The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Examples of problems and questions that develop conceptual understanding across the grade level include:

• In 7.RP.A.2a Hot Sauce!, students investigate heat ratings and discover that values must increase at the same rate, and ratios must be equivalent to each other in order to form a graph that is a straight line through the origin. 

• In 7.NS.A.1b Space Selfie, the Teacher Instruction includes, “What was the Trackometer reading right before the ship shut down?; In which direction were they headed when the ship shut down?; Should the distance from home be more or less than 50 parsecs?; What does the number 50 represent?; What does the number -32 represent?; What is 50 + -32?; How far are they from home?; How does a number line help you calculate this answer?”

• In 7.RP.A.2b Coffee Caravan, the Teacher Instruction includes, “Let’s take a deeper look at the constant of proportionality and how it is represented in various representations of proportional relationships. Let’s start with a verbal description of a proportional relationship. In a cookie recipe, for every 2 eggs there are 3 cups of flour. We could make a ratio table to show this relationship. We start with what we know, and then we can fill the rest in using the given proportion. For every 2 eggs, there are 3 cups of flour, which tells us, then, that for every 1 egg there will be $$1\frac{1}{2}$$ (or $$\frac{3}{2}$$) cups of flour.”

• In 7.EE.A.2a Taxing Problem, students rewrite equations and expressions in a variety of ways and decide between two sides of an argument. Students watch a video and try to determine, “Which dude is right?” about the cost of a bill. One dude argues, “It’s 0.085 times the bill, plus the bill” and the other says, “No dude, it’s 1.085 times the bill.” Teacher Instruction also provides other examples including calculating the cost of something at a discount.

Examples where students independently demonstrate conceptual understanding throughout the grade include:

• In 7.NS.A.2a Reverse Meditation, Practice Printable, Question 1 states, “Use a pattern to fill in each blank, and then explain the pattern.” In Part A, students create a table from 4 to -4, and multiply by 4. They should see that the products are decreasing by 4 each time, leading to the conclusion that a negative times a positive yields a negative product. In Part B, they do the same except multiply by -4 leading to a negative times a negative yields a positive product. 

• In 7.RP.A.2a Hot Sauce!, Practice Printable, Question 3 provides information about the cost of a gym membership at 2 gyms and students determine “For which company is the total cost proportional to the number of months? How do you know?”

• In 7.RP.A.2d Doggy Diet, Practice Printable, Question 3 states, “Plot and label the following points on the graph: a) $1.25 will buy 5 pencils.; b) 0 pencils cost $0.00.; c) The unit rate is $0.25 per pencil.; d) 8 pencils for $2.00.; Write three other points that could be on this graph if it were extended.” The graph shows the relationship between the number of pencils bought and the cost, in dollars, of the pencils.

• In 7.G.A.1 Build a Better Box, Practice Printable, Question 1 states, “Determine if each given scale factor would ENLARGE or REDUCE the size of the figure. a) 45%; b) $$\frac{6}{5}$$; c) 1.5; d) $$\frac{1}{3}$$; e) 110%; F) 0.8.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for attending to the standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level in the Math Simulator, examples in Teacher Instruction, Cluster Intensives, domain specific Test Trainer Pro and the Clicker Quiz. Examples include:

• In 7.NS.A.1d Bad Accounting, Teacher Instruction, the teacher prompt states, “Let’s rewrite the expression by replacing each subtraction with adding the additive inverse,” and “Now we can use the commutative property of addition to rearrange the terms by grouping positives and then negatives. We can then simplify.”

• In 7.EE.A.1 Mathmalian Logic, Teacher Instruction, the teacher prompt states, “Remember the order of operations when simplifying expressions.” The teacher works through three examples; the first example asks if two expressions are equivalent “$$3xy+4y-2x+8x-2xy-6y$$ and $$y(x+4)-6(y-x)$$”, the second involves simplifying an expression with fractional coefficients, and the third involves subtracting one expression from another. The Teacher’s Guide further prompts the teacher to discuss which properties might be used in each step, and walks though reordering, grouping, and combining like terms using the given example. In the Simulation Trainer, students are given an image of a large amount of land with the width being an integer, and the length divided into smaller lengths and labeled with variables. Students create two expressions that represent the total area.

• In 7.EE.B.4a Pen Perimeter, Teacher Instruction, the teacher discusses a real-world problem in which a jeweler makes a flat rate plus an additional $10 per sale. The teacher reasons through solving the problem.  Then he/she writes an equation and says, “We can solve the equation using inverse operations. We begin by subtracting the constant from both sides so we can isolate the variable.”

Examples of students independently demonstrating procedural skills and fluencies include:

• In 7.NS.A.1d Bad Accounting, the Practice Printable contains six expressions in Question 1, “Evaluate each expression. Indicate the properties of operations where appropriate, “ such as “1b.) $$22-8+(-3)+10$$” and “1d.) $$11.6-(-12.4)+15.3-9$$.”

• In 7.EE.A.1 Mathmalian Logic, Practice Printable, Question 1 states, “Simplify each expression. Combine all like terms when possible,” and contains a table of five complex expressions. Question 2, “For which value of $$m$$ would Expression 2 be equivalent to Expression 1?” A table with five pairs of expressions is provided. 

• In 7.EE.B.4a Pen Perimeter, Practice Printable, Question 7 states, “The sum of a number and 9 is multiplied by -2. The result is -8. What is the unknown number?”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine problems, presented in a context in which mathematics is applied. 

Examples of students engaging in routine application of skills and knowledge include:

• In 7.NS.A.1b Space Selfie, Practice Printable, Question 1 states, “The temperature at 6 a.m. was $$38\degree$$F. Throughout the day until 10 p.m., the temperature rose $$26\degree$$F and then fell $$23\degree$$F. What was the temperature at 10 p.m.?”

• In 7.NS.A.3 Chocolate Certified, an example is “Mark and his three friends went to the movies, where each ticket cost $11.25. They decided to share two large popcorns, which cost $4.25 each, and they each got a small soda for $3.25 each. Tax was 7.5% of the total. What was the total amount that they spent?”

• In 7.EE.B.3 Hay Talk, Practice Printable, Question 1 states, “After receiving the given raise at work, who will make the most money per hour? Malala $9.90 per hour,  6% raise; Aaliyah $9.75 per hour,  7% raise;  Phan $10 per hour, 4% raise.”

Examples of students engaging in non-routine application of skills and knowledge include:

• In 7.EE.B.4b The Fur Trader, the lesson narrative states, “It is important to not only know how to solve an inequality, but to also interpret an appropriate solution for the given context. In The Fur Trader, Professor Picklebottom decides to trade furs so he can make enough to survive the winter. He has already agreed to sell a large fur for $50, but needs to determine how many small furs he needs to obtain and sell, for $3 each. The data provided is two images -- one of Professor Picklebottom, contemplating his need to earn at least $100 to survive the winter and the other, an image of a large fur and small fur, showing the Trader’s payout amounts for each size.”

• In 7.RP.A.2b Coffee Caravan, Practice Printable, the Introduction Problem states, “At what rate are they traveling? Misha and Sonia decide to go on another road trip. Traveling always makes them remember their dad, which is one reason why they like to drink coffee. He loved coffee. To honor their father, the sisters like to measure their rate of travel in miles per cup of coffee, so Sonia keeps track of their trip in her notebook.This time, though, Misha distracted her with karaoke, so she missed writing down a few cups. Look at Sonia’s notes carefully, and determine their rate of travel.” The data provided is a table with four data points for the number of cups of coffee and miles traveled.

• In 7.RP.A.2c Food Factor, Practice Printable, the Introduction Problem states, "What equation should Ariel give to the guides? Mountain guide Ariel created an equation that has helped her fellow guides calculate the amount of food necessary based on the number of people on a trip. Since that equation has worked out so well, she wants an equation that the guides can use to determine the amount of water necessary for an excursion. She asks her guides to send another postcard with their water usage and number of people in the group. Help Ariel analyze the postcards, and write an equation that will calculate the liters of water necessary (w) based on the number of people in the group (n).” The data provided are three postcards with the requested information. 

• In 7.EE.B.3 Hay Talk, Practice Printable, the Introduction Problem states, “How many bales should Ron and Carlie buy? Ron and Carlie recently rescued five more horses who were abandoned by their previous owners: Prius, Creed, Dalla, Chibi and Drago. They need to go to Howard’s Hay again and purchase more hay. Use the information in the vet report to calculate how many bales Ron and Carlie should buy.” The data provided is a note from the vet, “When ordering hay bales, it’s important to purchase quality straw with a high moisture content. In our area we recommend Howard’s Hay which sells 80-pound blaes. Don’t forget horses eat 2% of theis weight in hay each day! Below you will find the latest weights of your rescues from their most recent check-ups. Need to purchase for: Oct, Nov, Dec, Jan, Feb, Mar.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Examples of the three aspects of rigor being present independently throughout the materials include:

• In 7.G.A.3 Doctor Dilim's Dimensions, students develop their conceptual understanding of two-dimensional plane sections by describing results from slicing three-dimensional figures. In the Practice Printable, Question 2 states, “Lloyd has two clay figures on a flat surface in front of him: a right square pyramid and a cube. He will make slices through each figure that are parallel and perpendicular to the flat surface. Determine which statements are true about the two-dimensional plane sections that could result from one of these slices. Place an ‘X’ in the appropriate column.” Students are given a chart with three statements for each of the shapes to identify if a cross-section could be triangular, square, or rectangular, but not square. 

• In 7.EE.A.2 A Taxing Problem, students develop procedural skill in determining if given expressions are equivalent. In the Practice Printable, Question 3, students, “Determine whether each pair of expressions is equivalent.” There are four sets of expressions to compare such as “$$3(a-4b) + 2a$$ and $$-12b + 5a$$.”

• In 7.G.B.6 Miracle Mural, students solve real-world problems involving area, surface area, and volume. In the Practice Printable, Question 2 states “Lauren’s grandma made her a birthday cake in the shape of an ‘L.’ She put frosting on all sides of the cake except for the bottom. a) How many square inches of cake did Grandma cover with frosting? b) How many cubic inches of space does Lauren’s cake take up?” 

Examples of multiple aspects of rigor being engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In 7.SP.A.1 Poll Position, students apply their conceptual understanding of sampling to solve real-world problems. In the Practice Printable, Question 3 states, “Darlene wants to know if sweetened tea or unsweetened tea is more popular in the United States. She posts a poll on social media that asks which one people prefer. 75% prefer unsweetened tea, and 25% prefer sweetened tea. Is this an accurate representation of the U.S. population? Why or why not?”

• In 7.EE.B.4b The Fur Trader, students develop skill in solving inequalities, then use conceptual understanding to match the inequalities with number lines that show the solutions. In the Practice Printable, Questions 4-8 state “Match each inequality with the correct graph of solutions: $$6x - 32 > 50$$; $$2x - 14 ≤ -29$$; $$118+\frac{2}{3}t≥160$$; $$-0.5x + 6 < 10$$ with corresponding graphs.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Examples where MP1 (Make sense of problems and persevere in solving them) is connected to grade-level content include:

• In 7.G.B.4 Crop Circle, Lesson Plan Overview, “It can be challenging for students to make sense of the formula for area of circle. Crop Circle creates a meaningful context for students to use area of circle, but they may not know the formula. In Immersion, teachers prompt students to connect to their prior knowledge of area of rectangles, and to explore the difference between that and a circle. In the Resolution phase, students discuss the relationship between area of rectangles, a circle, and pi, which will help their conceptual understanding of the formula.” 

• In 7.G.B.5 Guarding the Great Gate, Lesson Plan Overview, “In Immersion and Data & Computation, students will use the diagrams and what they know about straight angles to make sense of the relationship between angles and to identify angle measures. Using the ‘Math Circles’ protocol, students make sense of the problem by discussing five questions about straight lines and angles that help them plan and determine how to solve the problem. This lesson offers extra practice of planning how to solve a problem and making a problem of your own.”

• In 7.NS.A.1a Ghost Tamers!, Detailed Lesson Plan states, “In Immersion and Data & Computation, students will connect the idea of “neutralizing a charge” to the concept of zero pairs made from a negative value and positive value. This lesson provides an opportunity for students to struggle to make sense of the problem and persevere to solve through applying different approaches.”

Examples where MP2 (Reason abstractly and quantitatively) is connected to grade-level content include:

• In 7.NS.A.2a Reverse Meditation, Lesson Plan Overview, “Reverse Meditation offers context for reasoning abstractly by visualizing the outcome to encourage brain communication as described by Jo Boaler in her tip for SMP2. In Data & Computation, students are asked to practice ‘decontextualizing’ and ‘contextualizing’ a situation using multiplication with signed numbers. In Resolution, students have an opportunity to visualize and create their own problem representing the math concept. In Clicker Quiz and Practice Printable, students will interpret problems in context and translate these problems from a situation to an equation. They work with real life examples in order to strengthen their knowledge and understanding of multiplication with signed numbers.”

• In 7.RP.A.2d Doggie Diet, Detailed Lesson Plan, “SMP2: During Data & Computation, students analyze the abstract graphical representation and contextualize the parts of the graph as they relate to Simba’s diet situation.”

• In 7.EE.B.3 Hay Talk, Detailed Lesson Plan, Applying Standards for Mathematical Practice, “Hay Talk provides students with an opportunity to solve a real-life situation by modeling with mathematics. In Immersion, the problem is unstructured, and students must make assumptions and approximations to simplify the situation. They must determine how certain quantities, such as the number of horses and number of days, affects how much hay will be needed. In Data & Computation, students will likely see a different approach in the Data Artifact (using the weight of the horse) than they had taken, prompting them to think about revising their model to include additional details and/or revising their model to be more accurate. In Resolution, the full intent of the practice is met as students present different ways of approaching the problem, and consider final revisions to make the model more accurate and complete.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials include 10 protocols to support Mathematical Practices. Several of these protocols engage students in constructing arguments and analyzing the arguments of others. When they are included in a lesson, the materials provide directions or prompts for the teacher to support engaging students in MP3. These include: 

• “Lawyer Up! (12-17 min): When a task has the classroom divided between two answers or ideas, divide students into groups of four with two attorneys on each side. Tell each attorney team to prepare a defense for their ‘case’ (≈ 4 min). Instruct students to present their argument. Each attorney is given one minute to present their view, alternating sides (≈ 4 min). Together, the attorneys must decide which case is more defendable (≈ 1 min). Tally results of each group to determine which case wins (≈ 1-2 min). Complete the protocol with a ‘popcorn-style’ case summary (≈ 2-3 min).”

• “Math Circles (15-28 min): Prior to class, create 5 to 7 engaging questions at grade level, place on different table-tops. For example, Why does a circle have 360 degrees and a triangle 180 degrees? Assign groups to take turns at each table to discuss concepts (≈ 3-4 min each table).”

• “Quick Write (8-10 min): After showing an Immersion video, provide students with a unique prompt, such as: ‘I believe that the store owner should…’, or ‘The person on Mars should make the decision to…’ and include the prompt, ‘because…’ with blank space above and below. Quick writes are excellent for new concepts (≈ 8-10 min).”

• “Sketch It! (11-13 min): Tell students to draw a picture that includes both the story and math components that create a visual representation of the math concept (≈ 5-7 min). Choose two students with varying approaches to present their work (≈ 1 min each) to the class (via MidSchoolMath software platform or other method) and prepare the entire class to discuss the advantages of each model (≈ 5 min).” 

The materials include examples of prompting students to construct viable arguments and critique the arguments of others.

• In 7.RP.A.2a Hot Sauce!, Practice Printable, Introduction Problem, “Use Marty’s notes to determine if Mr. Davis’ perceived heat rating is proportional to the Scoville heat rating, and explain your reasoning.”

• In 7.SP.A.1 Poll Position, Practice Printable, Question 5, “In a poll of Mr. Grey’s English class at Harrington High, 66% percent of students say that English is their favorite subject. A school newspaper reporter in the class wants to write an article stating that English is the favored class among students at Harrington High. Explain why this population is not an accurate representation of the student body, and suggest a way to better gather data to determine which subject is favored by the entire student body.”

• In 7.NS.A.1d Bad Accounting, Practice Printable, Introduction Problem, “Cora Malone and her family have had issues with Mr. Skinner’s banking practices for as long as she can remember. She makes it a point to check Mr. Skinner’s calculations each time she goes to do business at the bank. He almost always has the incorrect balance. Determine if Mr. Skinner is swindling Miss Malone. If so, calculate Miss Malone’s actual account balance.”

• In 7.EE.A.1 Mathmalian Logic, Practice Printable, Introduction Problem, “Whose method is correct and why? Mathmalians Lumi and Dalek are working to purchase two lots on Earth. They have just decided on their lots and now wish to calculate the total cost of the lots. The price per yard is p dollars. As always, they each have a different idea on how to calculate the cost. Lumi wants to use the following method to determine total cost: $$70(x + y)p$$. Dalek wants to use the following method to determine total cost: $$70xp + 70yp$$. Determine whose method is correct and explain your reasoning.”

• In 7.EE.A.2 A Taxing Problem, Practice Printable, Introduction Problem, “The Giggle Barn has placed another order for Talking Giraffe 2, along with Singing Parrot. The quantities for each are equal, except the quantity is unknown. The dudes are again using different equations to determine the quantity. Dude #1: $$58x = 18,850$$; Dude #2: $$23x + 35x = 18,850.$$ Which dude is right? Explain your reasoning.” 

The materials provide guidance for teachers on how to engage students with MP3. In several lessons, the Detailed Lesson Plan identifies MP3 and provides prompts that support teachers in engaging students with MP3. Examples include:

• In 7.NS.A.1d Bad Accounting, “During the Immersion, students use the ‘Think-Pair-Share’ protocol to talk through what they need to know, with the teacher prompt (‘What are your ideas?’) provided. In Data & Computation, students are paired to use the ‘Lawyer Up!’ protocol to defend Ms. Malone’s or Mr. Skinner’s case. They are informed that one of them will be defending the ‘incorrect’ side but to try to make the strongest mathematical case possible. By arguing for a position that is incorrect, students learn to consider different perspectives and see how flawed arguments are formed. After a brief period, the student ‘attorneys’ must agree on which side they find most logical with best supporting evidence. This lesson encourages students to construct viable arguments and use reasoning while critiquing the arguments of others, including being able to see both strengths and weaknesses in arguments.”

• In 7.NS.A.3 Chocolate Certified, the materials include that this lesson “uses a role play protocol that provides an interesting way for teachers to engage students in critiquing the reasoning of other students. In Resolution, both the teacher and students engage in a feedback process that reinforces how their assumptions, variables, and visual representations support a constructive argument developed within a model and how they can be improved.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for supporting the intentional development of MP4 and MP5 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP4 to meet its full intent in connection to grade-level content include:

• In 7.EE.B.3 Hay Talk, the Detailed Lesson Plan states, “MP4: Model with mathematics. Hay Talk provides students with an opportunity to solve a real-life situation by modeling with mathematics. In Immersion, the problem is unstructured, and students must make assumptions and approximations to simplify the situation. They must determine how certain quantities, such as the number of horses and number of days, affects how much hay will be needed. In Data & Computation, students will likely see a different approach in the Data Artifact (using the weight of the horse) than they had taken, prompting them to think about revising their model to include additional details and/or revising their model to be more accurate. In Resolution, the full intent of the practice is met as students present different ways of approaching the problem, and consider final revisions to make the model more accurate and complete.”

• In 7.G.B.4 Crop Circle, Lesson Plan Overview, “MP4: Model with Mathematics. On Day 1, during the Data & Computation phase, students will use the formula for area of a circle to solve a real-world problem. During the Resolution phase, students will see the relationship between area of a circle and pi, which will help their conceptual understanding of the formula.”

• In 7.NS.A.3 Chocolate Certified, students calculate how much chocolate to bring on a group hike. The Detailed Lesson Plan states, “MP4: Model with mathematics. Chocolate Certified provides an opportunity for students to experience all aspects of Jo Boaler’s recommendations for this practice (open questions, make assumptions, create visuals, and revise work). This is a ‘deep’ modeling task with a role play protocol. Teachers are encouraged to spend additional time, in Immersion, to explore this multifaceted task and in debrief, during Resolution, to explore the process of modeling.”

Examples of the intentional development of MP5 to meet its full intent in connection to grade-level content include:

• In 7.G.A.2 Love Triangle, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. During Data & Computation, students strategically choose what tools they will use to help the designer construct a triangle and meet the client’s specifications, indicating the precise measurements. The protocol, which immerses students into the context of working as interns for the design company, enhances their thinking to consider a wider array of tools than they might if they maintained a student-only role.”

• In 7.SP.C.7b Break Time, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. In Resolution, students choose what tools to use in the development of a model that supports them in determining the likelihood of an outcome of an everyday life situation. Initial tools may include questionnaires or other observational tools. In Data & Computation, students may use spreadsheets, calculators, computational software, paper and pencil, rulers or other tools. In presentations, students may select a final medium, such as posters, animation software, slide decks, etc. Students are encouraged to choose tools that are appropriate and strategic to gather, calculate and communicate their data findings.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials provide instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include:

• Each Detailed Lesson Plan provides teachers with a list of vocabulary words and definitions that correspond to the language of the standard that is attached to the lesson; usually specific to content, but sometimes more general. For example, 7.RP.2c states, “Represent proportional relationships by equations.” The vocabulary provided to the teacher in 7.RP.A.2c Food Factor is, “Constant of proportionality: The constant value of the ratio of two proportional quantities, typically x and y; often written as $$k=\frac{y}{x}$$; also known as the rate of change.”

• The vocabulary provided for the teacher is highlighted in red in the student materials on the Practice Printable.

• Each Detailed Lesson Plan prompts teachers to “Look for opportunities to clarify vocabulary” while students work on the Immersion problem which includes, “As students explain their reasoning to you and to classmates, look for opportunities to clarify their vocabulary. Allow students to ‘get their idea out’ using their own language but when possible, make clarifying statements using precise vocabulary to say the same thing. This allows students to hear the vocabulary in context, which is among the strongest methods for learning vocabulary.” 

• Each Detailed Lesson Plan includes this reminder, “Vocabulary Protocols: In your math classroom, make a Word Wall to hang and refer to vocabulary words throughout the lesson. As a whole-class exercise, create a visual representation and definition once students have had time to use their new words throughout a lesson. In the Practice Printable, remind students that key vocabulary words are highlighted. Definitions are available at the upper right in their student account. In the Student Reflection, the rubric lists the key vocabulary words for the lesson. Students are required to use these vocabulary words to explain, in narrative form, the math experienced in this lesson. During ‘Gallery Walks,’ vocabulary can be a focus of the ‘I Wonder..., I Notice…’ protocol.”

• Each lesson includes student reflection. Students are provided with a list of vocabulary words from the lesson to help them include appropriate math vocabulary in the reflection. The rubric for the reflection includes, “I clearly described how the math is used in the story and used appropriate math vocabulary.” 

• Vocabulary for students is provided in the Glossary in the student workbook. “This glossary contains terms and definitions used in MidSchoolMath Comprehensive Curriculum, including 5th to 8th grades.” 

• The Teacher Instruction portion of each detailed lesson plan begins with, “Here are examples of statements you might make to the class:” which often, though not always, includes the vocabulary used in context. For example, the vocabulary provided for 7.RP.A.3 Sport Stats is “Proportional Relationship” and “Percent/Percentage.” The sample statements provided are, “In Sport Stats, we had to help Dave and Shannon calculate the win percentage for each unicycle hockey team so they could broadcast it on air; They calculated the win percentage by making a ratio of the number of games won to the total number of games played, dividing to create a decimal, and multiplying by 100; This is one way of solving this problem, but there are others. I’m wondering how does this relate to proportional relationships we’ve been studying?”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for supporting the intentional development of MP7 and MP8 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP7 to meet its full intent in connection to grade-level content include:

• In 7.G.B.6 Miracle Mural, the Detailed Lesson Plan states, “MP7: Look for and make use of structure. In Data & Computation, students have an opportunity to look for and use familiar structures (grids, 2-D figures within the mural) in the Data Artifact to determine the area. They are asked several prompting questions that help them look for and understand patterns and structures. In Practice Printable, students will make sense of structure by decomposing figures into simpler figures from which to calculate area and/or volume.”

• In 7.NS.A.1d Bad Accounting, the Detailed Lesson Plan states, “MP7: Look for and make use of structure. On Day 1, during both the Immersion and Data & Computation phases, students will recognize that they can use properties of operations to rewrite or rearrange expressions involving adding and subtracting rational numbers.”

• In 7.RP.A.2a Hot Sauce!, the Detailed Lesson Plan states, “MP7: Look for and make use of structure. In Data & Computation, students are asked to draw a visual representation that is connected to the data. As students look for patterns apparent in the visual, they begin to see the underlying structure (a graph) that helps them determine if the pattern is proportional (linear). In Resolution, students apply Jo Boaler’s tip for SMP7 as they share different ways they solved the problem. They are encouraged to look for structures that help illustrate proportional relationships.”

Examples of the intentional development of MP8 to meet its full intent in connection to grade-level content include:

• In 7.NS.A.1b Space Selfie, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. During Teacher Instruction, students experience repeated reasoning as they use the operation of addition with negative and positive numbers to solve each problem. This computation is further understood as repeated reasoning as they see the solution in a visual form on a number line. The full intent of the practice occurs as students are asked to generalize a rule in abstract form (using p and q as integers) through the prompts:  Explain when p + q is positive.;  Explain when p + q is negative.; Explain when p + q is neither positive nor negative.”

• In 7.RP.A.2c Food Factor, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. During Data & Computation, Practice Printable, and Clicker Quiz students have various opportunities to engage in MP8 as they look for the repeated reasoning within data provided either in a tabular format or in a verbal description. Students express regularity in the repeated reasoning shown in proportional relationships by writing an equation that relates the two variables in a general manner.”

The materials reviewed for Core Curriculum by MidSchoolMath 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.

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

• A Curriculum Overview provides a chart of the components and description for the lessons, assessments, and Domain Review. The curriculum components are described briefly in the Overview section. 

• A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines helps the teacher understand rubrics for the assessments.

• In the Teacher Guide, there is instruction on planning a lesson with a sample sequence for lessons and assessments. The materials provide pacing for the year.

• In the Teacher Guide, the instructional protocols used throughout the series are described and connected to the Mathematical Practices they support. 

• In the Detailed Lesson Plan, there is a section to help support Diverse Learners with a chart of Accommodations, Modifications, and Extensions, as well as Language Routines. 

• Common Misconceptions are listed in each Detailed Lesson Plan.

• Teachers are given suggestions for vocabulary incorporation such as, “In your math classroom, make a Word Wall to hang and refer to vocabulary words throughout the lesson. As a whole-class exercise, create a visual representation and definition once students have had time to use their new words throughout a lesson.”

• Guidance is given to teachers for applying and reinforcing math practices in the Teacher Guide and in Detailed Lesson Plans. For example, MP8: “This practice is reinforced by having the students watch a complimentary video in which Jo Boaler has students modeling how to look for and identify patterns in real-life scenarios.” Guidance shared directly from Jo Boaler states, “Students need time and space to develop their capacity to ‘look for and express regularity in repeated reasoning.’ When you provide tasks that are specific to supporting MP8, explicitly tell students that it’s ok to slow down, and to think deeply.” Several “tips” to address the MP are also shared.k to slow down, and to think deeply.”

• “Detailed Lesson Plans provide a step-by-step guide with specific learning objectives for the math standard, lesson summary, prerequisite standards, vocabulary and vocabulary protocols, applying Standards for Mathematical Practice, Jo Boaler's SMP Tips, cluster connection, common misconceptions, instruction at a glance, and day-by-day teaching instructions with time allotments. Also included are suggestions for differentiation, and instructional moves as well as tips for the English Language Learner student.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Throughout each lesson’s Detailed Lesson Plan, there is narrative information to assist the teacher in presenting student material throughout all phases. Examples include:

• 7.NS.A.2c The Mastermind, Teacher Instruction: “Remember that the properties of operations can be used as a tool when multiplying and dividing rational numbers. In The Mastermind, DJ Mastermind used the distributive property to quickly solve a problem involving mixed numbers in his head. We could have just multiplied these numbers on paper, but sometimes the structure of the properties of operations help us to rearrange problems or to break them down so we can simplify our calculations.”

• 7.G.B.6 Miracle Mural, Common Misconceptions: “Students may struggle with when to use square units or cubic units. Remind students that area is ‘covering,’ which involves 2-dimensions and square units, while volume is ‘filling,’ which involves 3-dimensions and cubic units.”

• 7.RP.A.2a Hot Sauce!, Instruction at a Glance Pro Tip: “Students may struggle with the difference between additive reasoning and multiplicative reasoning. Help them understand that although additive relationships can form a line, the line won't go through the origin and the ratios between values won't be equivalent.”

• 7.NS.A.1b Space Selfie, Part 3 Resolution: “1. Play Resolution video to the whole class, and have the students compare their solutions as they watch. 2. After the video, prompt students with the following questions: What did you do that was the same? What was different? What strategy do you think was more efficient to find the equation? Why? Students may respond aloud or in a journal.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Under the Resources Tab, there is a section dedicated to Adult-Level Resources. These contain adult-level explanations including examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. There are also professional articles provided on topics such as mathematical growth mindset, cultural diversity in math, and mathematical language routines.

The Teacher Guide contains a page at the beginning of each cluster section titled, “Cluster Refresher for the Teacher - Adult Level Explanation”. This provides a page of basic background information for the teacher including strategies to develop understanding. For example,

• “7.RP.A involves analyzing proportional relationships and using them to solve real-world and mathematical problems. The concept relies on the understanding of how numbers relate to each other in multiplicative and divisional ways and applying that understanding to real-world contexts. ... The importance of understanding proportionality is paramount to realizing how many uses it has in a real world context. Application of this skill assists with cooking, money applications, statistics, measurement, geometry, architecture, and a great deal in the field of science, just to name a few.”

The Adult-Level Explanations booklet under the Resources tab includes a progression through each domain from Grade 5 through High School. The last section is Beyond Grade 8, which explains how the middle grades learning connects to high school standards. For example: Beyond Grade 8: Ratios & Proportions:

• “Understanding the quantitative relationship between two quantities describes ratio, while equivalence between sets of ratios describes proportion. Making the connection that numbers relate to each other in the sense of multiplication and division is key to identifying and understanding proportional relationships. ... In the high school years, these concepts continue to build on each other. These concepts are essential in all facets of life, extending to topics such as speed, density, and acceleration in science, drawing conclusions about populations in statistics, adjusting recipes while cooking, and getting the best deal at the grocery store.”

The materials reviewed for Core Curriculum by MidSchoolMath 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.

• Each course in this series includes a document called Planning the Year that provides the standards and pacing for each lesson. 

• There are standards correlations in the Scope and Sequence Chart that lists each Lesson, Domain Review, and Major Cluster Lessons throughout a year.

• Each lesson is designed to address a single standard.

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

• The Teacher Guide contains a page at the beginning of each cluster section titled “Role of Mathematics” which clearly identifies the grade-level clusters and standards within a domain and describes the intent of the cluster. The Cluster Role Across Grade Levels describes the grade-level content in context of the domain progression from when the initial related skills were introduced to how the skills progress through high school. For example, “The 7.RP.A cluster involves analyzing proportional relationships and using them to solve real world and mathematical problems. Students compute unit rates, recognize and represent proportional relationships, and use them to solve multistep ratio and percent problems. Skills leading up to this cluster begin in Grade 4, where students multiply or divide to solve word problems, distinguishing multiplicative reasoning from additive reasoning (4.OA.A.2). ... These skills will continue to be applied in High School, as students create equations in two or more variables to represent relationships between quantities (HSA.CED.A.2), use units to understand and guide the solution of multi-step problems (HSN.Q.A.1) and apply concepts of density based on area and volume (HSG.MG.A.2).”

• The Detailed Lesson Plan for each lesson lists the Prerequisite Standards required for students to be successful in the lesson. For example, in 7.G.B.5 Guarding the Great Gate, the Prerequisite Standard listed is 4.MD.C.7.

• The Detailed Lesson Plan for each lesson includes Cluster Connections that identify connections between clusters and coherence across grade levels. For example, in 7.G.A.3 Doctor Dilim's Dimensions, Cross-Cluster Connection: “This activity connects 7.G.A to 8.G.C as students will build capacity for seeing 3-D figures as a collection of 2-D components, which provides a foundation to better understand how formulas are developed for surface area and volume of pyramids, cylinders and cones.”

The materials reviewed for Core Curriculum by MidSchoolMath 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.

• The Curriculum Overview in the Teacher's Guide states that the curriculum is designed to “STOP THE DROP.” The materials state, “Core Curriculum by MidSchoolMath is developed to fix this problem through a fundamentally different approach... MidSchoolMath emphasizes structured, conceptual learning to prepare students for Algebra I... MidSchoolMath is specifically designed to address the ‘The Mid School Math Cliff’.”

• In the Teacher Guide, the overview on Scoring Guidelines states, “In coordination with Dr. Jo Boaler, MidSchoolMath has developed an approach to using rubrics and scoring with an emphasis on making them useful and practical for helping teachers support student learning. This is in contrast to the use of scoring guidelines for the primary purpose of giving grades.” 

• In the Letter to the Parent, the instructional approaches are summarized, “MidSchoolMath strives to help students see that math is relevant and holds value and meaning in the world. The curriculum is designed not only to enhance student engagement, but also to provide stronger visual representation of concepts with focus on logic structures and mathematical thinking for long-term comprehension. ... Peer Teaching: Students learning from other students is a powerful mechanism, wherein both the ‘teachers’ and the ‘learners’' receive learning benefits.”

Materials reference relevant research sources:

• “Hattie, J. (2017) Visible Learning

• Cooney, J.B., Laidlaw, J. (2019) A curriculum structure with potential for higher than average gains in middle school math

• Tomlinson (2003) Differentiated Instruction

• Dweck (2016) Growth Mindset

• Carrier & Pashler (1992) The influence of retrieval on retention: the testing effect

• Boaler, J. (2016) Mathematical MindSets

• Rohrer, D., & Pashler, H. (2007) Increasing retention without increasing study time

• Kibble, J (2017) Best practices in summative assessment

• Laidlaw, J. (2019) Ongoing research in simulators and contextualized math

• Lave, J. (1988) Cognition in practice: Mind, mathematics and culture in everyday life

• Schmidt and Houang (2005) Lack of focus in mathematics curriculum: symptom or cause.”

Materials include research-based strategies. Examples include:

• “Detailed Lesson Plans (Research Indicator: Teacher pedagogy and efficacy remains the highest overall factor impacting student achievement. Multiple instructional models show greater gains than ‘stand and deliver’.)

• The Math Simulator (Research Indicator: On randomized controlled trials, The Math SimulatorTM elicited high effect sizes for achievement gains across educational interventions. Contextual learning and Productive Failure are likely influences contributing to the large achievement gains.)

• Teacher Instruction (Research Indicator: Clarity of teacher instruction shows a large effect on student achievement.)

• Practice Printable (Research Indicator: Differentiation of instruction leads to higher effect sizes compared to full-time ‘whole-group’ instruction. Varied instructional approaches support a growth mindset, an indicator for student success.)”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Teacher Guide includes Planning the Year, Comprehensive Supply List which provides a supply list of both required and recommended supplies for the grade. For example: “Required: Markers, Chart, Paper, Colored Pencils, Dry-erase Markers, Graph Paper, Ruler, Protractor, Compass; Recommended: Individual white boards/laminated alternative, Calculator, Dice, Playing cards, Algebra tiles.”

Each Detailed Lesson Plan includes a Materials List for each component of the lesson. For example in 7.SP.C.8.a-b Red Buffalo:

• “Immersion: Materials - Red Buffalo Immersion video; Chart paper/Interactive whiteboard; White paper

• Data & Computation: Materials - Copies of Red Buffalo Data Artifact, one per student 

• Resolution: Materials - Red Buffalo Resolution video

• Math Simulator: Materials - Red Buffalo Simulation Trainer; Student Devices; Paper and Pencil; Student Headphones

• Practice Printable: Materials - Red Buffalo Practice Printable

• Student Reflection: Materials - Copies of Student Reflection rubric, 1 per student; White Paper; Colored Pencils; Sticky notes

• Clicker Quiz: Materials - Red Buffalo Clicker Quiz; Student Devices; Paper and Pencil”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the standards and Mathematical Practices addressed by formal assessments.

• In the Teacher Guide, Curriculum Components lists several assessments: Clicker Quiz, Test Trainer Pro, and the summative Milestone Assessment. Each cluster has a Pre-assessment and a Post-assessment (Milestone Assessment) which clearly identifies the standard(s) being assessed. The standard is part of the title, for example, “Milestone Post-Assessment 7.SP.C.”; individual tasks and items are not identified on the actual assessment. However, each problem is identified with the standard being assessed in the teacher answer key. 

• Standards are identified accurately and are from the appropriate grade level. 

• Assessment problems are presented in the same order as the lessons. They are sequential according to Domain and Cluster headings.

• The Milestone Assessments include a chart that aligns Mathematical Practices to each question on the assessment, including identifying if the assessment is online, print, or both.

• The end of each lesson includes a student self-assessment rubric that has students evaluate their understanding of the content standard and the mathematical practices that align with the lesson.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 7 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. There is guidance provided to help interpret student performance and specific suggestions for following-up. 

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

• In the Teacher Guide, the Domain Curriculum Components lists two assessments: Test Trainer Pro (formative) and Milestone Assessment (summative). 

  • “Milestone Assessment is a summative evaluation following each cluster per grade. They are automatically graded, yielding the percentage of items answered correctly. The math items are crafted to include items of varying difficulty.” “Please note: Milestone Assessments should not be used to determine student growth. As summative assessments, they are not as sensitive nor as accurate as the adaptive tool, Test Trainer Pro, for providing individual student data for achievement gains over time.”

  • “Test Trainer Pro acts as a low-stakes, formative learning tool for students to practice testing under more relaxed and stress-free conditions. It is an adaptive tool and is designed to elicit the largest gains in student achievement possible in the shortest period of time.”

• The Teacher Guide contains a section titled “A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines.” This section provides several assessment rubrics: 

  • The MidSchoolMath Rubric and Scoring Framework aligns a percentage “raw score” with a 4-point rubric and proficiency levels.

  • The Milestone Assessment Rubric aligns a percentage “raw score” with a 4-point rubric and has suggestions for follow-up.

  • The Student Self Assessment has students reflect and identify understanding for each lesson component. 

• An article by Jo Boaler, “Assessing Students in a Growth Mindset Paradigm with Jo Boaler” provides “recommendations for assessment and grading practices to encourage growth mindsets.”

• Each Curricular cluster contains a tab for Assessments which has a Milestone Assessment Overview & Rubric. There is a rubric from 0 to 3 provided for the open response section of the assessment. To earn all 3 points, students must demonstrate accuracy, show work, and may only have minor mistakes.

  • “Recommended Scoring for Milestone Assessments: A 3-point response includes the correct solution(s) to the question and demonstrates a thorough understanding of the mathematical concepts and/or procedures in the task. This response: Indicates that the student has completed the task correctly, using mathematically sound procedures; Contains sufficient work to demonstrate a thorough understanding of the mathematical concepts and/or procedures; May contain inconsequential errors that to not detract from the correct solution(s) and the demonstration of a thorough understanding.”

• The Overview states, “All items in Milestone Assessments are at grade level and evaluate student understanding of the content at the ‘cluster’ level. Milestone Assessments should only be administered to students after all lessons are completed within the cluster, following recommended sequence and pacing.”

• The answer key for each Milestone Assessment provides examples of correct responses for each problem. There is a sample response for the open-ended questions. 

• Several of the other lesson components could be used as formative assessments or for progress monitoring such as the Clicker Quiz.

The assessment system provides task-specific suggestions for following-up with students. There are suggestions for follow-up that are generic strategies, and there are some that direct students to review specific content.

• The Milestone Assessment Rubric includes Recommendations for Follow Up. These are found in the front matter of the Teacher Guide. They are generic to all assessments and align with the 4 points of the rubric: 

  • “Review and correct any mistakes that were made. Participate in reteaching session led by teacher.

  • Review and correct any mistakes that were made. Identify common mistakes and create a ‘Top-3 Tips’ sheet for classmates. 

  • Review and correct any mistakes that were made. Participate in the tutorial session. 

  • Review and correct any mistakes that were made. Plan and host a tutorial session for the Nearing Proficient group.”

• The Milestone Assessment also includes suggestions based on which problems are missed. The guidance directs students to review the worked example and Clicker Quiz in the lessons that align to the missed problems and then revise the problems they missed in the assessment. This provides specific feedback to review the content of the lesson. 

• The Student Self-Assessment provides a generic strategy for follow-up: “Recommended follow-up: When students self-identify as ‘Don’t get it!’ Or ‘Getting there!’ on an assignment, is it essential for teachers to attempt to provide support for these students as soon as possible. Additionally, it is helpful for teachers to use scoring on Practice Printables and Clicker Quizzes to gauge student comprehension. Use the general scoring guidelines to determine approximate proficiency. It is highly recommended that all assignments may be revised by students, even those which are scored.”

• The Student Self-Assessment provides suggestions based on where the students rate themselves. Students are directed to review specific parts of the lesson to reinforce the parts they do not feel successful with. There are also more generic strategies suggested that go across lessons and grade levels.

• The materials state that “Test Trainer Pro automates assessment and recommendations for follow-up under the score. As an assessment, Test Trainer Pro is the most specific, and most accurate measure available in MidSchoolMath to determine how students are performing in terms of grade and domain level performance.” A teacher can view the Test Trainer Pro question bank; however, there is no way to review the specific follow-up recommendations provided since they are adapted to each student. 

• Exit Ticket results are sometimes used to suggest grouping for instructional activities the following day.

The materials reviewed for Core Curriculum by MidSchoolMath 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. Assessments include opportunities for students to demonstrate the full intent of grade-level standards and the mathematical practices across the series.

• Assessments are specific to each standard, so there is opportunity for students to demonstrate to the full intent of grade-level standards. 

• Considering both formative and summative assessments, there are a variety of item types offered including Exit Tickets, Clicker Quizzes, Test Trainer Pro, Lesson Reflection, Self-Reflection, and Milestone Assessments.

• Most assessments are online and multiple choice in format, though there is a print option for milestone assessments that includes open response. 

• Students have the opportunity to demonstrate the full intent of the practices in assessments; practices are aligned in Milestone assessments and addressed in the student self-assessments for each lesson.

The materials reviewed for Core Curriculum by MidSchoolMath 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/series mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In each Detailed Lesson Plan, Supporting Diverse Learners, there is a chart titled Accommodations, Modifications, and Extensions for English Learners (EL) and Special Populations that provides accommodations for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 7.EE.B.4a, Pen Perimeter, include:

• The Math Simulator Immersion: “Reinforce vocabulary in action: Quotient: a result obtained by dividing one number by another; Using the Frayer model, have students create vocabulary cards for each of the lesson’s vocabulary words, plus quotient, positive quotient, and negative quotient.”

• The Math Simulator Data & Computation: “Provide students with a graphic organizer with concrete examples as well as definitions and mathematical rules for the following words: integers, non-integers, rational numbers, undefined, quotients, positive quotients, and negative quotients. Give students real world examples of what the fractions mean (like the one in the problem) to help students make the fractions with a negative make more sense.  Another example would be negative temperatures.”

• Simulation Trainer: “Pair students to allow for peer teaching and support.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Pair students to allow for peer teaching and support. Consider allowing students to answer questions verbally to a scribe. Students may benefit from extended time.” The Practice Printable also has interactive buttons that allow students to complete work online through draw and text tools as well as a work pad that includes an opportunity to chat with the teacher. 

• Clicker Quiz: “Students may benefit from a printed copy of the Clicker Quiz pages. Read Clicker Quiz questions aloud to students, especially those questions that are text-intensive.”

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

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity. The Exit Ticket in each lesson provides a differentiation plan that includes extension. While some strategies are the same across lessons, there are a variety of tasks offered. Examples include:

• 7.G.A.3 Doctor Dilim's Dimensions, “Task the students with creating a match game with index cards. One pair of cards should have a visual model of a 3D figure with a line for a cross-section cut, and the other should have a visual model of 2D figure that matches.”

• 7.NS.A.1a Ghost Tamers, “Create a poster to explain the idea of opposite quantities. Include at least one diagram, at least one real-life example, and at least one mathematical example.”

• 7.NS.A.2d Grandpa's Journey, “Analyze these rational numbers and their decimal equivalents. Attempt to define a shortcut telling you and your classmates when a rational number will terminate or repeat just by looking at the fraction representation.”

In each Detailed Lesson Plan under Supporting Diverse Learners, there is a chart titled Accommodations, Modifications and Extensions for English Learners (EL) and Special Populations that provides extensions for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 7.EE.B.4a Pen Perimeter include:

• The Math Simulator Data & Computation: “Have students create additional clues to a map like the ones in the Immersion and trade with a partner to solve. Have them use a 20 × 20 map to do so.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Have students create problems like a)-d) using fractions. Have them exchange them with a partner and solve.”

• Clicker Quiz: “Task students with writing and solving their own ‘clicker quiz’ question.”

• Extensions are optional; there are no instances of advanced students doing more assignments than their classmates.

The materials reviewed for Core Curriculum by MidSchoolMath 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.

Materials consistently provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics. Examples include:

• In the Detailed Lesson Plan for every lesson, the same two strategies are suggested: “Access Closed Caption and Spanish Subtitles within the video.” and “Pair students to allow for peer teaching and support. Consider allowing EL students to write the narrative in their native language, then use a digital translator to help them transcribe it into English.” 

• Each Detailed Lesson Plan makes a connection with one of the eight identified Math Language Routines (MLR), listed and described in the Teacher Guide. The MLRs include: Stronger and Clearer Each Time, Collect and Display, Critique, Correct, and Clarify, Information Gap, Co-Craft Questions and Problems, Three Reads, Compare and Connect, Discussion Supports.

• All materials are available in Spanish. 

• The use of protocols such as Think-Pair-Share, Quick Write, and I Wonder I Notice provides opportunities for developing skills with speaking, reading, and writing.

• Vocabulary is provided at the beginning of each lesson and reinforced during practice and lesson reflection, “In the Practice Printable, remind students that key vocabulary words are highlighted.” In the Student Reflection, the rubric lists the key vocabulary words for the lesson. Students are required to use these vocabulary words to explain, in narrative form, the math experienced in the lesson. 

• There is teacher guidance under the Resources tab - Math Language Routines. “Principles for the Design of Mathematics Curricula: Promoting Language and Content Development”, from the Stanford University Graduate School of Education, provides background information, philosophy, four design principles, and eight math language routines with examples. 

• There are no strategies provided to differentiate the levels of student progress in language development.

The materials reviewed for Core Curriculum by MidSchoolMath 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.

Examples where manipulatives are accurate representations of mathematical objects include:

• The students have access to virtual manipulatives on the Work Pad which is available online in their Simulator Trainer, Practice Printable, Assessments, and Clicker Quiz. These include shapes, 2-color counters, base 10 blocks, algebra tiles, protractor, and ruler. In addition, there are different styles of digital graph paper and dot paper on the digital whiteboard.

• Throughout the materials, there are visual models with number lines, graphs, or bars, though these cannot be manipulated. 

• During the Immersion and Resolution videos, items from the real world are used to represent mathematical concepts. 

• The Teacher Guide has a section titled “Guidance on the Use of Virtual Manipulatives.” This section includes sub-sections titled: Overview, General Guidance, During Lessons, Manipulative Tools, and Examples of their Use & Connecting to Written Methods. The “Examples of their Use & Connecting to Written Methods” provides teachers with guidance about how to use and make connections with the manipulatives.

• In the Detailed Lesson Plan, Practice Printable, there is a “Manipulative Task!” where students use the virtual tools in the Work Pad and specifically connect manipulatives to written methods. For example, 7.RP.A.2b Coffee Caravan, Manipulative Task for Digital WorkPad: “Have students return to Problem #1d on the Practice Printable and use the WorkPad to graphically represent the data provided in the table. The students should aim to visually verify if the relation in the table has the same constant of proportionality as y = 1/3x. Encourage students to experiment with the different manipulatives provided in the WorkPad to complete the exercise. For example, students could use the grid background, Line tool, and the Text tool to create and label a coordinate plane, keeping in mind the scale necessary to easily plot the given data points. Finally, the students could use the 2-Color Counters to plot the given ordered pairs, and may connect them using the Line tool. Using this visual representation, students can explore how a graph can help them visually identify the constant of proportionality by observing patterns between each point of data.”