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Report Overview
Summary of Alignment & Usability: Math Nation | Math
Math 6-8
The materials reviewed for Math Nation Grades 6-8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for Usability.
6th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
7th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
8th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Report for 6th Grade
Alignment Summary
The materials reviewed for Math Nation Grade 6 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence, and in Gateway 2, the materials meet expectations for rigor and practice-content connections.
6th Grade
Alignment (Gateway 1 & 2)
Usability (Gateway 3)
Overview of Gateway 1
Focus & Coherence
The materials reviewed for Math Nation Grade 6 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.
Gateway 1
v1.5
Criterion 1.1: Focus
Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for Math Nation Grade 6 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.
Indicator 1A
Materials assess the grade-level content and, if applicable, content from earlier grades.
The materials reviewed for Math Nation Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.
The curriculum consists of nine units, including one optional unit. Assessments include Cool-down Tasks, Mid-Unit Assessments, and End-of-Unit Assessments. Examples of assessment items aligned to grade-level standards include:
Unit 4, Mid-Unit Assessment (B), Question 2, “Kiran has used of the pieces in his jigsaw puzzle. He has used 120 pieces. How many pieces are in the whole puzzle? A. 96; B. 125; C. 150; D. 216” (6.NS.1)
Unit 7, End-of-Unit Assessment (A), Question 2, “Diego’s dog weighs more than 10 kilograms and less than 15 kilograms. Select all the inequalities that must be true if w is the weight of Diego’s dog in kilograms. A. w>10; B. w<10; C. w>11; D. w<11 ; E. w>15; F. w<15.” (6.EE.5)
Unit 8, Lesson 7, Cool-down, Questions 1 and 2, “The two histograms show the points scored per game by a college basketball player in 2008 and 2016. 1. What is a typical number of points per game scored by this player in 2008? What about in 2016? Explain your reasoning. 2. Write 2–3 sentences that describe the spreads of the two distributions, including what spreads might tell us in this context.” (6.SP.5)
Indicator 1B
Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for Math Nation Grade 6 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials present opportunities for students to engage in extensive work and the full intent of most Grade 6 standards. Each lesson contains a Warm-Up, a minimum of one Exploration Activity, a Lesson Summary, Practice Problems, and three Check Your Understanding Questions. Each unit provides a Readiness Check and a Test Yourself! practice tool. Examples of full intent include:
Unit 2, Lesson 8, 2.8.6 Practice Problems, Question 1, engages students with the full intent of 6.RP.3b (Solve unit rate problems including those involving unit pricing and constant speed). Students use double number lines to solve unit pricing problems. “In 2016, the cost of 2 ounces of pure gold was $2,640. Complete the double number line to show the cost for 1, 3, and 4 ounces of gold.” Students are provided with a double number line with one labeled “cost in dollars” and the other labeled “ounces of gold.”
Unit 3, Lesson 6, 3.6.2 Exploration Activity, Question 1, engages students with the full intent of 6.RP.2 (Understand the concept of a unit rate a/b associated with a:b with b≠0, and use rate language in the context of a ratio relationship). Students solve real-world problems using ratios. “Priya, Han, Lin, and Diego are all on a camping trip with their families. The first morning, Priya and Han make oatmeal for the group. The instructions for a large batch say, ‘Bring 15 cups of water to a boil, and then add 6 cups of oats.’ Priya says, ‘The ratio of the cups of oats to the cups of water is 6:15. That’s 0.4 cups of oats per cup of water.’ Han says, ‘The ratio of the cups of water to the cups of oats is 15:6. That’s 2.5 cups of water per cup of oats.’ 1. Who is correct? Explain your reasoning. If you get stuck, consider using the table.” Students are provided with a table with one column labeled “Water (cups)” with rows 15, 1, and and the other labeled “Oats (cups)” with rows 6, , and 1.
Unit 6, Lesson 6, 6.6.6 Practice Problems, Question 3, engages students with the full intent of 6.EE.2a (Write expressions that record operations with numbers and with letters standing for numbers). Students solve problems involving variables. “A bottle holds 24 ounces of water. It has x ounces of water in it. a. What does represent in this situation? b. Write a question about this situation that has for the answer.”
The materials present opportunities for students to engage with extensive work with grade-level problems. Examples of extensive work include:
Unit 3, Lesson 3, Lesson 4, and Lesson 9 engage students in extensive work with 6.RP.3d (Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities). Lesson 3, 3.3.5 Practice Problems, Question 5, students convert units of measurement. “Clare wants to mail a package that weighs pounds. What could this weight be in kilograms?” Answer choices: 2.04, 4.5, 9.92, 4500. Lesson 4, 3.4.2 Exploration Activity, students use information provided in word problems in context to solve ratio reasoning application problems. “Elena and her mom are on a road trip outside the United States. Elena sees this road sign. (Image given stating, “Maximum 80.”) Elena’s mom is driving 75 miles per hour when she gets pulled over for speeding. 1. The police officer explains that 8 kilometers is approximately 5 miles. a. How many kilometers are in 1 mile? b. How many miles are in 1 kilometer? 2. If the speed limit is 80 kilometers per hour, and Elena’s mom was driving 75 miles per hour, was she speeding? By how much?” Unit 3, Lesson 9, 3.9.6 Practice Problems, Question 4, students use ratio reasoning to compare three jobs to find out which paid better. “Andre sometimes mows lawns on the weekend to make extra money. Two weeks ago, he mowed his neighbor’s lawn for hour and earned $10. Last week, he mowed his uncle’s lawn for hours and earned $30. This week, he mowed the lawn of a community center for 2 hours and earned $30. Which jobs paid better than others? Explain your reasoning.”
Unit 6, Lesson 16, Exploration Activity, Practice Problem, and Check Your Understanding engage students in extensive work with 6.EE.9 (Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation). In 6.16.2 Exploration Activity, Question 2, students solve problems identifying independent and dependent variables. “Lin notices that the number of cups of red paint is always of the total number of cups. She writes the equation to describe the relationship. Which is the independent variable? Which is the dependent variable? Explain how you know.” In 6.16.5 Practice Problems, Question 1, students analyze the relationship between two variables using graphs and tables and relate these to an equation. “Here is a graph that shows some values for the number of cups of sugar, s, required to make x batches of brownies. A. Complete the table so that the pair of numbers in each column represents the coordinates of a point on the graph. B. What does the point (8, 4) mean in terms of the amount of sugar and number of batches of brownies? C. Write an equation that shows the amount of sugar in terms of the number of batches.” Students are provided a graph that shows the “cups of sugar” on the y-axis and the “batches of brownies” on the x-axis. In 6.16.7 Check Your Understanding, Question 1, students write an equation to represent one quantity in terms of the other quantity. “At a local farm, you can buy 2 boxes of strawberries for $9.00. Which equation represents the relationship between the boxes of strawberries, x, and the cost in dollars, y ?” Answer choices: (A) ; (B) ; (C) ; (D) .”
Unit 7, Lessons 6 and 7, engage students in extensive work with 6.NS.7 (Understand ordering and absolute value of rational numbers). Lesson 6, 7.6.3 Exploration Activity, Questions 1 and 2, students use information from a word problem to describe what numbers mean in the context of the problem. “A part of the city of New Orleans is 6 feet below sea level. We can use ‘-6 feet’ to describe its elevation, and |-6| ‘feet’ to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe? A. 25 feet; B. |25| feet; C. -8 feet; D. |-8| feet 2. The elevation of a city is different from sea level by 10 feet. Name the two elevations that the city could have.” Lesson 7, 7.7.2 Exploration Activity, Questions 1 and 2, use information from a word problem to place people in order based on a description. “A submarine is at an elevation of −100 feet (100 feet below sea level). Let's compare the elevations of these four people to that of the submarine: Clare's elevation is greater than the elevation of the submarine. Clare is farther from sea level than the submarine. Andre's elevation is less than the elevation of the submarine. Andre is farther away from sea level than the submarine. Han's elevation is greater than the elevation of the submarine. Han is closer to sea level than is the submarine. Lin's elevation is the same distance away from sea level as the submarine's. 1. Complete the table as follows. A. Write a possible elevation for each person. B. Use <, >, or = to compare the elevation of that person to the submarine. C. Use absolute value to tell how far away that person is from sea level (elevation 0). 2. Priya says her elevation is less than the submarine’s and she is closer to sea level. Is this possible? Explain your reasoning.”
The materials do not provide opportunities for students to meet the full intent of the following standard:
While students engage with 6.RP.3a (Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.), students have no opportunities to work on plotting the pairs of values on the coordinate plane to meet the full intent of the grade-level standards.
The materials provide limited opportunities for all students to engage in extensive work with the following standard:
Unit 7, Lesson 11, 7.11.2 Exploration Activity, there are limited opportunities to engage with extensive work on standard 6.NS.6b (Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes). In this example students label, make observations about, and plot coordinates. This is the only opportunity for students to engage with this standard. “1. Label each point on the coordinate plane with an ordered pair. 2. What do you notice about the locations and ordered pairs of 𝐵, 𝐶, and 𝐷? How are they different from those for point A? 3. Plot a point at (-2,5). Label it 𝐸. Plot another point at (3, -4.5). Label it 𝐹. 4. The coordinate plane is divided into four quadrants, I, II, III, and IV, as shown here. A. In which quadrant is G located? H? I? B. A point has a positive y-coordinate. In which quadrant could it be?”
Criterion 1.2: Coherence
Each grade’s materials are coherent and consistent with the Standards.
The materials reviewed for Math Nation Grade 6 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.
Indicator 1C
When implemented as designed, the majority of the materials address the major clusters of each grade.
The materials reviewed for Math Nation Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.
To determine the amount of time spent on major work, the number of units, the number of lessons, and the number of days were examined. Assessment days are included. Any lesson, assessment, or unit marked optional was excluded.
The approximate number of units devoted to major work of the grade (including supporting work connected to the major work) is 5 out of 8, which is approximately 63%.
The approximate number of lessons devoted to major work (including assessments and supporting work connected to the major work) is 95 out of 142, which is approximately 67%.
The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 95 out of 142, which is approximately 67%.
A lesson-level analysis is most representative of the materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 67% of the materials focus on major work of the grade.
Indicator 1D
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The materials reviewed for Math Nation Grade 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
Materials are designed to connect supporting standards/clusters to the grade’s major standards/clusters. Examples of connections include:
Unit 2, Lesson 15, 2.15.7 Practice Problems, Question 5, connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems) and 6.RP.3b (Solve unit rate). Students interpret data from a number line to solve problems involving unit rates students use unit rates to solve real-world problems. “A cashier worked an 8-hour day, and earned $58.00. The double number line shows the amount she earned for working different numbers of hours. For each question, explain your reasoning. a. How much does the cashier earn per hour? b. How much does the cashier earn if she works 3 hours?”
Unit 4, Lesson 13, 4.13.5 Exploration Activity, connects the supporting work of 6.G.1(Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems) to the major work of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions). Students find the length of a tray given the width and area, calculate how many titles would be needed to cover the tray completely, and draw a diagram to represent their answer. “Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of inches and an area of square inches. 1. Find the length of the tray in inches. 2. If the tiles are inch by inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning. 3. Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile.”
Unit 5, Lesson 8, 5.8.7 Check Your Understanding, Question 1, connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions). Students multiply decimals to solve real-world problems. “Victor decided to take a taxi after watching a football game at Raymond James Stadium. The taxi charges a flat fee of $3.50 for pickup, $0.90 for each of the first 4 miles, and $0.75 for each additional mile. Victor’s apartment is 12 miles from the stadium. Complete the statement by typing a value into the blank space. The total fare for Victor’s taxi ride is $_________.”
Unit 6, Lesson 4, 6.4.2 Exploration Activity, connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form and for cases in which p, q and x are all nonnegative rational numbers). Students solve equations using the four operations. “Solve the equations in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren't the same, work together to find the error and correct it. [Column A ; Column B ] [Column A ; Column B ] [Column A ; Column B ].”
Indicator 1E
Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.
The materials reviewed for Math Nation Grade 6 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.
There are connections from supporting work to supporting work and/or major work to major work throughout the grade-level materials, when appropriate. Examples include:
Unit 4, Lesson 11, Cool-down, Question 2, connects the major work of 6.NS.A (Apply and extend previous understandings of multiplication and division to divide fractions by fractions) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Students use fractions to solve word problems. “If liters of water are enough to water of the plants in the house, how much water is necessary to water all the plants in the house? Write a multiplication equation and a division equation for the situation, then answer the question. Show your reasoning.”
Unit 5, Lesson 6, 5.6.3 Exploration Activity, connects the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples) to the supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume). Students use area diagrams to compute the products of decimals. “1. In the diagram, the side length of each square is 0.1 unit. a. Explain why the area of each square is not 0.1 square unit. b. How can you use the area of each square to find the area of the rectangle? Explain or show your reasoning. c. Explain how the diagram shows that the equation is true. 2. Label the squares with their side lengths so the area of this rectangle represents . a. What is the area of each square? b. Use the squares to help you find . Explain or show your reasoning. 3. Label the squares with their side lengths so the area of this rectangle represents . Next, use the diagram to help you find . Explain or show your reasoning.” Students are provided a rectangle divided into eight square boxes for each set of problems.
Unit 6, Lesson 7, 6.7.2 Exploration Activity, connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Students perform repeated calculations involving percentages and generalize to write algebraic expressions. “1. Answer each question and show your reasoning. A. Is 60% of 400 equal to 87? B. Is 60% of 200 equal to 87? C. Is 60% of 120 equal to 87? 2. 60% of x is equal to 87. Write an equation that expresses the relationship between 60%, x, and 87. Solve your equation. 3. Write an equation to help you find the value of each variable. Solve the equation. A. 60% of c is 43.2 B. 38% of e is 190.”
Unit 8, Lesson 4, 8.4.2 Exploration Activity, connects the supporting work of 6.SP.A (Develop understanding of statistical variability) to the supporting work of 6.SP.B (Summarize and describe distributions). Students construct a dot plot from a set of data, make observations about the dot plot, and summarize their observations. “1. Use the tables from the warm-up to display the number of toppings as a dot plot. Label your drawing clearly. 2. Use your dot plot to study the distribution for number of toppings. What do you notice about the number of toppings that this group of customers ordered? Write 2-3 sentences summarizing your observations.”
Indicator 1F
Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
The materials reviewed for Math Nation Grade 6 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
Grade-level concepts related explicitly to prior knowledge from earlier grades along with content from future grades is identified and related to grade-level work in the Teacher Edition. Generally, explicit connections are found in the Course Guide or the Full Lesson Plan.
Examples of connections to future grades include:
Unit 3, Lesson 15, Full Lesson Plan, Lesson Narrative, connects 6.RP.3 to 7.RPA. “This lesson focuses on finding ‘A% of B’ as efficiently as possible…The third activity hints at work students will do in grade 7, namely finding a constant of proportionality and writing an equation to represent a proportional relationship.”
Unit 6, Lesson 13, Full Lesson Plan, 6.13.2 Classroom activity, connects 6.EE.4 to 7.EE.A, A-SSE.A and A-SSE.B. “The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number. For students who are accustomed to viewing the equal sign as a directive that means ‘perform an operation,’ tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, ‘The expressions on either side have the same value.’”
Unit 7, Lesson 14, Full Lesson Plan, Lesson Narrative, connects 6.NS.6 to 7.NS.1 and 8.EE.B, “In this lesson, students explore ways to find vertical and horizontal distances in the coordinate plane. [...] Students will use these skills in Grade 7 to find distances on maps. In Grade 8, they will use these skills to draw slope triangles in the coordinate plane and find the lengths of their sides when considering graphs of proportional and nonproportional relationships.”
Examples of connections to prior knowledge include:
Unit 2, Lesson 2, Full Lesson Plan, 2.2.1 Warm-Up, builds on 4.NF.4 and 5.NF.3 and connects to 6.RP.1. “This number talk helps students recall that dividing by a number is the same as multiplying by its reciprocal. Four problems are given, however, they do not all require the same amount of time. Consider spending 6 minutes on the first three questions and 4 on the fourth question. In grade 4, students multiplied a fraction by a whole number, using their understanding of multiplication as groups of a number as the basis for their reasoning. In grade 5, students multiply fractions by whole numbers, reasoning in terms of taking a part of a part, either by using division or partitioning a whole…Two important ideas that follow from this work and that will be relevant to future work should be emphasized during discussions: Dividing by a number is the same as multiplying by its reciprocal. We can multiply numbers in any order if it makes it easier to find the answer.”
Unit 4, Lesson 2, Full Lesson Plan, Lesson Narrative and 4.2.1 Warm-Up, builds on 3.OA.2 and connects to 6.NS.A. Lesson Narrative, “In this lesson, students revisit the relationship between multiplication and division that they learned in prior grades. Specifically, students recall that we can think of multiplication as expressing the number of equal-size groups, and that we can find a product if we know the number of groups and the size of each group. They interpret division as a way of finding a missing factor, which can either be the number of groups, or the size of one group. They do so in the context of concrete situations and by using diagrams and equations to support their reasoning.”... 4.2.1 Warm-up, “The purpose of this warm-up is to review students' prior understanding of division and elicit the ways in which they interpret a division expression. This review prepares them to explore the meanings of division in the lesson.”
Unit 7, Lesson 4, Full Lesson Plan, 7.4.1 Warm-Up, builds on 4.NBT.2 and 5.NBT.3b and connects to 6.NS.C. “The purpose of this warm-up is for students to review strategies for comparing whole numbers, decimal numbers, and fractions as well as the use of inequality symbols. The numbers in each pair have been purposefully chosen based on misunderstandings students typically have when comparing.”
Indicator 1G
In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.
The materials reviewed for Math Nation Grade 6 foster coherence between grades and can be completed within a regular school year with little to no modification.
The materials will require a little modification to ensure there is content for the entire school year.
The materials contain nine total units with the last unit being optional. Each unit contains between 15-19 lessons and begins with an optional Check-Your-Readiness Assessment and concludes with an End-of-Unit Assessment. Each lesson includes: A warm-up (5-10 minutes in length), one to three Exploration Activities (10-30 minutes in length), Lesson Synthesis (5-10 minutes in length), and Cool down (5 minutes in length). Lessons include “Are you ready for more?” extensions, but do not have specified time allotments explicitly stated in the materials. It is unclear whether the specified time allotted for the “Are you ready for more?” extension fits within the exploration activity it is paired with or if additional time would be needed beyond what is stated in the Full Lesson Plan. Five units include a Mid-Unit Assessment (three are optional).
There are approximately 28.4 weeks of instruction which includes 142 lesson days, including assessments.
Overview of Gateway 2
Rigor & the Mathematical Practices
The materials reviewed for Math Nation Grade 6 meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. The materials make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Gateway 2
v1.5
Criterion 2.1: Rigor and Balance
Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.
The materials reviewed for Math Nation Grade 6 meet expectations for Rigor and Balance. The materials meet expectations for providing students opportunities in developing conceptual understanding, procedural skills, and application, and the materials also meet expectations for balancing the three aspects of Rigor.
Indicator 2A
Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The materials reviewed for Math Nation Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Materials develop conceptual understanding throughout the grade level and materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. There are opportunities for students to develop their conceptual understanding in the various parts of each lesson: Warm-up, Exploration Activities, Lesson Synthesis, Cool Down, Check Your Understanding, and Practice Problems. Additionally, students’ conceptual understanding was assessed on Mid-Unit Assessments and End-of-Unit Assessments.
Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Course Guide, “Concepts Develop from Concrete to Abstract, Mathematical concepts are introduced simply, concretely, and repeatedly, with complexity and abstraction developing over time. Students begin with concrete examples, and transition to diagrams and tables before relying exclusively on symbols to represent the mathematics they encounter.” Examples include:
Unit 2, Lesson 6, 2.6.4 Exploration Activity, students develop conceptual understanding by using ratio and rate reasoning to solve real-world problems (6.RP.3). “Here is a diagram showing Elena's recipe for light blue paint. 1. Complete the double number line diagram to show the amounts of white paint and blue paint in different-sized batches of light blue paint. 2. Compare your double number line diagram with your partner. Discuss your thinking. If needed, revise your diagram. 3. How many cups of white paint should Elena mix with 12 tablespoons of blue paint? How many batches would this make? 4. How many tablespoons of blue paint should Elena mix with 6 cups of white paint? How many batches would this make? 5. Use your double number line diagram to find another amount of white paint and blue paint that would make the same shade of light blue paint. 6. How do you know that these mixtures would make the same shade of light blue paint?”
Unit 4, Lesson 6, 4.6.1 Warm-Up, students develop conceptual understanding as they interpret and compute quotients of fractions (6.NS.1). “We can think of the division expression as the answer to the question: ‘How many groups of ’s are in 10? Complete the tape diagram to represent the question. Then find the answer.”
Unit 6, Lesson 2, Cool-Down, students develop conceptual understanding by explaining how they know value makes an equation true(6.EE.5). “Explain how you know that 88 is a solution to the equation by completing the sentences: The word ‘solution’ means . . . 88 is a solution to because . . .”
The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:
Unit 2, End-of-Unit Assessment (A), Question 1, students develop conceptual understanding as they use ratio language to determine true relationships between quantities (6.RP.1). “Select all the true statements. A. The ratio of triangles to squares is 2 to 4. B. The ratio of squares to smiley faces is 6:4. C. The ratio of smiley faces to triangles is 6 to 4. D. There are two squares for every triangle. E. There are two triangles for every smiley face. F. There are three smiley faces for every triangle.” Provided is a picture of six smiley faces, two triangles, and four squares.
Unit 3, Lesson 11, 3.11.6 Practice Problems, Question 3, students develop conceptual understanding by using ratio reasoning to solve real-world problems (6.RP.3). “At a school, 40% of the sixth-grade students said that hip-hop is their favorite kind of music. If 100 sixth grade students prefer hip hop music how many sixth grade students are at the school? Explain or show your reasoning.”
Unit 6, Lesson 1, 6.1.6 Practice Problems, Question 1, students develop conceptual understanding by drawing a tape diagram to represent an equation and then interpret how parts of the equation are represented in the tape diagram (6.EE.6). “Here is an equation . A. Draw a tape diagram to represent the equation. B. Which part of the diagram shows the quantity x? What about 4? What about 17? C. How does the diagram show that has the same value as 17?”
Indicator 2B
Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.
The materials reviewed for Math Nation Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
There are opportunities for students to develop their procedural skills and fluency throughout the grade levels in each lesson, these opportunities can be found in the Warm-up, Exploration Activities, and Practice Problems. Examples include:
Unit 2, Lesson 8, 2.8.4 Exploration Activity, Questions 1-3, students develop procedural skills and fluency solving problems involving unit rate (6.RP.3b). "1. Four bags of chips cost $6. a. What is the cost per bag? b. At this rate, how much will 7 bags of chips cost? 2. At a used book sale, 5 books cost $15. a. What is the cost per book? b. At this rate, how many books can you buy for $21? 3. Neon bracelets cost $1 for 4. a. What is the cost per bracelet? b. At this rate, how much will 11 neon bracelets cost?"
Unit 4, Lesson 4, 4.4.4 Practice Problems, Question 7, students develop procedural skills and fluency as they solve problems with percents (6.RP.3c). “Find each unknown number. a. 12 is 150% of what number? b. 5 is 50% of what number? c. 10% of what number is 300? d. 5% of what number is 72? e. 20 is 80% of what number?”
Unit 5, Lesson 11, 5.11.3 Exploration Activity, Question 2, students develop procedural skills and fluency as they solve problems using long-division (6.NS.2). “Use long division to find the value of each expression. Then pause so your teacher can review your work. a. ; b. ”
The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:
Unit 4, Lesson 11, 4.11. 7 Check Your Understanding, Question 1, students independently demonstrate procedural skills and fluency while dividing two mixed fractions (6.NS.1). “What is the quotient of ? (A) ; (B) ; (C) ; (D) ”
Unit 6, Lesson 12, 6.12.6 Practice Problems, Question 6, students independently demonstrate procedural skills and fluency by solving one-step equations (6.EE.7). “Solve each equation. A. B. C. D. ”
Unit 7, End-of-Unit Assessment (A), Question 6, students independently demonstrate procedural skills and fluency by positioning pairs of integers on a coordinate plane to draw a polygon (6.NS.6c). “Draw polygon ABCDEF in this coordinate plane, given its vertices A = (-2,-3), B = (0,3), C = (0,1), D = (3,1), E = (3,3), F = (-2,3).”
Indicator 2C
Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.
The materials reviewed for Math Nation Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Materials provide opportunities for students to work with multiple routine and non-routine applications of mathematics throughout the grade level and independently. Applications of mathematics occur throughout a lesson in the exploration activities, practice problems, and assessments.
Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:
Unit 4, Lesson 11, 4.11.3 Exploration Activity, Question 2, students solve a routine word problem involving the division of fractions (6.NS.1). “After biking miles, Jada has traveled of the length of the trip. How long (in miles) is the entire length of her trip? Write an equation to represent the situation, and find the answer using your preferred strategy.”
Unit 5, Lesson 8, 5.8.6 Practice Problems, Question 4, students solve a routine word problem involving adding decimals (6.NS.3). “A pound of blueberries costs $3.98 and a pound of clementines costs $2.49. What is the combined cost of 0.6 pound of blueberries and 1.8 pounds of clementines? Round your answer to the nearest cent.”
Unit 6, Lesson 17, 6.17.1 Warm-Up, students solve a non-routine word problem involving unit rate (6.RP.3b). “Lin and Jada each walk at a steady rate from school to the library. Lin can walk 13 miles in 5 hours, and Jada can walk 25 miles in 10 hours. They each leave school at 3:00 and walk miles to the library. What time do they each arrive?”
Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:
Unit 2, Lesson 9, 2.9.6 Practice Problems, Question 1, students solve a routine word problem independently involving unit rate (6.RP.3b). “Han ran 10 meters in 2.7 seconds. Priya ran 10 meters in 2.4 seconds. a. Who ran faster? Explain how you know. b. At this rate, how long would it take each person to run 50 meters? Explain or show your reasoning.”
Unit 4, Lesson 4, 4.4.4 Practice Problems, Question 1, students solve a non-routine word problem by drawing a diagram and writing a multiplication or division equation to represent the situation (6.NS.1). “Consider the problem: A shopper buys cat food in bags of 3 lbs. Her cat eats lb each week. How many weeks does one bag last? a. Draw a diagram to represent the situation and label your diagram so it can be followed by others. Answer the question. b. Write a multiplication or division equation to represent the situation. c. Multiply your answer in the first question (the number of weeks) by . Did you get 3 as a result? If not, revise your previous work.”
Unit 6, Lesson 7, 6.7.3 Exploration Activity, students solve a routine word problem involving percentages by writing an equation (6.RP.3c and 6.EE.7). “1. Puppy A weighs 8 pounds, which is about 25% of its adult weight. What will be the adult weight of Puppy A? 2. Puppy B weighs 8 pounds, which is about 75% of its adult weight. What will be the adult weight of Puppy B? 3. If you haven't already, write an equation for each situation. Then, show how you could find the adult weight of each puppy by solving the equation.”
Indicator 2D
The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.
The materials reviewed for Math Nation Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade level.
All three aspects of rigor are present independently throughout each grade level. Examples include:
Unit 1, Lesson 15, Cool Down, students develop conceptual understanding as they solve problems with three-dimensional figures (6.G.4). “In this net, the two triangles are right triangles. All quadrilaterals are rectangles. What is its surface area in square units? Show your reasoning. 2. If the net is assembled, which of the following polyhedra would it make?”
Unit 3, Lesson 1, 3.1.6 Practice Problems, Question 5, students develop procedural skills and fluency as they use unit rates to answer questions about sandwiches (6.RP.2). “A sandwich shop serves 4 ounces of meat and 3 ounces of cheese on each sandwich. After making sandwiches for an hour, the shop owner has used 91 combined ounces of meat and cheese. a. How many combined ounces of meat and cheese are used on each sandwich? b. How many sandwiches were made in the hour? c. How many ounces of meat were used? d. How many ounces of cheese were used?”
Unit 6, Lesson 12, 6.12.2 Exploration Activity, students apply their understanding of exponents to write equivalent expressions and evaluate numerical expressions with whole-number exponents (6.EE.1). ”You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one: $50,000 or A magical $1 coin. The coin will turn into two coins on the first day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days. 1. The number of coins on the third day will be 2⋅2⋅2. Write an equivalent expression using exponents. 2. What do and represent in this situation? Evaluate and without a calculator. Pause for discussion. 3. How many days would it take for the number of magical coins to exceed $50,000? 4. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.”
Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study. Examples include:
Unit 2, Lesson 13, 2.13.6 Practice Problems, Question 2, students build conceptual understanding as they become more fluent using ratio and rate reasoning to find equivalent ratios on double number lines and ratio tables (6.RP.3).“A bread recipe uses 3 tablespoons of olive oil for every 2 cloves of crushed garlic. a. Complete the table to show different-sized batches of bread that taste the same as the recipe. b. Draw a double number line that represents the same situation. c. Which representation do you think works better in this situation? Explain why.”
Unit 5, Lesson 10, 5.10.2 Exploration Activity, students discuss with a partner the similarities and differences between different division methods and then divide numbers using one of the methods (6.NS.2). “Lin has a method of calculating quotients that is different from Elena’s method and Andre’s method. Here is how she found the quotient of : 1. Discuss with your partner how Lin’s method is similar to and different from drawing base-ten diagrams or using the partial quotients method. a. Lin subtracted , then , and lastly . Earlier, Andre subtracted , then , and lastly . Why did they have the same quotient? b. In the third step, why do you think Lin wrote the 7 next to the remainder of 2 rather than adding 7 and 2 to get 9? 2. Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor. a. b. c. ”
Unit 8, Lesson 10, 8.10.5 Practice Problems, Question 1, students use the real-world problem of walking to school to build conceptual understanding as they fluently calculate means (6.SP.3). “On school days, Kiran walks to school. Here are the lengths of time, in minutes, for Kiran’s walks on 5 school days. 16 11 18 12 13 A. Create a dot plot for Kiran’s data. B. Without calculating, decide if 15 minutes would be a good estimate of the mean. If you think it is a good estimate, explain your reasoning. If not, give a better estimate and explain your reasoning. C. Calculate the mean for Kiran’s data. D. In the table, record the distance of each data point from the mean and its location relative to the mean. E. Calculate the sum of all distances to the left of the mean, then calculate the sum of distances to the right of the mean. Explain how these sums show that the mean is a balance point for the values in the data set.”
Criterion 2.2: Math Practices
Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).
The materials reviewed for Math Nation Grade 6 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Indicator 2E
Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Nation Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative, and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide).
There is intentional development of MP1 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to analyze and make sense of problems, work to understand the information in problems, and use a variety of strategies to make sense of problems. Examples include:
Unit 1, Lesson 12, 1.12.1 Warm-Up, students estimate the surface area of a cabinet. “Your teacher will show you a video about a cabinet or some pictures of it. Estimate an answer to the question: How many sticky notes would it take to cover the cabinet, excluding the bottom?” This activity attends to the full intent of MP1 as students need to make sense of the problem and persevere in solving it as they are given no specific techniques for how to calculate surface area ahead of time.
Unit 2, Lesson 14, 2.14.2 Exploration Activity, students engage in an Info-Gap activity with a partner. “Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the problem card: a. Read your card silently and think about what you need to know to be able to answer the questions. b. Ask your partner for the specific information that you need. c. Explain how you are using the information to solve the problem. d. Solve the problem and show your reasoning to your partner. If your teacher gives you the data card: a. Read your card silently. b. Ask your partner ‘What specific information do you need?’ and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. c. Have them explain ‘Why do you need that information?’ before telling them the information. d. After your partner solves the problem, ask them to explain their reasoning, even if you understand what they have done. Both you and your partner should record a solution to each problem.” This activity attends to the full intent of MP1 as students need to make sense of the problems by determining what information is necessary and know what questions to ask to receive the information they need to solve the problem.
Unit 3, Lesson 9, 3.9.6 Practice Problems, Question 2, students solve a unit rate problem about a copy machine. “A copy machine can print 480 copies every 4 minutes. For each question, explain or show your reasoning. a. How many copies can it print in 10 minutes? b. A teacher printed 720 copies. How long did it take to print?” This problem attends to the full intent of MP1 as students must make sense of the problem in order to answer the questions about copies and time.
There is intentional development of MP2 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to represent situations symbolically, attend to the meaning of quantities, and understand relationships between problem scenarios and mathematical representations. Examples include:
Unit 2, Lesson 4, 2.4.6 Practice Problems, Question 3, students use reasoning to explain how to apply the ratio concepts learned to mix varying shades of blue paint. “To make 1 batch of sky blue paint, Clare mixes 2 cups of blue paint with 1 gallon of white paint. a. Explain how Clare can make 2 batches of sky blue paint. b. Explain how to make a mixture that is a darker shade of blue than the sky blue. c. Explain how to make a mixture that is a lighter shade of blue than the sky blue.” This problem attends to the full intent of MP2 as students need to reason abstractly and quantitatively about how to make the different batches of paint.
Unit 7, Lesson 13, 7.13.2 Exploration Activity, students reason quantitatively about a graph and answer questions based on the information in the graph. “The graph shows the balance in a bank account over a period of 14 days. The axis labeled b represents account balance in dollars. The axis labeled d represents the day. 1. Estimate the greatest account balance. On which day did it occur. 2. Estimate the least account balance. On which day did it occur? 3. What does the point (6,-50) tell you about the account balance? 4. How can we interpret |-50| in the context?” This problem intentionally develops the full intent of MP2 as students need to reason about the meaning of quantities given the context of the problem.
Unit 8, Lesson 14, Cool-Down, students reason quantitatively and abstractly regarding the measures of center for given data sets. “For each dot plot or histogram: 1. Predict if the mean is greater than, less than, or approximately equal to the median. Explain your reasoning. 2. Which measure of center—the mean or the median— better describes a typical value for the following distributions? 1. Heights of 50 NBA basketball players 2. Backpack weights of 55 sixth-grade students 3. Ages of 30 people at a family dinner party.” This problem attends to the full intent of MP2 as students reason quantitatively and abstractly about the means and medians of data.
Indicator 2F
Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Nation Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials provide opportunities for students to construct viable arguments and critique the reasoning of others in whole class and small group settings (i.e. exploration activities) and independent work settings (i.e. practice problems and assessments).
There is intentional development of MP3 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to construct mathematical arguments by explaining/justifying their strategies and thinking, performing error analysis of provided student work/solutions, listening to the arguments of others and deciding if it makes sense, asking useful questions to better understand, and critiquing the reasoning of others. Examples include:
Unit 1, Lesson 2, 1.2.6 Practice Problems, Question 5, students explain why a statement about area is incorrect. “A student said, ‘We can't find the area of the shaded region because the shape has many different measurements, instead of just a length and a width that we could multiply.’ Explain why the student's statement about area is incorrect.” This question intentionally develops MP3 as students critique the reasoning of others and explain why the statement is incorrect.
Unit 2, Lesson 8, 2.8.1 Warm-Up, students mentally find a quotient and then explain their strategy. “Find the quotient mentally. ” Full Lesson Plan, Teacher Guidance: “Invite students to share their strategies. Record and display student explanations for all to see. Ask students to explain if or how the dividend or divisor impacted their choice of strategy and how they decided to write their remainder. To involve more students in the conversation, consider asking: • ‘Who can restate ___’s reasoning in a different way?’ • ‘Did anyone solve the problem the same way but would explain it differently?’ • ‘Did anyone solve the problem in a different way?’ • ‘Does anyone want to add on to _____’s strategy?’ • ‘Do you agree or disagree? Why?’” This activity intentionally develops MP3 as students explain their strategy for agreeing or disagreeing with their classmates' strategies.
Unit 3, Lesson 6, 3.6.6 Practice Problems, Question 4, students calculate unit rate for a real-world situation. “A large art project requires enough paint to cover 1,750 square feet. Each gallon of paint can cover 350 square feet. Each square foot requires of a gallon of paint. Andre thinks he should use the rate gallons of paint per square foot to find how much paint they need. Do you agree with Andre? Explain or show your reasoning.” This question intentionally develops MP3 as students critique the reasoning of others while constructing arguments to justify their conclusions.
Indicator 2G
Materials support the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Nation Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide).
Examples where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:
Unit 6, Lesson 1, 6.1.1 Warm-Up, students identify equations that match diagrams and draw a diagram that represents equations. “Here are two diagrams. One represents . The other represents . Which is which? Label the length of each diagram. Draw a diagram that represents each equation. 1. ; 2. " This activity intentionally develops (MP4), model with mathematics, and (MP5) use appropriate tools strategically as students choose which tools to use as they create their model.
Unit 6, Lesson 6, 6.6.6 Practice Problems, Question 1, students write and evaluate expressions with numbers and variables. “Instructions for a craft project say that the length of a piece of red ribbon should be 7 inches less than the length of a piece of blue ribbon. How long is the red ribbon if the length of the blue ribbon is: 10 inches? 27 Inches? x inches? How long is the blue ribbon if the red ribbon is 12 inches?” Students are given the option to represent expressions with tape diagrams. This activity intentionally develops (MP4), model with mathematics, and(MP5) use appropriate tools strategically as students can choose to use an appropriate tool to set-up the expressions based on the scenario.
Unit 7, Lesson 9, 7.9.1 Warm-up, Question 1, given a number line with several points students complete blank inequality statements with the points to make the inequality true. “1. Fill in each blank with a letter so that the inequality statements are true A. ___ > ___ B. ___ < ___ 2. Jada says that she found three different ways to complete the first question correctly. Do you think this is possible? Explain your reasoning. 3. List a possible value for each letter on the number line based on its location.“ This activity intentionally develops (MP4), model with mathematics, and (MP5) use appropriate tools strategically as students use the number line strategically in order to answer the questions.
Unit 8, Lesson 3, 8.3.1 Warm-Up, students create a statistical question about a given scenario and explain their reasoning. ”Clare collects bottle caps and keeps them in plastic containers. Write one statistical question that someone could ask Clare about her collection. Be prepared to explain your reasoning.” This activity intentionally develops MP4, model with mathematics.
Indicator 2H
Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Nation Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide).
There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students communicate using grade-level appropriate vocabulary and conventions and formulate clear explanations when engaging with course materials. Students must calculate accurately and efficiently, specify units of measure, and use and label tables and graphs appropriately when engaging with course materials. Teacher guidance very clearly develops the specialized language of mathematics as teachers are explicitly prompted when to introduce content-related vocabulary and use accurate definitions when communicating mathematically. Examples include:
Unit 1, Lesson 5, 1.5.1 Warm-Up, students compare and contrast two strategies for finding the area of a parallelogram. “Elena and Tyler were finding the area of this parallelogram: Move the slider to see how Tyler did it: [An applet shows you how Tyler decomposed his parallelogram] Move the slider to see how Elena did it: [An applet shows you how Elena decomposed his parallelogram How are the two strategies for finding the area of a parallelogram the same? How are they different?” Full Lesson Plan, Teacher Guidance: “The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a base. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a height that corresponds to that base.” This activity attends to the specialized language of mathematics as students learn and are encouraged to use the correct terms to describe Elena and Tyler's strategies.
Unit 2, Lesson 5, 2.5.6 Practice Problems, Question 1, students explain why given ratios are equivalent. “Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a diagram that shows why they are equivalent ratios. 1. 4:5 and 8:10; 2. 18:3 and 6:1; 3. 2:7 and 10,000:35,000,” This activity attends to the full intent of MP6 and the specialized language of mathematics as students communicate using grade-level appropriate vocabulary and conventions.
Unit 4, Lesson 5, 4.5.6 Practice Problems, Question 3, students use a ruler to write multiplication and division equations. ”Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answer the question. How many s are in 7? How many s are in 6? How many s are in ?” This activity attends to the full intent of MP6 and the specialized language of mathematics as students communicate using grade-level appropriate vocabulary such as halves, eighths…etc.
Unit 6, Lesson 2, 6.2.1 Warm-Up, students determine if an equation is true or false when substituting in chosen values. “The equation could be true or false. a. If a is 3, b is 4, and c is 5, is the equation true or false? b. Find new values of a, b, and c that make the equation true. c. Find new values of a, b, and c that make the equation false.” This activity attends to the full intent of MP6 as students would need to attend to precision to get the correct value.
Indicator 2I
Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math Nation Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 6, Course Guide).
There is intentional development of MP7 to meet its full intent in connection to grade-level content. Materials provide opportunities for students to look for patterns or structures to make generalizations and solve problems, look for and explain the structure of mathematical representations, and look at and decompose “complicated” into “simpler.” Examples include:
Unit 1, Lesson 3, 1.3.2 Exploration Activity, students find the area of the shaded region in different diagrams by decomposing, rearranging, subtracting, and enclosing figures. “Each grid square is 1 square unit. Find the area, in square units, of each shaded region without counting every square. Be prepared to explain your reasoning. (Students are given four figures.)” This activity attends to the full intent of MP7 as students use the structure of the shape to decompose them into simpler ones.
Unit 4, Lesson 8, 4.8.1 Warm-Up, students interpret a division statement and write a question in which the equation represents the scenario. “1. Think of a situation with a question that can be represented by = ? Describe the situation and the question. 2. Trade descriptions with your partner, and answer your partner's question.” This activity intentionally develops MP7 as students must look at the structure of the equation in order to write a problem that represents it.
Unit 5, Lesson 5, 5.5.1 Warm-up, students compare the same variable in each equation to determine which value is the largest, multiplying by 10 to consider the effect on how the decimal point moves. “1. In which equation is the value of x the largest? A. B. C. D. 2. How many times the size of 0.81 is 810?” This activity intentionally develops MP7 as students use the structure of the equations to multiply other decimal products.
There is intentional development of MP8 to meet its full intent in connection to grade-level content.
Materials provide opportunities for students to notice repeated calculations to understand algorithms and make generalizations or create shortcuts, evaluate the reasonableness of their answers and their thinking, and create, describe, or explain a general method/formula/process/algorithm. Examples include:
Unit 1, Lesson 9, 1.9.1 Warm-Up, students identify base-height measurements of triangles, use them to determine area, and look for a pattern in their reasoning to help them write a general formula for finding area. “Study the examples and non-examples of bases and heights in a triangle. Answer the questions that follow. These dashed segments represent heights of the triangle. These dashed segments do not represent heights of the triangle. Select all the statements that are true about bases and heights in a triangle.” The following are the statements: Any side of a triangle can be a base; There is only one possible height; A height is always one of the sides of a triangle; A height that corresponds to a base must be drawn at an acute angle to the base; A height that corresponds to a base must be drawn at a right angle to the base; Once we choose a base, there is only one segment that represents the corresponding height; and A segment representing a height must go through a vertex. This warm-up attends to the full intent of MP8 as students create, describe, and explain a general formula, process, method, algorithm, model, etc. This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students must repeatedly reference the information given to determine which of the statements are correct.
Unit 3, Lesson 7, 3.7.2 Exploration Activity, Question 1, students calculate the unit price of a burrito and then generalize to find the cost for any number of burritos. “Two burritos cost $14.00. Complete the table to show the cost of 4, 5, and 10 burritos at that rate. Next, find the cost for a single burrito in each case.” This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students use repeated calculations to fill out the table in order to create a general method.
Unit 6, Lesson 10, 6.10.3 Exploration Activity, students complete a table representing the width, length, and area of several rectangles. “For each rectangle, write expressions for the length and width of two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.” Students are given six rectangles labeled A - F. This activity attends to the full intent of MP8 as students notice repeated calculations to understand algorithms and make generalizations or create shortcuts. This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students repeatedly write the expressions for area of the rectangle, they formulate a general method for the distributive property.
Overview of Gateway 3
Usability
The materials reviewed for Math Nation Grade 6 series meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports, Criterion 2, Assessment, and Criterion 3, Student Supports.
Gateway 3
v1.5
Criterion 3.1: Teacher Supports
The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.
The materials reviewed for Math Nation Grade 6 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, include standards correlation information that explains the role of the standards in the context of the overall series, provide explanations of the instructional approaches of the program and identification of the research-based strategies, and provide a comprehensive list of supplies needed to support instructional activities. The materials partially contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject.
Indicator 3A
Materials provide 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.
The materials reviewed for Math Nation 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 to guide their mathematical development.
Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:
Course Overview: A Course Overview (Unit 0) is found at the beginning of each course. Within each Course Overview there is a Course Narrative, which contains a summary of the mathematical content contained in each course, and a Course Guide. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Resources, Corrections, and Cool-Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the About These Materials section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool-Down. The How to Use These Materials section contains guidance about the three phases of classroom activities (Launch-Work-Synthesize) and utilizing instructional routines. In the Scope and Sequence section, teachers will find a Pacing Guide which contains time estimates for coverage of each of the units.
Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessments (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Student Task Statements Cool-downs, Practice Problems, Blackline Masters, and My Reflections all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Full Lesson Plan downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.
Full Lesson Plan: Within each Teacher Edition lesson component, teachers can find a Full Lesson Plan that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.
Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Course Guide, Lesson Plans, Lesson Narratives, Overviews, and Warm-up provide useful annotations. Examples include:
Course Guide, Assessments Overview, “Pre-Unit Diagnostic Assessments At the start of each unit is a pre-unit diagnostic assessment. These assessments vary in length. Most of the problems in the pre-unit diagnostic assessment address prerequisite concepts and skills for the unit. Teachers can use these problems to identify students with particular below-grade needs, or topics to carefully address during the unit. Teachers are encouraged to address below-grade skills while continuing to work through the on-grade tasks and concepts of each unit, instead of abandoning the current work in favor of material that only addresses below-grade skills…What if a large number of students can’t do the same pre-unit assessment problem? Look for opportunities within the upcoming unit where the target skill could be addressed in context…What if all students do really well on the pre-unit diagnostic assessment? Great! That means they are ready for the work ahead, and special attention likely doesn’t need to be paid to below-grade skills.”
Unit 1, Lesson 9, Full Lesson Plan, 1.9.3 Exploration Activity, “Anticipated Misconceptions The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the warm-up activity.”
Unit 5, Lesson 8, Full Lesson Plan, “Lesson Narrative In this culminating lesson on multiplication, students continue to use the structure of base-ten numbers to make sense of calculations (MP7) and consolidate their understanding of the themes from the previous lessons. They see that multiplication of decimals can be accomplished by: thinking of the decimals as products of whole numbers and fractions; writing the non-zero digits of the factors as whole numbers, multiplying them, and moving the decimal point in the product; representing the multiplication with an area diagram and finding partial products; and using a multiplication algorithm, the steps of which can be explained with the reasonings above.”
Indicator 3B
Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.
The materials reviewed for Math Nation Grade 6 partially meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current courses that teachers can improve their own knowledge of the subject. The materials do not contain adult-level explanations and examples of concepts beyond the current grade so that teachers can improve their own knowledge of the subject.
Each lesson includes a Teacher Prep Video and a Full Lesson Plan resource that contains adult-level explanations and examples of the more complex grade-level concepts. Examples include:
A 5-10 minute Teacher Prep Video that provides an overview of the lesson, including content and pedagogy tips is provided for each lesson. During the video a Math Nation Instructor goes through the lesson, highlighting grade-level concepts and showing examples, while also giving suggestions that teachers can use during the lesson to support students.
Unit 4, Lesson 3, Full Lesson Plan, Lesson 3 Synthesis, “In this lesson, we solved problems that involved multiplication and division. Reiterate to students that in division situations that involve equal-size groups, we are not always looking for the same unknown. There are typically three pieces of information involved: the number of groups, the size of each group, and the total amount. Knowing what information we have and what is missing can help us answer questions.”
Unit 7, Lesson 6, Full Lesson Plan, Lesson 6 Synthesis, “Students should see that the order remained the same for the positive numbers but reversed for the negative numbers. They should be able to explain that as numbers move to the left on the number line, their absolute value gets larger because they are further from 0. This realization should help solidify the thinking that has been building for the past several lessons about the ordering and magnitude of rational numbers.”
Indicator 3C
Materials include standards correlation information that explains the role of the standards in the context of the overall series.
The materials reviewed for Math Nation Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.
The Course Guide, About These Materials sections, states the following note about standards alignment, “There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on.’ When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards.’ When a task is focused on the grade-level work, the alignment is indicated as ‘addressing.’” All lessons in the materials have this correlation information. An example:
Unit 7, Lesson 4, Full Lesson Plan, Lesson Standards Alignment, Building on 4.NBT.2, 5.NBT.3b; Addressing 6.NS.C, 6.NS.6, 6.NS.6a, 6.NS.7; Building Towards 6.NS.7.
Explanations of the role of the specific grade-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative, and Full Lesson Plan. Examples include:
Course Guide, Scope and Sequence, Unit 1: Area and Surface Area, “Work with area in grade 6 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area…In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths…In grade 8, students will understand ‘identical copy of’ as ‘congruent to’ and understand congruence in terms of rigid motions, that is, motions such as reflection, rotation, and translation. In grade 6, students do not have any way to check for congruence except by inspection, but it is not practical to cut out and stack every pair of figures one sees…”
Course Guide, Scope and Sequence, Unit 2: Introducing Ratios, “Work with ratios in grade 6 draws on earlier work with numbers and operations. In elementary school, students worked to understand, represent, and solve arithmetic problems involving quantities with the same units. In grade 4, students began to use two-column tables, e.g., to record conversions between measurements in inches and yards. In grade 5, they began to plot points on the coordinate plane, building on their work with length and area…Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond. Because of this, students should learn to use tables to solve all kinds of ratio problems, but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking…The terms proportion and proportional relationship are not used anywhere in the grade 6 materials. A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. In high school-after their study of ratios, rates, and proportional relationships-students discard the term ‘unit rate,’ referring to a to b, a:b, and ab as ‘ratios.’”
Indicator 3D
Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.
The materials reviewed for Math Nation Grade 6 provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.
Unit Overview videos, available through the Math Nation website, and unit lesson summary videos, links to Vimeo and YouTube, outline the mathematics that students will be learning in that unit. Family Support materials are available for each unit (available digitally and can be printed; available in English and Spanish). These provide a brief overview of some of the main concepts taught within each unit followed by tasks, with worked solutions, for parents/caregivers to work on with their student. Examples include:
Student Edition, Unit 1, Family Support: Area and Surface Area, “Here are the video lesson summaries for Grade 6 Unit 1, Area and Surface Area. Each video highlights key concepts and vocabulary that students learn across one or more lessons in the unit. The content of these video lesson summaries is based on the written Lesson Summaries found at the end of lessons in the curriculum. The goal of these videos is to support students in reviewing and checking their understanding of important concepts and vocabulary. Here are some possible ways families can use these videos:
Keep informed on concepts and vocabulary students are learning about in class.
Watch with their student and pause at key points to predict what comes next or think up other examples of vocabulary terms (the bolded words).
Consider following the Connecting to Other Units links to review the math concepts that led up to this unit or to preview where the concepts in this unit lead to in future units.”
Five videos are provided (via Vimeo or Youtube) that take families through the lessons in the unit.
Student Edition, Unit 4, Family Support: Dividing Fractions, Algorithm for Fraction Division, “Many people have learned that to divide a fraction, we ‘invert and multiply.’ This week, your student will learn why this works by studying a series of division statements and diagrams such as these…”
Unit 6, Family Materials, Relationships Between Quantities, Lessons 16-18, “This week your student will study relationships between two quantities…Here is a task to try with your student: A shopper is buying granola bars. The cost of each granola bar is $0.75. 1. Write an equation that shows the cost of the granola bars, c, in terms of the number of bars purchased, n. 2. Create a graph representing associated values of c and n. 3. What are the coordinates of some points on your graph? What do they represent?” Solutions with explanations are provided for families.
Indicator 3E
Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.
The materials reviewed for Math Nation Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials sections.
The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”
Examples of materials including and referencing research-based strategies include:
“The Five Practices Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem…”
“Supporting English Language Learners This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012).”
“Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”
Within the Course Guide, How to Use These Materials, a Reference section is included.
Indicator 3F
Materials provide a comprehensive list of supplies needed to support instructional activities.
The materials reviewed for Math Nation Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Resources), Teacher Editions, for each lesson, under Lesson Preparation (Required Material(s)), and in Teacher Guides for specific lessons. Examples include:
Unit 1, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.1, Cool-down, copies of blackline master, geometry toolkits (tracing paper, graph paper, colored pencils, scissors, and an index card)”
Unit 3, Lesson 3, Lesson Preparation, Required Materials: “Blackline master for Activity 3.2, Cool-down, base-ten blocks, blank paper, cuisenaire rods, gallon-sized jug, graduated cylinders, household items, inch cubes, internet-enabled device, liter-sized bottle, materials assembled from the blackline master, metal paper fasteners, meter sticks, pre-assembled polyhedra, quart-sized bottle, rulers, salt, scale, straightedges, teaspoon, tray”
Unit 7, Lesson 7, Lesson Preparation, Required Materials: “Cool-down, sticky notes”
Indicator 3G
This is not an assessed indicator in Mathematics.
Indicator 3H
This is not an assessed indicator in Mathematics.
Criterion 3.2: Assessment
The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.
The materials reviewed for Math Nation Grade 6 meet expectations for Assessment. The materials include an assessment system that provides multiple opportunities throughout the courses to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up and provide assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices. The materials partially include assessment information that indicates which standards and practices are assessed.
Indicator 3I
Assessment information is included in the materials to indicate which standards are assessed.
The materials reviewed for Math Nation Grade 6 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed.
The materials consistently identify the standards assessed for each of the problems in each of the following formal assessments: Mid-Unit Assessment, End-of-Unit Assessment, and Cool-Downs. All assessments are available as Word or PDF downloads in English or Spanish versions. Materials do not identify the practices assessed for any of the formal assessments.
Examples of how the materials consistently identify the standards for assessment include:
Unit 2, Lesson 8, Cool-down, “Here is a double number line showing that it costs $3 to buy 2 bags of rice: At this rate, how many bags of rice can you buy with $12? 1. Find the cost per bag. 2. How much do 20 bags of rice cost?” The Full Lesson Plan identifies the standard alignment as 6.RP.3b.
Unit 4, End-of-Unit Assessment (A), Question 5, “Andre draws this tape diagram for 3 ÷ : Andre says that 3 ÷ = 4 because there are 4 groups of and left. Do you agree with Andre? Explain your reasoning.” Aligned Standard: 6.NS.1.
Unit 6, Mid-Unit Assessment (B), Question 4, “ of the students in a school are in sixth grade. 1. How many sixth graders are there if the school has 70 students? 2. How many sixth graders are there if the school has 28 students? 3. If the school has x students, write an expression for the number of sixth graders in terms of x. 4. How many students are in the school if 63 of them are sixth graders?” Aligned Standards: 6.EE.6 and 6.EE.7.
Indicator 3J
Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
The materials reviewed for Math Nation Grade 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses.
Suggestions to teachers for following up with students are provided throughout the materials via the Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.
Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:
Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses.
Restricted Constructed Response
Tier 1 response: Work is complete and correct.
Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.
Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.
Extended Response
Tier 1 response: Work is complete and correct, with complete explanation or justification.
Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors.
Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”
Unit 5, End-of-Unit Assessment (B), Question 6, “A sign in a bakery gives these options: 12 muffins for $25; 24 muffins for $46; 50 muffins for $94 1. Find each unit price to the nearest cent, and show your reasoning. 2. Which option gives the lowest unit price? Solution 1. The unit prices are $2.08, $1.92, and $1.88. Reasoning varies, but long division is a reasonable approach. 2. 50 muffins for $94 has the lowest unit price. Minimal Tier 1 response: Work is complete and correct. Sample: (Accompanied by work showing long division or other calculation methods.) The unit prices are $2.08, $1.92, and $1.88. 50 muffins for $94 is the best deal. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Acceptable errors: incorrect selection of lowest unit rate comes from errors in calculation of unit rates. Sample errors: one or two errors in long division; incorrect selection of the best deal despite having calculated all three unit rates correctly. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: calculation of unit rates involves a conceptual error, such as dividing the number of muffins by the price; three or more errors in long division; correct selection of the best deal without calculation of each unit rate.”
Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following up with students include:
Course Guide, Cool-Down Guidance states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”
Unit 1, Check-Your-Readiness (A), Question 5, “The content assessed in this problem is first encountered in Lesson 5: Bases and Heights of Parallelograms. In this unit, students will find the area of parallelograms and triangles by decomposing them into shapes with perpendicular sides and rearranging the pieces. Students will need to be familiar with perpendicular lines in order to make sense of the ‘height’ of a parallelogram or triangle. If most students struggle with this item, plan to start Lesson 5 Activity 2 by amplifying the term perpendicular for the students. Students may need some visual cues to support this concept.”
Unit 6, Lesson 15, Cool-down Guidance, “Support Level 2. Points to Emphasize. Notes If students struggle with using properties of exponents strategically in the cool-down, plan to focus on this idea when opportunities arise over the next several lessons. For example, in the practice problem set for Lesson 17, consider inviting students to reflect on the reasoning behind prompt 3.”
Indicator 3K
Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.
The materials reviewed for Math Nation Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.
All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in-classroom. Examples Include:
Unit 1, Lesson 2, Cool-down, demonstrates the full intent of 6.G.1. “The square in the middle has an area of 1 square unit. What is the area of the entire rectangle in square units? Explain your reasoning.” A figure is given that is composed of many different shapes with a square in the middle of it.
Unit 3, Lesson 6, Cool-down, demonstrates the full intent of 6.RP.3 and MP2. “Two pounds of grapes cost $6. 1. Complete the table showing the price of different amounts of grapes at this rate. 2. Explain the meaning of each of the numbers you found.” A table is provided with two columns one labeled “grapes (pounds)” and the other labeled “price (dollars)”.
Unit 7, End-of-Unit Assessment (A), Question 4, demonstrates the full intent of 6.NS.7. “Given , mark and place these expressions on the same number line. , , , , , ”
Indicator 3L
Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.
The materials reviewed for Math Nation Grade 6 do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.
Assessments are available in English and Spanish and are designed to be downloaded as Word documents or PDFs and administered in class. There is no modification or guidance given to teachers within the materials on how to administer the assessment with accommodations.
Criterion 3.3: Student Supports
The program includes materials designed for each student’s regular and active participation in grade-level/grade-band/series content.
The materials reviewed for Math Nation Grade 6 meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations and for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics; multiple extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity; and manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.
Indicator 3M
Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.
The materials reviewed for Math Nation 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 mathematics.
Course Guide, How to Use These Materials, Supporting Students with Disabilities sections states the following: “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many children who struggle to access rigorous, grade-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Suggested supports are identified for teachers in the Full Lesson Plan to support learners of all levels. Lesson and activity-level supports, identified as “Support for Students with Disabilities,” are aligned to an area of cognitive functioning and are paired with a suggested strategy aimed to increase access and eliminate barriers. Supports are classified under the following categories: eliminate barriers, processing time, peer tutors, assistive technology, visual aids, graphic organizers, and brain breaks. Examples include:
Eliminate Barriers: “Eliminate any barriers that students may encounter that prevent them from engaging with the important mathematical work of a lesson. This requires flexibility and attention to areas such as the physical environment of the classroom, access to tools, organization of lesson activities, and means of communication.” Unit 2, Lesson 15, Full Lesson Plan, 2.15.3 Exploration Activity, “Support for Students with Disabilities Executive Functioning: Eliminate Barriers. Chunk this task into more manageable parts (e.g., presenting one question at a time), which will aid students who benefit from support with organizational skills in problem solving”.
Peer Tutors: “Develop peer tutors to help struggling students access content and solve problems. This support keeps all students engaged in the material by helping students who struggle and deepening the understanding of both the tutor and the tutee. For students with disabilities, peer tutor relationships with non-disabled peers can help them develop authentic, age-appropriate communication skills, and allow them to rely on a natural support while increasing independence.” Unit 6, Lesson 6, Full Lesson Plan, 6.6.3 Exploration Activity, “Support for Students with Disabilities… Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors…”
Processing Time: “Increased time engaged in thinking and learning leads to mastery of grade level content for all students, including students with disabilities. Frequent switching between topics creates confusion and does not allow for content to deeply embed in the mind of the learner. Mathematical ideas and representations are carefully introduced in the materials in a gradual, purposeful way to establish a base of conceptual understanding. Some students may need additional time, which should be provided as required.” Unit 4, Lesson 12, Full Lesson Plan, 4.12.1 Warm Up, “Support for Students with Disabilities Memory: Processing Time. Provide sticky notes or mini whiteboards to aid students with working memory challenges. Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.”
There are several accessibility options (accessed via the wrench icon in the lower left-hand corner of the screen) available to help students navigate the materials. Examples include:
Tools Menu allow students to change the language, and access a Demos Scientific and Graphing Calculator.
Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, screen reader support.
UserWay, allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), Saturation (4 settings).
Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.
Indicator 3N
Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.
The materials reviewed for Math Nation Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.
Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. Every extension problem is made available to all students with the heading ‘Are You Ready for More?’ These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts at grade level or that are outside of the standard K-12 curriculum. They are not routine or procedural, and intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students would complete these optional activities, then they might be doing more assignments than their classmates.
Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:
Unit 1, Lesson 15, 1.15.4 Exploration Extension: Are you Ready for More?, “1. Figure C shows a net of a cube. Draw a different net of a cube. Draw another one. And then another one. Show your work. 2. How many different nets can be drawn and assembled into a cube?”
Unit 2, Lesson 1, 2.1.4 Exploration Extension: Are you Ready for More?, “Use two colors to shade the rectangle so there are 2 square units of one color for every 1 square unit of the other color. 2. The rectangle you just colored has an area of 24 square units. Draw a different shape that does not have an area of 24 square units, but that can also be shaded with two colors in a 2:1 ratio. Shade your new shape using two colors.”
Unit 8, Lesson 13, 8.13.3 Exploration Extension: Are you Ready for More?, “Invent a data set with a mean that is significantly lower than what you would consider a typical value for the data set.”
Indicator 3O
Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.
The materials reviewed for Math Nation Grade 6 provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.
The Course Guide, About These Materials, Design Principles section states the following: “Developing Conceptual Understanding and Procedural Fluency Each unit begins with a pre-assessment that helps teachers ascertain what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. Distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.”
Examples of where materials provide varied approaches to learning tasks over time and variety of how students are expected to demonstrate their learning include:
Unit 1, Lesson 19, 1.19.2 Exploration Activity, students compare their tent design with other students and reflect on their measurements and choices in design. “1. Explain your tent design and fabric estimate to your partner or partners. Be sure to explain why you chose this design and how you found your fabric estimate. 2. Compare the estimated fabric necessary for each tent in your group. Discuss the following questions: A. Which tent design used the least fabric? Why? B. Which tent design used the most fabric? Why? C. Which change in design most impacted the amount of fabric needed for the tent? Why?” Full Lesson Plan, Classroom Activity Narrative: “This activity gives students a chance to explain and reflect on their work. In groups of 2–3, they share drawings of their tent design, an estimate of the amount of fabric needed, and the justification. They compare their creations with one or more peers. Students discuss not only the amount of fabric required, but also the effects that different designs have on that amount.”
Unit 6, Lesson 9, 6.9.6 Practice Problems, Question 2, students draw diagrams showing two different ways to apply the distributive property when multiplying two multi-digit integers. “Draw and label diagrams that show these two methods for calculating 19 ⋅ 50. A. First find 10 ⋅ 50 and then add 9 ⋅ 50. 2. First find 20 ⋅ 50 and then take away 50.”
Unit 8, Lesson 17, 8.17.4 Cool-Down, Question 3, students have to make sense of box plots to determine whether they agree or disagree with interpretation statements. “Humpback whales are one of the larger species of whales that can be seen off the coast of California. Suppose that researchers measured the lengths, in feet, of 20 male humpback whales and 20 female humpback whales. Then, the researchers drew two box plots to summarize the data. 3. Do you agree with each of the following statements about the whales that were measured? Explain your reasoning. a. More than half of male humpback whales measured were longer than 46 feet. b. The male humpback whales tended to be longer than female humpback whales. c. The lengths of the male humpback whales tended to vary more than the lengths of the female humpback whales.”
Students can monitor their learning in the following ways: The “Check Your Understanding” provides three questions at the end of each lesson that covers the standards from the lesson and is auto-scored. Students are able to get feedback about the correct solution(s). The “Test Yourself! practice tool” provides ten questions (of different item types) taken at the end of the unit and is composed of the entire unit standards. It is also auto-scored, students can see what they got correct and incorrect, and a solution video for any question they choose.
Indicator 3P
Materials provide opportunities for teachers to use a variety of grouping strategies.
The materials reviewed for Math Nation Grade 6 provide opportunities for teachers to use a variety of grouping strategies.
The Course Guide, How to Use These Materials, states the following about groups: “Group Presentations Some activities instruct students to work in small groups to solve a problem with mathematical modeling, invent a new problem, design something, or organize and display data, and then create a visual display of their work. Teachers need to help groups organize their work so that others can follow it, and then facilitate different groups' presentation of work to the class.” Additionally, “the launch for an activity frequently includes suggestions for grouping students. This gives students the opportunity to work individually, with a partner, or in small groups.” However, the guidance is general and is not targeted based on the needs of individual students. Examples include:
Unit 4, Lesson 9, Full Lesson Plan, 4.9.2 Exploration Activity, “Launch…Arrange students in groups of 3–4. Show the short video. Ask students what questions we could ask about the amount of water in this situation that would require working with fractions to determine the answers. Give groups a moment to think about their questions. If needed, show the video again, or refer to the photos to identify the fractions.”
Unit 5, Lesson 4, Full Lesson Plan, 5.4.3 Exploration Activity, “Launch Arrange students in groups of 2. Give students 8–10 minutes of quiet work time and 2–3 minutes to discuss their answers with a partner. Follow with a whole-class discussion.”
Unit 7, Lesson 16, Full Lesson Plan, 7.16.2 Exploration Activity, “Launch Arrange students in groups of 2. Give students 10 minutes work time followed by whole-class discussion. Encourage students to check in with their partner after each question to make sure they get every possible combination of bags.”
Indicator 3Q
Materials provide 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.
The materials reviewed for Math Nation 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.
The Course Guide, How to Use These Materials section states the following: “The framework for supporting English language learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The four design principles are, support sense-making, optimize output, cultivate conversation, and maximize meta-awareness. Each design principle has an explanation that goes into more detail about how teachers can use it to support students. The routines are the Mathematical Language Routines (MLRs), the materials state, “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. Each MLR facilitates attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment for all learners. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others' ideas.” These design principles and routines are referenced under Instructional Routines, in the Full Lesson Plan for lesson, to assist teachers with lesson planning. The “Supports for English Language Learners” section within the Full Lesson Plan contains explanations of how to implement the MLRs.
Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:
Unit 1, Lesson 18, Full Lesson Plan, 1.18.3 Exploration Activity, “Support for English Language Learners Conversing: MLR 3 Clarify, Critique, Correct. Present an incorrect response such as, ‘If the cube has edge length s, then the area of each face is 2s because ‘.’ Ask students to identify the error and to offer a correct argument to write an expression for the area of each face. This will help students to use symbolic representations while generalizing calculations related to surface area. Design Principle(s): Optimize output (for generalization); Maximize meta-awareness”
Unit 2, Lesson 16, Full Lesson Plan, 2.16.2 Exploration Activity, “Support for English Language Learners Speaking: MLR 8 Discussion Supports. Provide sentence frames for students to state their reasoning (e.g., ‘I liked this method of solving the problem because _______’; ‘This way worked best because ________’; ‘The _____ strategy is the same as / different from the _____ strategy because _____ ‘). The helps students place extra attention on the language used to engage in mathematical communication and reasoning. Design Principle(s): Maximize meta-awareness; Optimize output (for generalization)”
Unit 6, Lesson 4, Full Lesson Plan, 6.4.3 Exploration Activity, “Support for English Language Learners Reading: MLR 6 Three Reads. Demonstrate this routine with the first situation to support reading comprehension. Use the first read to help students understand context. Ask, ‘What is this situation about?’ (e.g., Clare and Mai each have a different number of books). After the second read, ask students ‘What are the quantities in the situation?’ (e.g., the number of books Mai has, the number of books Clare has). After the third read, ask students to brainstorm possible strategies to connect the situation with the appropriate equation(s). Encourage students to repeat this routine themselves for each situation. This helps students connect the language in the word problem with the equation(s) while keeping the intended level of cognitive demand in the task. Design Principle(s): Support sense-making”
Indicator 3R
Materials provide a balance of images or information about people, representing various demographic and physical characteristics.
The materials reviewed for Math Nation Grade 6 provide a balance of images or information about people, representing various demographic and physical characteristics.
The materials include problems depicting persons of different genders, races, ethnicities, and those with various physical characteristics. Instructional videos contain a diverse group of presenters of various races and/or ethnicities. Included in the lesson activities is a balance of positive portrayals of persons representing various demographic groups. This is indicated by the names used in problems and the images shown in some of the problems. The materials also reference various countries and regions, historical figures and works of art that contain mathematical designs, and contributions of ancient mathematicians within the problems. Examples include:
Unit 2, End-of-Unit Assessment (B), Question 7, “To make fruit punch, Priya mixes 3 scoops of powder with 5 cups of water. Mai mixes 4 scoops of powder with 6 cups of water. 1. Create a double number line or a table that shows different amounts of powder and water that taste the same as Priya’s mixture. 2. Create a double number line or a table that shows different amounts of powder and water that taste the same as Mai’s mixture. 3. How do their two mixtures compare in taste? Explain your reasoning.”
Unit 5, Lesson 3, 5.3.7 Check Your Understanding, “Lauren is clearing a path in the woods to work on. She has cleared 0.94 miles so far. The path will be 1.25 miles when she is done. How much more does she have to clear?”
Unit 7, Lesson 12, 7.12.6 Practice Problems, “Diego was asked to plot these points: (-50, 0), (150, 100), (200, -100), (350, 50), (-250, 0) What interval could he use for each axis? Explain your reasoning.”
Indicator 3S
Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.
The materials reviewed for Math Nation Grade 6 partially provide guidance to encourage teachers to draw upon student home language to facilitate learning.
Materials can be accessed in different languages by clicking on the wrench icon in the lower left-hand corner of the Teacher and Student Edition web pages. The web page content is then displayed in the selected language (135 options available). All Unit-level downloadable files (For example: Assessments and Unit Level Downloads) are available in English and Spanish. All Lesson-level downloadable files are only available in English. The lesson videos for students can be viewed in English and Spanish.
Additionally, the first time glossary terms are introduced in the materials they have a video attached to them, the video is available in five languages: English, Spanish, Haitian Creole, Portuguese, and American Sign Language. Students have access to all the glossary terms and videos in the Glossary section under Student Resources.
The materials do not provide guidance to encourage teachers to draw upon student home language to facilitate learning.
Indicator 3T
Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.
The materials reviewed for Math Nation Grade 6 do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning. Although, throughout the materials, references are made to other cultures and different social backgrounds, no guidance is provided to teachers to draw upon students’ cultural and social backgrounds to facilitate learning.
Indicator 3U
Materials provide supports for different reading levels to ensure accessibility for students.
The materials reviewed for Math Nation Grade 6 provide supports for different reading levels to ensure accessibility for students.
In the Full Lesson Plan, some of the supports identified as “Supports for Students with Disabilities,” could assist students who struggle with reading to access the mathematics of the lesson. The videos embedded within each lesson narrate the problem and may help struggling readers in accessing the mathematics of the exploration activity or practice problems. The materials provide Math Language Routines (MLRs) that are specifically geared directly to different reading levels to ensure accessibility for students. Detailed explanations of how to use these routines are included in the Full Lesson Plan in the “Supports for English Language Learners” section. However, none of these supports directly address different student reading levels. Examples include:
Unit 2, Lesson 5, Full Lesson Plan, 2.5.2 Exploration Activity, “Support for Students with Disabilities Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.”
Unit 4, Lesson 4, Full Lesson Plan, 4.4.2 Exploration Activity, “Support for English Language Learners Representing, Writing: MLR 3 Clarify, Critique, Correct. Ask students to share their responses to the first question before moving on to the second. In this discussion, present an incorrect response that reflects a possible common misunderstanding. For example, ‘The area of the rhombus is 3 because 3 fit inside the hexagon.’ Prompt discussion by asking, ‘Do you agree with the statement? Why or why not?’ Ask students to identify the error, correct the statement, and draw a diagram to represent the situation. Improved statements should include fractional language and direct connections to the diagram. This helps students evaluate, and improve on, the written mathematical arguments of others. Design Principle(s): Maximize meta-awareness; Optimize output (for justification)“
Unit 6, Lesson 10, Full Lesson Plan, 6.10.2 Exploration Activity, “Support for English Language Learners Reading: MLR 6 Three Reads. Use this routine to support reading comprehension. In the first read, students read the situation with the goal of comprehending the text (e.g., the problem is about two rectangles with some dimensions given). In the second read, ask students to analyze the text to understand the mathematical structure (e.g., the width of both rectangles is 5. The length of one rectangle is 8 and the other rectangle’s length is x). In the third read, ask students to brainstorm possible strategies to answer the follow-up questions. This routine helps students in reading comprehension and negotiating information in the text with a partner through mathematical language. Design Principle(s): Support sense-making”
Indicator 3V
Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.
The materials reviewed for Math Nation Grade 6 meet expectations for providing manipulatives, both virtual and physical, that are representations of the mathematical objects they represent and, when appropriate, are connected to written methods.
Virtual and physical manipulatives support student understanding throughout the materials. Examples include:
Unit 1, Lesson 4, 1.4.2 Exploration Activity, students use an applet to find the area of parallelograms by decomposing a parallelogram into pieces to find the area. “1. Find the area of each parallelogram. Show your reasoning. 2. Change the parallelogram by dragging the green points at its vertices. Find its area and explain your reasoning. 3. If you used the polygons on the side, how were they helpful? If you did not, could you use one or more of the polygons to show another way to find the area of the parallelogram?” Two Geogebra applets are provided with seven parallelograms on each, which students can manipulate to find the area.
Unit 4, Lesson 14, 4.14.3 Exploration Activity, Questions 1 and 2, students use physical cubes or an applet of snap cubes to model the volume of a rectangular prism. “Use cubes or the applet to help you answer the following questions. 1. Here is a drawing of a cube with an edge length of 1 inch. a. How many cubes with an edge length of inch are needed to fill this cube? b. What is the volume, in cubic inches, of a cube with an edge length of inch? Explain or show your reasoning. 2. Four cubes are piled in a single stack to make a prism. Each cube has an edge length of inch. Sketch the prism, and find its volume in cubic inches.” A GeoGebra applet is available that allows students to make prisms by manipulating the block length, width, and height.
Unit 6, Lesson 8, 6.8.6 Practice Problems, Question 1, students use an applet to show and explain when two equations are equivalent. A. Draw a diagram of and a diagram of 2x when x is 1. B. Draw a diagram of and 2x when x is 2. C. Draw a diagram of and 2x when x is 3. D. Draw a diagram of and 2x when x is 4…” Students have access to an applet with a grid where they can draw the diagrams.
Criterion 3.4: Intentional Design
The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.
The materials reviewed for Math Nation Grade 6 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards; and have a visual design that supports students in engaging thoughtfully with the subject that is neither distracting nor chaotic. The materials do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, and do not provide teacher guidance for the use of embedded technology to support and enhance student learning.
Indicator 3W
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.
The materials reviewed for Math Nation Grade 6 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards, when applicable.
All lessons have a Desmos Calculator and Desmos Graphing Calculator for students to use as they wish. Additionally, lessons contain multiple interactive activities embedded throughout the series to support students' engagement in mathematics. Examples include:
Unit 1, Lesson 2, 1.2.2 Exploration Activity, Question 1, students create shapes with specified areas, using an applet. “This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit. Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it. 1. Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.”
Unit 4, Lesson 3, 4.3.2 Exploration Activity, Question 2, students draw diagrams using an applet and write equations to represent simple division situations involving making jam. “Here is an applet to use if you choose to. The toolbar includes buttons that represent 1 whole and fractional parts, as shown here. Click a button to choose a quantity, and then click in the workspace of the applet window to drop it. When you're done choosing pieces, use the Move tool (the arrow) to drag them into the jars. The jars in this applet are shown as stacked to make it easier to combine the jam and find out how much you have. You can always go back and get more pieces, or delete them with the Trash Can tool. 2. Priya filled 5 jars, using a total of 7 cups of strawberry jam. How many cups of jam are in each jar?”
Unit 8, Lesson 9, 8.9.2 Exploration Activity, Question 1c, students have the option to use an applet to draw at least two different distributions that meet certain criteria. “C. Another room in the shelter has 6 crates. No two crates contain the same number of kittens, and there is an average of 3 kittens per crate. Draw or describe at least two different arrangements of kittens that match this description. You may choose to use the applet to help.“ Provided is an interactive applet, where the student can pick the number of crates, the max cats per crate, and drag the picture of cats into crates.
Indicator 3X
Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.
The materials reviewed for Math Nation Grade 6 partially include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.
In the Teacher Edition, Lesson Preparation, Community Created Resource section, teachers are able to leave their names and comments on a Google Sheet that provides teachers access to resources created by other teachers as well as their comments and/or questions. There is no opportunity for students to collaborate with teachers or other students using digital technology.
Indicator 3Y
The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.
The materials reviewed for Math Nation Grade 6 have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.
There is a consistent design within units and lessons that support student understanding of mathematics. Examples include:
Each unit contains the following components: Unit Introduction, Assessments (In English or Spanish), and Unit Level Downloads (In English or Spanish). All assessments and unit-level downloads are available as either PDFs or Word documents.
Lessons begin with the Learning Target(s) which let students know the objective(s) of the lesson. Each lesson uses a consistent format with the following components: Warm-Up, followed by Exploration and Extension Activities, a Lesson Summary, Practice Problems, and Check Your Understanding (2-3 problems that review lesson concepts).
Teacher and student edition: Lesson outlines are always on the left and lesson content is always on the right of the screen. Tab to jump to the top when needed. Videos are highlighted in blue ovals labeled “Videos.” When students need to respond to questions it is either a blue rectangle that says “free response”, a blue oval that says “show your work”, or a pencil icon in a blue box.
Indicator 3Z
Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.
The materials reviewed for Math Nation Grade 6 partially provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.
In the Lesson Preparation, Full Lesson Plans are available for download either as Word documents or PDFs. Some lesson plans provide guidance for using embedded technology to support and enhance student learning. Examples include:
Unit 1, Lesson 4, Full Lesson Plan, 1.4.2 Exploration Activity, Launch, “For digital classrooms, project the applet to introduce it. Ask students to experiment with the given polygons to find the area of the parallelograms. For the second question, students are given the same starting parallelogram as in the first question. They will need to move the vertices to change it into a different parallelogram before finding its area.”
Unit 2, Lesson 10, Full Lesson Plan, 2.10.2 Exploration Activity, Launch, “If students have digital access, they can use an applet to explore the problem and justify their reasoning before sharing with a partner. If students have not used the number line applet in previous activities or need a refresher as to how to use it, demonstrate the treadmill problem with the applet.”
Unit 6, Lesson 12, Full Lesson Plan, 6.12.2 Exploration Activity, Launch, “Ask students to close their books or devices. Display the scenario above for all to see, or explain it verbally. Ask students, “What do you notice? What do you wonder?” It is natural to wonder which is the better option. Poll the class and record the results. If possible, show the first few screens from the applet at https://ggbm.at/wetnwfkf to help students see how the coins double each day, keeping the “Count” hidden. Use the Play and Pause buttons in the lower left corner of the screen. If it cannot be projected for all to see, ask students to describe what the first four days of the second offer would look like. Draw their descriptions for all to see.”