6th Grade - Gateway 2

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.”