planning & Assessment pages offer access to both general course and unit-specific instructional information, ensuring teachers have the necessary materials for lesson execution. The PD Library includes written and video-based professional development on implementing Teacher Gems, Communication Breaks, Fluency Boards & Routines, and the 5E Instructional Model, equipping teachers with techniques for effective instruction. Additionally, the ELL Supports Guide provides strategies for ELL Proficiency Levels, Instructional Design, Mathematical Language Routines (MLRs), and Scaffolding Techniques. This guide includes resources such as a Word Problem Graphic Organizer, Target Trackers, Math Practice Trackers, a Math Self-Assessment Rubric, and a Vocabulary Journal Format, ensuring multilingual learners receive appropriate language supports.

• Lesson planning guidance is structured through unit resources that outline daily instructional expectations. The Unit Launch Guide provides a two-day lesson plan for introducing each unit, detailing required and optional components with class time allocations and facilitation instructions. These components include the Target Tracker Launch, Storyboard Launch, Fluency Board Launch, Readiness Check, and Unit Launch Teacher Gem, all designed to establish foundational knowledge. The Unit Finale Guide supports teachers in unit review, differentiation, and assessment through a three-day lesson plan incorporating the Unit Review, Unit Finale Teacher Gem, Fluency Board Finale, Storyboard Finale, and Assessments, along with explanations of assessment options.

• Lesson implementation support is embedded within the Teacher Guides, which contain detailed two-day lesson plans with structured guidance on instruction and differentiation. The At a Glance section provides a one-page lesson summary covering Standards, Materials, Starter Choice Board, Lesson Planning Overview, and Learning Outcomes. The Deep Dive section offers explicit lesson planning guidance, outlining both required and optional components with recommended class time. Day 1 lessons include the Starter Choice Board, Explore! Activity, Lesson Presentation, and Independent Practice, while Day 2 includes the Starter Choice Board, Teacher Gem options, Exit Card & Target Tracker, and additional Independent Practice. The Deep Dive also incorporates formative assessments, Focus Math Practices, Math Practices: Teacher and Student Moves, and Supports for Students with Learning and Language Differences, ensuring teachers have clear implementation strategies for diverse learners.

Materials include sufficient and useful annotations and suggestions that are embedded within specific learning objectives to support effective lesson implementation. Preparation materials, lesson narratives, and instructional supports provide teachers with structured lesson planning guidance, differentiation strategies, formative assessment recommendations, and opportunities for student engagement. These supports are found in resources such as the Unit Launch Guides, Unit Finale Guides, Lesson Planning Guidance, Teacher Guides, Deep Dive sections, Starter Choice Boards, and Small Group Instruction recommendations.

• Unit 3, Planning & Assessment, Unit Finale Guide, Lesson Planning Guidance: Day 3, “Assessments (40-45 minutes) Four summative assessment options are available, providing teachers with flexibility to select an assessment type that meets their needs. When selecting an assessment, teachers can consider whether they (a) desire a print-ready assessment versus a digital option, (b) prefer a traditional test versus a work sample and (c) if students require particular accommodations. Unit Assessment: The Unit Assessment is a print-ready traditional test which assesses all of the Focus Standards covered in the unit. This assessment offers constructed response items that range from Depth of Knowledge Levels 1 to 3 (Recall + Reproduce, Basic Skills + Concepts and Strategic Thinking). A ‘Form A’ and ‘Form B’ are provided, in which the questions are nearly identical but with different numbers. Tiered Unit Assessment: Like the Unit Assessment, the Tiered Assessment is a print-ready traditional test which assesses all of the Focus Standards covered in the unit using a constructed response format. This version of the assessment includes items that are nearly identical to the Unit Assessment, while providing common accommodations for students with special needs, such as fewer items, a lower reading level, friendlier numbers and more space to work. A ‘Form AT’ and ‘Form BT’ are provided, in which the questions are nearly identical but with different numbers. Online Unit Assessment: Like the Unit Assessment, the Online Unit Assessment assesses all of the Focus Standards covered in the unit at a variety of Depth of Knowledge levels. Unlike the Unit Assessment, which includes constructed response items, the Online Unit Assessment includes selected response items, such as multiple choice, select all that apply, true/false and yes/no. Teachers have the option to provide a print version of the Online Unit Assessment, found in the Editable Resources spreadsheet, to encourage students to show their work before submitting their answers digitally. A ‘Form A’ and ‘Form B’ are available, in which the questions are nearly identical but with different numbers. Performance Assessment: The Performance Assessment is a print-ready assessment in the style of a work sample, providing a Depth of Knowledge Level 4 (Extended Thinking) experience to demonstrate mastery of the unit’s standards. In the Unit 3 Performance Assessment, Students will attend to precision (SMP6) as they explore changing quantities using functions, proportional relationships and slope (8.F.A.1, 8.EE.B.5-6) to consider the impact of unit prices when designing fruit baskets. A grading rubric is also included.”

• Unit 5, Lesson 5.3, Teacher Guide, Deep Dive, Lesson Planning Guidance: Day 1, “Lesson Presentation (15-20 minutes) Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Communication Break–Think, Ink, Pair, Square: Use the prompt ‘Why do you think this method is called substitution?’ Have students think and write independently before joining with a partner to share. Then have two partner sets join together. Ask one group to start and the other group respond using the sentence stems provided. Communication Break–Heads Together: Using Example 2, have students put pencils down and look at the question. Consider reading it together. Ask partner sets or small groups of students to examine the question and think about how it is similar or different to what they have already done in the lesson/unit and how that will affect solving the problem. Have students use the sentence stems to share out in groups and full class, if desired.”

• Unit 7, Lesson 7.4, Teacher Guide, Deep Dive, Math Practices: Teacher and Student Moves, “SMP1 Make sense of problems and persevere in solving them. Teacher Moves Talk with students about when you need to graph a figure to determine the solution to a question or when you can determine it without graphing. Some students may automatically resort to graphing to determine if the figure is a reduction or enlargement. Student Moves Explain how to determine if a dilation is an enlargement or reduction without graphing. SMP2 Reason abstractly and quantitatively. Teacher Moves Ask students to identify and define all quantities. Ask students to explain how the solution is related to the context. Prompt students to continually ask, “]’Does this make sense? How do you know?’ Consider asking students, ‘How is a dilation similar to and/or different from the other types of transformations in this unit?’ Student Moves For Student Lesson Exercises #13-14, determine the meaning of all quantities and distances in each task and how they relate to the situations. After finding a solution to each exercise, relate the solution back to the original context. SMP3 Construct viable arguments and critique the reasoning of others. Teacher Moves Develop students’ ability to justify methods for completing tasks involving transformations and compare their responses to the responses of the peers using the Teacher Gem Stations. Student Moves Communicate mathematical ideas, vocabulary, and methods effectively by comparing answers with peers and justifying the work of others using the Teacher Gem activity Stations.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Each Unit’s Planning & Assessment page includes a PD Library that provides teachers with access to Achieve the Core open-source publications from Student Achievement Partners. These documents offer adult-level explanations of mathematical content, organized by vertical progression within each domain. Additionally, the Planning & Assessment page contains a Unit Overview with the following information:

The Content Analysis section explains the major mathematical concepts taught in the unit, providing examples and explanations to enhance teachers’ understanding of both the content and its vertical progression within the standards. It also illustrates the types of tasks and procedures students will encounter. For example: 

• Unit 3, Planning & Assessment, Unit Overview, Content Analysis states, “Unit 3, Planning & Assessment, Unit Overview, Content Analysis states, “In this unit, students will build upon their work with two-variable equations, tables and graphs in earlier courses to explore relationships in which one quantity is a function of another. Previously in Unit 1 Equations, students worked exclusively with equations in one variable. This unit begins the exploration of equations in two variables in this course, with a specific focus on determining whether relationships between variables presented in tables or graphs represent functions. Though this unit introduces students to the concept of a function, a formal understanding of domain, range and function notation will not occur in this course.” Visual student examples of Relationships that are Functions and Relationships that are not Functions are provided for teachers to review. “This unit revisits students’ work with proportional relationships from Grade 7 as an introduction to linear functions. Students represent and compare functions of proportional relationships in equations, tables and graphs with a focus on identifying the unit rate (or constant of proportionality). By the end of the unit, students are introduced to the concept of slope, which they will calculate for both proportional and non-proportional linear relationships. The slope of a proportional relationship has a constant rate, whereas the slope of a non-proportional linear relationship has a constant rate of change.” Visual student examples of proportional and nonproportional graphs are provided for teachers to review. “Students learn about the four classifications of slope and make the connection that a linear relationship in the form x = a is a linear equation but not a linear function. They build upon their work with calculating distance on the coordinate plane from Unit 2 Pythagorean Theorem to develop the slope formula.” Visual student examples of finding slope from a table and using the slope formula are provided for teachers to review. “While students are introduced to slopes of non-proportional linear relationships in this unit, they will not specifically work with equations in the form y = mx + b, where b ≠ 0, until Unit 4 Functions.”

The Learning Progression section explains and provides specific examples of the vertical progression of standards within the unit’s targeted domains. These examples include diagrams, models, numerical or algebraic representations, sample problems, and solution pathways. The Learning Progression is structured under the headings: ‘Previously, students have…, ‘In this unit, students will…,’ and ‘In the future, students will…’ with corresponding standards identified. For example:

• Unit 6, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states, “In the future, students will… Prove theorems about lines and angles. HS.G-CO-C.9, Prove theorems about triangles. HS.G-CO.C.10, Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. HS.G-GPE.B.5, Use properties of similarity transformations to establish AA criterion for two triangles to be similar. HS.G-SRT.B.4”

• Unit 9, Planning & Assessment, Unit Overview, Content Analysis, Readiness Check & Learning Progression states, “In this unit, students will… Know the formula for volume of a cone and use it to solve real-world problems. 8.G.C.9, Know the formula for volume of a cylinder and use it to solve real-world problems. 8.G.C.9, Know the formula for volume of a sphere and use it to solve real-world problems. 8.G.C.9”

Each lesson’s Teacher Guide includes a Common Misconceptions section, which identifies common errors and provides explanations and recommendations to help students develop a stronger understanding. For example: 

• Unit 4, Lesson 4.2, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “One of the most common errors students make when graphing lines from slope-intercept form is confusing the rise and run in the ratio for slope. This mistake is easily observed as students calculate slope. Vocabulary foldables using the terms rise and run may help students remember the differences. Some students may graph the y-intercept as the x-intercept. Have students circle the slope and y-intercept in the linear equation and then label each as such to reinforce the slope being the coefficient of x and the constant representing the y-intercept (not the x-intercept). If a linear equation is written in the form y = b + mx, students may be confused by the order. Reinforce the use of the Commutative Property of Addition with students and help them see how they can rewrite the equation (if desired) to be in y = mx + b form. Students may forget that not every linear function is a proportional relationship (going through the origin). Explain that proportional relationships belong to the linear function family and are a special case of linear functions.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Standards correlation information is included to support teachers in making connections from grade-level content to prior and future content. Standards can be found in multiple places throughout the course, including the Course Level, Unit Level, and Lesson Level of the program. Examples include:

• Each Unit’s Planning and Assessment section includes a Pacing Guide & Correlations, where the EdGems Math Course 3 Content Standards Alignment lists all grade-level standards along with the specific lessons where they are addressed. The program provides a structured approach to standards alignment through its Focus and Connecting Standards framework. A correlation chart is included, organizing standards into columns that indicate where each standard is taught as a Focus Standard in specific lessons and as a Connecting Standard across different units. This structure helps ensure that concepts are reinforced and revisited throughout the course.

  • “EdGems Math supports students’ proficiency in the Common Core State Standards through a program-design which supports the interconnectivity of mathematical ideas while providing clear learning objectives. This is achieved by designating Focus Standards in each lesson and Connecting Standards in each unit. The qualifiers of Focus and Connecting Standards were developed by the EdGems Math authoring team to design a scope and sequence in which mathematical ideas build upon each other and are revisited throughout the course. Each EdGems Math lesson identifies one or more standards as a Focus Standard to provide a focal point for the lesson objectives. The unit then provides opportunities for further connections to other standards across clusters and domains. These Connecting Standards offer opportunities for students to draw up and apply many mathematical ideas throughout the unit. The following chart shows when each standard is aligned as a Focus Standard or Connecting Standard throughout the course. Further explanations of the Focus and Connecting Standards are available within each Unit Overview.”

• Unit 4, Planning and Assessment, Unit Overview, Standards Correlation, Focus Content Standards states, “This unit incorporates Focus Standards across two domains and three major clusters. Standard 8.EE.B.6 was previously a Focus Standard in Unit 3 Proportional Relationships and Slope and will reappear again as a Focus Standard in Unit 6 Angle Relationships. 8.F.A.2 was also previously targeted in Unit 3. It reappears in this unit alongside 8.F.A.3 to conclude instruction on the cluster. The other major cluster in the Functions domain is also targeted in this unit, completing the concentration on the Functions domain for the remainder of the course. All standards in this unit are formatively assessed throughout the unit and summatively assessed in the unit’s Test Prep, Performance Assessment and Unit Assessments.”

• Each Unit's Planning & Assessment section includes a PD Library with resources from Achieve the Core to support professional learning and instructional planning. These resources offer in-depth explanations of mathematical progressions aligned with the Common Core State Standards.

  • “CCSS Math Learning Progressions: Student Achievement Partners, a nonprofit organization, developed Achieve The Core to provide free professional learning and planning resources to teachers and districts across the country. The narrative documents below provide adult-level descriptions of the progression of mathematical ideas within domains or topics within the Common Core State standards for Mathematics.”

The Planning & Assessment sections within each unit provide coherence by summarizing content connections across grades. These sections highlight how mathematical concepts build upon prior knowledge and prepare students for future learning. Examples of where explanations of the role of specific grade-level mathematics appear in the context of the series include:

• Unit 2, Planning & Assessment, Unit Overview, Connecting Content Standards states, “As students develop their understanding of the Pythagorean Theorem, they will have ample opportunities to solve multi-step linear equations (8.EE.C.7) and equations involving exponents (8.EE.A.2). The Pythagorean Theorem offers students natural connections to irrational numbers (8.NS.A.1-2), and the concept of Pythagorean triples is a building block towards future work with proportions and similar triangles. Students will also make connections to similarity (8.G.A.4-5) when multiplying common Pythagorean triples by a constant to identify other sets of side lengths that form right triangles.”

• Unit 2, Planning and Assessment, Unit Overview, Readiness Check & Learning Progression includes a structured progression of learning, outlining prior knowledge, current instructional goals, and future learning connections to reinforce coherence across grades. It states, "Previously, students have… Graphed polygons in the coordinate plane and determined their side lengths (5.G.A.1-2, 6.G.A.3), solved problems involving area, volume, and surface area (6.G.A.1-2, 7.G.B.6), solved equations involving square or cube roots (8.EE.A.2), and understood the concept of irrational numbers and compared their sizes (8.NS.A.1-2). In this unit, students will… Understand and explain the Pythagorean Theorem (8.G.B.6), apply the Pythagorean Theorem to determine side lengths in right triangles (8.G.B.7), and use the Pythagorean Theorem in the coordinate plane (8.G.B.8). In the future, students will… Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to an understanding of trigonometric ratios (HS.G-SRT.C.6), use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems (HS.G.SRT.C.8), and use coordinates to compute the perimeters of polygons using the distance formula (HS.B-GPE.B.7)."

• Unit 6, Lesson 6.2, Teacher Guide, Standards Overview states, “Focus Content Standard(s): 8.G.A.5 (Major)” Starter Choice Board Overview, “Storyboard: Describe alternate interior and alternate exterior angles (8.G.A.5) Building Blocks: Solve equations (8.EE.C.7) Blast from the Past: Blast from the Past: Determine solutions for inequalities Fluency Board Skills: Plot approximations on a number line, graph linear equations in slope-intercept form, multiply and divide fractions.” 

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly within each unit’s Planning & Assessment, About EdGems Math, Research Guide. 

Research Guide states, “Middle school is a critical stage for math instruction. Students form conclusions about their mathematical abilities, interests, and motivation.^1-10 Middle school students in the United States are falling behind compared to other countries in their math performance.² Studies have shown that struggles with math are particularly acute in middle school grades. The transition from elementary to middle school can lead to students falling behind with accumulated learning gaps.^3-5 Research shows that the mathematical achievement of middle schoolers has a direct impact on the likelihood that they will persist through the challenging material in pathways that can prepare them for the broadest range of options in high school and beyond.^6-7 Within this crucial time frame, a principal goal for middle school math teachers is to create a learning environment in which students are encouraged to see themselves as capable thinkers and doers of mathematics. Research demonstrates that to do this successfully, instructional materials must provide teachers with opportunities to 1) build upon and expand students’ cultural knowledge bases, identities, and experiences, 2) actively support students’ conceptual understanding, engagement, and motivation, 3) provide relevant, problem-oriented tasks that enables them to combine explicit instruction about key ideas with well-designed inquiry opportunities, and 4) spark student peer-to-peer discussion, perseverance and curiosity as they think and reason mathematically to solve problems in mathematical and real-world contexts.⁸ EdGems Math has been intentionally designed to support the diverse mathematical journeys of middle school students as they grow in their learning, critical thinking, and reasoning abilities. To reach the goal of higher order thinking for all, the EdGems Math curriculum connects each grade’s foundational math concepts to authentic, real-world contexts taught in multi-dimensional ways that meet a variety of learning needs. EdGems Math empowers teachers to adjust the content and instructional strategy and tailor outcomes of how learning is assessed.^9-10 EdGems Math curriculum is comprehensive, rigorous, and focused. It draws on decades of research exploring the best methods for teaching and learning math.”

The Unit Planning & Assessment, About EdGems Math, Research Guide incorporates multiple research-based strategies to support student learning:

• “Explore! Activities: The lesson-based Explore! Activities engage students in scaffolded tasks, guiding students as they begin grappling with the big ideas of the lesson and discovering new concepts (Small, M. and Lin, A.). The steps of the Explore! Activities move students through ‘Comprehension Checkpoints’ (National Council of Teachers of Mathematics) to guide information processing, ensure prior knowledge is activated, and discover patterns, big ideas, and relationships. Utilizing a student-centered approach, Explore! Activities engage students in the Standards of Mathematical Practice, which allows teachers to better facilitate learning using effective mathematical teaching practices (McCullum, W.). Every Explore! Activity provides students with ways to connect to key concepts through investigative, discovery-based tasks, culminating in an opportunity to generalize or transfer learning and move toward procedural and strategic proficiencies (California Department of Education).”

• “Lesson Presentation Communication Breaks: Communication Breaks are integrated into each Lesson Presentation as an opportunity for students to make sense of their learning. Each Lesson Presentation features two of seven structures to support students in the communication of their ideas or questions directly with their peers. The use of sentence stems in each Communication Break increases accessibility, enabling students to develop both social and academic language as they reflect on their learning (Smith et al.). The structures of Communication Breaks allow teachers to elicit student thinking, provide multiple entry points, focus students’ attention on structure, and facilitate student discourse (Chapin, S.H., O’Connor, C., Anderson, N.). As a result, students engage in the Standards of Mathematical Practice and gradually become more secure in their understanding and abilities to develop their knowledge (Bay-Williams, J.M., & Livers, S.).”

• “Mathematical Language Routines: Within every Teacher Lesson Guide are instructional supports and practices called Mathematical Language Routines to help teachers recognize and support students’ language development in the context of mathematical sense-making when planning and delivering lessons (Aguirre, J. M. & Bunch, G. C.). While Mathematical Language Routines can support all students when reading, writing, listening, conversing, and representing in math, they are particularly well-suited to meet the needs of linguistically and culturally diverse students. When students use language in ways that are purposeful and meaningful for themselves, they are motivated to attend to ways in which language can be both clarified and clarifying (Mondada, L. & Pekarek Doehler, S.). These routines help teachers ‘amplify, assess, and develop students’ language in math class’ (Zwiers, J. et al).”

• “Lesson Presentation: The editable Lesson Presentation enables teachers to shape lessons of balanced instruction on mathematical content and practice as they guide students through productive perseverance, small group instruction, and growth mindset activities. Components of the Lesson Presentation include Fluency Routines to develop number sense, vocabulary terms with definitions, examples with solution pathways, extra examples, the Explore! activity, Communication Breaks, and a formative assessment Exit Card. Teaching tips provide guidance on independent, group, and whole-class instruction. Studies identify explicit attention to concepts and students’ opportunity to struggle (as during the Explore! activity and Communication Breaks) as key teaching features that foster conceptual understanding (Hiebert, J., & Grouws, D. A.).”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Comprehensive lists of materials needed for instruction can be found in the PD Library link on the Unit Planning & Assessments page. The Required & Recommended Materials document provides a lesson-by-lesson breakdown of necessary resources for the course. Additionally, the Teacher Guide for each lesson includes a list of required materials. Examples include:

• Unit 1, Lesson 1.1, Teacher Guide, Materials states, “Required: Equation mats and algebra tiles (for the Explore! activity) Optional: Unifix cubes, base ten blocks, colored pencils, graph paper, and balances”

• Unit 7, Lesson 7.3, Teacher Guide, Materials states, “Required: Patty paper or tracing paper, graph paper (4 quadrants), protractor (for the Explore! activity) Optional: Markers, grid paper, geometric shapes”

• Unit 9, Lesson 9.2, Teacher Guide, Materials states,“Required: Cylinders and cones, rice, beans or popcorn kernels (for the Explore! activity) Optional: Geometric software, sand, measuring tools, grid paper”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

Each Unit Overview outlines standards alignment for formal assessments, including the Readiness Check, Performance Assessment, and Fluency Pre- and Post-Assessments. The EdGems website provides grade-level standards alignment for all formal assessments, including Assessments, Tiered Assessments, Performance Assessment, Unit Review (Print and Online), and Review & Assessments, accessible via the information (“i”) button at the bottom right corner of the icon. Each question's assessed standard(s) is listed for teachers, while only the Performance Assessment and Performance Task include practice standards. Examples include:

• Unit 2, Planning & Assessment, Unit Overview, Performance Task states, “In ‘Cross Country Race,’ students will apply the Pythagorean Theorem to determine the distance jogged by a participant before the race. Students will construct viable arguments (SMP3) as they discover shape and space using distance on the coordinate plane and rates (8.G.G.7-8, 7.RP.A.1) to predict if the participant will make it back in time before the race begins."

• Unit 2, Planning & Assessment, Unit Overview, Performance Assessment states, “In this Performance Assessment, students will analyze the map of a lake and plan a family fishing trip. Students will use appropriate tools strategically (SMP5) and discover shape and space as they apply the Pythagorean Theorem and the distance formula (8.G.B.6-8) to make sense of the connections between right triangles and distance on the coordinate plane.”

• Unit 4, Planning & Assessment, Assessments, Exercise 12 states, “Last week, Chet rented a beach bike from Top Notch Bike Rental Company. They charge an initial fee plus $1 for each hour the bike is rented. Chet rented a bike for 6 hours and was charged $10. This week, he wants to try a new bike rental shop, Cruiser Rentals. Cruiser Rentals charges their customers according to the equation $y=1.5x$, where x represents the number of hours and y represents the total cost of the rental. Which company charges more per hour?” The assessment information identifies the standard alignment as 8.EE.B.6/8.F.A.2/8.F.B.4.

• Unit 10, Materials, Online Review & Assessments, Online Unit Review, Item 4 states, “Michelle sells flowers for weddings. The price she charges (P) can be approximated by the equation for the line of fit: $P=40+12d$, where d represents how many dozens of flowers are ordered. Ann paid Michelle $400 for the flowers at her wedding. How many dozens of flowers did Ann order?” The assessment information identifies the standard alignment as 8.SP.A.3.

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each Lesson At A Glance in the Teacher Guide outlines specific learning goals, ensuring that teachers know the math standards that students should be able to demonstrate by the end of the lesson. Exit Cards serve as formative assessments, providing a real-time snapshot of student understanding, while a related Student Lesson exercise offers another checkpoint to identify students needing additional support. The EdGems website provides rubrics for scoring Exit Cards, ensuring consistency in evaluation.

The Online Class Results page generates automatic proficiency ratings based on student performance in Online Practice, Online Challenge, Test Prep, and Online Unit Reviews. These ratings align with state assessment benchmarks, helping teachers interpret mastery levels. Assessment Scoring Guides for Unit Assessments and Tiered Assessments follow the same ranking system, allowing teachers to track progress across multiple assessment formats.

Teachers receive real-time student performance insights through Teacher Gem activities, which embed informal assessment opportunities. For example, in Relay, teachers track how often a team revises an answer before moving forward, and in Ticket Time, class dot plots allow teachers to identify common errors. The PD Library provides written and video-based facilitation guides to support teachers in implementing these strategies effectively.

The Lesson Guide Deep Dive helps teachers analyze assessment data and adjust instruction accordingly. Exit Card results inform assignments for Leveled Practice and Differentiation Days, while the Differentiation Day guides provide self-assessments and targeted rotations to meet diverse learning needs. The Deep Dive section also identifies common misconceptions, equipping teachers with strategies to proactively address misunderstandings.

Performance assessments include rubrics for teacher grading and student self-reflection prompts, reinforcing the Standards for Mathematical Practice (SMP). The Unit Overview and Lesson Guide At A Glance ensure that all assessments and activities align with content and practice standards, with detailed mapping to Readiness Check skills, Storyboards, Performance Tasks, Fluency Boards, and Tiered Assessments.

To further support follow-up instruction, the Online Class Results tool provides recommended activities based on student proficiency levels, allowing teachers to tailor instructional strategies. By incorporating structured assessments, clear proficiency guidance, real-time monitoring tools, and differentiation strategies, EdGems Math ensures teachers have the necessary resources to assess student learning, interpret performance data, and provide targeted follow-up instruction.

Examples include: 

• Unit 4, Planning & Assessment, includes Form A and Form B of the Unit 4 Assessment, along with an Answer Key and Assessment Scoring Guidelines for each question. The materials state, “3-point Items: #12, 13, 17 Items that are each worth three points consist primarily of Depth of Knowledge Level 3 items considered 'Strategic Thinking'. Students may earn partial credit on items when showing progress on a solution pathway that connects to the concept being assessed with one or more errors. Students can earn either 0, 1, 2 or 3 points for items in this category. 0 points: An incorrect solution is given with no work or with work that does not show understanding of the concept. 1 point: Progress is made towards a correct solution, but multiple errors have been made. OR A correct solution is given with no supporting work or explanation. 2 points: Progress is made towards a correct solution, but one small error is made. OR A correct solution is given with partial supporting work provided. 3 points: The correct solution is given and is supported by necessary work or explanation. Total Points Possible: 33 Not Yet Met 0-19, Nearly Meets 20-23, Meets 24-29, Exceeds 30-33” Similar guidance is provided for 1 and 2 point assessment items.

• Unit 10, Planning & Assessment, Performance Assessment, Performance Assessment Rubric ~ Student Reflection states, “Describe at least two ways you demonstrated the Focus Math practice below while completing this performance assessment. SMP7 I can look for and use structure to improve my understanding. I made a connection between different representations. I noticed a structure that helped me understand a concept. I made a rule for when something is always, sometimes or never true.” The Performance Assessment Rubric ~ Teacher Grading Rubric consists of four categories rated on a scale of 4 to 1 and a space for Comments. The four categories are “Making Sense of the Problem: Interpret the concepts of the assessment and translate them into mathematics. Representing and Solving the Problem: Select an effective strategy that uses models, pictures, diagrams, and/or symbols to represent and solve problems. Communicating Reasoning: Effectively communicate mathematical reasoning and clearly use mathematical language. Accuracy: Solutions are correct and supported.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

Assessments align with grade-level content and practice standards through various item types, including multiple-choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. They are available as downloadable PDFs for in-class printing and administration or can be completed through the online platform. Examples include:

• Unit 2, Materials, Online Review & Assessments, Unit Assessment, Form A, includes two items that demonstrate the full intent of 8.G.7. The materials state, “1. What is the length of the hypotenuse in the triangle? a. $\approx 4.8$ b. 17 c. 23 d. 289” An image of a right triangle with legs 8 and 15 is provided. “4. Seth bikes 3 miles south then 2 miles west. If Seth walks directly back to his starting point, how far will he walk? a. $\approx 1$ mile b. 6 miles c. $\approx 5$ miles d. $\approx 13$ miles”

• Unit 5, Planning & Assessment, Performance Assessment includes five items that demonstrate the full intent of 8.EE.8 and SMP3. The materials state, “Olaf wants to build a small rectangular concrete patio in his backyard. Two separate builders have given him bids. Builder #1 charges an initial fee of $50 plus $1.50 per square foot. Builder #2 does not have an initial fee and charges $2.50 per square foot. 1. Olaf wants the patio to form a rectangle measuring 7 feet by 10 feet. Which builder will be the least expensive in this case? Show all work necessary to justify your answer. 2. Olaf believes that if he builds a 64 square foot patio, it will cost $146 no matter who he chooses. Write Olaf’s prediction as an ordered pair. Do you agree or disagree with Olaf? Show all work necessary to justify your answer. 3. Write a system of equations that represents Olaf’s options for the cost of building his patio. Let 𝑥 represent the number of square feet of concrete and 𝑦𝑦 represent the total cost. 4. Solve the system of equations you wrote using either substitution or elimination. Explain your solution in the context of the problem. 5. Graph the system of equations. Does your graph verify your solution?”

• Unit 8, Planning & Assessment, Assessments, Form A, includes two items that demonstrate the full intent of 8.EE.1, Exercise 2. The materials state, “Simplify. $\frac{3^{13}}{3^{6}}$” Exercise 12, “Esther wrote the equation at the right. a. Give a possible value for a and b that would make the equation true. b. If the value of a is 11, what is the value of b? c. Write an equation relating a, b and 7 that is true for all values in Esther’s equation.” The equation $\frac{5^{b}}{5^{a}}=5^{7}$ is provided.

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

For each lesson, teacher guidance is provided alongside the Teacher Gem activity, which includes strategies to support instruction and student engagement. Printable PDF resources, such as the Student Lesson Textbook and Interactive consumable, include tools like graphic organizers, sentence stems, number lines, and coordinate planes. Lessons are also available as e-books with features including adjustable font sizes, text highlighting, text-to-speech, and note-taking tools.

Resources that support students in special populations to actively participate in learning grade level mathematics include:

• Differentiation Days: Differentiation Days are designed to provide teachers with structured opportunities to work with small groups based on specific learning targets. During these sessions, other students participate in mixed-ability group rotations, including the Teacher Small Group Rotation, Additional Practice Rotation, Application Rotation, and Tech Rotation. 

• Leveled Practice: The program includes three levels of leveled practice to address varying student needs. “Leveled Practice-T” is structured for students with learning and language differences, offering shorter problem sets, additional workspace, and simplified terminology and numbers to align with accessibility needs while maintaining grade-level alignment.

• ELL Supports: ELL supports are provided in the Planning and Assessment menu of each unit. These include explanations of Mathematical Language Routines (MLRs) and specific directions for incorporating these routines into lessons and activities. 

Each lesson includes Spanish translations of the student lesson, Explore! activities, leveled practice, and Exit tickets. Accompanying videos are included to guide students through the lesson content. These resources are structured to support student learning and accessibility.

Examples of the materials providing strategies and support for students in special populations include:

• Unit 3, Lesson 3.3, Lesson Presentation, Slide 15, Communication Break - Heads Together states, “Graph a line with a slope of 2. How is the problem similar or different to other problems we have done? One way the problem is similar is… One way this problem is different is… How do you think this will affect solving the problem? This might affect solving the problem by… To solve this problem, we will need to…” Lesson Guide, Teacher Presentation “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse. Heads Together: Using Example 3, have students put pencils down and look at the question. Consider reading it together. Ask partner sets or small groups of students to examine the question and think about how it is similar or different to what they have already done in the lesson/unit and how that will affect solving the problem. Have students use the sentence stems to share out in groups and full class, if desired. ”

• Unit 6, Lesson 6.3, Teacher Gem, Partner Math, Partner Math Instructions state, “1. Students should be separated into two groups using formative or self-assessment ratings on the standard addressed in the Partner Math activity. Students still struggling with the content should be given one color of the Partner Math template while students who have shown proficiency in the standard should be given a second color of the template. The two groups need to be equal in size. 2. Next, students find a partner with a different color template and sit next to them. Two problems (Task A and B) addressing the standard are projected or written on the board. Students work with their partner to solve the problems. Both students write on their own papers but work together to reach the same solutions.”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Each unit includes multiple opportunities for students to engage with grade-level mathematics at increasing levels of complexity. These opportunities are embedded within lessons and available to all students, supporting a range of learners in exploring mathematical concepts at higher levels of complexity.

Several resources and features within the program provide opportunities for extended mathematical exploration.

• Performance Task: “The Performance Task provides applications of most or all of the standards addressed in the unit. This task contains Depth of Knowledge Level 3 and 4 strategic and extended thinking questions where students apply multiple standards in a non-routine manner to solve. These tasks provide entry points for all levels of learners and encourage students to explain their thought processes or critique the reasoning of others.”

• Performance Assessment: These non-routine problems require students to engage in higher-level thinking while applying their knowledge of the standards.

• Leveled Practice: The “C” in each lesson is designed for students who have already demonstrated proficiency. It extends their learning by making connections to future standards and incorporating Depth of Knowledge (DOK) Level 3 or 4 exercises.

•  Tic-Tac-Toe Boards: According to the EdGems Math Program Components state, “Each Tic-Tac-Toe Board includes nine activities that extend or look at the content of the unit in different ways. The Tic-Tac-Toe Boards include activities that make use of a variety of multiple intelligences.”

• Teacher Gems: Teacher Gems include problem sets at multiple levels of complexity, allowing for differentiated problem-solving experiences. For example, activities such as Four Corners, Relay, and Stations include multiple levels of complexity within the tasks.

• Online Practice & Exit Card Resource: This resource offers five options for each lesson: Two Online Practice sets (A and B), each containing six items at the proficient level. Two Online Challenge sets (A and B), each containing four challenge questions. Attempt A provides immediate feedback on correctness, while Attempt B includes worked-out solution pathways to help students identify errors in their work.

The materials include structured activities that provide opportunities for students to engage with mathematical concepts at increasing levels of complexity.

• Unit 1, Materials, Performance Task states, “Jaylee owns a coffee shop and wants to have a mural painted on one of the walls. Two separate artists have given her bids: Artist 1. Design Fee: $84. Cost per Square Foot: $1. Artist 2. Design Fee: $0. Cost per Square Foot: $2.50. Part 3: Jaylee decides that she wants to spend around $120 on the mural and thinks that the mural would look best if it were square-shaped. 5. Which artist could create the largest square-shaped mural for $120? 6. Based on your work above, what is the difference in height between the two $120 square shaped murals? Round to the nearest hundredth.”

• Unit 8, Lesson 8.4, Teacher Gem, Relay, Directions state, “Print the two sets of relay cards. The first set, numbered 1 through 8, provide students practice in the standard at a proficiency level. A challenge set of cards, A through H, provide opportunities for students to extend and apply their thinking around the standard. The questions in each set of relay cards increase in difficulty throughout the activity.” Relay 1, “$(1.46\times 10^{4})(2\times 10^{4})$" Relay H, “The diameter of Jupiter is $1.428\times 10^{5}$ kilometers. The diameter of Mercury is 4.879 million meters. Which has the larger diameter? Approximately how many times larger is it?”

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the ELL Supports Guide, “We view the background knowledge, experiences, and insights that English Learners bring to the classroom as strengths to be leveraged, and we are committed to ensuring that they receive academic success with rigorous grade-level curriculum. In recognition of the unique needs of learners, including those with diverse levels of mathematical proficiency, our curriculum includes research-based guidance for differentiated English Language Learner (ELL) instruction."

• The ELL Supports Guide outlines strategies for students who read, write, and/or speak in a language other than English to engage with grade-level mathematics. Key areas of focus include scaffolding tasks, fostering mathematical discourse, and incorporating instructional strategies informed by research. Tasks include scaffolds and language supports designed to facilitate mathematical understanding. The instructional design integrates opportunities for students to express their mathematical thinking both orally and in writing.

• The ELL Supports Guide contain recommendations related to student assessments. Additional resources in the materials include Target Trackers and Math Practice Trackers, which align with structured conferencing planned three times per unit. A Math Self-Assessment Rubric is included to support student reflection, along with a Sample Vocabulary Journal Format that provides space for root words, home-language translations, definitions, images, and sentence frames.

• Each lesson’s Teacher Guide includes three lesson-specific Mathematical Language Routines (MLRs), with two MLRs suggested for implementation per lesson. Strategies described in the materials include language modeling through think-alouds, the use of visual aids featuring key vocabulary, and a multilingual glossary with online vocabulary available in ten languages. Videos within the ELL Supports Guide provide examples of teachers breaking down tasks, using cognates, and prompting students to explain their thinking. Language functions are also included to structure discussions.

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 3, Lesson 3.2, Teacher Guide, Supports for Students with Learning and Language Differences, Mathematical Language Routines states, “MLR 4 – Information Gap: In partner sets, students use information from Challenge Leveled Practice #4 to work with each other to piece information together about the given situation. Prepare in advance a Question Card and an Info Card for each partner set by breaking the problem apart, as shown below: Question Card: Philip is putting books into boxes. How much does each novel weigh? Does this represent a proportional relationship? Info Card: Philip is putting books into boxes. All of the books are novels, and they each weigh $\frac{3}{4}$ pound. Each box holds two dozen novels. Students bridge the gap between their information orally and visually (on a common recording sheet). Encourage students to focus on using mathematical language while discussing the Question and Info Card provided. Emphasize an environment where students are working together to consider the information they need to know and efficient strategies for solving the given question.”

• Unit 5, Lesson 5-5, Teacher Guide, Lesson Presentation states, “Have students utilize their interactive textbooks or composition notebooks to participate in guided note taking using the Lesson Presentation. Have students attempt Extra Examples with partners, in small groups or independently. Use ‘Communication Break’ slides as opportunities for meaningful discourse.Communication Break–Estimation Pause: Use an ‘Estimation Pause’ for Extra Example 1. Have students work with partners or small groups to complete one of the sentence frames provided on the slide and be ready to share their reasoning. Then work together to solve the task and compare to initial estimates.“ Lesson Presentation, Extra Example 1, “Communication Break–Estimation Pause 1. Examine the Problem. 2. Without writing, estimate the answer or range of answers. I think the answer will be between and because… I think the answer will be more/less than because… I think the answer may be about because….”

• Unit 10, Lesson 10.3, Teacher Guide, Supports for Students with Learning and Language Differences, Common Misconceptions states, “Many students have difficulty finding the ‘best’ equation for a set of graphed points. (i.e., some may not understand the importance of having the same number of points above and below the line of fit; some will provide the wrong sign for the slope). In their journals or notebooks, teachers may want to have students map out each of the steps of finding a linear equation for a line of fit, using a situation that is engaging/interesting to all the students in the class. Have them write down common errors for each step, so that they can refer back to their notebooks when given similar problems. Some students may think that correlation will always imply causation. Ask them for examples of a situation where other factors or a third correlation can contribute to an outcome. Using data sets that are associative but do not imply causal relationships can help clarify this misconception. Students should be exposed to scatter plots with different correlations, no correlations, tight correlations, and loose correlations.”

The materials reviewed for EdGems Grade 8 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Materials consistently include suggestions and links to manipulatives to support grade-level math concepts. The Teacher and Student Moves for Math Practice 5 and Explore! Activities incorporate physical manipulatives when appropriate, with required materials listed in the full course materials list under the Teacher Guide At A Glance section. Student Gems in each lesson provide virtual manipulatives, such as Desmos and Geogebra, to help students make sense of concepts and procedures. Examples include:

• Unit 5, Lesson 5.2, Student Gems, Desmos states, “What does it mean for a point to be a solution to a system of equations?” Resource Info states, “This activity will help students understand what it means for a point to be a solution to a system of equations–both graphically and algebraically.”

• Unit 7, Lesson 7.1, Teacher Guide, Explore! Activity: Mirror! Mirror! "In ‘Mirror, Mirror,’ students create reflections using tracing paper (or patty paper) as well as on a grid. This activity allows for students to build upon their experiences with reflections from everyday life and progress into representing reflections on a coordinate plane. The activity concludes by asking students to consider if the pre-image and image are congruent after a reflection has occurred.”

• Unit 9, Lesson 9.1, Teacher Guide, Math Practices: Teacher and Student Moves, SMP5 Teacher Moves states, “Provide 3-dimensional shapes, measuring tools, 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.”