6th Grade - Gateway 2

The instructional materials reviewed for Agile Mind Grade 6 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 6.RP.A addresses understanding ratio concepts and using ratio reasoning to solve problems.

• In Topic 2 “Understanding Ratio and Proportion” and Topic 3 “Introduction to Rates” students develop their understanding of ratios and rates through real-world, interactive examples. Students represent ratios and rates in various forms, including ratio tables, bar models, equivalent fractions, and points on a line or coordinate plane. Students are also given opportunities to solve problems using these concepts in multiple topics within the materials.
• In Topic 4 students understand the connections between fractions, decimals, and percents, and during the topic, they are reminded that each of these forms represent ratios of rational numbers.

Cluster 6.NS.C calls for applying and extending previous understandings of numbers to the system of rational numbers.

• In Topic 8 “Extending the Number System” students are introduced to rational numbers, integers, absolute value, and opposites. The animations allow students to develop an understanding of these terms visually, and many of the animations include the use of number lines and a coordinate grid. Through the topic, students have the opportunity to understand that the absolute value of a number is its distance from 0 on a number line. Students also have an opportunity to explore rational numbers in real-world contexts such as banking, sea level, and traveling distances. One aspect of 6.NS.C for which students are given limited opportunities to develop an understanding is that the opposite of the opposite of a number is the number itself.

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, there are opportunities for students to practice dividing multi-digit numbers using the standard algorithm, and students are given opportunities to develop fluency with decimal operations.

Standard 6.NS.2 addresses students being able to fluently divide multi-digit numbers using the standard algorithm.

• In Topic 1 Blocks 1-3 there are multiple opportunities for students to engage with dividing multi-digit numbers within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.

Standard 6.NS.3 addresses students being able to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

• In Topic 5 there are multiple opportunities for students to develop fluency with adding and subtracting decimals within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.
• In Topic 6 there are multiple opportunities for students to develop fluency with multiplying and dividing decimals within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.

The materials reviewed for Agile Mind Grade 6 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 6.RP.3 addresses using ratio and rate reasoning to solve real-world and mathematical problems.

• In Topic 2 Constructed Response 1 students are expected to use rate reasoning to determine the number of candles in boxes of different sizes or the price of boxes of different sizes. This problem does not include any questions or prompts for scaffolding, and the context is unique when compared to other contexts used in the Topic.
• In Topic 7 Constructed Response 2 students determine how long it will take for two people to paint a wall when they work together. Students calculate the area of the wall and then calculate how much of the wall each person paints per minute.

Standards 6.EE.7 and 6.EE.9 address students writing and solving linear equations in order to solve real-world and mathematical problems.

• In Topic 9 Constructed Response 1 students write linear equations, along with tables and graphs for the equations, in order to answer questions about the type of growth for two different plants. The context is unique for this Topic.
• In Topic 11 Constructed Response 1 students write linear equations in order to answer questions about how much time is needed to save a certain amount of money or how much money needs to be saved if there is only a certain amount of time. This context is the same as one that is used in the Topic, and there are scaffolded questions that let students know how to define the variables and how many equations they need to write.

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately. Most Topics attempt to 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 Topics 14 and 15. In Topic 14 students begin to understand how statistical questions involve variability, and they also begin to develop procedural skills with creating different plots that represent data distributions. In Topic 15 students further their understanding of statistical variability as they summarize and describe data distributions with measures of center and measures of spread, and they also explain which measures best represent the distributions. These understandings and skills are developed in conjunction with sets of data that allow students to apply them in real-world contexts.

The instructional materials reviewed for Agile Mind Grade 6 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 four 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 7 Block 5 while students are working individually on Constructed Response 1, the teacher is asked to encourage the practice of modeling (MP4) and sense‐making with mathematics (MP1) by helping students make connections between what they are exploring about rates in graphs with questions such as: “How is this situation similar to the ones we have been studying?; What information do you need in order to determine a rate?; and What operations help you determine a unit rate?”

The instructional materials reviewed for Agile Mind Grade 6 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 to analyze the arguments of others.

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

• In Topic 3 Block 2 students are matching dimensions of photographs to the corresponding coordinate on a graphical representation. The Student Activity Sheet asks the student to explain the process used to match the name with the picture.
• In Topic 12 Block 12 students are given the coordinates of a parallelogram and are asked to determine whether statements are true or false and record their reasoning.
• In Topic 15 Block 5 during the MARS Task “Suzi’s Company,” students are given a table depicting jobs and annual salaries of fifteen people in a small company. Students connect their understanding of measures of center as they explain their solution to several questions.

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

• In Topic 1 Block 8 Constructed Response 2 students engage in analyzing the arguments of fictitious students. Even though students are told in which step the error occurs, this is an opportunity for students to analyze the mathematical arguments that are presented to them and justify their suggested corrections.
• In Topic 15 Block 5 one part of the MARS Task “Suzi’s Company” asks students to analyze the statement made by a person in the problem, identify the mistake that was made, and present the correct mode for the problem.

The instructional materials reviewed for Agile Mind Grade 6 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 1 Block 5 students are determining all possible dimensions for a 225 square-foot school banner. The teacher is instructed as follows: “As pairs work to find all the possible dimensions for this banner, they will develop the practice of constructing viable arguments and critiquing the reasoning of the members of their group. Encourage students to justify the reasoning they used to decide when they found all possible dimensions.”
• In Topic 1 Block 8 Constructed Response 2 teachers are told that two parts of the problem “provide an opportunity to promote the practice of constructing arguments and responding to the ideas of others.”
• In Topic 3 Block 5 students are analyzing batting averages of two softball games at a family reunion. The teacher is instructed to “Promote the habit of communicating mathematical ideas by critiquing the reasoning of others with this activity. Allow students plenty of time to engage with each written argument and discuss their ideas with a partner.”
• In Topic 12 Block 1 teachers are assisted with “as they share strategies, ask students to consider which strategies may be more or less accurate and have them justify their responses. This discussion promotes the mathematical practice of constructing viable arguments and critiquing the reasoning of others.”
• In Topic 12 Block 6 students are finding the area of an irregular shape, a map of the United States. The teacher is instructed as follows: “As students share and discuss their strategies for finding the area of the unusual shape, promote the mathematical practice of constructing viable arguments and critiquing the reasoning of others. You may need to model this strategy to get the discussion started, “Who agrees with Monique’s strategy? Tell us why her strategy is effective.”
• In Topic 15 Block 11 in the MARS Task “TV Hours” teachers are given the following assistance: “Encourage students to be active listeners during the debriefing of this task. For example, after a student has shared her strategy summarizing the results of Mrs. Campbell’s letters you can engage students by saying; “Raise your hand if you understand Sarah’s conclusion.” Then call on one or more of those students to restate the first student’s conclusion in their own words. ... As students or pairs of students are sharing their answers and strategies with the rest of the class, encourage students to critique each other’s reasoning and to compare the various strategies.”

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 6 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 7: "Reinforce the use of precise mathematical language as you introduce the Distributive Property. Most students are familiar with the words product, addend, factor, and sum. However, this is a new context and application for those words. Provide additional number sentences with which to identify factors, addends, products and sums (with and without common factors).”
• Topic 9 Block 2: “Students at this grade level may have had very few experiences with using variables to write equations or expressions. Pause with each panel to clarify for students how the variables, x and y, are being used to take the place of the numbers.”

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.