8th Grade - Gateway 1

The materials reviewed for i-Ready Classroom Mathematics Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Within the i-Ready Classroom Mathematics materials, the Unit Assessments are found in the Teacher Toolbox and include two forms for Unit Assessment, Form A and Form B. Both Forms contain similar problems for each unit. The Unit Assessments can be found at the end of each unit in the materials. 

Examples of assessment items in i-Ready Classroom Mathematics include:

• Unit 3, Unit Assessment, Form A, Problem 1, assesses 8.EE.5 as students determine the slope of a line. “Seth owns a printing shop that makes customized T-shirts. The cost to make T-shirts is proportional to the number of T-shirts he makes. Seth graphs a line that shows the cost per customized T-shirt. Two points on the line are (2,8) and (4,16). What does the slope of the line mean in this situation? A.The cost is $0.25 per T-shirt. B The cost is $2 per T-shirt. C. The cost is $4 per T-shirt. D. The cost is $8 per T-shirt.”

• Unit 4, Unit Assessment, Form A, Problem 1, assesses 8.F.1 as students understand that a function is a rule that assigns exactly one output to one input. “Jackson goes to a basketball game. He pays $3 for a student ticket and $1 per item bought at the concession stand. Is the total cost to attend the basketball game a function of the number of items purchased at the concession stand? Use a graph to help explain your answer. Show your work.”

• Unit 5, Unit Assessment, Form B, Problem 11, assesses 8.EE.1 as students generate equivalent numerical expressions.. “The average estimated distance Mars is from the sun is $$2.3×10^{11}$$ meters. The average estimated distance Pluto is from the sun is $$5.9×10^{12}$$  meters. About how many times farther is Pluto from the sun than Mars? A. $$2.57×10^1$$ B. $$3.90×10^2$$ C. $$3.90×10^{-1}$$ D. $$2.57×10^{23}$$”

• Unit 6, Unit Assessment, Form A, Problem 8, assesses 8.G.6 as students use a proof to explain the Pythagorean Theorem. “A right triangle with leg lengths of 8 and 15 has a hypotenuse of length 17. Show that a triangle with side lengths $$\frac{8}{k}, \frac{15}{k},$$ and $$\frac{17}{k}$$ is also a right triangle. Explain your reasoning.”

• Unit 6, Unit Assessment, Form B, Problem 14, assesses 8.G.7 as students apply the Pythagorean Theorem to determine unknown side lengths. “Andre has a plastic storage box in the shape of a cube with side lengths of 6 in. Can he put the lid on the box with a 12-in. paint brush inside it? Explain your reasoning.”

The materials reviewed for i-Ready Classroom Mathematics Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. In the materials, there are ample opportunities for students to work with grade level problems. This includes:  

• Lessons contain multiple opportunities for students to work with grade-level problems in the “Try It”, “Discuss It”, “Connect It”, “Apply It”, and “Practice” sections of the lessons. 

• Differentiation of grade-level concepts for small groups are found in the “Reteach”, “Reinforce”, and “Extend” sections of each lesson. 

• Fluency and Skills Practice problems are included in the Math Toolkit in addition to the lessons.

• Interactive tutorials for the majority of the lessons include a 17 minute interactive skill tutorial as an option for the teacher to assign to students. 

Examples of extensive work with grade-level problems to meet the full intent of grade-level standards include:

• Unit 2, Lesson 5, Session 2, Fluency and Skills Practice, Problems 1-3, students perform dilations, translations, rotations, and reflections on figures in the coordinate plane and give the coordinates of the image after each step of the sequence of transformations is performed (8.G.3). A diagram of the coordinate plane and the original figure is included for each problem. After each problem, there is a table for students to enter the coordinates of each image.

  • “Dilate $$\triangle$$ABC by a scale factor of 2 with the center of dilation at the origin to form $$\triangle$$A'B'C. Then translate the image 4 units up to form $$\triangle$$A"B"C".

  • Reflect rectangle DEFG across the y-axis to form D'E'F'G'. Then dilate the image by a scale factor of $$\frac{1}{3}$$ with the center of dilation at the origin to form D"E"F"G". 

  • Rotate $$\triangle$$XYZ 90$$\degree$$ clockwise around the origin to form $$\triangle$$X'Y'Z'. Then dilate the image by a scale factor of $$\frac{1}{2}$$ with the center of dilation at the origin to form $$\triangle$$X"Y"Z".”

• Unit 3, Lesson 10, Interactive Tutorial.  Students solve linear equations in one variable using the Interactive Tutorials (8.EE.7). In Lesson 10, there are five Interactive Tutorials included for students to practice additional work.  The titles of these five 17-minute Interactive Tutorials for students include Solving Linear Equations- Level H, Solving Linear Equations with Rational Coefficients- Level H, Solve Multi-Step Equations- Part 1, Solve Multi-step Equations- Part 2, and Write and Solve Multi-Step Equations.  

• Unit 4, Lesson 16, Session 4, Connect It, Problems 1-7, students “use the problem from the previous page to help you understand how to write an equation for a linear function from a verbal description.” (8.F.4) 

  • What does the point (60,80) represent? 

  • Use the graph in Picture It to estimate the y-intercept. Check your estimate by solving the equation in Model It to find the value of b. What does this value represent in this situation? 

  • Write the equation for the page Kadeem is working on as a function of minutes he spends reading. 

  • What is the rate of change in pages per hour? Write an equation for the page Kadeem is on as a function of hours spent reading. 

  • Use the equations you wrote in problems 3 and 4 to find the page that Kadeem was on after 3 hours, or 180 minutes, of reading. Are your answers the same?

  • Do the different equations in problems 3 and 4 represent the same function? Explain. 

  • Reflect, Think about all the models and strategies you have discussed today. Describe how one of them helped you better understand how to solve the Try It problem. 

• Unit 5, Lesson 21, Session 3, Apply It, Problem 9, students compare two values written as a single digit time with an integer power of 10 and express how many times greater one value is than the other (8.EE.3). “A deer’s hair has a diameter of about $$4×10^{-4}m$$. A dog’s hair has a diameter of about $$8×10^{-5}m$$. Which animal's hair has a greater diameter? How many times as great? Show your work.”

• Unit 6, Lesson 27, Session 3, Apply It, Problems 6 and 8, students 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 (8.G.7). “Find the length of the diagonal from P to Q in this right rectangular prism. Show your work.” A right rectangular prism is shown with dimensions $$9×12×8$$ labeled. “Anica is shipping a poster to a customer. When the poster is rolled up it measures 6 feet long. She will use a box that is a right rectangular prism with a base that is 3 feet by 4 feet. What whole number could be the shortest height of the box that will hold the poster? Show your work.” The rectangular prism is shown with the 3ft and 4 ft length and width labeled and the 6 ft space diagonal labeled.

The materials reviewed for i-Ready Classroom Mathematics Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. Materials were analyzed from three different perspectives: units, lessons, and days. Each analysis includes assessments and supporting work connected to major work of the grade.  

• The approximate number of units devoted to major work of the grade is 5.5 out of 7 units, which is approximately 79%. 

• The number of lessons, including end of unit assessments, devoted to major work of the grade is 36.5 out of 46 lessons, which is approximately 79%. 

• The number of days, including end of unit assessments, devoted to major work of the grade is 119.5 out of 154 days, which is approximately 78%.

A day-level analysis is the most representative of the materials because the number of sessions within each topic and lesson can vary. When reviewing the number of instructional days for i-Ready Classroom Mathematics Grade 8, approximately 78% of the days focus on major work of the grade.

he instructional materials reviewed for i-Ready Classroom Mathematics Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Throughout the materials, supporting standards/clusters are connected to the major standards/ clusters of the grade. The following are examples of the connections between supporting work and major work in the materials: 

• Unit 4, Lesson 16, Session 2,Apply It, Problem 6, connects major work of 8.F.4 to supporting work with interpreting rate of change in 8.SP.3. “The graph shows distance in feet as a function of time in seconds. Write an equation for the function and describe a situation that it could represent. Include the initial value and rate of change for the function and what each quantity represents in this situation.” 

• Unit 6, Lesson 24, Session 2, Connect It, Problem 4a, connects supporting work of 8.NS.1  to major work in 8.EE.7b as students solve linear equations to convert a decimal expansion which repeats into a rational number. “To use an equation to write $$0.\bar{0.36}$$ as a fraction, multiply both sides of the equation by 100. Show how to write $$0.\bar{0.36}$$ as a fraction.”

• Unit 6, Lesson 28, Session 3, Apply It, Problem 7, connects supporting work of knowing formulas 8.G.9 with major work of using the Pythagorean theorem 8.G.7 to find radius and volume. “The cone and sphere have equal volumes. What is the radius of the sphere? Show your work.”

• Unit 7, Lesson 29, Session 2, Apply It, Problem 7, connects supporting work of scatter plots in 8.SP.1 and 8.SP. 2 with major work in describing the functional relationship between two quantities 8.F.5. “Does the scatter plot show a linear association, a nonlinear association, or no association between the variables? Explain.”

• Unit 7, Lesson 30, Session 4, Apply It, Problem 8, connects the supporting work of 8.SP.3 to major work of 8.F.4 when students construct a function to model a linear relationship between two quantities. They use their model to respond to the prompt. “A middle school science teacher surveys his students each year. He asks how many families have a landline telephone. The scatter plot shows the percent of families with a landline in each year since 2010. A good line of fit is drawn through the data. Write an equation of the line. Then predict the percent of families that will have a landline in 2020.”

The materials reviewed for i-Ready Classroom Mathematics 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. 

Examples of problems and activities that serve to connect two or more major clusters or domains in a grade are: 

• Unit 3, Lesson 8, Session 2, Model It and Connect it, Problems 2 and 3, connect the major work of 8.G.A and 8.EE.B as students construct similar triangles using dilations to explain why slope is the same between two points on a line in the coordinate plane. Model It: “You can use dilations of right triangles. Again, choose any two points, A and B, on the line and draw a slope triangle. Dilate this triangle along the line using different scale factors to determine if the slope between any two points changes along the line. Connect It, Problem 2 , “Look at the second Model It. What is the slope of the line between points A and B? What is the slope of the line between points A and D?” Problem 3, “Both models started by choosing points A and B. How do you know the slope of a line is the same between any two points on the line?” In the TE, Monitor and Confirm Understanding provides teacher guidance to help develop this concept, “Because the dilated triangles are similar, the simplified slope (unit rate) will be the same.”

• Unit 4, Lesson 15, Session 2, Model It:Equations, Problem 3, connects the major work of 8.F.A with the major work of 8.EE.B when students define a linear function from linear equations. “Many functions can be represented by equations that show how to calculate the output y for the input x. a. Determine whether each equation represents a linear function. Show your work. a. y=2x-1; $$y=-x^2$$; y=-x. b. Explain how you know that equations of the form y=mx+b always represent linear functions. 

An example of a problem and activity that serves to connect two or more supporting clusters or domains in a grade is: 

• Unit 6, Lesson 28, Session 2, Apply It, Problem 6, connects the supporting work of 8.G.C with the supporting work of 8.NS.A when students use approximations of $$\pi$$ to find the volume of a cylinder. “A cylindrical swimming pool has a radius of 9 feet. Daniel fills the pool with water at a rate of 72 cubic feet per hour. How long will it take for the depth of water in the pool to reach 4 feet? Show your work Use 3.14 for  $$\pi$$ . 

It is mathematically reasonable for at least one 8th grade supporting work domain and cluster to not be connected to other supporting work domains and clusters. For example: 

• Examples of the 8.SP.A connections to other supporting work domains and clusters are not found in the materials. This is mathematically reasonable for investigating patterns of associations in bivariate data.

The materials reviewed for i-Ready Classroom Mathematics 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. Each Unit contains the Teacher’s Guide which includes a Unit Flow and Progression video, a Lesson Progression, a Math Background, and a Lesson Overview that contains prior and future grade-level connections to the lessons in the unit. Examples include:

• Unit 2, Beginning of Unit, Lesson Progression, includes a chart, “Which lessons are students building upon?” connecting prior work to the lessons of this unit. Lesson 6 Describe Angle Relationships 8.G.5 is connected to the prior work, Grade 7 Lesson 29 Draw Plane Figures with Given Conditions 7.G.2.

• Unit 3, Math Background, Linear Relationships, Prior Knowledge includes “Students should: know that solving an equation means finding a value that makes the equation true, be able to solve equations in the form px + q = r and p(x + q) = r, understand the idea of unit rate and constant of proportionality, be able to graph proportional relationships.” (7.RP.A and 7.EE.B) Future Learning states, “Students will: interpret and represent linear and nonlinear functions, explore and compare increasing and decreasing functions, compare functions written in different forms, and describe functions qualitatively based on their graphs.” (F.IF.C)

• Unit 4, Beginning of Unit, Lesson Progression, includes the chart “Which lessons are students preparing for?” connecting the lessons from this unit to future work. In Lesson 15, Understand Functions is connected to the future work of Functions - Interpreting Functions, F.IF.1.

• Unit 6, Beginning of Unit, Math Background, Future Learning, describes the future work of students connected to the unit. “Students will move on to extend their understanding of real numbers. Students will: solve quadratic equations; use properties of rational and irrational numbers to solve problems; prove theorems about triangles; further explore triangles to learn about trigonometric functions; give informal arguments for the volume formulas of figures.” (G.SRT.8)

• Unit 7, Lesson 31, Overview, Learning Progression, “In high school, students will use relative frequencies and probability, including conditional probability, to analyze relationships displayed in two-way tables.” (S.CP.A)