7th Grade - Gateway 3

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include: 

• The Guide to Implementing AF Math provides a Program Overview for the teacher with information on the program components and scope and sequence. This includes descriptions of the types of lessons, Skill Fluency, Mixed Practice, and Problem of the Day.

• The Teacher’s Guide supports whole group/partner discussion, ask/listen fors, common misconceptions and errors, etc. 

• Each lesson includes a table identifying the steps and actions for the teacher which helps in planning the lesson and is intended to be reviewed with a coach.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 6, Ratios and Proportional Relationships: Understanding and Applying Proportional Relationships to Percents, Lesson 6 include:

• “What do we want every student to take away or do as a result of this lesson? How will a teacher know if students have met this goal? Understand: Students understand that more than one expression can be used to determine the amount after the change. Students understand that percentages are additive and can be rewritten as the sum or difference of two percentages. Students understand that the change +/- the original amount is used to determine the amount after the change. Do: Students write two equivalent expressions that can be used to find the amount after the change given a percent increase or decrease and the whole or original value.”

• “Conjecture: Multiple expressions can be used to solve the same percent increase and decrease problem. Let’s form our conjecture for today. With your partner, come up with a conjecture about what we learned about writing expressions to represent and solve percent increase and decrease problems. Students might only specify a percent increase. If this is the case, move on to the TTC #1 (Test the Conjecture) which is a percent decrease problem and come back to revise the conjecture to include percent decrease.”

• “Frame - You have just formed our conjecture for today. A percent change will increase or decrease a number and we can use this understanding to write and solve different expressions that will have the same value. Post the Conjecture in a visible place for student reference.” 

• “What will we be able to do if our conjecture is true? What is the question asking us to do? How can we apply our conjecture to solve the problem? How can we represent this on a double number line? What is one expression that we can write? Why? What is another equivalent expression that we can write? Why? How can we prove that our conjecture worked? Why does this make sense? So far, does our conjecture hold up?” 

• “Anticipated Misconceptions and Errors: Students might not convert the percent change into a decimal when evaluating. Students might only find the change and not the amount after the change. Students might write the expression using addition instead of subtraction (and vice versa).”

Each lesson includes a “How” section that lists the key strategies of the lesson and delineates what “top quality” work should include. Examples from Unit 6, Ratios and Proportional Relationships: Understanding and Applying Proportional Relationships to Percents, Lesson 6 include:

• “Key Strategy: Annotate the problem for quantity, whole, and percent increase/decrease. Problem is represented using a double number line. One expression is written as the whole +/- the percentage increase/decrease of the whole. Another expression is written as the sum or difference of the whole and the percent of the whole. Expressions are evaluated to show equivalence (if needed).”

• “CFS (Criterion for Success) for top quality work: Problem is annotated for quantity, whole, and percent change. DNL created and labeled. Two equivalent expressions are written to find the amount after the change. Expressions are tested to be equivalent.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. The Unit Overview includes Appendix: Teacher Background Knowledge which provides “clear links, excerpts, and specific pages from the Common Core the Number System, 6-8 Progression related to the unit content.” Examples include:

• The Unit Overview Appendix also often includes an excerpt from an unknown source which provides a teacher with an understanding of grade-level standards progression. Unit 9, Appendix A: Teacher Background Knowledge 7.G, 2D and 3D Shares: Area, Surface Area and Volume, “Students will understand that the surface area of a three-dimensional figure is the sum of the areas of its surfaces and apply this reasoning to write equations to represent and determine the surface area of rectangular prisms, cubes and right triangular prisms. The surface area of a rectangular prism is the sum of the areas of the six rectangles in its net, which can be represented either as 2lw+2lh+2wh or 2(lw+lh+wh). The surface area of a right triangular prism is the sum of the areas of the two triangles and three rectangles in its net, which can be represented as (2)(\frac{1}{2}bh)+2(lw)+bl or bh+2lw+bl.”

Materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. Examples include:

• Unit 3 Overview, The Number System: Expressions and Rational Number Operations, Identify the Narrative, “Following Unit 3, students continue to develop their ability to think algebraically by applying what they learned about expressions in conjunction to understandings developed during previous grade levels to write, interpret and solve equations and inequalities (7.EE.4). Later, students learn to represent and solve problems involving proportional relationships (7.RP.A), percentages (7.RP.3) and geometric applications (7.G.1, 7.G.B) using algebraic thinking. In 8th grade, students continue to develop their ability to think algebraically as they apply their understanding of expressions, equations and number properties to work with radicals and integer exponents (8.EE.A), graphs (8.EE.B), linear equations and pairs of linear equations (8.EE.C) and functions (8F). Eighth graders utilize their skills working with expressions and equations to solve geometric and statistical problems as well. Fluency with number properties and operations provides students the boundaries within which they can operate while manipulating expressions so that the meaning of the expression is preserved. This is especially important as students apply the skills, they have acquired to increasingly complex functions and other equations throughout high school and college. The manipulation of such relationships allows students to determine critical values regardless of the form the relationship is presented in.”

• Unit 7 Overview, Statistics and Probability: Probability, Identify the Narrative, “Looking ahead, unit 7 sets students up for learning about statistics in unit 8. In particular, students will apply their understanding of probability to understand and apply the concept and skill of random sampling to collect data to answer a statistical question. Later, in HS, probability is particularly relevant because probability reinforces key understandings about ratio and proportion that are prerequisites for everything from rate of change, to rational equations, rational exponents, and trigonometry. Students will also build on their basic computations of probabilities in Geometry by examining more complex situations than they are exposed to in middle school. Once studying statistics, (in Algebra 1, Algebra 2, A2PC, as well as AP Statistics) students apply their learning about the likelihood of an event happening to concepts such as certainty and margin of error.”

• Unit 9 Overview, Geometry – Area, Surface Area and Volume, Identify the Narrative, “While the study of area, surface area and volume culminate primarily in 7th grade, students continue their study of volume in 8th grade as it applies to cylinders, cones and spheres. In High School, students’ understanding of these concepts is very important as students will use their understanding of volume, area, and surface area to solve application problems throughout Algebra 1 and 2. These applications are particularly prevalent when studying polynomials as the length of sides can be given using variable expressions. In Geometry, students more directly apply their understanding of these basic measurements, but at a new level of complexity. For example, a cone-shaped cup is filled to half its capacity with water. What is the height of the water?”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series. Examples include:

• Guide to Implementing AF Grade 7, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

• The Program Overview informs teachers “about how to ensure scholars have sufficient practice with all of the Common Core State Standards. Standards or parts thereof that are bolded are addressed within a lesson but with limited exposure. It is recommended that teachers supplement the lessons addressing these standards by using the AF Practice Workbooks to ensure mastery for all students. Recommendations for when to revisit these standards during Math Practice and Friday Cumulative Review are noted in the Practice section of each unit.”

• The Unit Overview includes a section called Identify Desired Results: Identify the Standards which lists the standards addressed within the unit and previously addressed standards that relate to the content of the unit.

• In the Unit Overview, the Identify The Narrative provides rationale about the unit connections to previous standards for each of the lessons. Future grade-level content is also identified.

• The Unit Overview provides a table listing Mathematical Practices connected to the lessons and identifies whether the MP is a major focus of the unit.

• At the beginning of each lesson, each standard is identified. 

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work. 

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series. Examples include:

In the Unit Overview, the Identify the Narrative section provides the teacher with information to unpack the learning progressions and make connections between key concepts. Lesson Support includes information about connections to previous lessons and identifies the important concepts within those lessons. Examples include:

• In the Unit Overview, How do the MPs live across lessons?, “Unit 8 emphasizes MP3 by pushing students to apply their understanding of the calculations and concepts of statistics in order to compare two populations in a meaningful way. In lesson 1, students begin the unit by making arguments about the need for random sampling, designing data collection that is valid, and the process of answering a statistical question. This underlying reasoning is the foundation upon which the rest of the unit relies. In lessons 6 and 7 students begin to apply their measures of center and variability to compare two populations and must critique the reasoning of others to ensure these comparisons are accurate.”

• Unit 10 Overview, connection to previous skills is identified. “Students wrap up the year with a geometry unit on angle relationships and triangle constructions. Prior to this unit, students have learned about the concept of angles as well as how to measure and construct angles with a protractor in 4th grade. Students studied properties of 2D shapes in 5th grade and revisited those properties in 6th grade through their study of area and surface area. In 7th grade, students revisited the topic of surface area and they also learned how to write and solve multi-step equations. These previously studied topics serve as the foundation needed to be successful in Unit 10.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program. Examples include:

• The Implementation Guide states, "Our program aims to see the mathematical practices come to life through the shifts (focus, coherence, rigor) called for by the standards. For students to engage daily with all 3 tenets, we structure our program into two main daily components: math lesson and math cumulative review. The math lessons are divided into three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson. On a given day students will be engaging in EITHER a conjecture-based, exercise-based lesson or less often an error analysis lesson. The math cumulative review component has three sub-components: skill fluency, mixed practice, and problem of the day. Three of the five school days students engage with all three sub-components of the math cumulative review. The last two days of the week have time reserved for lessons, reteach lessons, and assessments. See the diagram below followed by each category overview for more information.”

Research-based strategies are cited and described within the Program Overview, Guide to Implementing AF Math: Grade 5-8, Instructional Approach and Research Background and References. Examples of research-based strategies include:

• Concrete-Representational-Abstract Instructional Approach, Access Center: Improving Outcomes for All Students K-8, OESP, “Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.”

• Introduction to the Math Shifts, by Achieve the Core, 2013, “According to the National Council of Teachers of Mathematics, Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.”

• Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell by Common Core Tools, “According to the National Mathematics Advisory Panel (2008), explicit instruction includes ‘teachers providing clear models for solving a particular problem type using an array of examples,’ students receiving extensive practice, including many opportunities to think aloud or verbalize their strategies as they work, and students being provided with extensive feedback.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Program Overview, Guide to Implementing AF Math: Grade 7, Scope and Sequence Detail, Supplies List includes a breakdown of materials needed for each Achievement First Mathematics Program. Examples include:

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2Mind Plastic Two-Color Counters, plastic, set of 200.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2Mind Manipulite Dot Dice, Set of 72.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2Mind TI-30X-IIS Calculator (Set of 30 also available), Need enough for State Testing.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

Unit Assessments consistently and accurately identify grade-level content standards along with the mathematical practices within each unit. Examples from unit assessments include:

• Unit 3 Overview, Unit 3 Assessment: Expressions and Rational Number Operations, denotes the aligned grade-level standards and mathematical practices. Question 5, “Simplify the expression: (-0.2 + (-4x) – 5) – 2(1.5x – d) Answer:___.” (7.EE.1, MP6, MP7, MP8)

• Unit 5 Overview, Unit 5 Assessment: Understanding and Applying Proportional Relationships, denotes the aligned grade-level standards and mathematical practices. Question 4, “A new shade of neon dye is being designed for True Religion jeans. The mixture calls for 2 pints of blue dye and 6 pints of green dye. The designer is making a large batch of the dye mixture and pours in 5 pints of blue dye and 9 pints of green dye. Would the large batch of the dye mixture come out correctly? If not, would it be bluer or greener than what the designer has intended? Explain your reasoning.” (7.RP.2, MP3, MP4)

• Unit 8 Overview, Unit 8 Assessment: Statistics, denotes the aligned grade-level standards and mathematical practices. Question 5, “A researcher chose a random sample of registered voters in Kentsville. He found that 3 out of every 5 voters surveyed said that they would vote for Miguel Miller for mayor. If there are 800 eligible voters in Kentsville, predict how many of those voters will choose Miguel Miller for mayor. Show all your steps.” (7.SP.2, MP2, MP6)

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance. Examples include:

• Assessments include an informal Exit Ticket in each lesson and a formal Unit Assessment for every unit. 

• There is guidance, or “look-fors,” to teachers about what the student should be able to do on the assessments.

• All Unit Assessments include an answer key with exemplar student responses.

• The is a rubric for exit tickets that indicates, “You mastered the learning objective today; You are almost there; You need more practice and feedback.” 

Program Overview, Guide to Implementing AF Math: Grade 7, Differentiation, Unit-Level Errors, Misconceptions, and Response, “Every unit plan includes an ‘Evaluating and Responding to Student Learning Outcomes’ section after the post-unit assessment. The purpose of this section is to provide teachers with the most common 1-2 errors as observed on the questions related to each standard, the anticipated misconceptions associated with those errors, and a variety of possible responses that could be taken to address those misconceptions as outlined with possible critical thinking, strategic practice problems, or additional resources.” Examples include: 

• Unit 3 Overview, Unit 3 Assessment: Expressions and Rational Number Operations, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 7.EE.2 / 7.EE.3, “Suggested re-teach activities by question group: Q: 9-15: Show call the most common error and compare with an exemplar. For each given problem, students should be able to articulate the process for modeling a mathematical problem with an algebraic expression: (1) define the unknown variable, (2) identify the signs of the quantities, (3) identify operations and grouping symbols, and (4) solve. They should also know how to re-write (namely combining like terms, factoring, or using the distributive property) expressions to illuminate different information about the problem at hand. Lessons for possible re-teach focus: Lessons 5-7: focus on master’s/PhD level problems, and use the guiding questions above and ensure students get plenty of at bats to practice in order to gain fluency and application skills.”

• Unit 5 Overview, Unit 5 Assessment: Understanding Proportional Relationships, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 7.RP.1, “Suggested re-teach activities by question group: Q3-8: Show call the most common error and compare with an exemplar. Students should be able to articulate why the question at hand is asking them to find a unit rate, and then develop a method for computing the appropriate unit rate (including placing the units in the right order). There are multiple solution pathways given each unit rate problem (they can divide in the order they are looking–’per 1 unit’ is in the denominator OR they can set up an equivalent ratio statement and find the factor that gets one unit to 1 and apply it to the other given unit). You can ask the following guiding questions in your debrief: What is this question asking us to find? This S computed the unit rate.  How did he/she know to do so? How do we find the unit rate? Does the order of our units matter? Why? Is there a different way we can find this solution? Lessons for possible re-teach focus: Lessons 10; focus on iP #1-4, and use the guiding questions above and ensure students understand the solution pathway and why they are finding a unit rate to solve each problem.”

• Unit 9 Overview, Unit 9 Assessment: Area, Surface Area and Volume, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 7.G.6, “Suggested re-teach activities by question group: Q1-9, 11, 12: Show call the most common error and compare with an exemplar. For problems in context, students should not only be able to model the appropriate operations with a numerical/algebraic expression, but should be able to differentiate clearly that a problem involving surface area typically involves covering the outside whereas volume involves filling. Students will also need to be familiar with taking a fraction of a given area/volume as well as multiplying and dividing given rates when it comes to these problems as well. They will also need to appropriately model and solve for missing dimensions in order to find solutions. When debriefing, you can ask the following guiding questions: Whose work do we agree with in terms of how the shape was broken up?  Why did this allow us to find the dimensions we needed? Why does this algebraic expression match the given problem? What information/operations do we need to solve the problem? How do we know that this problem involves SA/Volume? Do you agree with the rest of this solution pathway? How can we check the reasonableness of the given solution? Lessons for possible re-teach focus: Lessons 1-5, 9-11; focus on master level IP, and use the guiding questions above when debriefing problems to ensure students are able to get more at bats solving area, surface area, and volume problems.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems. 

Assessments include opportunities for students to demonstrate the full intent of grade-level standards across the series. Examples include:

• The Unit 3 Assessment contributes to the full intent of 7.EE.3 (solve multi-step real-life and mathematical problems posed with positive and negative rational numbers). Item 13, “Nashia is in debt to her mother. Her debt can be represented as -$150. In order to pay off her debt, she got a job at Kennedy’s shoe store. Blue shoes cost $45.00 a pair and red shoes cost $50.00 a pair. Kennedy makes a commission that earns her \frac{1}{20} of the  purchase price for each pair of shoes sold. Over the course of the last couple weeks, Kenney sold 40 pairs of blue shoes. If she only sells red shoes from this point forward, how many pairs does she need to sell to be able to fully pay her mother back?” 

• Unit 6, Lesson 5, Exit Ticket, Problem 2 contributes to the full intent of 7.RP.3 (use proportional relationships to solve multistep ratio and percent problems). “The weight of a dolphin increased by 17% when it became pregnant. What is the approximate pregnant weight of the dolphin if it was 278 lbs. before it became pregnant? Round to the nearest tenth of a pound.” 

• The Unit 7 Assessment contributes to the full intent of 7.SP.7 (develop a probability model and use it to find probabilities of events). Item 3, “A bag of marbles has the following contents: 9 blue marbles, 4 red marbles, 5 white marbles, 2 black marbles. Determine the probabilities for each of the following events: a) Selecting a blue marble. b) Selecting a blue or red marble. c) Selecting a yellow marble. d) Selecting a blue, red, white, or black marble.”

Assessments include opportunities for students to demonstrate the full intent of grade-level practices across the series. Examples include:

• Unit 2 Assessment, Item 3, supports the full development of MP7 (look for and make use of structure). “Will the product of (2) and (-7) be positive or negative? Justify your answer with reference to applicable number properties.“

• Unit 4 Assessment, Item 9, supports the full development of MP2 (reason abstractly and quantitatively). "Consider the following mathematical statements: x + 9.25 = 20 and x  + 9.25 > 20. Are the two solutions the same or different? Justify your response below.”

• Unit 6 Assessment, Item 9, supports the full development of MP1 (make sense of problems and persevere in solving them). “Apple bank is offering 0.9% interest on savings accounts while Chase bank is offering 0.5% with an initial gift of a $500 deposit into new accounts. Silvia wants to deposit $24,000 into a new savings account and leave it there to earn interest for five years.  Which bank is offering a better investment opportunity? How much more money will she have in total after 5 years if she invests in the bank with the better deal?”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each unit overview. According to the Program Overview, Guide to Implementing AF Math: Grade 7, Differentiation, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning and not feel successful. Therefore, it is critical that strong curricular materials provide support for all student learners, but are created in a way to support students who have diagnosed disabilities. Our program has been designed to align to the elements identified by the Council for Learning Disabilities that should be used in successful curriculum and instruction: Specific and clear teacher models, Examples that are sequenced in level of difficulty, Scaffolding, Consistent Feedback, and Frequent opportunity for cumulative review. Unit Overviews and lesson level materials include guidance around working with students with disabilities, including daily suggested interventions in the Workshop Section of the daily lesson plan.” 

Examples of supports for special populations include: 

• Unit 3 Overview, The Number System: Expressions and Rational Number Operations, Differentiating for Learning Needs, “Previous Grade Content: Students have worked with expressions in Grade 6’s Unit 6 - Expressions and Equations, where students focus on writing, simplifying and evaluating expressions with integers. In this unit, students apply the distributive property to create equivalent expressions. The following lessons may be useful for differentiation of pre-skill content: Grade 6, Unit 6 - Lesson 2, 5, 11, 13.” Responding to Student Learning Outcomes, “See the Unit Assessment ‘Evaluating and Responding to Student Learning Outcomes’ at the end of the Overview for suggestions on uniti-level common errors, misconceptions, and suggestions on how to respond. These can be useful for supporting struggling learners proactively throughout the unit.” Student Grouping Suggestions, “Pre-Test: Use the 7th Grade, Unit 3 Pre-test and Key to identify student strengths and weaknesses when it comes to understanding equivalent expressions, combining like terms and the distributive property. Identify specific problems to sequence through cumulative review and create groupings of students for small group instruction during that period. Consider changing student seating so that students who struggled with the pre-test are seated next to students who had higher mastery for support throughout the unit or strategically group students who struggle together for teacher support or small group instruction. Exit Tickets: Closely analyze student mastery of the Exit Tickets for Lessons 1, 2, and 3 as these are critical for mastery of generating equivalent expressions with rational numbers. In these lessons, students master the skills of combining like terms and using the distributive property to create equivalent expressions. Students who have struggled with these should be prioritized for small group instruction and/or more support during independent practice prior to the end of the unit.”

• Unit 5 Overview, Ratios and Proportional Relationships: Understanding and Applying Proportional Relationships, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to gain mastery of graphing tables and points to prove relationships are proportional. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed for each proportional relationship along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example in Lesson 3 to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: - PP1, 2, IP 1, 2, 3. These problems would ensure students have had practice with calculating the constant of proportionality from tables to justify relationships proportional or not.”

• Unit 10 Overview, Geometry: Constructing with Angles, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to gain mastery of calculating various angle measurements using simple equations and key vocabulary. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example in Lesson 3 to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: - PP1, 2, IP 1, 2, 3, 4. These problems would ensure students have had practice with utilizing key angle vocabulary to write and solve simple equations representing angle relationships.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. According to the Program Overview, Guide to Implementing AF Math: Grade 7, Differentiation, Supporting Advanced Students, “Part of supporting all learners is ensuring that advanced students also have opportunities to learn and grow by engaging with the grade level content at higher levels of complexity. A problem-based approach is naturally differentiated as students choose the strategies they use to model and solve the problem. Teachers highlight particular strategies for the class, but they always affirm any strategy that works, regardless of its level of complexity. In a classroom implementing the Achievement First Mathematics program, students are expected to work with a variety of tools and strategies even as they work through the same set of problems; this allows advanced students to engage with the content at higher levels of complexity. Daily lessons resources (DLRs) also provide differentiated problems labeled by difficulty. Teachers should differentiate for student needs by assigning the most challenging problems to advanced students while allowing them to skip some of the simpler ones, so that they can engage with the same number of problems, but at the appropriate difficulty level.” Independent Practice in each lesson provides three levels of rigor in the lesson for student work: Bachelor, Master, and PhD work. Examples include:

• Unit 3, Lesson 6, Independent Practice Bachelor Level, “Every flavor of macaroon at Mac’s Macs costs d dollars each. If Jeremy buys 3 pistachio, 12 vanilla, and 7 red apple macaroons, what are two possible expressions for his total cost?”

• Unit 3, Lesson 6, Independent Practice Master Level, “Which expression does not represent b – 0.05b? a) 0.95b d) Multiply 0.95 by b d) Subtract 0.05 from b e) A decrease of 0.05.”

• Unit 3, Lesson 6, Independent Practice PhD Level, “At Yankees Stadium, hot dogs and cheese burgers both cost $8.50. Write two different expressions that could represent the total amount someone spends onh hot dogs and cburgers. Explain how you know that both expressions could be used to find the total amount spent.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Program Overview, Guide to Implementing AF Math: Grade 7, Differentiation, Supporting Multilingual and English Language Learners, “The Achievement First Mathematics Program appreciates the importance of creating a classroom environment in which Multilingual and English language learners (MLLs/ ELLs) can thrive socially, emotionally, and academically. MLLs/ ELLs have the double-task of learning mathematics while continuing to build their language mastery. Therefore, additional support and thoughtful curriculum is often needed to ensure their mastery and support in learning. Our materials are designed to help teachers recognize and serve the unique educational needs of MLLs/ ELLs while also celebrating the assets they bring to the learning environment, both culturally and linguistically. Our three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson all build on the four design principles for promoting mathematical language use and development in curriculum and instructions outlined by Stanford’s Graduate School of Education, Understanding Language/SCALE.” The series provides the following principles that promote mathematical language use and development: 

• “Design Principle 1: Support sense-making - Daily lesson resources (DLRs) are designed to promote student sense-making with an initial ‘Think About It’ task that engages students with a meaningful task upon which they can build connections. Students have time to read and understand the problem individually and the debrief of these tasks include clear definitions of new terminology and/or key ideas or conjectures…Additionally, teachers are provided with student-friendly vocabulary definitions for all new vocabulary terms in the unit plan that can support MLLs/ELLs further.”

• “Design Principle 2: Optimize output - Lessons are strategically built to focus on student thinking. Students engage in each new task individually, have opportunities to discuss with partners, and then analyze student work samples during a whole class debrief…All students benefit from the focus on the mathematical discourse and revising their own thinking, but this is especially true of MLLs/ELLs who will benefit from hearing other students thinking and reasoning on the concepts and/or different methods of solving.”

• “Design Principle 3: Cultivate conversation - A key element of all lesson types is student discussion. Daily lesson resources (DLRs) rely heavily on the use of individual think/write time, turn-and-talks with partners, and whole class discussion to answer key questions throughout the lesson script. The rationale for this is that all learners, but especially MLLs/ELLs benefit from multiple opportunities to engage with the content. Students that are building their mastery of the language may struggle more with following a whole-class discussion; however, having an opportunity to ask questions and discuss with a strategic partner beforehand can help deepen their understanding and empower them to engage further in the class discussion….”

• “Design Principle 4: Maximize linguistic and cognitive meta-awareness - The curriculum is strategically designed to build on previous lesson mastery. Students are given opportunities to discuss different methods to solve similar problems and/or how these concepts build on each other. The focus of the ‘Think About It’ portion of the Exercise-Based lesson is to help students build on their current understanding of mathematics in order to make a new key point for the day’s lesson. The entire focus of the Test the Conjecture lesson is for students to create their own conjecture about the new learning and then to test this by applying it to an additional problem(s). Students focus on building their own mathematical claims and conjectures and see mathematics as a subject that involves active participation of all learners. By ending each lesson type with this meta-awareness, all learners, but especially MLLs/ELLs benefit by building deeper connections.”

The series also provides Mathematical Language Routines in each unit. According to the Program Overview, Guide to Implementing AF Math: Grade 7, Differentiation, Supporting Multilingual and English Language Learners, “Beyond these design principles, our program outlines for teachers in every unit plan the most appropriate mathematical language routines (MLRs) to support language and content development of MLLs/ELLs with their learning within the specific unit.” Examples include:

• Unit 4 Overview, Expressions and Equations: Equations and Inequalities, Differentiating for Learning Needs, Supporting MLLs/ELLs, 

  • Vocabulary: “MLLs/ELLs should be provided with a student-friendly vocabulary handout throughout the unit that is either completed for them and/or that they add to each day. All terms included in the ‘Vocabulary’ section below should be included. This scaffold can be incredibly helpful for other learners to help them see a verbal and visual definition for each term. Each of the terms, definitions, and examples should be translated into the students preferred language using Google Translate or a translator (Spanish in the example provided).”

  • Sentence Frames: “MLLs/ELLs and all students can greatly benefit from specific guidance around sentence frames for standard justifications or explanation within the unit. For this unit, Lesson 7 focuses on building conceptual understanding for solving equations in the form p(x + q) = r by dividing both sides of the equation by p or applying the distributive property. Students must justify why both strategies produce solutions. Teachers can provide students with the following sentence frames to use throughout these problems: Justifying Different Strategies: ‘First I _________, then I _________.’ ‘Dividing first is easier because __________’ ‘Distributing first is easier because __________.’”

  • Language Development Routines: “Throughout the unit, teachers should focus on student discussion and use of critical thinking when analyzing student work samples. See the ‘Implementing Language Routines’ of the Implementation Guide for the course for further detail on how these routines live within all lessons. Within this unit, students should specifically focus on the following Mathematical Language Routines.

    • MLR1: Stronger and Clearer Each Time - Students will focus in ALL lessons on analyzing student work and revising their thinking either during the Think About It or Test the Conjecture portion of each lesson. 

    • MLR 3: Critique, Correct, and Clarify - In Lesson 4 and 13.2,  Students will analyze common misconceptions using student work that is not their own. During this lesson students will work independently and in pairs to identify and fix common errors in multiplying and dividing rational numbers.

    • MLR7: Compare and Connect - Throughout this unit, use this routine when students explain how they solved their equations and inequalities, specifically in Lessons 4, 6, and 7. Ask students, ‘What is the same and what is different?’ about their strategies. Draw students’ attention to the connection between the different approaches in finding the solutions to complex equations and inequalities. These exchanges strengthen students’ mathematical language use and reasoning based on ways to solve equations and inequalities that involve rational numbers.

    • MLR8: Discussion Supports - Students will focus in ALL lessons on class discussions to revise their thinking, different representations, and strategies during the Think About It, Interaction with New Material, or Test the Conjecture portion of each lesson.”

The materials reviewed for Achievement First Mathematics Grade 7 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are described as accurate representations of mathematical objects in the narrative of the Unit Overviews, and although there is little guidance for teachers or students about the use of manipulatives in the lessons, the use of manipulatives can be connected to written methods. Examples include:

• In Unit 7, Lesson 7, Interaction with New Material Example 1, students predict sample space of a probability experiment using manipulatives, “T(eacher) should have an envelope ready with the following color distribution to use as a check at the end (1 red, 2 purple, 2 green, 5  blue). Ms. Fleck conducted a probability experiment in a previous class where scholars picked a color paper out of an envelope and recorded the frequency in the table below. She doesn’t remember the amount of each color in the envelope but she does know that there are 10 pieces of paper. Determine the number of each color without opening the envelope.” 

• In Unit 10 Overview, “In both lessons 3 and 9, students must choose the appropriate tools to explain their reasoning for angle relationships and constructions with triangles. Students may use a variety of tools to do this, such as sketching and labeling a diagram, applying definitions (such as complementary angles or SAS), and/or using  construction tools such as a ruler or protractor.”