7th Grade - Gateway 2

The instructional materials reviewed for Agile Mind Grade 7 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 7.NS.A addresses applying and extending previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

• In Topics 5, 6, and 7 students develop their understanding of rational number operations, including integers and fractions, through several real-world, interactive examples. Students use algebra tiles, number lines, and other manipulatives to represent various problems and their solutions. Fact families and patterns are also used to build understanding of rational number operations. Real world situations include change of elevation, debt, and traveling across time zones.

Cluster 7.EE.A addresses understanding how the properties of operations can be used to generate equivalent expressions.

• In Topic 8 students begin by looking at growing patterns in “virtual creatures.” Their understanding develops as they transition from the interactive applet in the materials to modeling equivalent expressions with algebra tiles. Furthermore, equivalent expressions leads to solving equations and inequalities, which are modeled with different representations as students transition from a concrete to an abstract understanding of equivalent expressions.

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skills. Students are given multiple opportunities to develop procedural skills with rational numbers, expressions, equations, and inequalities.

Cluster 7.NS.A addresses students developing procedural skills with adding, subtracting, multiplying, and dividing rational numbers.

• In Topic 5 students have multiple opportunities to develop procedural skills with 7.NS.1, including combining quantities to make zero and addition/subtraction of integers. As students generalize patterns through basic practice and verbal description, they make connections between conceptual understanding and procedural skills.
• In Topic 6 students have multiple opportunities to multiply and divide integers through Practice problems and Student Activity Sheets. Some problems are strictly procedural while others are built into constructed response assessment items.

Standard 7.EE.1 addresses students being able to add, subtract, factor, and expand linear expressions with rational coefficients, and 7.EE.4 expects students to develop procedural skills with solving linear equations and inequalities.

• Topics 7 and 8 give students ample opportunities to combine linear expressions with rational coefficients along with solving equations and inequalities. In both topics, various rational numbers (fractions and integers, both positive and negative) are embedded into the expressions, equations, and inequalities.

The materials reviewed for Agile Mind Grade 7 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).

Cluster 7.RP.A addresses students using proportional relationships to solve real-world and mathematical problems.

• In Topic 1 Constructed Response 1 students use proportional reasoning to modify a vegetable soup recipe based on a given number of people or a specific amount of one of the ingredients. 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 2 Constructed Response 1 students use proportional reasoning to determine the best buy among different packages of sports drinks. This problem provides some scaffolding for the students as the first part of the problem directs students to find the unit price for each of the three different packages.
• In Topic 4 Constructed Response 2 students use proportional reasoning to determine the new price of an item after a sale and an additional reduction in price. Students must create a model to represent the situation and determine the full discount without any scaffolding questions or prompts.

Standard 7.NS.3 addresses solving real-world and mathematical problems involving the four operations with rational numbers.

• In Topic 7 Constructed Response 1 students apply their knowledge of operations with rational numbers when modifying a recipe for trail mix. However, the context for this problem is almost identical to a context that is used during the Topic, and the students are provided with partially completed tables of values that help them determine which operations to use when modifying the recipe.
• In Topic 7 Constructed Response 2, students answer multiple questions about the costs incurred when different numbers of people participate in different activities, and they are provided a table that lists the activities and their associated costs. In this problem, students are not provided with questions or prompts that lead them toward answers, and the context is unique from others used in the Topic.

The instructional materials reviewed for Agile Mind Grade 7 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 5 in which students apply and extend previous understanding of addition/subtraction of integers using several models. Balance is further evidenced in Topic 13 in which students apply and extend previous understanding of area and volume in a series of animations using number cubes to fill a rectangular prism and cutting the box open to determine its surface area.

The instructional materials reviewed for Agile Mind Grade 7 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 three to eight 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.” Several opportunities exist within the Grade 7 materials where the MPs enrich student learning.
• 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 3 Block 8 the teacher guides the student to complete the MARS task “Tiling Squares.” In Deliver Instruction, the questions and activities help the teacher to guide students to identify and generalize patterns verbally and with an algebraic rule. The investigation allows students to engage in MPs 1, 4, and 8.
• In Topic 9 Block 3 provides an opportunity to reason abstractly and quantitatively (MP2) as students relate experimental probability to theoretical probability through probability experiments with flipping coins.

The instructional materials reviewed for Agile Mind Grade 7 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 to analyze the arguments of others.

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

• In Topic 1 Block 5 during the MARS task “Mixing paints” students use proportional reasoning to answer several questions in the context of paint mixtures. They are prompted to explain their answer.
• In Topic 1 Block 7 during the MARS task “Cereal” students explain their reasoning when determining which cereal has the higher ratio of protein.
• In Topic 10 Block 8 during the MARS task “Best Guess” students determine which student is the best at guessing how long 30 seconds is with a given scenario and data set. They are then asked to justify their own reasoning.
• In Topic 12 Block 7 students construct the triangles with given angle measurements and side lengths. Students then compare their answers with a partner. There is a missed opportunity for students to defend their own answers and critique the answers of their partners.

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

• In Topic 4 Block 8 during the MARS task “25% Sale” students are presented with a statement made by a person in the problem, requiring them to identify, analyze, and explain why the person is wrong.
• In Topic 8 Block 6 Constructed Response 1 students are presented a situation where a teacher is taking his students to a pizza parlor and asks them to to write an equation to represent the total cost for taking any number of students to BB’s Park and Pizza. Then, the students are prompted to evaluate four possible equations with the following two questions. “Which equation(s) correctly represent the cost for BB’s Park and Pizza? Explain why each equation is or is not correct. Which equations are equivalent? Provide evidence that the equations are equivalent.”

The instructional materials reviewed for Agile Mind Grade 7 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 the MARS task “Mixing Paints” assists teachers with “Encourage students to be active listeners during the debrief of this task. For example, after a student has shared his or her answer or strategy you can engage students by saying; “Raise your hand if you understand Michael’s strategy.” Then call on one or more of those students to restate the strategy in their own words. ... encourage students to critique each other’s reasoning and to compare the various strategies.”
• InTopic 2 Block 4 students are asked to solve a punch recipe problem and then play an animation to compare their answer with Briana’s. The teacher is directed to “have them critique Briana’s strategy and discuss how effective it is and how it compares to other strategies used by classmates. Emphasize that there are many different ways that students can use a rate table to reason about the situation.”
• In Topic 3 Block 3 students work to determine what the cost will be for having medium pizzas delivered. The assistance that teachers are provided with is as follows: “This presents a great opportunity to reinforce the mathematical practice of constructing viable arguments and critiquing the reasoning of others.”
• In Topic 8 Block 6 as students complete Constructed Response 1, they get the opportunity to analyze the solutions of fictitious students that are presented to them. The assistance teachers are given for engaging students in analyzing each other’s critiques of the fictitious students is as follows: “After students have had 10-12 minutes to work on the task, have them pair up and share the solutions and strategies. Encourage students to critique others' arguments for which equations are equivalent.”
• In Topic 10 Block 8 the MARS Task “Best Guess” assists teachers with “Students will also need to provide strong evidence for the arguments, so their written responses should include a description of how they applied statistical reasoning and measures of center and spread to make their decisions. ... After students have formulated their arguments, have students pair up and share their responses and rationale. Encourage them to critique each other’s reasoning. ... During the class debrief, continue to encourage students' careful analysis of the various arguments. Some students may be uncomfortable with the fact that there is not a single correct answer. Be sure that measures of center and variability are both discussed, especially when analyzing the guesses of Ben.”
• In Topic 12 Block 7 students construct the triangles with given angle measurements and side lengths. Students are then asked to compare their answers to a partner. The teacher is instructed to ask students to present their work to the class and given the following assistance: “Doing so will get the students to think about how to articulate their reasoning and defend their work in front of their peers. It will also allow students the opportunity to ask questions of one another so that the student presenting the ideas can justify their thinking.”

The Advice for Instruction misses the opportunity to assist teachers to understand the construct of problems and situations that lend themselves to these prompts in any lesson throughout the course, instilling the habit for communicating mathematical ideas that construct viable arguments and analyze the arguments of others.

The instructional materials reviewed for Agile Mind Grade 7 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 4 Block 1: "This animation reinforces that equivalencies among ratios, percents, and decimals exist because each can be written as an equivalent ratio. Its purpose is not to demonstrate the most efficient process for translating among the representations of ratios, decimals, and percents. The animation builds on the idea that any rational number, including compound ratios, can be expressed as a ratio with a denominator of 1.”
• Topic 7 Block 1: "Review the different types of numbers: natural, whole, integer, and rational…. Ask students to share their own versions of each definition, correcting any misconceptions as needed.”
• Topic 12 Block 3: “The animation develops the Angle Sum Theorem for triangles. For this course, the theorem is neither formalized nor proven, but students should be fluent with the fact that the measures of the interior angles in a triangle sum to 180°, and be able to explain why.”

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.