The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.EE.A.3 Provision Problem states, “How many equivalent expressions can we create from this expression? $$24n-16n+12$$.” Suggested answers include: “Combine like terms. $$8n +12$$; Factor out GCF. $$4(6n-4n+3)$$; Factor out GCF & combine like terms $$4(2n + 3)$$; Factor out common factor (not GCF) $$2(12n-8n+6)$$; Factor out common factor (not GCF) & combine terms $$2(4n + 6)$$”.

• In 6.G.A.3 Fuel Factor, the teacher is prompted to ask students questions to further their thinking such as, “What is the direction for the course? What do you notice about the coordinates of the endpoints of horizontal line segments? What do you notice about the coordinates of the endpoints of vertical line segments? Imagine the coordinates on a grid. How might you find the length of a line segment connecting them? Since there is no grid, how might you find the length of the line segment anyway?”

• In 6.NS.A.1 Mr. Mung’s Ice Cream, the teacher is prompted to complete this example using a bar diagram. “$$3\frac{1}{2}$$ is divided by $$1\frac{3}{4}$$. What is the quotient? We could draw a visual fraction model. We start by drawing a representation of $$3\frac{1}{2}$$. Then we separate the diagram into fourths because of the denominator of the divisor. We then ask ourselves how many groups of $$1\frac{3}{4}$$ are in $$3\frac{1}{2}$$? Then separate the diagram into groups of $$1\frac{3}{4}$$. We can see two groups of $$1\frac{3}{4}$$, so the quotient is 2.”

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

• In 6.EE.C.9 Sister Act, Practice Printable, Question 1 states, “A worker earns $17 per hour.  a.Write an equation to show the relationship between the hours she works (h) and the amount she is paid (p).  b. What is the independent variable? What is the dependent variable?”

• In 6.RP.A.1 For Every Day, Practice Printable, Question 4 states, “To make a deep orange color, Regina mixed 8 drops of red paint and 2 drops of yellow paint. Describe the relationship between the red paint and yellow paint in at least 4 different ways.”

• In 6.RP.A.2 Road Trip Ratios, Practice Printable, Questions 1-4 each provide a ratio and ask students to produce two unit rates, “1) 8 cats eat 4 large cans of cat food. _ cans per cat _ cats per can.”

• In 6.SP.A.1 Statistical Friends, Practice Printable, Question 1 states, “Determine whether each question below is statistical or non-statistical.” In Question 9, students demonstrate conceptual understanding when they, “Write a statistical question that could be answered by collecting data from your classmates.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.NS.B.2 Which Way, the Teacher Instruction directs teachers to say, “perform long division and make sure to line up columns.” Three different example problems are provided. Steps for the division algorithm are shown three days in a row during the lesson. Students also see the completed algorithm for at least 15 practice problems. 6.NS.2 is practiced in two other units and in the game Ko’s Journey. 

• In 6.NS.B.3 Enter the Dragon, the Teacher Instruction has the teacher walk through the steps for the four operations, repeatedly asking, “What is the process to (add, subtract, multiply, divide) decimals?” The Math Simulator provides five sets of three problems for students to solve. After each set, the correct answer and process is worked out for the student. There are decimal calculations in eight other lessons. 

• In 6.EE.A.1 I Dream of Djinni, the Teacher Instruction supports students in procedural skills related to exponents. The terms base and exponent are introduced, and the teacher walks through examples where the base and exponent are unknown. Then, the teacher walks through evaluating an expression that involves an exponent and requires using the order of operations.

• In 6.EE.A.2c Real Stories of the AIF (Accident Investigation Force), the teacher provides instruction for using an expression for drag factor, a formula to convert between Farenheit and Celsius, and an expression with multiple variables. The teacher prompt states, “It’s important to substitute values carefully; many mathematicians put parentheses around each value to make sure they have substituted it correctly and in the right spot,” and later, “When substituting in for variables in formulas or expressions, it’s often helpful to put the values in parentheses to help keep them separated and to remain clear on the operations to be used on each value.”

Examples of students independently demonstrating procedural skills and fluencies include:

• In 6.NS.B.2 Which Way, the Clicker Quiz contains six questions that involve long division, including five word problems and one problem that requires interpreting a quotient in multiple ways (remainder, fraction, decimal). In the Practice Printable, there are four problems in Question 1, “Find each quotient” and four word problems that require long division. For example, Practice Printable Question 1a states, “$$40,584 ÷ 76$$”; Question 5 states, “The city of Vine View is building a new rectangular park for the townspeople. The park will have an area of 8,925 square yards. If the width of the park is to be 84 yards, how much fencing does the city need to surround the park?”

• In 6.NS.B.3 Enter the Dragon, the Practice Printable contains four problems each for adding, subtracting, multiplying and dividing multi-digit decimal numbers. The Clicker Quiz contains six problems, one problem each for each of the four operations, and two word problems that require using multiple operations to solve. For example, Practice Printable, Question 2 states, “$$70.64 + 0.0059$$”;  Question 6 states “$$43.02-0.0078$$”; Question 10 states “$$48.5 ⋅ 1.604$$”; Question 14 states “$$0.5208 ÷ 6.2$$”.

• In 6.EE.A.1 I Dream of Djinni, one question in the Clicker Quiz shows an image of a man thinking $$7^6$$ and a woman thinking $$6^7$$ and states, “$$7×7×7×7×7×7$$ Ryan and Jane are thinking about writing this expression using an exponent. Who is correct?” In the Practice Printable, Question 4 states, “Fill in the missing information for each row.” A three column table with the headings “exponential form, expanded form and standard form,” is provided.

• In 6.EE.A.2c Real Stories of the AIF, Practice Printable, Question 1 states, “Complete the chart using the formula for area of a triangle, $$A=\frac{1}{2}bh$$”; students are given a table with 5 pairs of base and height values, and calculate the area. In Practice Printable, Question 2 states, “Evaluate each expression in the chart if $$a=2, b=4, c=6$$ and $$d=3$$”; the chart contains six expressions using the variables.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.RP.A.3c Stealing Home, an example states “Country music makes up 75% of Ashley’s music collection. If she has 33 albums that are by country artists, how many albums does she have in her entire music collection? 

• In 6.NS.A.1 Mr. Mung’s Ice Cream, Practice Printable, Question 6 states, “Milo bags groceries at a local market. The plastic bags he uses are designed to hold 25 pounds. If a typical water bottle weighs $$1\frac{1}{4}$$ pounds, how many bottles could Milo put in a bag (assuming they all fit)?”

• In 6.EE.7 The Sign of Zero, Practice Printable, Question 2 states, “The map at the right shows points A, B, and C. Say the distance from point A to point C is three times the distance from point A to point B, and the distance from point A to point C is 105 miles. What is the distance from point A to point B?”

• In 6.RP.A.3a Clone Wars, Practice Printable, Question 4 states, “Kennedy thinks the best orange juice is made using 3 cups of water and 5 cups of juice concentrate. How many cups of water and juice concentrate will she need to make 40 cups of juice? Create a table or diagram to show your reasoning.” 

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

• In 6.SP.B.4 & 5 Shoot for the Moon!, Practice Printable, students write a newspaper article based on data from a survey including “a headline, graphical display, the number of observations, at least one graphic, a description of how the survey was conducted, the measures of center including mean, median and mode. Report all measures of variability and striking deviations.” 

• In 6.NS.C.7c Day by Day, the Practice Printable includes: At Wonder Toys, new employees receive a 30-day evaluation that ranks bad days and good days on a scale of -10 to 10. “Miss Brooks has a new assistant at Wonder Toys named Mary Smithson. The time has come for Mary’s 30-day evaluation. Based on the number of bad days, Mary thinks she may lose her job. Miss Brooks explains that, along with the number of good and bad days, she has to look at the magnitude of the good and bad days to determine job performance. Use Mary Smithson’s evaluation to explain what Miss Brooks is talking about and to determine whether Mary has a good evaluation or a poor evaluation.”

• In 6.RP.A.2 Road Trip Ratios, Practice Printable, Question 7 states, “Pareesa bought two new aquariums, each holding exactly 200 gallons of water. One aquarium will hold only small fish and the other will hold large fish. She will buy 5 small fish for every 10 gallons of water in the aquarium. She will buy 8 large fish for every 40 gallons of water in the aquarium. How many total fish will Paressa have? What will be the ratio of large fish to small fish?”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.NS.C.5 Weather Bear, students develop conceptual understanding of the meaning of positive and negative numbers. During Teacher Instruction, these examples are provided:  “Let’s look at a different example of walking forwards and backwards. What is the meaning of taking 5 steps forward? What is the meaning of taking 2 steps backward? What is the meaning of zero in this case?; Now, let’s look at gaining and losing yards in football. What is the meaning of a gain of 6 yards? What is the meaning of a loss of 4 yards? What is the meaning of zero in this case?”

• In 6.NS.C.6c Special Intelligence, students develop procedural skill in plotting points. In the Practice Printable, Question 4, students “Plot each point on the coordinate plane, and label it with the corresponding letter.” Students are given nine points to plot, including some with $$\frac{1}{2}$$, to ensure they have multiple opportunities to plot points on the coordinate plane.

• In 6.EE.A.2c Real Stories of the AIF, students evaluate expressions at specific values of their variables that arise from formulas used in real-world problems. In the Practice Printable, Question 3 states “The cost of a pass to the amusement park for 5 days or less is $$50+10n$$, where n is the number of days you are visiting. The cost for a pass to the amusement park for more than 5 days is $$45+10(n-1)$$, where n is the number of days you are visiting. a) If you plan on visiting for 5 days, what is the cost of the pass? b) What would be the cost for visiting for 6 days? c) Is it a better deal to visit for 5 or 6 days? Explain. d) What would be the cost to visit for one week?”

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

• In 6.G.A.1 The Lilliput Regatta, students use their conceptual understanding of area to find the area of right triangles, other triangles, special quadrilaterals, and polygons in application problems by composing into rectangles or decomposing into triangles and other shapes. The teacher shows students how to decompose figures, find missing dimensions, and calculate the area of each region. Students also practice procedural skills while finding the areas of figures throughout the lesson. For example, Practice Printable, Question 2 states, “What is the area of figure ABCD, in square centimeters?” (figure ABCD is a kite).

• In 6.EE.B.6 Land in Lama, students use their understanding of variables to represent numbers as they develop skill in writing equations that represent real-world problems. In the Practice Printable, Question 2 states, “David went into a floral shop to buy his mother some flowers. Depending on the season, carnations cost c dollars; roses cost r dollars; and tulips cost t dollars. Vases are $12. Write an expression to represent David’s cost for 4 carnations, 5 roses, 3 tulips, and a vase.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 meet expectations for supporting the intentional development of MP1 and MP2 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP1 to meet its full intent in connection to grade-level content include:

• In 6.NS.C.8 The Mark of Zero, Detailed Lesson Plan, “During Immersion and Data & Computation, students receive information that initially seems vague. As they explore the statements provided to them in conjunction with the ‘map’ (coordinate plane), they begin to see more value in the statements, can infer more specific details, and consequently change course as needed. The ‘Think-Pair-Share’ protocol aids students in making sense of the problem, as they look for entry points to its solution.”

• In 6.RP.A.3b Vacation Day, Lesson Plan Overview, Applying Standards for Mathematical Practice, “During Immersion, students use a ‘Think-Pair-Share’ protocol to determine what they need to know and begin a solution pathway. In Data & Computation, students recognize that equivalent ratios and the unit rate can give them important information about the guard’s wages. Students can use ratio tables, double number lines, and other strategies to solve the problem, and have the opportunity to share their strategies.” 

Examples of the intentional development of MP2 to meet its full intent in connection to grade-level content include:

• In 6.NS.C.5 Weather Bear, Detailed Lesson Plan states, “During Immersion and Data & Computation, students will recognize that a positive number, a negative number, and zero have specific meanings within a context. Students will make sense of real-world quantities and their relationships when looking at altitudes. Students will also recognize that numbers, such as 7 and -7, are opposite values and are in opposite direction from zero on a number line.”

• In 6.RP.A.1 For Every Day, Lesson Plan Overview, Applying Standards for Mathematical Practice, “During Data & Computation, students compute the quantitative ratio of two quantities, contextualize it to make meaning of a ‘real-world’ situation, then express it using ratio language.”

• In 6.EE.B.7 The Sign Of Zero, Lesson Plan Overview, Applying Standards for Mathematical Practice, “During Immersion, students have the opportunity to make initial sense of what is being asked, by talking with a partner about what was presented in the video, specifically a diagram that is shown a second time. The ‘Think-Pair-Share’ protocol allows students to gain a perspective other than their own. During Data & Computation, students receive additional quantitative information whose meaning must be attended to when building symbolic equations of the mathematical relationship. After solving the equation, students must attend fully to its meaning in the situation, as it's not the final answer; an additional operation must be performed to get the intended value.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 meet expectations for supporting the intentional development of MP3 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. Examples 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 6.EE.A.2c Real Stories of the AIF, Practice Printable, Question 3c, “The cost of a pass to the amusement park for 5 days or less is $$50 + 10n$$, where n is the number of days you are visiting. The cost for a pass to the amusement park for more than 5 days is $$45+10(n-1)$$, where n is the number of days you are visiting. Is it a better deal to visit for 5 or 6 days? Explain.”

• In 6.SP.B.4 & 5 Shoot for the Moon!, Practice Printable, Introduction Problem, “What could the newspaper article look like? Be sure to include a headline, graphical display, the number of observations, a description of how the survey was conducted, the measures of center including mean, median and mode. Report all measures of variability and striking deviations. Choose the most appropriate measure of center and measure of variability and defend your choices; include a closing comment.” In the Simulator question, “Choose the most appropriate measure of center and measure of variability and defend your choices.”, and in Practice Printable, Questions 2c-d states, “What is the better measure of center for this data set? Why? Which is the better measure of variation of this data set? Why?”

• In 6.RP.A.3c Stealing Home, Practice Printable, Question 3, “Tyrell took a history test. He answered 21 of the 25 questions correctly. In order to get an ‘A’ on the test he needs to get at least a 90%. Did Tyrell get an ‘A’ on his history test? Explain your reasoning.”

• In 6.SP.A.3 Periodontal Pockets, Practice Printable, Question 1, “All sixth graders at Madison Middle School were given a math and reading placement test at the beginning of the year. a) If you wanted to know on average if sixth grade students scored better on the math test or reading test, would you consider the measure of center of the data or the measure of variability of the data? Explain your reasoning. b) If you wanted to see how consistent (or similar to each other) the scores on the respective tests were, would you focus on the measure of center of the data or the measure of variability of the data? Explain your reasoning.”

• In 6.NS.C.7c Day by Day, Practice Printable, Introduction Problem, “Use Mary Smithson’s evaluation to explain what Miss Brooks is talking about and to determine whether Mary has a good evaluation or a poor evaluation.”

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 6.NS.C.7d Coffee Accounting, “In Data & Computation, students take the practice test by themselves, then work with another student to justify their conclusions in the ‘Study Hall’ protocol. Because the order of the wording impacts the meaning of the statements, students practice a logical progression of statements. Paired students explore the truth of their partner’s conjectures, and ask rich questions and critique the reasoning of other students. The following Teacher Prompts encourage students to explain their reasoning and examine their partner’s reasoning and logic. Did your study hall partners present a logical argument? Can you repeat what another student’s logic is? Did you notice any flaws in their logic? Can you draw a picture to explain your reasoning? Is there another way to explain your own logic?”

• In 6.NS.C.7c Day by Day, Data & Computation “includes prompts that support students in developing their own arguments and critiquing those of others: 2. Use the ‘Quick Write’ protocol, where students are prompted to write down ideas about whether Rob is having good days or bad days, and prompted to make a conclusion with supporting evidence. It is important that students are not given too much information, or prompted with guiding questions at this stage. 3. Have students join with two other students. Each student has 2 minutes to present their ‘Quick Write.’ During which the other two students act as supervisors, and are there to provide feedback they feel would be helpful in strengthening the conclusion. Use the following  prompts with students to encourage the critique process: ‘I was confused when you ____.  ‘It might be more clear if you said ________. ‘Can you re-state that in a different way?’"

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.EE.A.2c Real Stories of the AIF, Lesson Plan Overview, “MP4: Model with mathematics. On Day 1, during the Data & Computation phase, the students will decide how fast the driver was driving using the ‘$$30df$$’ formula. The students will use the formula to identify the rate of speed on a specific road surface.” 

• In 6.EE.B.6 Land in Lama, the Detailed Lesson Plan states, “MP4: Model with mathematics. In Land in Lama, students are tasked with representing the cost of the land through an expression, which is, in essence, a modeling task. Further supporting the practice, during Immersion, the problem is relatively unstructured, requiring students to determine what they need to know, and analyzing how the problem might be solved while making assumptions about the relationships between unknown quantities. Visual representation the students develop supports clarity of thinking about their model and assumptions. Students refine their model as more information is given during Data & Computation. The full intent of the practice occurs as students create their own variables and include them as part of the expression.”

• In 6.G.A.2 River Rescue, the Detailed Lesson Plan states, “MP4: Model with mathematics. River Rescue opens with a unique protocol that leads students to begin modeling with mathematics right away in Immersion. Students imagine a flowing river and must try to think of a way to determine the amount of water that is flowing per second. They team in small groups, using their intuition to guide them in an early attempt to model the situation. They draw pictures and discuss ideas in an attempt to find an entry point into the upcoming task. In Data & Computation, students calculate the flow rate of the river (modeled as volume of a rectangular prism with fractional side lengths). In Resolution, students revise their thinking, comparing not only their answer, but their original conceptual ideas of how to calculate flow rate.”

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

• In 6.RP.A.3c Stealing Home, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. During Data & Computation and Practice Printable, students discover that two parts of a percent problem are given (whole, part, or percent) and a third unknown part must be determined. Students can use different tools that help them see that 100% splits up into parts (double number lines, tape diagrams, ratio tables, etc.).” In the Practice Printable, Question 5 states, “Solve each problem below by using a table of equivalent ratios, a tape diagram, a double number line or an equation. a) 75 is 15% of what number? b) What is 60% of 210? c) 120 is 30% of what number? d) 160 is 20% of what number?” 

• In 6.SP.B.4&5 Shoot for the Moon!, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. In Data & Computation, students are asked "What could the newspaper article look like?" This general question requires students to consider the tools available to them and to make personal choices as to how to use them. These include mathematical tools (graphs, tables, mathematical graphics, etc.) and also physical tools ( rulers, graph paper, pencils, etc.). Technology tools (computers, tablets, calculators, spreadsheets, and graphical display software) may also be considered.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 meet expectations for supporting the intentional development of MP6 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, 6.NS.3 states “Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.” The vocabulary provided to the teacher in 6.NS.B.3, Enter the Dragon is, “Decimal number: A number that can show place value less than 1; represents values such as tenths, hundredths, thousandths, etc.” 

• 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 with a brief definition or used in context. For example, the vocabulary provided for 6.RP.A.3c, Stealing Home is “Part”, “Whole”, and “Percent.” The sample statements provided are, “Remember there are always two parts of the percent problem given from the part, whole, or percent; Remember that the percent of a quantity is per 100; In Stealing Home, we had to help find the number of runs Jackie Robinson would score during the 1948 season; We can convert the percent to a rate per 100, so 52% is $$\frac{52}{100}$$; A ratio table can be created using the percent as a rate of 100, and then other helpful equivalent ratios can be identified.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.NS.B.4 The Castle Guard, the materials state, “MP7: Look for and make use of structure. On Day 1, during both the Immersion and Data & Computation phases, students will understand the differences on how to calculate the greatest common factor and least common multiple between two numbers.” Optional teacher prompts include, “What do we know about the number of days that each guard works? How can we use a number line as a tool to show when the guards work? Does the LCM or GCF need to be found when the guards work together again? How do we find the LCM? What are the multiples of 2? What are the multiples of 4? What are the multiples of 5? What is the LCM of 2, 4, and 5? When will the guards work together again?”

• In 6.RP.A.3d Saffron Shuffle, Lesson Plan Overview, “MP7: Look for and make use of structure. During Data & Computation, students work together to notice that ratios can be used to convert a measurement from one unit to another. By using ratios written in fraction form as conversion factors, students recognize the structure of the fraction, where a common numerator and denominator make 1 (cancel each other out).Students use this structure repeatedly to keep track of units during conversion and to cancel them out as needed to end with the appropriate unit.”

• In 6.G.A.3 Fuel Factor, the Detailed Lesson Plan states, “MP7: Look for and make use of structure. During Immersion and Data & Computation, students will recognize that endpoints for horizontal line segments have the same y-coordinate, and the length of such segments can be found by subtracting the x-coordinates because the grid structure shows the lengths to be the distance between x-coordinates. Similarly, endpoints for vertical line segments have the same x-coordinate, and the length of such segments can be found by subtracting the y-coordinates because the grid structure shows the lengths to be the distance between y-coordinates. Students are able to make use of these structures for the practical purpose of determining the length of the race course.” The prompts provided for teachers include: “What is the direction for the course? What do you notice about the coordinates of the endpoints of horizontal line segments? What do you notice about the coordinates of the endpoints of vertical line segments? Imagine the coordinates on a grid. How might you find the length of a line segment connecting them? Since there is no grid, how might you find the length of the line segment anyway? What is the total distance of the course? For how many megaspans do the sisters think the ship will last?”

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

• In 6.NS.C.6b Treasure Trail, the Detailed Lesson Plan states, “MP8: Look for and make use of structure. During Data & Computation, students have opportunity to recognize the coordinate plane as a structure that aids them in seeing a repeated pattern for coordinates that are reflected. During Resolution, teacher prompts during the ‘Number Talk’ ask students to identify the constant pattern of how coordinates are affected by reflection and to explain how the grid lines in the coordinate plane aided them in realizing the general rule.”

• In 6.EE.A.1 I Dream of Djinni, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. On Day 1, during the Data & Computation phase, students will try to determine which prize (option 1 or 2) will have the greatest number value. The students will create a chart to write down the information they already know about options 1 & 2 and then move towards using repeated multiplication to trigger other tools or strategies that will produce the correct solution.”

• In 6.G.A.1 The Lilliput Regatta, Lesson Plan Overview, “MP8: Look for and express regularity in repeated reasoning. During Data & Computation and Practice Printable, as students repeatedly calculate the area for each geometrical shape, they are able to see that they can manipulate the shape to find faster and easier ways to determine the area. They can repeatedly cut and re-form the shape into parts, or can double its size and divide by two, or use other methods to determine the area. The regularity in the repeated reasoning is that the area is always the same, no matter how they manipulate the shape so long as its size is not changed.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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.

• “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:

• 6.RP.A.2 Road Trip Ratios, Common Misconceptions: “Students may use additive reasoning versus multiplicative reasoning to scale or simplify ratios. Encourage students to draw visuals to represent the ratios to concretely see the multiplicative reasoning.”

• 6.SP.A.2 Build a Better Forest, Cluster Connection: “This activity connects 6.SP.A and 6.SP.B as students calculate simple variation calculations, such as range (referred to here as ‘spread’).”

• 6.NS.C.6c Special Intelligence, Teacher Instruction: “Here are some examples you might make to the class. Before plotting any points on the coordinate plane, review the scale for each axis. The interval may or may not be one and may or may not be the same for each axis; Remember that when plotting on a coordinate plane, we first move horizontally and then vertically; In Special Intelligence, we plotted points on a coordinate plane that had a scale of 6 rather than 1. The larger scale allowed for the larger coordinate values to fit on the image of the map.”

• 6.EE.C.9 Sister Act, 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 6 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,

• “6.EE.B develops an understanding of the relationship between an equation and the arithmetic process. The algebraic approach becomes a more efficient path to the solution for more complex problems. By seeing the relationship of inverse operations, we can use an algebraic approach to solving problems rather than work backward arithmetically. We build an understanding of solving real-world and mathematical problems by writing and solving equations. ... These number line diagrams provide information such as which numbers are part of the solution through the inclusive closed dot and the exclusive open circle. This visual reinforces an understanding of the number line and the possible infinite answers that make the inequality true.”

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: Expressions & Equations:

• “Beginning in the early years of school, expressions are an integral part of the math curriculum. An expression is a statement that can include numbers, variables, operations, exponents and radicals, or a multitude of combinations of such. In the early years, arithmetic expressions were explored and mastered. In the middle grades, algebraic expressions are introduced, first with whole numbers, then whole numbers with exponents, and then integers with exponents. Following the practice with these, radical and trigonometric expressions are introduced and explored. ... The concepts of functions are vast and will be studied throughout high school and beyond. Having a strong foundation in properties of operations, writing and manipulating expressions, solving for equations with one or more variables, and understanding what answers are part of the solution(s) will allow for deeper understanding of functions and math beyond middle school.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.EE.A cluster involves applying and extending previous understandings of arithmetic to algebraic expressions. Students write and evaluate numerical and algebraic expressions, some of which involve whole-number exponents. The basic skills for this understanding begin in Grade 3, where students fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (3.OA.C.7). In Grade 4, students gain familiarity with factors and multiples, recognizing that a whole number is a multiple of each of its factors (4.OA.B.4). In Grade 5, students explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10 (5.NBT.A.2). In Grade 6, students further apply skills in this cluster as they transition to working with equations and inequalities in 6.EE.B. In Grade 8, students extend their knowledge of exponents in numerical expressions when they create equivalent expressions when working with integer exponents (8.EE.A.1) and while performing operations with numbers expressed in scientific notation (8.EE.A.4). In Algebra these will prove to be essential when students work with polynomials, understanding that they form a system analogous to the integers (HSA.APR.A.1), and when they begin work with logarithms (HSF.BF.B.5).”

• The Detailed Lesson Plan for each lesson lists the Prerequisite Standards required for students to be successful in the lesson. For example, in 6.NS.A.1 Mr. Mung’s Ice Cream, the Prerequisite Standards listed are 3.OA.B.6 and 5.NF.B.7.

• The Detailed Lesson Plan for each lesson includes Cluster Connections that identify connections between clusters and coherence across grade levels. For example, in 6.RP.A.1 For Every Day, Cross-Cluster Connection, “This activity connects 6.RP to 7.RP.A and 8.EE.B, as unit rate is the basis for work involving constant of proportionality and slope of linear equations.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6 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; Recommended: Individual white boards/laminated alternative, Calculator, Algebra tiles.”

Each Detailed Lesson Plan includes a Materials List for each component of the lesson. For example in 6.NS.B.3 Enter the Dragon:

• “Immersion: Materials -  Enter the Dragon Immersion video; Poster paper/butcher paper; Markers/colored pencils

• Data & Computation: Materials - Copies of Enter the Dragon Data Artifact, one per pair  

• Resolution: Materials - Enter the Dragon Resolution video

• Math Simulator: Materials - Enter the Dragon Simulation Trainer; Student Devices; Paper and Pencil; Student Headphones

• Practice Printable: Materials - Copies of Enter the Dragon Practice Printable, 1 per student

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

• Clicker Quiz: Materials - Enter the Dragon Clicker Quiz; Student Devices; Paper and Pencil”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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 6.NS.A.”; 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 6 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 6 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 6 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 6.EE.A.3, Provision Problem, include:

• The Math Simulator Immersion: “Reinforce lesson vocabulary  and ensure students understand the meaning and function of each word.”

• The Math Simulator Data & Computation: “Create vocabulary flash cards with each measure of center or variability with its definition, how to find it, and an example..”

• Simulation Trainer: “Pair students to allow for peer teaching and support. Allow students to use vocabulary flash cards.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Allow students to use vocabulary flash cards and calculators. Consider decoding the word problems together, circling the important numbers, and identifying important words.” 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. 

• Student Reflection: “Pair students to allow for peer teaching and support.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 6 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:

• 6.RP.A.3 Vacation Day Trails, “Bring in and distribute multiple local grocery store ads, and give students a shopping list of common items likely to be found in those ads. Tell students they each have $200 to spend on groceries. The goal is to pay the least for the most groceries. (They will justify their spending by calculating unit prices.)”

• 6.RP.A.2 Road Trip Ratios, “Research the distance to travel from your cities to other cities. Estimate the number of hours that the trip would take. Then find the unit rate to get to each city.”

• 6.G.A.2 River Rescue, “Have students create their own shipping boxes that are right rectangular prisms. Find the volume of each box.”

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 6.SP.A.3 Periodontal Pockets include:

• The Math Simulator Data & Computation: “Task the students with answering this follow up question: ‘The mean of Mr. Novak’s pockets is 3.2. If they measure them again after a month, and the mean is now 3.5, describe what happened with the data. What could be a possible data set now to have a mean of 3.4?’”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Task the students with comparing measures of center and variability of 2 similar data sets. They must explain the similarities and differences in each of the measures of center and variability.”

• 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 6 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 6 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, in 6.NS.C.6b Treasure Trail, Manipulative Task for Digital WorkPad: “Have students return to Problem #4a-c on the Practice Printable and use the WorkPad to graph and reflect the given point. Encourage students to experiment with the different manipulatives provided in the WorkPad to complete the exercise. For example, students could use the grid background and the Line tool to create and label the coordinate plane. They could then use the 2-Color Counters to plot the original point. Students could then use any strategy to reflect the point according to each question prompt, labeling/indicating the reflection method and the new coordinates using the Text tool. The physical act of reflecting on the graph may help students to more clearly see the connection between the type of reflection and the corresponding change in the sign of the coordinates.”