8th Grade - Gateway 1

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Within the materials, print-based and digital assessments are included. Each unit has the following assessment types: Assessments that are available in two forms (A and B), Tiered Assessments available in two forms (AT and BT), Online Assessments available in two forms (A and B), and a Performance Assessment.

Examples of grade-level assessments include: 

• Unit 1, Tiered Assessments, Form BT, Problem 7, “Nieka stated that there is no solution to the equation 5(x + 3) = 5x + 15. Do you agree or disagree? Explain your reasoning.” (8.EE.7a)

• Unit 4, Performance Assessment, Problem 1, “The total cost of an order of gift baskets from Gift Baskets Galore includes the cost of each gift basket plus a one-time delivery fee. The cost of each gift basket is the same regardless of how many the customer orders. One company bought 40 gift baskets for their employees for a total of $715. Another company bought 400 gift baskets for $5,935. 1. Based on the information given, what is the cost per gift basket? Show all work necessary to justify your answer.” (8.F.4)

• Unit 9, Assessments, Form A, Problem 7, “A cylindrical can of soda pop is 12 cm tall and has a diameter of 6 cm. The box for a 12-pack of soda pop has a length of 25 cm, a width of 19 cm and is 12.2 cm tall. The cans are placed in the box in three rows of 4. What is the approximate volume of space that is not used when a 12-pack of soda pop is full?” (8.G.9)

There are above grade-level assessment items that could be modified or omitted without impacting the underlying structure of the materials. Examples include, but are not limited to:

• Unit 3, Assessments, Form A, Problems 1 and 2, “Give the domain and range of each relationship. Determine whether or not each relationship is a function. 1. Domain: ___ Range: ___ Function: ___ 2. Domain: ___ Range: ___ Function: ___ .” Students are given two tables, one with columns labeled Input and Output and one with columns labeled x and y, (F-IF.1)

• Unit 10, Assessments, Form A, Problems 6b and 6c, “Two hundred random students were asked whether or not they bought music on CDs and whether or not they bought music as digital downloads…6b. Write the conditional frequencies for students who buy music as digital downloads. 6c. If a student buys digital downloads are they less likely to buy CDs? Explain your reasoning.” (S.ID.5)

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Each unit has a Storyboard that includes a Launch and a Finale. These tasks incorporate real-world applications and provide opportunities for students to apply unit concepts. Explore! activities provide students with an opportunity to discover mathematical concepts in a variety of methods. Teacher Gems are teacher-led activities that engage students with the main concepts of the lesson. Student lesson tasks fall into four categories (Practice My Skills, Reason and Communicate, Apply to the World Around Me and Spiral Review) in which students engage in grade-level content.

Materials engage all students in extensive work with grade-level problems to meet the full intent of grade-level standards. Examples include:

• Unit 1, Lesson 1.5, students engage in extensive work with grade-level problems to meet the full intent of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.) In Student Lesson, Exercise 7, students evaluate expressions with irrational numbers and locate them approximately on a number line. One problem states, “Match each of the values below with one of the points on the number line. a. -2\sqrt{2} b. \sqrt[3]{-20} + 3 c. 1-\pi d. \sqrt{32} - 6” A number line is given labeled with integers from -3 to 2 and marked with points labeled L, M, N, P, R and S. L is just to the right of -3, M is just to the left of -2, N is just to the left of -1, P is just to the left of 0, R is in the middle of 0 and 1, S is just to the right of 1. In Climb the Ladder, Exercises 1-5, students use inequality symbols to compare rational and irrational values. “Compare the two numbers using <, > or =. 1. \sqrt{65} \square 8 2. 2\pi \square \frac{15}{2} 3. -\sqrt{29}\square-5\frac{4}{5}4. -\sqrt{16} \square \sqrt[3]{-64} 5. \frac{\pi }{3} \square \sqrt{6}.

• Unit 3, Lesson 3.3, and Unit 4, Lessons 4.2 and 4.3, students engage in extensive work with grade-level problems to meet the full intent of 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.) In Lesson 3.3, Teacher Gems, Climb the Ladder, Ladder 4, Problem 2, students use slope triangles to explain why the slope is the same between two points. The problem states, “Carla and Jim both drew slope triangles on the same line. Carla’s triangle had a height of 4 and a base of 3. Jim’s triangle had a height of 8 and a base of 6. Is it possible for both of these slope triangles to be drawn on the same line? Explain your reasoning.” In Lesson 4.2, Student Lesson, Exercise 9, students write equations for lines in y=mx and y=mx+b form. One example states, “Use the graph below to answer the following: a. Find the slope-intercept equations for lines m, n and p. b. What do the three equations have in common? c. What geometry term can be used to describe the relationship between these three lines?” An image of a graph with three lines is given. Line m has the points (0,2) and (3,3) plotted. Line n has the points (0,0) and (3,1) plotted. Line p has the points (0, -4) and (3, -3) plotted. In Lesson 4.3, Student Lesson, Exercise 11, students find the equations of lines on a graph. “Four line segments make the four sides of a quadrilateral on the coordinate plane shown below. a. Find the equations of the lines containing each side: \overline{AB} \overline{BC} , \overline{CD} , \overline{AD} b. Are there any similarities in the equations for lines \overline{AB} and \overline{CD} ? How about \overline{BC} and \overline{AD} ?” An image of a graph is provided of a parallelogram with vertices marked A (-4,7), B (4,5), C (8,-4) and D (0,-2).

• Unit 7, Lesson 7.5, students engage in extensive work with grade-level problems to meet the full intent of 8.G.4 (Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.) In Teacher Gems, Task Rotation, Rotation 2, students translate and dilate a figure and then explain why they are similar. “Rectangle WXYZ have vertices at W(2, 2), X(2, 8), Y(6, 8) and Z(6, 2). The rectangle is translated 8 units to the left and 5 units down. The rectangle is then dilated by a factor of \frac{3}{4} with a center at the origin to form W’X’Y’Z’. a. Graph WXYZ and W’X’Y’Z’. b. Are the rectangles similar? Explain.” In Leveled Practice-P, Exercise 8, students perform two transformations to make a similar figure. “Write a series of two transformations that would form a similar but not congruent image. Explain how you can know the figures will be similar without graphing.” Unit 7, Planning and Assessment, Launch and Finale Teacher Gems, Pathways Finale, Skill 4C, students write a transformation rule to map a given figure to a second figure. It states, “Prove that the figures shown are either similar or congruent by writing a transformation rule that maps ΔABC onto ΔDEF.” An image of the coordinate plane is given with two triangles ABC and DEF with vertices A(4,2), B(2, 4), C(0,2) and D(0,0), E(-2, 2), and F(-4,0). 

• Unit 10, Lesson 10.4, students engage in extensive work with grade-level problems to meet the full intent of 8.SP.4 (Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between two variables.) In Student Lesson, Exercise 3, students construct a two-way frequency table from data given on two variables. “Two hundred random seniors from a high school were asked if they had passed Algebra II and if they had passed a college readiness exam. Of the 120 who had passed Algebra II, 110 had passed the exam. Of the 80 who had not passed Algebra II, 30 had passed the exam. a. Formulate a statistical question that could be investigated using his data. b. Construct a two-way frequency table showing this information.” In Student Lesson, Exercise 8, students create a relative frequency table from a two-way table, make observations from the relative frequencies and write conditional frequencies. “A phone company was trying to determine if their customer base still had home phones. Potential customers were asked if they have a cell phone and if they have a home phone. The table at right shows the responses. a. Create a relative frequency table for the given two-way table. b. Make one observation from the relative frequencies. c. Write the conditional frequencies for people with cell phones. If a person has a cell phone, are they likely to also have a home phone?” A two-way table is given with a horizontal label of “Home Phone”. The columns underneath are Yes and No. The vertical label is “Cell Phone” and the rows are labeled Yes, followed by 20 and 50, and No, followed by 5 and 35.

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

When implemented as designed, the majority of the materials address the major clusters at least 65% of the time. Materials were considered from three perspectives; units, lessons, and instructional time (days).

• The approximate number of units devoted to major work of the grade is 8 out of 10, which is approximately 80%.

• The approximate number of lessons devoted to major work is 37 out of 46, which is approximately 80%.

• The approximate number of days devoted to instructional time, including assessments, of major work is 123.5 out of 156, which is approximately 79% of the time.

The lesson instructional time (days) are considered the best representation of the materials because these represent the time students are engaged with major work, supporting work connected to major work, and include assessment of major work. Based on this analysis, approximately 79% of the instructional materials focus on the major work of the grade.

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Each unit contains a Unit Overview with information regarding standards correlation and how standards are connected in a unit. Specific examples are provided as well.

Materials connect supporting work to major work throughout the grade level, when appropriate, to enhance major grade-level work. Examples include:

• Unit 1, Lesson 1.4, Student Lesson, Exercise 24, connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number) to the major work of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations of the form x^{2}=p and x^{3}=p , where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that \sqrt{2} is irrational). Students represent the value of a non-perfect square as a decimal and with the radical symbol. An example provided is, “Monique designed a square patio. The area of the patio is 294 ft^{2} . Express the side length as a decimal and as an exact root in simplified form.”

• Unit 6, Lesson 6.2, Leveled Practice - P, Exercise 7, connects the supporting work of 8.G.5 (Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.) to the major work on 8.EE.7 (Solve linear equations in one variable). Students use their knowledge of angle relationships to write an equation to find each angle measure. An example provided is as follows, “Solve for x. Then find the measure of each identified angle.” A diagram of two parallel lines cut by a transversal is given. Two corresponding angles are labeled (15x+1)^{\circ} and (10+12x)^{\circ} . 

• Unit 10, Lesson 10.3, Student Lesson, Exercise 8, connects the supporting work of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line) to the major work of 8.F.4 (Construct a function to model a linear relationship between two quantities. Determine the rate of change and the initial value of the function from the description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values). Students informally fit a straight line in a scatter plot and write a linear equation for the line of fit. An example is as follows, “Nolan works at a ski resort. He noticed there was an association between the number of inches of new snow that day and the number of skiers per hour on a particular ski-run. Create a scatter plot of the data set and draw a line of fit. Find the equation of your line of fit.” A table of values is given with x values representing the “Number of Inches of New Snow (x)” and y values representing the “Number of Skiers per Hour (y)”.

The following connection is absent from the materials:

• No connection is made between the supporting work of 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems) to the major work of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.)

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Each unit contains a Unit Overview with a section titled, Connecting Content Standards where information regarding connections as well as specific examples is provided when applicable.

There are connections from supporting work to supporting work and/or major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 2, Lesson 2.2, Exit Card, Item 2 connects the major work of 8.EE.A (Expressions and Equations: Work with radicals and integer exponents) to the major work of 8.G.B (Understand and apply the Pythagorean Theorem). Students find square roots when using the Pythagorean Theorem to find the interior diagonal of a rectangular prism: “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. Round to the nearest tenth.”

• Unit 4, Lesson 4.5, Student Lesson, Exercise 13 connects the major work of 8.F.A (Define, evaluate, and compare functions) to the major work of 8.F.B (Use functions to model relationships between quantities). Students construct a function and analyze its multiple representations to determine if the function is linear: “Harrison designed a miniature shoe that he can print with a 3D printer. The 3D printer can complete one shoe every 30 seconds. a. Create a table of values showing the number of shoes printed after minutes 1, 2 and 3. b. Is this relationship linear or non-linear? Explain your reasoning.”

• Unit 9, Lesson 9.2, Explore!, Step 8 connects the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers) to the supporting work of 8.G.C (Solve real-world and mathematical problems involving the volume of cylinders, cones, and spheres). Students use the approximation of 3.14 for \pi in their formula to find the volume of a cone: “Use your formula to find the volume of each cone. Use 3.14 for \pi. Round to the nearest hundredth.” Students are given three cones: one with the radius labeled as 6 in and the height labeled as 15 in, one with a diameter of 18 ft and a height of 7.5 ft, and another with a radius of 2.1mm and a height of 9.4 mm.

The materials reviewed for EdGems Math (2024) Grade 8 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

EdGems Math provides teachers with evidence that the content addressed within each unit is related to both previous and future learning. This information is first outlined in the Content Analysis section of the Unit Overview. The Unit Overview then provides a Learning Progressions table for each unit, illustrating the vertical alignment of the topics and standards present in the unit. This vertical progression of mathematical concepts and standards is further elaborated throughout each unit. Each unit includes a Readiness Check and Starter Choice Boards that focus on prerequisite skills. Each Readiness Check reviews three to five skills from a previous grade level, which represent prerequisite skills for the unit. The Unit Overview outlines the skills targeted within the Readiness Checks. Starter Choice Boards offer three options: "Storyboard," "Building Blocks," and "Blast from the Past." The Building Blocks warm-up focuses on a prerequisite skill that directly relates to the current lesson. The standards alignment for Building Blocks is provided in the Teacher Guide for each lesson. Finally, "Explore!" activities build upon students' prior knowledge and experiences to scaffold the discovery of grade-level concepts or skills. The Teacher Guide provides an overview of the activity, including connections to previous grades.

Materials identify content from future grades and relate it to grade-level work. Examples include:

• Unit 1, Planning and Assessment, Unit Overview, Readiness Check & Learning Progression, “In this unit, students will… Solve multi-step linear equations in one variable (8.EE.C.7), understand what an irrational number is and compare their sizes (8.NS.A.1-2), and solve equations with square or cube roots (8.EE.A.2)", connecting it to “In the future, students will… Create equations and inequalities and use them to solve problems (HS.A-CED.A.1). Solve multi-step inequalities and quadratics in one variable (HS.A-REI.B.3-4). Solve systems of linear equations (HS.A-REI.C.6). Graph solutions to linear inequalities and systems of linear inequalities in two variables (HS.A-REI.D.12).” Examples are given for each skill.

• Unit 3, Planning and Assessment, Unit Overview, Readiness Check & Learning Progression, “In this unit, students will… Understand that a function is a rule that assigns one input to exactly one output. 8.F.A.1. Compare properties of two functions represented in different ways. 8.F.A.2. Graph proportional relationships and interpret unit rate as the slope. 8.EE.B.5. Explain why the slope is the same between distinct points on a non-vertical line and derive y=mx+b. 8.EE.B.6", connecting it to “In the future, students will… Understand that a function assigns one element from the domain to exactly one element of the range. HS.F-IF.A.1. Interpret key features of graphs and tables of functions and sketch functions showing key features. HS.F-IF.B.4. Calculate and interpret the average rate of change of a function over a specified interval. HS.F-IF.B.6.” Examples are given for each skill.

• Unit 5, Planning and Assessment, Unit Overview, Content Analysis, “The unit begins with an opportunity for students to discover that systems of linear equations may have one solution, infinitely many solutions, or no solution. Students will learn to solve systems of linear equations by graphing, substitution, and elimination. Solving systems using different graphical and algebraic methods will bolster students’ conceptual understanding, paving the way for success in future courses when they solve systems of inequalities and systems of linear and non-linear equations.”

Materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include:

• Unit 1, Lesson 1.1, Teacher Guide, Starter Choice Board Overview identifies prior grade-level skills with their standards: “Building Blocks: Substitute values into one- or two-step equations to determine if it is a solution (6.EE.B.5)” and “Blast from the Past: Compare a percent and a fraction (7.RP.A.3).” The Lesson Planning Guidance for Day 1 prepares teachers to help assess students' previous knowledge of substituting values- or two-step equations to determine if they are solutions. “In this lesson, the 'Building Blocks' task asks students to access background knowledge on substituting values into one- or two-step equations to determine if they are a solution. Use this activity if many of your students need support in recalling this skill. Consider using Expert Crayons to have students move around the room supporting each other. Choose the Starter Choice Board’s 'Blast from the Past' task to give students an opportunity to utilize problem solving skills involving comparing percents and fractions.”

• Unit 2, Lesson 2.1, Teacher Guide, Starter Choice Board Overview identifies prior grade-level skills with their standards: Storyboard: Find the area of a rectangle (7.G.B.6), Building Blocks: Solve equations in the form x^{2}=p (8.EE.A.2), Blast from the Past: Solve application problems with rates (7.RP.A.1), Fluency Board Skills: Plot rational number approximations on a number line, solve equations, evaluate expressions. The Lesson Planning Guidance: Day 1 supports teachers in selecting activities that meet the needs of their students: “In this lesson, the ‘Building Blocks’ task asks students to access background knowledge on solving equations in the form x^{2}=p. Use this activity if many of your students need support recalling this skill. Consider using Expert Crayons to encourage students to move around the room supporting each other. Choose the Starter Choice Board’s ‘Blast from the Past’ task to give students an opportunity to apply problem-solving skills to rate application problems.” Lesson Planning Guidance: Day 2 states, “In this Storyboard Starter, students find the area of a rectangle. Consider having students read the storyboard frame and take 1–2 minutes of silent think time before working with table partners. Another option is to have one or two students in each group read the characters’ statements aloud and then work together to answer the prompt.”

• Unit 6, Planning and Assessment, Unit Overview, Content Analysis highlights prior grade-level skills, such as discovering relationships between angles in two-dimensional figures and supplementary angles, aligned with standards 7.G.A.2 and 7.G.B.5. “The unit begins by exploring alternate interior, alternate exterior, corresponding, and same-side interior angles as a continuation of the angle relationships learned in the previous course. This work then informs the development of triangle relationships, including the Triangle Sum Theorem, the Angle-Angle Similarity Theorem, and the Exterior Angles Theorem. Later in the unit, students will connect to supplementary angles from Grade 7 by arranging the angles of a triangle to form a straight angle, before finally applying the relationship of corresponding angles as yet another informal proof that the interior angles of a triangle have a sum of 180°.”