8th Grade - Gateway 2

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding and include the use of visual representations, interactive examples, and different strategies.

Cluster 8.EE.B addresses understanding the connections between proportional relationships, lines, and linear equations, and standards 8.F.2,3 address comparing and defining functions.

• In Topic 8 students are given opportunities to connect verbal descriptions of situations to graphs and write situations that can be represented by a specific graph. The MARS task “Vacations” gives students the opportunity to compare slopes on two different graphs. Students also use proportional and nonproportional reasoning to derive a two-variable equation representing a situation. Overall, students are responsible for interpreting the given real-world situation and representing it in multiple ways, i.e. tables, graphs, verbal descriptions, and equations.

Standard 8.F.1 addresses understanding that a function is a rule that assigns to each input exactly one output.

• In Topic 7 the definition of a function is developed in multiple ways. Students understand functions through the use of verbal description, input-output machines, real-world situations, graphs, and mappings.

Cluster 8.G.A addresses understanding congruence and similarity through different tools.

• In Topic 1 students are given multiple opportunities to describe the effects of a transformation of a shape on the coordinate plane. Students demonstrate understanding by moving points on the coordinate plane given a specific transformation and by describing the movement of a point algebraically. Although there are a few opportunities to describe a sequence of transformations and to explain the effects of change on the figure as a whole, greater focus is placed on individual ordered pairs. In this topic, angle measures and side lengths are referenced in dilations.

• In Topic 14 students further develop their understanding of how angle relationships are affected by transformations through the use of verbal descriptions, animations, real-world examples with maps, and the use of geometric tools.

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skills. Overall, students are given opportunities to develop procedural skills within clusters 8.EE.C and 8.G.C.

Cluster 8.EE.C addresses students developing procedural skills with analyzing and solving linear equations and pairs of linear equations in one variable.

• In Topic 11 students have opportunities to practice solving multi-step equations in one variable using the Distributive Property and combining like terms through the Practice and Assessment sections, along with the Student Activity Sheets. Also, within this topic there are opportunities to develop procedural skills with solving equations resulting in infinitely many or no solutions.

• In Topic 12 students have opportunities to solve systems of equations both graphically and algebraically. Also, within this Topic there are opportunities to develop procedural skills with solving systems of linear equations resulting in infinitely many or no solutions.

Cluster 8.G.C addresses developing procedural skills with the formulas for the volume of cylinders, cones, and spheres.

• In Topic 15, 3-dimensional animations are provided as assistance for students in understanding the development of the volume formulas. There is limited practice in using the formulas to find the volume of cones, cylinders, and spheres. There are problems within the Practice and Assessment questions, which are multiple choice, that involve finding the volume of cones, cylinders, and spheres, and there are three Constructed Response items that involve these formulas as well.

The materials reviewed for Agile Mind Grade 8 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Overall, students are given opportunities to solve application problems that include multiple steps, real-world contexts, and are non-routine.

Application problems allowing students to make their own assumptions in order to apply their mathematical knowledge can be found in different parts of the materials, including MARS Tasks, Constructed Response items, and occasionally within the Student Activity Sheets (SAS).

Standard 8.EE.8c addresses students solving real-world and mathematical problems leading to two linear equations in two variables.

• In Topic 12 Constructed Response 1 students are given tables of data for two students walking in front of a motion detector. The tables include the distances in feet from the motion detector after different amounts of time in seconds. Students are ultimately asked to interpret the point of intersection for the graphs of the two sets of data. Students are provided with scaffolded steps in this problem that lead them to interpreting the point of intersection.

• Topic 12 Constructed Response 2 is a scaffolded problem that leads students through the steps of creating a solution that contains a certain percentage of pure acid.

• In Topic 12 Constructed Response 3 students are presented a problem that has them create a solution containing a certain percentage of pure acid, just as in Constructed Response 2 of Topic 12, but the problem does not provide them any scaffolded questions to help them obtain the answer.

• In Topic 12 the MARS task, Pathways, allows students to write and solve a system of equations that will yield the appropriate dimensions of a paving stone based on a desired design design of the pathway. This problem does not include any questions or prompts for scaffolding, and the context is unique to the topic, which makes the problem non-routine.

• In Topic 13 Constructed Response 1 students write and solve a system of equations that will result in the dimensions of three horse pens. This problem does not include any questions or prompts for scaffolding, and the context is unique to the topic, which makes the problem non-routine. Furthermore, students must alter the equations and recompute the dimensions based on a change in feet of fencing used.

Cluster 8.F.B addresses students being able to use functions to model relationships between quantities.

• In Topic 5 Constructed Response 2 students are given a graph that shows the volume of four different gas tanks and how much time is needed for each gas tank to become empty. Students have to answer different questions using the graph, and the questions involve analyzing the volumes of the tanks, the time needed to empty the tanks, distance traveled, and gas mileage. There are no questions or prompts that provide scaffolding to lead students toward the answers, and although the context is similar to one students encountered during the Topic, the use of four gas tanks as opposed to one gives students the opportunity to apply their mathematical knowledge in a non-routine way.

• In Topic 6 Constructed Response 1 students are presented with a graph that shows distance traveled from a motion detector over time. Students are expected to answer different questions about the graph, but the context used is exactly the same as the context used with many other problems throughout the Topic. Also, the questions used in this problem are the same in wording and structure as other questions posed in the Topic.

• In Topic 7 Constructed Response 2 students are presented with the first three steps of a tile pattern and must answer questions about different steps in the pattern. Students are provided with some scaffolding during the problem as they are instructed to include a general function rule and a description of what is constant and what changes in the tile pattern as they respond. The context of tiles is unique for this problem in comparison to the other contexts used in the Topic.

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately. Most Topics provide opportunities through lessons and assessments for students to connect conceptual understanding, procedural skill and fluency, and application when appropriate or engage with them separately as needed.

Balance is displayed in Topic 4 when students apply and extend previous understanding of the Pythagorean Theorem as they complete an activity. Balance is further evidenced in Topic 11 where students conceptually solve linear equations using different models.

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for the Standards for Mathematical Practices (MPs) being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials for the teacher identify the MPs, and students using the materials as intended will engage in the MPs along with the content standards for the grade.

• The Practice Standards Connections are found within the Professional Support section for the teacher. The eight MPs are listed with six to ten examples for each. According to the Practice Standards Connections, “each citation is intended to show how the materials provide students with ongoing opportunities to develop and demonstrate proficiency with the Standards for Mathematical Practice.”

• Deliver Instruction is located within Advice for Instruction under Professional Support in the teacher material. Occasionally, there will be information within the Deliver Instruction section giving some guidance on how to implement the MP within the task/activity.

  • In Topic 8 Block 8 the teacher leads the class through the “Connecting Representations” pages. With the use of teacher questioning and activities found in the Deliver Instruction teacher material, the teacher helps the students understand that relationships can be expressed abstractly and quantitatively. Throughout this lesson, the students are engaging in MP2.

  • Topic 12 Block 6 Deliver Instruction suggests teachers tell students to make sense of the problem by explaining it to their partner. By doing this, students are engaging in MP1; however, that is not noted within the teacher information.

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials prompt students to construct viable arguments and present opportunities for students to analyze the arguments of others.

The instructional materials provide opportunities for students to construct viable arguments.

• In Topic 4 Block 4 the class is investigating the converse of the Pythagorean Theorem and must answer, “What do the conclusions you have reached so far tell you about the triangles and ∠?????”

• In Topic 5 during the MARS task “Graphs” students are told to “Explain how you made your choices” after matching several equations and graphs.

• In Topic 8 Block 4 students compare data in a table during a class discussion and must answer, “Which amount grows at a faster rate—the amount paid or the amount collected? How do you know?”

• In Topic 14 Block 2 the class is introduced to parallel lines cut by a transversal and the related angles. The class is asked “Use what you have learned about parallel lines, supplementary angles and corresponding angles to explain why ????∠2 = ????∠7.”

The instructional materials provide opportunities for students to analyze the arguments of others.

• In Topic 2 Constructed Response 3 students create a cube out of modeling clay that has a volume of 20 cubic centimeters. In the last part of the problem, students are instructed to “Compare your group's estimates and results with another group by reading your written responses to another group. Did your methods differ for finding an approximation for the edge length? Did your explanations differ? Revise your writing after reflection and feedback from the other group.”

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. In Deliver Instruction, classroom strategies and question prompts are provided to assist teachers in engaging students to construct viable arguments or analyzing the arguments of others.

The following are examples of assistance provided to the teachers to promote the construction of viable arguments and analysis of other’s thinking, including prompts, sample questions to ask, and guidance for discussions.

• In Topic 2 after the students complete Constructed Response 1, there is a class discussion and the teacher is instructed, “The debrief of this task provides an opportunity for students to construct viable arguments and critique the reasoning of others. As students are explaining their solution methods, they should also attend to precision. Encourage the audience members to ask questions of the presenters if information is not clear, either in vocabulary or logic. Push students to clarify their thinking and to use precise vocabulary when explaining their solution method(s).”

• In Topic 3 Block 1 Professional Support Deliver Instruction, teachers are guided to listen for misconceptions related to exponents, use questions as needed that specifically ask students to analyze why or how a hypothetical person thinks about an answer or gets an answer, and identify whether the person correctly interpreted the meaning of the mathematics in the problem. The intent of these questions is to help students analyze the arguments presented by others and determine how those arguments support the mathematics in this specific problem.

• In Topic 4 Block 3 students watch animations to help them understand the Pythagorean Theorem, and the teacher is instructed to pose these questions: “Consider one of the right triangles on your Patty Paper. What is the base of the triangle? What is the height of the triangle? Do all four right triangles have the same area? How do you know?” These questions assist teachers in engaging students in constructing viable arguments.

• In Topic 4 Block 4 students investigate the converse of the Pythagorean Theorem. Teachers are provided with the following assistance, “This may be the first time students have been pushed to make precise mathematical arguments about a geometric relationship. ... Give students the opportunity to make their own arguments before showing the final Check button. Then let students compare their arguments with the one shown online. Discuss the precision of the language and the logical reasoning used. The intent of this page is not for students in this course to develop strict mathematical proofs, but instead to expose students to the reasoning and language used in such arguments.” This assistance is specific in that teachers can draw students’ attention to specific aspects of the solutions provided, which helps in constructing an argument. Also, students can use the correct solutions as a way to analyze their own arguments and improve them as needed.

• In Topic 5 of the MARS Task “Graphs” teachers are provided with the following assistance, “As the majority of students seem to be finishing the task, put students into pairs and assign one of the four graphs to the each pair of students by counting off student pairs by four. ... This can help students grow in their ability to construct sound arguments and provide meaningful critiques of others’ arguments.”

• In Topic 5 as the students work on Constructed Response 1, the teacher is instructed to have “students verify their graphs and written responses with a partner. Provide students an opportunity to revise their work as needed but ask that they note any modifications and justification for changes.”

• In Topic 5 as the students work on Constructed Response 2, the teacher is given the following Classroom Strategy: “Divide the class into two “teams.” Then have each team come to a consensus on their responses for each part. Have team leaders present their team’s answers. Allow time for the other team to ask for clarifications. This will provide students with an opportunity to practice constructing viable arguments and critiquing the reasoning of others as they justify their conclusions, communicate them to others, and respond to the arguments of others.” The assistance provided for the teacher helps create an environment where MP3 can occur.

• In Topic 14 Block 2 students construct an argument to show that the measures of two angles are equal. The assistance provided to the teacher is as follows: “Encourage students to begin their proving process by measuring or tracing angles. They can write algebraic equations using the variables labeling each angle. Students may need to state their reasons verbally before recording their ideas on paper. These are a great opportunity to promote the mathematical practice of constructing viable arguments and critiquing the reasoning of others. Engage students in each others' arguments by asking them to restate key arguments in their own words or describe how those arguments are related to angles within each image.” This assistance gives teachers specific strategies for helping students construct a viable argument, and it also provides specific ways in which students can begin to analyze the arguments of others.

• In Topic 14 Constructed Response 1 students construct an argument to prove that two rays are parallel. The assistance that is provided to teachers with this problem is to “Ask a few students to share their explanation. Again, encourage students to critique the reasoning of the students, in a respectful way, and come to a class consensus on a strong explanation.%"

In the Advice for Instruction there is a missed opportunity to provide support for teachers that explains and identifies where and when problems, tasks, examples, and situations lend themselves to these types of questions. Additional guidance is needed to broaden the application of these questions throughout the course so that students routinely construct viable arguments and analyze the arguments of others.

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for explicitly attending to the specialized language of mathematics. Overall, the materials appropriately use the specialized language of mathematics and expect students and teachers to use it appropriately as well.

Occasionally, there are suggestions within Deliver Instruction as to how teachers can reinforce mathematical language during instruction.

• Topic 1 Block 2: “Discuss the term corresponding points. Make sure students understand this, as it is part of all three transformations.”

• Topic 3 Block 2: “...encourage them to use technical vocabulary to describe what is happening: base, exponent, and sum.”

• Topic 11 Block 1: “Language strategy. Students may have trouble at first telling the difference between a function and an equation, and may need to review some core vocabulary from previous topics: function rule, function, equation, input, output, domain, and range may need further review before, during, and after the lessons. Stress the relationship between a function, which describes the relationship between two varying quantities, and an equation, which represents a specific instance of the functional relationship.”

In the student materials, vocabulary terms can be found in bold print within the lesson pages, and these terms are used in context during instruction, practice, and assessment. Vocabulary terms are also available to the students at all time through My Glossary within the materials. For teachers, vocabulary terms for each Topic can be found under Language Support, which is within Advice for Instruction. Both core vocabulary and reinforced vocabulary are listed for each unit.