The instructional materials for Open Up Resources 6-8 Math, Grade 7 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding. Students access concepts from a number of perspectives and independently demonstrate conceptual understanding throughout the grade.

Cluster 7.EE.A addresses understanding how the properties of operations can be used to generate equivalent expressions. In Unit 6, students develop the conceptual understanding of equivalence needed to manipulate expressions, using the properties of operations in order to identify and generate equivalent expressions.

• Lessons 2 (Reasoning about Contexts with Tape Diagrams) through 5 (Reasoning About Equations and Tape Diagrams) introduce students to equations in the forms px + q = r and p(x + q) = r. Students develop an understanding of equations as they manipulate equal parts, different parts, and the total parts related; label terms in a model to represent a given context; and write their own contexts and develop the corresponding equation.

• In Lesson 7, students use “hanger diagrams” introduced in Grade 6 as a way to model and maintain equivalency in equations. Students use the hanger diagrams to reason about solving equations (7.EE.3 and 7.EE.4).

• In Lessons 13 through 15, students use number lines to find solutions to inequalities. For example, in Lesson 13 Activity 1, students represent solutions on number lines and in the next activity look for patterns. In the Lesson 14 Cool-Down, students use the given context to write and explain an appropriate equation, variables, and possible solutions (7.EE.4).

• In Lessons 20 through 22, students use multiple strategies including expanded form with variables and partitioned rectangle areas to apply various properties of operations, simplify expressions, and identify equivalent expressions (7.EE.1). In Lesson 20 Activity 2, students are prompted to “Replace each ? with an expression that will make the left side of the equation equivalent to the right side. (Set A: 6x+?=10x, 6x+?=2x, 6x+?=-10x, 6x+?=0, 6x+?=10).” Students then respond to questions: “Why didn't you combine x terms and numbers?” “How did you decide on the components of the missing term?” “Did you use the commutative property?” “Did you use the distributive property?” “What are some ways we can tell that 7x+2 is not equivalent to 9x?” “Someone is doubtful that 3b−8b is equivalent to -5b, but they do understand the distributive property. How could you convince them that these expressions are equivalent?” “What are some ways we could rearrange the terms in the expression -2x+6y−6x+15y and create an equivalent expression?” This activity engages students in both developing and articulating their understanding of equivalent expressions.

Cluster 7.NS.A, presented in Unit 5, addresses applying and extending previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers:

• Lesson 2 introduces students to addition and subtraction of rational numbers on the number line (7.NS.1). In the first Activity, students show understanding by drawing number lines, writing equations, and verbally explaining number line movement. A thermometer applet, which students can manipulate to represent the given situation, is included in the second Activity. Arrows are applied within the applet, and the thermometer context serves as a vertical number line representation on which students add integers.

• In Lesson 3, conceptual understanding is extended as students apply the information from given situations to visually model elevation change (7.NS.1).

• The Lesson 4 Introduction explains that students will use money as a way “to practice performing operations on signed values, but the emphasis is really on noticing that money can be represented with positive and negative values.” The Overview further presses conceptual understanding with the following explanation: “Any situation in which we use a negative number to represent a debt (for example), we could equally well just use a positive number and distinguish it by calling it a debt. The reason we use signed numbers in this context is that it allows us to represent a whole class of problems with the same expression. For example, if a person has $50 in the bank and writes a $20 check, we can represent the balance as 50−20. If they had written an $80 check, we can still write the balance as 50−80, as long as we have adopted the convention that negative numbers represent what the person owes the bank (and assuming the bank allows overdrafts).”

• Lessons 8 through 12 focus on multiplication and division (7.NS.2). Lesson 8 begins the cluster of lessons with exploring multiplication of a negative and a positive value in terms of speed in a certain direction over a number of seconds. Students use a number line and a table to establish a pattern. For example, one row in the table states “starting at zero” “left” “4 units per second” “6 seconds”. Students use a number line to come up with the equation. In Lesson 9, through explorations of contexts students determine the pattern for multiplying two negative values (7.NS.2). Building on the understandings developed in Lessons 8 and 9, students use the relationship between multiplication and division to develop rules for dividing signed numbers in Lesson 11.

The instructional materials for Open Up Resources 6-8 Math, Grade 7 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 7 expected procedural skills, particularly those related to rational numbers and solving equations.

Procedural skills and fluencies are built on conceptual understanding and the work students have accomplished with operations and equations from prior grades. Opportunities to formally practice developed procedures are found throughout practice problem sets that follow the units, including opportunities to use and practice emerging fluencies in the context of solving problems. According to the Design Principles within the Grade 7 Course Guide, “As the unit progresses, students are systematically introduced to representations, contexts, concepts, language, and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The practice problems give students ongoing practice, which also supports developing procedural proficiency.” Number Talks included in many Warm-Ups often revisit fluencies developed in earlier grades and specifically relate to the Activities found in the lessons. Additionally, students demonstrate procedural skill and fluency throughout the year in a variety of practice problems. Examples of lesson practice problems follow:

• In Unit 6 Lesson 10 Items 3 through 5, students solve equations such as “7(x + 2) = 91” (7.EE.4a).

• 7.NS.A is found in Unit 5. Students perform operations with signed numbers in Lessons 2 through 12 and compute all forms of rational numbers within expressions and equations in Lessons 13 and 17:
  • Lesson 5 introduces subtracting integers with number lines, the coordinate plane, tables, and the relationship between addition and subtraction of integers. Students then solve expressions with rational numbers. In Lesson 7, students apply their knowledge of operations with integers to real world applications and problems. Multiplying integers is introduced in Lesson 8 with additional opportunities to build procedural skill in Lessons 9 through 11.
  • Lesson 11 Items 1-4 provide practice with multiplication and division with rational numbers.
  • The Lesson 13 Warm-Up includes true/false reasoning with signed numbers. The first Activity, a Card Sort, states: “In this activity students continue to build fluency operating with signed numbers as they match different expressions that have the same value.” The cards include both integers and fractions.

• In Unit 6, procedural fluencies with grade-level operations are revisited in the Lesson 18 and 19 Warm-Ups in preparation for working with equations including rational numbers. In Lessons 18 through 22, students work with rational numbers and negative terms (7.EE.1) in ways that continue to build fluency when computing with signed numbers.

7.EE.1 is found in Unit 6 when students apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

• In Lesson 19 Activity 1, students factor and expand expressions with signed numbers in the given table to develop procedural fluency. The following Cool-Down Activity includes two more opportunities to expand and factor equations with rational values.

• In Lesson 20 Activity 1, students use properties of operations to justify equivalence of expressions. In Activity 2, students fill in missing terms to balance equations and are provided questions such as, “How did you decide on the missing components of the term?” to develop the procedures of combining like terms.

• The Lesson 22 Overview states: “In this lesson, students have an opportunity to demonstrate fluency in combining like terms and look for and make use of structure (MP7) to apply the distributive property in more sophisticated ways.”

The instructional materials for Open Up Resources 6-8 Math, Grade 7 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, both routine and non-routine, presented in a context in which the mathematics is applied.

Work with applications of mathematics occurs throughout the materials in ways that enhance the focus on major work and when standards call for application in real-world or mathematical contexts. The materials have multiple opportunities for students to work on this aspect of rigor in ways that become more complex over time and with routine and non-routine contexts. The Grade 7 Course Guide states: “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts. The first unit on geometry is an example of this.”

Cluster 7.RP.A addresses students using proportional relationships to solve real-world and mathematical problems and is found in Units 2, 4, 5, 7, and 8. Students work regularly with speed and water filling/draining contexts in increasingly complex ways but also use proportional reasoning skills in more complex and non-routine ways alongside work with surface area/volume and later with probability. For example:

• In Unit 2 Lesson 5, students are given a rate of water filling an object and find multiple equations that model the situation and identify and interpret the rates in context. The lesson introduction states: “Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and generalize from repeated calculations to arrive at an equation.”

• In Unit 2 Lesson 6, students work to determine the constant of proportionality in the context of concert ticket sales, recipes, and recycling. In Lesson 7, students make adjustments to quantities used in recipes, compare costs for various group sizes to enter a state park, and determine if several runners are moving at a constant pace. In the last context, students begin to reason about making determinations about proportionality in a discrete context (7.RP.2a) and their inability to establish proportionality given a continuous relationship when considering the limitations of intervals in the table.

• In Unit 4 Lesson 10, students apply percentage to taxes and tips. Activity 1 provides students with two tables giving the sales tax charges on the same items in two different cities. Students must complete the table, finding the tax rate in each city, and compare them. Activity 2 adds another city to the problem, and students realize through their calculations that the tax amount must be rounded.

• In Unit 4 Lesson 16, students sort actual newspaper clippings, decide if they are percents of increase or decrease, choose a clipping that is interesting to them, and “create a visual display that includes: a title that describes the situation, the news clipping, your diagram of the situation, the two questions you asked about the situation, the answers to each of your questions, [and] an explanation of how you calculated each answer.”

• In the Unit 5 Lesson 8 Warm-Up, three problem scenarios are given in which students apply understanding of proportional relationships in a constant speed context. Students first find the total distance a plane travels in prompt 1; the speed of a train in prompt 2; and the time a car traveled in prompt 3. This work prepares students for thinking about contexts involving negative numbers in context of velocity, time, and position.

• In Unit 7 Lesson 16, students apply understanding of proportional relationships to find surface area and volume and then find the total cost of constructing a ramp in a non-routine problem context. Activity 2 presents another non-routine problem involving surface area, volume, and using proportional relationships: “The daycare has two sandboxes that are both prisms with regular hexagons as their bases. The smaller sandbox has a base area of 1,146 square inches and is filled 10 inches deep with sand.” The following prompts accompany the given real-world scenario: “It took 14 bags of sand to fill the small sandbox to this depth. What volume of sand comes in one bag? (Round to the nearest whole cubic inch.); The daycare manager wants to add three more inches to the depth of the sand in the small sandbox. How many bags of sand will they need to buy?; The daycare manager also wants to add three more inches to the depth of the sand in the large sandbox. The base of the large sandbox is a scaled copy of the base of the small sandbox, with a scale factor of 1.5. How many bags of sand will they need to buy for the large sandbox?; A lawn and garden store is selling six bags of sand for $19.50. How much will they spend to buy all the new sand for both sandboxes?”

Standard 7.EE.B.3 addresses solving multi-step real-life and mathematical problems with rational numbers in various forms and is primarily explored in Unit 6. Examples follow:

• In Lesson 2, students encounter many routine problems that require algebraic thinking to solve: “Noah’s family bought some fruit bars to put in the gift bags. They bought one box each of four flavors: apple, strawberry, blueberry, and peach. The boxes all had the same number of bars. Noah wanted to taste the flavors and ate one bar from each box. There were 28 bars left for the gift bags.” Students are encouraged to use the familiar tape diagrams to establish a context for equations of the forms p(x+q)=r and px+q=r.

• In Lesson 12 Activity 1, students use the given tape diagram and sample student responses to make connections between the context (change in temperature over three days) and equations that model contexts involving percent increase. In Activity 2, students solve four multi-step real-world problems. In the Cool-Down, students write and solve an equation for this problem: “The track team is trying to reduce their time for a relay race. First, they reduce their time by 2.1 minutes. Then, they are able to reduce that time by 1/10. If their final time is 3.96 minutes, what was their beginning time? Show or explain your reasoning.”

The instructional materials for Open Up Resources 6-8 Math, Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

There is evidence that the curriculum addresses standards, when called for, with specific and separate aspects of rigor and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized.

Unit 6 lessons illustrate how the materials attend to each aspect of rigor independently and together. Examples of conceptual understanding in Unit 6 include:

• In Lesson 18 Activity 1, students use area models to develop conceptual understanding of how the distributive property works with both positive and negative terms. In the Lesson Synthesis, students develop a conceptual understanding on how the commutative property works with addition but not subtraction.

• Lessons 7 and 8 intentionally isolate the conceptual understanding related to solving equations in the form of px + q = r and p(x+q) = r by using hanger diagram models to visualize balance between the sides of an equality (7.EE.4a).

Examples of procedural skill in Unit 6 include:

• In Lesson 10, all Activities are focused on developing the procedural skill required in 7.EE.4 as students share their solution strategies. The Warm-Up involves an Algebra Talk where students have 30 seconds to think about methods of solving and solutions to five different problems in p(x+q) = r form. In Activity 1, students perform an error analysis of given students’ methods. In Activity 2, students solve five problems using two different methods and compare them. This task also prompts students to evaluate more inefficient methods and explain why they are not effective.

• In the Unit 6 Lesson 22 Overview, it states that “students have an opportunity to demonstrate fluency in combining like terms and look for and make use of structure (MP7) to apply the distributive property in more sophisticated ways.” In Activity 1, students match equivalent expressions which include negative terms with a partner.

Examples of application include:

• In Unit 5 Lesson 17, students apply their understanding of percent increase and decrease and work with rational numbers to calculate changes in stock value and total value of investment portfolios in non-routine stock market situations. This lesson is focused on the application of standards 7.NS.3 and 7.EE.3.

• In Unit 2 Lesson 15, students engage in the application of 7.RP.A as they work to answer whether baths or showers use more water. Students determine what information they would need to solve the problem, gather the data/information they need, and make assumptions using what they find.

All three aspects of rigor are balanced throughout the course, including the unit assessments. There are multiple lessons where two or all three of the aspects are connected. For example:

• In Unit 2 Lesson 14, students apply conceptual understanding of proportional relationships. Students are given lists of items (creatures, length units, time units, volume units, body parts, area units, etc.) and must make up a situation that shows a proportional relationship between two different things in the list, then describe a situation where two things are not proportional and create a table, graph, and equation that represent the proportional/nonproportional relationship.

• Unit 5 Lessons 1 and 2 build upon the conceptual development of negative numbers in Grade 6, including placing them on the number line, comparing and ordering them, and interpreting them in the contexts of temperature and elevation. Procedural skill and fluency are integrated within the conceptual and application practice when students draw diagrams to represent temperature changes, write equations to represent the context, and solve by adding and/or subtracting integers. In Lesson 3, students represent quantities on number lines, identify opposites, and add integers. In Lesson 4, students apply their knowledge of integers to real-world scenarios with banking. Students begin “to understand that when representing a debt with a negative number, the additive inverse tells how much money is needed to pay off the debt.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All eight MPs are clearly identified throughout the materials. The Math Practices are initially identified in the Teacher Guide under the narrative descriptions of each unit within the Course Information. For example:

• In Unit 4 Proportional Relationships and Percentages, the narrative states, “In small groups, students identify important quantities in a situation described in a news item, use diagrams to map the relationship of of the quantities, and reason mathematically to draw conclusions (MP4).”

• In Unit 7 Angles, Triangles and Prisms, an excerpt from the Teacher Guide states, “[Students] understand and use the formula for the volume of a right rectangular prism, and solve problems involving area, surface area, and volume (MP1, MP4). Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically (MP5).”

Within a lesson, the MPs are identified within the teacher narratives accompanying the lesson in general or before each of the activities. Lesson narratives often highlight when a Math Practice is particularly important for a concept or when a task may exemplify the identified Practice. For example:

• MP8: In Unit 2 Lesson 4, the introduction states, “In this lesson, students build on their work with tables and represent proportional relationships using equations of the form y=kx. The activities revisit contexts from the previous two lessons, presenting values in tables and focusing on the idea that for each table, there is a number k, so that all values in the table satisfy the equation y=kx. By expressing the regularity of repeated calculations of values in the table with the equations, students are engaging in MP8.”

• MP7: The Unit 3 Lesson 4 introductory narrative connects the application of the circumference formula to two math practices, “Students think strategically about how to decompose and recompose complex shapes (MP7) and need to choose an appropriate level of precision for π and for their final calculations (MP6).”

• MP3: The Unit 8 Lesson 5 narrative accompanying the first Activity states, “Students have a chance to construct arguments (MP3) about why probability estimates based on carrying out the experiment many times might differ from the expected probability.”

The MPs are used to enrich the mathematical content and are not treated separately from the content in stand-alone lessons. MPs are are discussed within narratives as pertaining to the learning target or specific task at hand. The narratives are used to support deepening a teacher’s understanding of the standard itself as the teacher is provided direction regarding how the content is connected to the MP. For example:

• MP8: In Unit 5 Lesson 11, the introduction to the first Activity states, “The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).”

• MP1: In the first Activity of Unit 2 Lesson 7, students use a table to explore the cost of parking and admission per person. “These diagrams may be helpful in illustrating to them that their resulting prices are including more than one vehicle. This gives them an opportunity to make sense of problems and persevere in solving them (MP1).”

The MPs are not identified in the student materials, however, they are highlighted in the Teacher Edition in the narrative provided with each Activity. For example, in the first Activity of Unit 2 Lesson 7, the student facing prompt related to MP1 (see previous bullet for teacher facing information) includes, “How might you determine the entrance cost for a bus with 50 people? Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the unit and lesson narratives, as appropriate, when they relate to the overall work. They are also explained within individual activities, when necessary. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP1 Make sense of problems and persevere in solving them.

• In Unit 2 Lesson 2, students build on their understanding of scale and trying to find the scale factor. The narrative states, “Students grapple with finding missing values for ratios of whole numbers presented in a table where identifying a usable scale factor.” Students must make sense of new terms and apply them to their learning specifically the constant of proportionality and proportional relationship.

• The Unit 4 Lesson 2 Introduction highlights MP1: “The Mona Lisa task has more than one reasonable answer, and students must make sense of the situation in order to choose one.”

MP2 Reason abstractly and quantitatively.

• In the Unit 2 Lesson 3 second Activity, students make connections between constant speed and proportional relationships, with special attention to the constant of proportionality. Students “need opportunities to make the connection to proportional relationships; students who successfully make this connection are reasoning abstractly about contexts with constant speed.”

• In Unit 4 Lesson 2, students are given the following optional problem: “In real life, the Mona Lisa measures 2 ½ feet by 1 ¾ feet. A company that makes office supplies wants to print a scaled copy of the Mona Lisa on the cover of a notebook that measures 11 inches by 9 inches. Students are asked, 1. What size should they use for the scaled copy of the Mona Lisa on the notebook cover? 2. What is the scale factor from the real painting to its copy on the notebook cover? 3. Discuss your thinking with your partner. Did you use the same scale factor? If not, is one more reasonable than the other?” An applet is provided for students to experiment and understand the problem by changing the size of the Mona Lisa (abstract). The applet will display the new dimensions of the picture, but the information given for the picture cannot be scaled to the exact size of the notebook. Further investigation with equivalent ratios will lead students to one of the many possible dimensions and scale factors appropriate for the picture (quantitative).

MP4 Model with mathematics.

• In Unit 1 Lesson 9, students apply the mathematics they know to solve problems arising in everyday life. They create their own scale drawing of a floor plan, two different scale drawings of the state of Utah, noticing how the scale impacts the drawing, and make a scale drawing of a swimming pool.

• In Unit 4 Lesson 16, students work in groups to collect news clippings that mention percentages and sort them according to whether they are about percent increase or percent decrease, formulate questions about them, and then share their questions with other groups in a gallery walk. The purpose is for students to apply percentages in a real-world context.

MP5 Use appropriate tools strategically.

• In the Unit 1 Lesson 3 optional Activity, students use a digital platform to create scaled copies of an original figure by increasing or shrinking the dimensions with the use of a fractional scale factor. In the Activity, “Students continue to work with scaled copies of simple geometric figures, this time on a grid. When trying to scale non-horizontal and non-vertical segments, students may think of using tracing paper or a ruler to measure lengths and a protractor to measure angles.”

• Throughout Units 1 and 3, the lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor, and an index card to use as a straightedge or to mark right angles. Providing students with these toolkits gives opportunities for students to select appropriate tools and use them strategically to solve problems.

MP7 Look for and make use of structure.

• In the Unit 1 Lesson 2 Warm-Up Number Talk, students review multiplication strategies and develop the idea that multiplying by a unit fraction is the same as dividing by its whole number reciprocal. Students find 7.2 x 1/9 mentally. Students are guided to think about 729 and then consider what occurs to the decimal. Students are encouraged to use the structure of base ten numbers and the properties of operations.

• In Unit 4 Lesson 4, students work with the distributive property. It is noted in the lesson that when students look for opportunities to use the distributive property to write equations in a simpler way, they are looking for and making use of structure.

MP8 Look for and express regularity in repeated reasoning.

• In Unit 1 Lesson 2, students continue their work with corresponding parts to develop the understanding and use of scale factors. The narrative states, “Students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy - hence the scale factor.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to both construct viable arguments and/or analyze the arguments of others. Students are consistently asked to explain their reasoning and compare their strategies for solving in small group and whole class settings.

• In the Unit 5 Lesson 3 Optional Activity 1, students work with a partner to analyze a number line using variable expressions in order to compare two expressions with an inequality or an equal sign. They must also give an explanation to support their answer. For example, “-a ___ -b” based on their given locations on the number line.

• In the Unit 5 Lesson 5 first Activity, students find the length of the missing arrow on a number line and analyze two equations (both are correct) written to represent the situation. Students then write equations in the same form as the original two and explain their solution.

• In Unit 4 Lesson 16, students work in groups to collect news clippings that mention percentages and sort them according to whether they reflect percent increase or percent decrease. They formulate questions about them and then share their questions with other groups in a gallery walk. During the gallery walk, students use sticky notes to ask questions about the information presented on each poster.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and/or analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate, explaining their reasoning to each other.

• The Unit 2 Lesson 1 Warm-Up provides teacher guidance as students complete a double number line and invent a situation. Teachers are instructed, “As students discuss their answers with their partner, select students to share their answers during the whole-class discussion...Invite selected students to explain how they reasoned about possible labels for each of the number lines and the units of each. After each student shares, invite others to agree, disagree, or question the reasonableness of the number line descriptions.”

• The Unit 5 Lesson 6 Warm-Up Activity Synthesis provides teachers with questions to encourage constructing viable arguments. This strategy is used repeatedly throughout the series. “To involve more students in the conversation, use some of the following questions: Who can restate ___’s reasoning in a different way? Did anyone find the value of n the same way, but would explain it differently? Did anyone find the value of n in a different way? Does anyone want to add on to _____’s strategy? Do you agree or disagree? Why?”

• In Unit 2 Lesson 4 Number Talk states, “This Number Talk encourages students to think about the numbers in division problems and how they can use the result of one division problem to find the answer to a similar problem with a different, but related, divisor...Each problem is chosen to elicit a slightly different reasoning, so, as students explain their strategies, ask how the factors impacted their product.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations that the materials attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the teacher materials, the Grade 7 Glossary is located in the Teacher Guide within the Course Information section. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples. In the student materials, the Grade 7 Glossary is accessible by a tab within each Unit or in the bottom margin of each lesson page. Lesson-specific vocabulary can be found in bold when used within the lesson and at the bottom of each lesson page with a drop-down accessible definition with examples.

• Both the unit and the lesson narratives contain specific guidance for the teacher as to best methods to support students to communicate mathematically. Within the lesson narratives, new terms are in bold print and explained as related to the context of the material.

• Unit 4 Lesson 11 introduces students to a variety of percentage contexts: markups, markdowns, tax, tip, and commission. Examples involving these contexts are the focus for several lessons. In the Optional Activity provided in this lesson, students are paired and given the task of completing a “card sort.” Taking turns, the students match the term with the percentage scenario and must explain their reasoning for doing so. If the other student disagrees, they must explain why, and the pair must work to an agreement. The focus is on explaining the definition of these terms and relating them to specific scenarios.

• Unit 3 builds on students' understanding of a circle. It is a concept 7th grade students are familiar with; however, in Lesson 2 the concept is developed to include the formal definition of a circle (the set of points that are equally distant from the center, enclosing a circular region). Also in this lesson, students develop the idea that the size of a circle can be measured by its diameter, radius, circumference, or the enclosed area, depending on the context.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when the students are solving problems to learn and when they are applying their skills to build mastery.

Lessons include Warm-Up, Activities, and an Activity Synthesis. Practice Problems are in a separate section of the instructional materials, distinguishing between problems students complete and exercises in the lessons. Warm-Ups serve to either connect prior learning or prime students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Activity Synthesis, students have opportunities to build on their understanding of the new concept. Each activity lesson ends with a ""Cool-Down in which students have opportunities to apply what they have learned from the activities in the lesson and either provide preliminary practice or an introduction to skills they may need in the next lesson.

Practice problems are consistently found in the “Practice Problem” sets that accompany each lesson. These sets of problems include problems that support students in developing mastery of the current lesson and unit concepts, in addition to review of material from previous units. When Practice Problems contain content from previous lessons, students apply their skills and understandings in different ways that deepen understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for not being haphazard; exercises are given in intentional sequences.

Overall, clusters of lessons within units and activities within lessons are intentionally sequenced so students develop understanding leading to content mastery. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skill and fluency, and engage with multiple activities that are sequenced from concrete to abstract or increase in complexity. Lessons close with a “Cool-Down” which is typically 1-2 activities aligned to the daily lesson objective. Unit sequences consistently follow the progressions outlined in the CCSSM Standards to support students' conceptual and skill development.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a plethora of ways. Students not only produce answers and solutions within Activities and Practice Problems, but also in class, group and partner discussions. Students have opportunities to construct viable arguments and critique the reasoning of their peers in the instructional materials. Students use a digital platform (applets) and paper-pencil to conduct and present their work. Materials consistently call for student solutions that represent the language and intent of the standards. Students use representations such as tables, number lines, double-number lines, tape diagrams, and graphs (MP4), as well as strategically choose tools to complete their work (MP5). Lesson activities and tasks are varied within and across lessons.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The series includes a variety of virtual manipulatives and integrates hands-on activities that allow the use of physical manipulatives. For example:

• Manipulatives and other mathematical representations are consistently aligned to the expectations and concepts in the standards. The majority of manipulatives used are commonly accessible measurement and geometry tools.
• The curriculum also provides digital applets for manipulating geometric shapes, such as GeoGebra applets, tailored to the lesson content and tasks. When physical, pictorial, or virtual manipulatives are used, they are aligned to the mathematical concepts they represent. In Unit 6 Lesson 7, hanger diagrams are used to represent and support the conceptual development of balance as it relates to equality in the virtual applet practice.
• Examples of manipulatives for Grade 7 include:
  • Tangram kits (or digital Tangram applet)
  • Geometry toolkits containing tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
  • GeoGebra applets are used for both investigating the characteristics of shapes and area/perimeter as well as exploring coordinate and isometric grids.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each section of each lesson contains an opening and closing narrative for the teacher. Included in these narratives are the objectives of the lesson, as well as suggested questions for discussion and guiding questions designed to increase classroom discourse and ensure understanding of the concepts. For example, in Unit 2 Lesson 3, the following questions are included: “Which quantities are in a proportional relationship? How do you know?” The narratives and the questions for discussion support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectations for containing a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

• Each lesson opens with a table including Learning Goals written in both teacher and student language, learning targets written in student language, a list of Print-Formatted Word/PDF documents that can be downloaded, CCSSM Standards that are either being “built upon” or “addressed” for the lesson, and any instructional routines to be implemented. Within the technology, there are expandable links to standards and instructional routines.

• Lessons include detailed guidance for teachers for the Warm-Up, Activities and the Lesson Synthesis.

• Each lesson activity contains an overview and launch narrative, guidance for teachers and student facing materials, anticipated misconceptions, “Are you Ready for More?” and an activity synthesis. Included within these narratives are guiding questions and additional supports for students.

• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. A "Launch" section follows which explains how to set up the activity and what to tell students. After the activity is complete there is often an “Anticipated Misconceptions” section, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misunderstandings.

• The materials are available in both print and digital forms. The digital format has an embed GeoGebra applet. Guidance is provided to both the teacher and the student on how to manipulate the applet. For example, in Unit 7 Lesson 6, both teachers and students are provided with these directions on how to build a polygon: “1. Use the segments in the applet to build several polygons, including at least one triangle and one quadrilateral. 2. After you finish building several polygons, select one triangle and one quadrilateral that you have made. a.) Measure all the angles in the two shapes you selected. Note: select points in order counterclockwise, like a protractor. b.) Using these measurements along with the side lengths as marked, draw your triangle and quadrilateral as accurately as possible on separate paper.”

The instructional materials reviewed for the Open Up Resources 6-8 Math, Grade 7 meet expectations for the teacher’ edition containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.

The narratives provided for each unit provide information on the mathematical connections of the concepts being taught. Previous and future grade levels are also referenced to show the progression of the mathematics over time. Important vocabulary is included when it relates to the “big picture” of the unit.

Lesson narratives provide specific information of the mathematical content within the lesson and are presented in adult language. These narratives contextualize the mathematics of the lesson to build teacher understanding, as well as guidance on what to expect from students and important vocabulary.

The Course Information and Scope and Sequence, Unit 2: Introducing Proportional Relationships states, “A unit rate is the numerical part of a rate per 1 unit, e.g., the 6 in 6 miles per hour. The fractions a/b and b/a are never called ratios. The fractions a/b and b/a are identified as “unit rates” for the ratio a:b. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to a to b, a:b, and a/b as “ratios.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Teacher Guide fully explains how mathematical concepts are built from previous grade-level and lesson material. For example, the Unit 5 Overview states the following regarding rational number arithmetic: “In Grade 6, students learned that the rational numbers comprise positive and negative fractions.”

There are limited explanations given for how the grade-level concepts fit into future grade-level work. For example, the Unit 6 Overview concludes with a note that states: “In these materials an expression is built from numbers, variables, operation symbols (+,-,*, ÷), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit. Students work with integer exponents in Grade 8 and non-integer exponents in high school.)”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels.

• Prior grade-level standards are indicated in the instructional materials. The lesson Warm-Up is designed to engage students thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills.

• Prior knowledge is gathered about students through the pre-unit assessments. In these assessments, prerequisite skills necessary for understanding the topics in the unit are assessed. Commentary for each question as to why the question is relevant to the topics in the unit, and exactly which standards are assessed is provided for the teacher. For example, the Unit 5 Pre-Unit Assessment Problem 6 states: “Graphing in the coordinate plane requires a different kind of visual interpretation of signed numbers: right or left as well as up or down. Students learned to graph signed numbers on the coordinate plane in Grade 6. 6.NS.C.8 6.NS.C.6.b”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions.

• Lesson Activities include “Anticipated Misconceptions” that identify where students may make a mistake or struggle. There is a rationale that explains why the mistake could have been made, suggestions for teachers to make instructional adjustments for students, as well as steps teachers can take to help clear up the misconceptions. For example, in Unit 2 Lesson 6, the Anticipated Misconception section gives the following guidance: “If students have trouble getting started, encourage them to create representations of the relationships, like a diagram or a table. If they are still stuck, suggest that they first find the weight and dollar value of one can.”

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The lesson structure consisting of a Warm-Up, Activities, Lesson Synthesis, and Cool-Down provide students with opportunities to connect prior knowledge to new learning, engage with content, and synthesize their learning. Throughout the lesson, students have opportunities to work independently, with partners and in groups where review, practice, and feedback are embedded into the instructional routine. In addition, Practice problems for each lesson activity reinforce learning concepts and skills and enable them to engage with the content and receive timely feedback. In addition, discussion prompts in the Teacher Guide provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for assessments clearly denoting which standards are being emphasized.

Assessments are located on a separate tab at the top of the grade-level page and can be accessed at any time. For each unit there is a Pre-Unit Assessment and an End-Unit Assessment. Assessments begin with guidance for teachers on each problem followed by the student facing problem, solution(s), and the standard targeted. Unit 6 also includes a Mid-Unit Assessment.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson is intentionally designed with a Warm-Up that reviews prior knowledge and/or prepares all students for the following activities. The Cool-Down following lesson activities solidifies the concepts of the lesson.

• Within a lesson, narratives provide explicit instructional supports for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis sections. This information assists a teacher in making the content accessible to all learners.

• Lesson narratives often include guidance on where to focus questions in Activities or in the Lesson Synthesis portions.

• Optional activities are often included that can be used for additional practice or support before moving on to the next activity or lesson.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for providing teachers with strategies for meeting the needs of a range of learners.

The lesson structure: Warm-Up, Activities, Lesson Synthesis, and Cool-Down all include guidance for the teacher on the mathematics of the lesson, possible misconceptions, and specific strategies to address the needs of a range of learners. Embedded supports include:

• Mathematical Language Routines to support a range of learners to be successful are provided for the teacher throughout lessons to maximize output and cultivate conversation. For example:
  • MLR1: Stronger and Clearer Each Time, in which “students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response.
  • MLR4, Information Gap, which “allows teachers to facilitate meaningful interactions by giving partners or team members different pieces of necessary information that must be used together to solve a problem or play a game...[S]tudents need to orally (and/or visually) share their ideas and information in order to bridge the gap.”
  • MLR6, Three Reads, in order to ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented,… and [support] negotiating information in a text with a partner in mathematical conversation.”
• Sidebar text features appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These text-boxes call out specific needs addressed in a recommended strategy that are relevant to the given task and include supports for Conceptual Processing, Expressive & Receptive Language, Visual-Spatial Processing, Executive Functioning, Memory, Social-Emotional Functioning, and Fine-motor Skills. For each support there are multiple strategies teachers can employ; for example, Conceptual Processing includes strategies to Eliminate Barriers, Processing Time, Peer Tutors, Assistive Technology, Visual Aids, Graphic Organizers, and Brain Breaks.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

The problem-based curriculum design engages students with rigorous tasks multiple times each lesson. The Warm-Up, Activities, and Cool-Down all provide opportunity for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” sections as well as “Which One Doesn’t Belong.” The lesson and task narratives provided for teachers offer possible solution paths and presentation strategies from various levels. For example:

• In Unit 2 Lesson 2, students are involved in a “Notice and Wonder” activity involving a table that shows the paper towels a store receives when they order different amounts of cases. The teacher is provided different examples of noticings and wonderings students may have with guidance on how to focus the conversation on the relationship between the cases ordered and paper towels received if it does not come from student responses.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation that the materials include support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The ELL Design is highlighted in the Teacher Guide and embodies the Understanding Language/SCALE Framework from the Stanford University Graduate School of Education and consists of four principles: Support Sense-Making, Optimize Outputs, Cultivate Conversation, and Maximize Meta-Awareness. In addition, there are eight Mathematical Language Routines (MLR) that were included “because they are the most effective and practical for simultaneously learning mathematical practices, content, and language.” "A Mathematical Language Routine refers to a structured but adaptable format for amplifying, accessing, and developing students’ language.”

In addition, “ELL Enhanced Lessons” are identified in the Unit Overview. These lessons highlight specific strategies for students who have a language barrier which affects their ability to participate in a given task. Throughout lessons, the use of one of a variety of instructional routines designed to assist students in developing full understanding of math concepts and terminology. These Mathematical Language Routines include:

• MLR2, Collect and Display, in which “the teacher listens for, and scribes, the student output using written words, diagrams and pictures; this collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use.”
• MLR5, Co-Craft Questions and Problems, which “[allows] students to get inside of a context before feeling pressure to produce answers, and to create space for students to produce the language of mathematical questions themselves.”
• MLR7, Compare and Connect, which “[fosters] students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language.”

In addition, lesson narratives include strategies designed to assist other special populations of students in completing specific tasks. Examples of these supports for students with disabilities include:

• Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.
• Conceptual Processing: Eliminate Barriers. Assist students in seeing the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
• Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.
• Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.
• Memory: Processing Time. Provide students with a number line that includes rational numbers.
• Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth.

All students complete the same lessons and activities; however, there are some optional lessons and activities that a teacher may choose to implement with students. In addition, Unit 9 “Putting It All Together” is an optional unit. Lessons in this unit tend to be multi-day, complex applications of the mathematics covered over the year.

“Are you ready for more?” is included in some some lessons to provide students additional interactions with the key concepts of the lesson. Some of these tasks would be considered investigations at greater depth, while others are additional practice.

It should be noted that there is no clear guidance for the teacher on how to specifically engage advanced students in going deeper.

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 7 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics.

• The lessons contain a variety of tasks that interest students of various demographic and personal characteristics. All names and wording are chosen with diversity in mind, and the materials do not contain gender biases.

• The Grade 7 materials include a set number of names used throughout the problems and examples (e.g., Elena, Tyler, Lin, Noah, Diego, Kiran, Mia, Priya, Han, Jada, Andre, Clare). These names are presented repeatedly and in a way that does not appear to stereotype characters by gender, race, or ethnicity.

• Characters are often presented in pairs with different solution strategies. There does not appear to be a pattern in one character using more/less sophisticated strategies.

• When multiple characters are involved in a scenario they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias. For example, in Unit 2 Lesson 7 Activity 7.3, Han and Clare are running laps and are timed by their coach. The times in the table do not suggest any gendered stereotypes around athletic ability.