6th Grade - Gateway 3

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

• ORIGO Stepping Stones 2.0 Comprehensive Mathematics, Teacher Edition, Program Overview, The Stepping Stone structure, provides a program that is interconnected to allow major, supporting, and additional clusters to be coherently developed. “One of the most unique things about ORIGO Stepping Stones is the approach to sequencing content and practice. Stepping Stones uses a spaced teaching and practice approach in which each content area is spaced apart, the key ideas and skills of these topics have been identified and placed in smaller blocks (modules) over time. In the actual lessons, work is included to help students fully comprehend what is taught alongside the other content development. Consequently, when students come to a new topic, it can be easily connected to previous work.”

• Module 1, Resources, Preparing for the module, Focus, provides an overview of content and expectations for the module. “In this module, students begin with a review of whole number place value, focusing on multi-digit numbers. They explore patterns in place-value charts and examine numbers written in expanded form, at times including the use of exponents. Review of the place-value chart includes a look to the right of the decimal point to extend the patterns into decimal fractions. These explorations set the stage for decimal-format abbreviations for multi-digit numbers. Students think in terms of billions of dollars, rounding to the nearest 100 million for the digit to the right of the decimal point. Next, students explore exponents, initially in the context of area of squares and volume of cubes. The models show students what powers of 2 and 3 look like and provide opportunities for re-writing exponent expressions as repeated multiplication in a familiar context. Students are formally introduced to integers in Lesson 1.5. In this lesson, students brainstorm instances of integers in daily life and make connections between what they already know about the positive portion of the number line and the newly introduced negative portion of the number line. Students focus on reasoning about the position of numbers on the number line and on extending what they know to this new realm. This supports interpreting the negative sign in Lesson 1.6. Here, students are introduced to the idea of the negative sign as “the opposite of” and use their reasoning about symmetry from Lesson 1.5 to understand that −2 is the opposite of + 2 or −(−2) is the opposite of −2 or 2. These experiences lead naturally to comparing integers in Lesson 1.7. This work is based on the idea that the number farthest right on the number line has the greatest value. Absolute value is introduced in Lesson 1.8 as the distance from 0. Here, students are asked to separate the two components of an integer — how far the number is from 0 (the absolute value), and the direction the number is from 0 (the sign: positive or negative). In the last four lessons of this module, students use their new learning about negative numbers to plot points, first in quadrants 1 and 2 (where all y-coordinates are positive) and then in all four quadrants. Students move among tables of values, graphs, and lists of coordinates in these exercises. In Lesson 1.11, students are asked to find side lengths of rectangles whose coordinates cross an axis. This requires application of absolute value as students realize one vertex is three units above the axis and the opposite vertex (the other end of the length) is six units below the axis. Finally, students revisit the idea of reflecting a shape to draw a reflection of a given polygon and list the coordinates of the vertices. Prime notation is introduced in Lesson 1.12 as a way to indicate corresponding vertices of a transformed figure.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson, such as the Step In, Step Up, Step Ahead, Lesson Slides, Step 1 Preparing the Lesson, while other components, like the Step 2 Starting the lesson, Step 3 Teaching the lesson, and Step 4 Reflecting on the work, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” Lesson notes can also highlight potential misconceptions to support teacher planning and practice. Examples include:

• Module 1, Lesson 5, Number: Introducing positive and negative numbers, Step 2 Starting the lesson, teachers are provided context about positive and negative integers. “Project slide one and ask, What two numbers on this number line have a difference of 5? Encourage students to give pairs of numbers. Decimal fractions or common fractions may be included. Write each  pair of numbers on the board.”

• Module 4, Lesson 2, Algebra: Writing equations to match word problems, Step 3 Teaching the lesson, provides teachers guidance about how to set up equations to match word problems. “Project slide 2 as shown, and discuss the points below. Organize the students into pairs to write an equation to match the word problem. Invite one pair to write their equation on the board and explain what each part represents. Continue until all variations of the equation (for example, 65 – (5 • 5.50) = t, (5 • 5.50) + t = 65, 65 – t = (5 • 5.50), or t + (5 • 5.50) = 65) are written on the board. Project the word problem (slide 3) and read it aloud. Say, Three students wrote these equations to match this word problem. Have the pairs determine which of the equations is correct. Afterward, invite pairs to share and justify their findings, including why the other equations are incorrect. Encourage the other students to ask questions to clarify, if necessary, and to critique their reasoning.”

• Module 8, Lesson 6, Ratio: Percentages of collections, Lesson overview and focus, Misconceptions, include guidance to address common misconceptions as students work with percentages. “Because these examples focus on percentages less than 100%, students may develop the misconception that percentages are only used in this range. When examples arise in the news or through classroom conversation, discuss percentages greater than 100% to help students recognize that these are valid values as well.”

The materials reviewed for Origo Stepping Stones 2.0 Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Within Module Resources, Preparing for the module, there are sections entitled “Research into practice” and “Focus” that consistently link research to pedagogy. There are adult-level explanations including examples of the more complex grade-level concepts so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. There are also professional learning videos, called MathEd, embedded across the curriculum to support teachers in building their knowledge of key mathematical concepts. Examples include:

• Module 2, Preparing for the module, Focus, Operations, explains concepts connected to operations with decimal fractions. “In the last part of Module 2, students review addition and subtraction of decimal fractions before exploring multiplication of decimals. In Lesson 2.9, students add and subtract decimals with regrouping, emphasizing the need to align decimal points to ensure place-value columns are correct. In Lesson 2.10, students are encouraged to use partial products for multiplying a decimal fraction by a whole number. The situation can be described by the expression 4 × 8.2, and students partition this to 4 × 8 and 4 × 0.2. This strategy is compared with the standard algorithm for multiplication. It is important to note that students place the decimal point in these lessons by reasoning about its position, not by counting digits to the right of the decimal point. If the product of 4 × 8.2 is 328 (without the decimal point), it makes sense that the product would be 32.8 because it would be a bit more than 32. In Lessons 2.11 and 2.12, students work with greater factors. They are still asked to reason about an estimated product and then to place the decimal point using their estimation. The standard multiplication algorithm is used for calculations. Some of the problems given require regrouping.”

• Module 4, Preparing for the module, Research into practice, Algebra, includes explanations and examples connected to expressions, equations, and formal algebraic notation. To learn more includes additional adult-level explanations for teachers. “As students begin to understand algebra, it is essential that they make connections to arithmetic. The expressions and equations students write using variables and more formal algebraic notation are the same problems they solved in the elementary grades when they learned about the various situations each of the four operations describes. This is a start unknown subtraction problem — students begin solving these problems with whole numbers in Grades 1 and 2. As students learn to write expressions and equations to describe situations, they can see the connections between algebra and arithmetic. Students must be able to make meaning from the symbols of arithmetic (+, −, ×, ÷, and =) in order for algebraic notation to make sense. The equals sign is particularly important. Many students understand the equals sign operationally, as a cue that the answer is next. These students will say that 5 = 1 + 4 is written backwards or will complete the number sentence 3 + 5 = __ + 2 with 8 because 3 + 5 = 8. To be successful in algebra, students must understand the equals sign relationally, as equivalence. Both sides of the equation (3 + 5 and 6 + 2) represent the same value (8) and so the two are equal. Both 5 and 1 + 4 represent the same value so they are equal, regardless of order. The symbol shows the relationship between the expressions. Algebraic notation is a powerful tool in mathematics and variables are a key component of algebraic notation. Variables can be used in many different ways, depending on the situation. Many students are first introduced to variables that represent a specific unknown value (for example, 3 × __ = 9 so __ = 3). Another use of variables is as a term that can vary, and can take on multiple values in a given situation. This latter idea of a variable builds on patterning when students see a pattern in a mathematical situation and use variables to describe what is happening. In this context, variables allow a relationship to be shown in an efficient form. If the cat eats 3 cans of food per day, then the equation f = 3d describes the number of cans of food eaten in a given number of days. Some students look for patterns on tables by working with each row or column separately. More algebraic reasoning requires looking for the relationship that connects the two.” To learn more, “Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Portsmouth, NH: Heinemann.”

• Module 5, Research into Practice, Ratio, supports teachers with concepts for work beyond the grade. “As the Mathematics Focus suggests, this work with ratio sets the stage for the ongoing work with ratio (Modules 6, 8, 11, and 12) and provides a foundation for learning about proportional relationships in Grade 7 in the major cluster 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. In preparation, provide every opportunity for students to investigate, represent, and describe part-part and part-whole relationships in context. Read more in the Research into Practice sections of Modules 6, 8, 11, and 12.”

• Module 11, Preparing for the module, Research in practice, Ratio, supports teachers with concepts for work beyond the grade. “As the Mathematics Focus suggests, the work in this and previous modules serves as a foundation to the introduction of percentages greater than 100% and percentage discounts/markups (Module 12). It provides the groundwork for learning about proportional relationships in Grade 7 in the major cluster 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems. Provide many opportunities for students to investigate and discuss different ways of representing a problem in context as they work toward a solution. For example, drawing a picture to help their understanding, or using different symbolic representations (decimal and/or common fractions, or percentages) to make the calculations easier (30% of 4⁄5 = 0.3•0.8). Read more in the Research into Practice section of Module 12.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the curriculum front matter and program overview, module overview and resources, and within each lesson. Examples include:

• Front Matter, Grade 6 and the CCSS by Lesson includes a table with each grade level lesson (in columns) and aligned grade level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Front Matter, Grade 6 and the Common Core Standards, includes all Grade 6 standards and the modules and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Module 6, Module Overview Resources, Lesson Content and Learning Targets, outlines standards, learning targets and the lesson where they appear. This is present for all modules and allows teachers to identify targeted standards for any lesson.

• Module 5, Lesson 1, Division: Interpreting division situations, the Core Standard is identified as 6.NS.A.1. The Prior Learning Standard is identified 4.MD.A.2. Lessons contain a consistent structure that includes Lesson Focus, Topic progression, Formative assessment opportunity, Misconceptions, Step 1 Preparing the lesson, Step 2 Starting the lesson, Step 3 Teaching the lesson, Step 4 Reflecting on the work, and Maintaining concepts and skills. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each module includes a Mathematics Overview that includes content standards addressed within the module as well as a narrative outlining relevant prior and future content connections. Each lesson includes a Topic Progression that also includes relevant prior and future learning connections. Examples include:

• Module 5, Mathematics Overview, The Number System, includes an overview of how the math of this module builds from previous work in math. “Module 5 begins with an exploration of fraction division. Students have experience dividing a whole number and a fraction from Grade 5. They extend their learning now to fraction by fraction division. Lessons 5.1 and 5.2 ask students to write division expressions for problem situations and to draw models of those situations. Students should recognize that the actual division problems are similar to those they have solved in the past, except now there are more fractions. Making this connection to whole number division helps many students reason effectively about these problems. Students solve the problems using visual models and reasoning in this module. They do not need the algorithm at this stage.”

• Module 10, Mathematics Overview, Coherence, includes an overview of how the content in sixth grade connects to mathematics students will learn in seventh grade. “Lessons 10.8-10.12 focus on calculating volume of rectangular-based prisms with up to three fractional side lengths. This work extends from finding and calculating volume of prisms (5.2.7- 5.2.12) and solving volume word problems (5.11.12) and serves as a foundation for solving real-world problems involving angle measure, area, surface area, and volume in Grade 7.”

• Module 12, Lesson 3, Ratio: Using complementary percentages, Topic progression, “Prior learning: In Lesson 6.12.2, students review percentages greater than 100%. They assess situations to determine where it makes sense to have a percentage greater than 100%, and complete equivalency statements to match. 6.RP.A.3, 6.RP.A.3c; Current focus: In this lesson, students relate a percentage reduction of an item to the amount to be paid (for example, 25% off an item means 75% of the total will be paid). They then calculate an answer using a preferred method. 6.RP.A.3, 6.RP.A.3c; Future learning: In Lesson 6.12.4, students relate a percentage increase of an item to the amount to be paid (for example, a 20% increase in the cost price of an item means that 120% of the cost price will be paid). 6.RP.A.3, 6.RP.A.3c” Each lesson provides a correlation to standards and a chart relating the target standard(s) to prior learning and future learning.

The materials reviewed for ORIGO Stepping Stones 2.0 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 are described within the Pedagogy section of the Program Overview at each grade. Examples include:

• Program Overview, Pedagogy, The Stepping Stones approach to teaching concepts includes the mission of the program as well as a description of the core beliefs. “Mathematics involves the use of symbols, and a major goal of a program is to prepare students to read, write, and interpret these symbols. ORIGO Stepping Stones introduces symbols gradually after students have had many meaningful experiences with models ranging from real objects, classroom materials and 2D pictures, as shown on the left side of the diagram below. Symbols are also abstract representations of verbal words, so students move through distinct language stages (see right side of diagram), which are described in further detail below. The emphasis of both material and language development summarizes ORIGO's unique, holistic approach to concept development. A description of each language stage is provided in the next section. This approach serves to build a deeper understanding of the concepts underlying abstract symbols. In this way, Stepping Stones better equips students with the confidence and ability to apply mathematics in new and unfamiliar situations.”

• Program Overview, Pedagogy, The Stepping Stones approach to teaching skills helps to outline how to teach a lesson. “In Stepping Stones, students master skills over time as they engage in four distinctly different types of activities. 1. Introduce. In the first stage, students are introduced to the skill using contextual situations, concrete materials, and pictorial representations to help them make sense of the mathematics. 2. Reinforce. In the second stage, the concept or skill is reinforced through activities or games. This stage provides students with the opportunity to understand the concepts and skills as it connects the concrete and pictorial models of the introductory stage to the abstract symbols of the practice stage. 3. Practice. When students are confident with the concept or skill, they move to the third stage where visual models are no longer used. This stage develops accuracy and speed of recall. Written and oral activities are used to practice the skill to develop fluency. 4. Extend. Finally, as the name suggests, students extend their understanding of the concept or skill in the last stage. For example, the use-tens thinking strategy for multiplication can be extended beyond the number fact range to include computation with greater whole numbers and eventually to decimal fractions.” 

• Program Overview, Pedagogy, The Stepping Stones structure outlines the learning experiences. “The scope and sequence of learning experiences carefully focuses on the major clusters in each grade to ensure students gain conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply this knowledge to solve problems inside and outside the mathematics classroom. Mathematics contains many concepts and skills that are closely interconnected. A strong curriculum will carefully build the structure, so that all of the major, supporting, and additional clusters are appropriately addressed and coherently developed. One of the most unique things about ORIGO Stepping Stones is the approach to sequencing content and practice. Stepping Stones uses a spaced teaching and practice approach in which each content area is spaced apart, the key ideas and skills of these topics have been identified and placed in smaller blocks (modules) over time. In the actual lessons, work is included to help students fully comprehend what is taught alongside the other content development. Consequently, when students come to a new topic, it can be easily connected to previous work. For example, within one module students may work on addition, time, and shapes, addressing some of the grade level content for each, and returning to each one later in the year. This allows students to make connections across content and helps students master content and skills with less practice, allowing more time for instruction.”

Research-based strategies within the program are cited and described regularly within each module, within the Research into practice section inside Preparing for the module. Examples of research- based strategies include:

• Module 2, Preparing for the module, Research into practice, “Algebra: In Grade 6, students are expected to understand the order of operations more deeply. For many students, the order of operations is an acronym (PEMDAS) rather than a meaningful way of thinking about mathematics. There are four common misconceptions about the order of operations: Multiplication must come before division; Addition must come before subtraction; Operations must be performed from left to right; and Parentheses come first. Students must understand that multiplication and division are done in the order they appear from left to right, as are addition and subtraction. A multiplication to the right of an addition (2 + 3 × 4) is done before the addition — operations are not all done in one left to right sequence. Finally, grouping symbols (not just parentheses) are used to re-sequence operations. In (2 + 3) × 4, the parentheses push the addition to the beginning of the sequence. In 3 × 4 ÷ (1 + 2), the multiplication of 3 and 4 or the addition of 1 and 2 could happen first. The parentheses mean to add before you divide. Number: Students learn about factors as the numbers (often prime) that are multiplied together to make a product. While the procedures of factoring are simple, students are sometimes challenged to use what they know about factors to solve problems, including finding the Greatest Common Factor (GCF) or Least Common Multiple (LCM). Students may not realize that a number with both 2 and 3 as prime factors also has 6 as a factor. They may work only with the factors that are visible, ignoring factors that are products of prime factors. Operations: Students use their understanding of place value to add and subtract decimal fractions. This understanding of place value also supports multiplication of decimals. Students reason about the approximate size of a product in order to place the decimal point correctly in a product. This supports developing strong operation sense around decimal multiplication. Decimal multiplication challenges the common misconception (present even in many adults) that multiplication makes things bigger and division makes things smaller. To learn more: Dupree, Kami M. 2016. “Questioning the Order of Operations.” Mathematics Teaching in the Middle School 22 (3): 152–59. Feldman, Ziv. 2014. “Rethinking Factors.” Mathematics Teaching in the Middle School 20 (4): 230–36. Jeon, Kyungsoon. 2012. “Reflecting on PEMDAS.” Teaching Children Mathematics 18 (6): 370–77. Empson, Susan B. and Linda Levi. 2011. Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann. References: Vamvakoussi, Xenia, Wim Van Dooren, and Lieven Verschaffel. 2013. “Educated Adults Are Still Affected by Intuitions about the Effect of Arithmetical Operations: Evidence from a Reaction-time Study.” Educational Studies in Mathematics 82, no. 2: 323–30.”

• Module 6, Preparing for the module, Research into practice, “Area: Just as students have learned to compose and decompose numbers, they can also compose and decompose shapes. Once students understand that measuring area is about counting the number of square unit tiles inside a shape, formulas can be figured out by reasoning based on what students already know. The area of a rectangle is a simple calculation. The use of visual images allows students to develop and use the formulas for the area of parallelograms and right triangles. They can then extend this understanding to reason about any quadrilateral or triangle — in fact, any polygon. Even if students have access to a formula list during high- stakes assessment, these experiences reasoning about the formulas will minimize confusion between area and perimeter as well as use them more effectively (and accurately) because they understand them. Ratio: Unit rate is one of the first ways students reason proportionally. By asking, “How much for one?” students are able to figure out solutions to complex problems in ways that make sense to them. Rates are composed units (e.g. miles per hour or dollars per pound) that describe a relationship. Unit rates have been simplified to include a one in the relationship. Depending on the situation, it may be appropriate to leave the rate in whole numbers other than one. Students build proportional reasoning skills by using these skills to solve problems. They must understand that there are multiple strategies for solving a problem and they will come to understand that these different strategies are related to one another. For example, students can use a common multiple or a unit rate when comparing prices. Both methods work because common multiples build on unit rates. To learn more: Beigie, Darin. 2016. “Dare to Compare.” Mathematics Teaching in the Middle School 21 (8): 460–69. Sinclair, Nathalie, David Pimm, Melanie Skelin, and Rose Mary Zbiek. 2012. Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 6-8. Reston, VA: National Council of Teachers of Mathematics. References: Cramer, Kathleen and Thomas Post. 1993. “Making Connections: A Case for Proportionality.” The Arithmetic Teacher 40 (6): 342–46. de la Cruz, Jessica A. and Sandra Garney. 2016. “Saving Money Using Proportional Reasoning.” Mathematics Teaching in the Middle School 21 (9): 552–61. Van de Walle, John, Karen Karp, and Jenny Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally (8th Edition). Boston: Pearson.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. In the Program Overview, Program components, Preparing for the module, “Resource overview - provides a comprehensive view of the materials used within the module to assist with planning and preparation.” Each module includes a Resource overview to outline supplies needed for each lesson within the module. Additionally, specific lessons include notes about supplies needed to support instructional activities, often within Step 1 Preparing the lesson. Examples include:

• Module 2, Preparing for the module, According to the Resource overview, teachers need, “ten-sided dice in lesson 11 and The Number Case in lesson 6. Each student uses a blank card in lesson 9, counters in lesson 5, Student Journal in each lesson, Support 8 in lesson 11, and Support 12 in lesson 5.” 

• Module 2, Lesson 5, Number: Reviewing factors and multiples, Lesson notes, Step 1 Preparing the lesson, “Each student will need: 1 copy of Support 8, counters, and Student Journal 2.5.” Step 3 Teaching the lesson, “Distribute the support page. Explain that you are going to read some clues and that you want them to use the hundred chart to find the mystery number. They can place their counters on top of numbers that may match the clues below: Clue 1: I am a composite number. Clue 2: I am a multiple of 8. Clue 3: I have 7 factors. Clue 4: I am greater than 60 but less than 70.”

• Module 5, Preparing for the module, According to the Resource overview, teachers need, “non-permanent markers in lesson 9 and The Number Case in lesson 9. Each pair of students needs access to water, blue and red paint cotton tips, eye droppers, a non-permanent marker, a paint tray or plastic lid, and The Number Case in lesson 8. Each student will need a strip of paper or ribbon in lesson 1.”

• Module 7, Lesson 9, Algebra: Solving division equations, Lesson notes, Step 1 Preparing the lesson, “Each pair of students will need: 1 pan balance card from The Number Case; Each student will need: 1 set of algebra tiles (Note: If not available, use cut-outs from Support 34.) Student Journal 7.9”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative Assessments include Pre-test, Observations and discussions, and Journals and Portfolios. Summative Assessments include Check-ups, Interviews, Performance tasks, and Quarterly tests. All assessments regularly demonstrate the full intent of grade level content and practice standards through a variety of item types: multiple choice, short answer, and constructed response. Examples include:

• Module 1, Check-up 1 and Performance task, develop the full intent of standard 6.NS.7, understand ordering and absolute value of rational numbers. Check-up 1, Question 4, “Color the bubble beside the statement that is true. a. -7 > -2, -7 > 2, -2 > -7, 2 < -7. b. -2 = 2, -7 < 2, 9 < -7, -2 > 9.” Performance task, ”Olivia is housesitting for a friend in Canada. She notices that both the fridge and freezer show a temperature of 0°C. She decides to change the temperature on the controls. A few hours later, the temperature of the fridge is 3°C, while the temperature of the freezer is -9°C. Question 1, Circle the part of the fridge that records the greatest change in temperature. fridge; freezer. Question 2, Use this open number line to prove your thinking. Question 3, Olivia writes this statement to compare the magnitude of the temperature change. -9 < 3.”

• Module 6, Quarterly test questions support the full intent of MP2, reason abstractly and quantitatively, as students reason abstractly and quantitatively about ratios. Question 3, “Circle the best offer. Show your thinking. Image shows 4 spaghettis for $2.80 or 6 spaghetti for $3.90.” Question 6, “Which watermelon has a greater ratio of seeds to slices? Show your thinking. Tyler eats 3 slices of watermelon and counts 8 seeds. Beatrice eats 4 slices of watermelon and counts 10 seeds.”

• Module 9, Interview 1, develops the full intent of 6.SP.2, understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. “Steps: Provide the student with the support page. Explain that screen time is the amount of time that is spent each day in front of any screen. Ask the student to describe what the results show. If necessary, prompt them to make reference to the shape and spread of data. Ask the student to identify the mode, median, and mean. Encourage students to verbalize their thinking as they calculate each measure of center. Draw a ✔ beside the learning the student has successfully.”

• Module 12, Quarterly test questions support the full intent of MP6, attend to precision, as students identify and calculate the mean average deviation for a data set. For example, Question 23, “Calculate the mean average deviation (MAD) for the data set. Round your answer to the nearest hundredth. 1, 3, 5, 7, 9, 11, 13.”

The materials reviewed for ORIGO Stepping Stones 2.0 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.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In each Module Lesson, Differentiation notes, there is a document titled Extra help, Extra practice, and Extra challenge that provides accommodations for an activity of the lesson. For example, the components of Module 5, Lesson 5, Division: Whole numbers by common fractions (with remainders), include:

• Extra help, “Activity: Have each student cut out their three foldables, then fold the ends in so the expression is on the front as shown. Instruct them to open one foldable, and draw an area model picture to help solve the expression. They then write the mixed number quotient in the answer box on the front. Repeat for the other two foldables. Then have them share and compare their completed foldables to check each other’s work.”

• Extra practice, “Activity: Have the students work individually to write a word problem to match each equation in Question 3 of the Student Journal. Explain that they must make sure the mixed number solution makes sense. They then exchanged problems with another student to check each other’s work.”

• Extra challenge, “Activity: Organize the students into pairs. They take turns to write an equivalent whole number expression for one expression on the game board. For example, for 7\div\frac{5}{3} they would write 21\div5 on a blank card. Allow them to make notes as needed. They continue until they have written an equivalent expression for each expression on the game board. The cards are then mixed and placed facedown in a pile. Students take turns to flip a card and place a counter on the equivalent expression on the game board. They then say the quotient. If the quotient is not correct they must remove their counter. Otherwise play continues until one student places 3 adjacent counters in one row or column to win the game, or until all the cards are used. The cards can be remixed and the game played again.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities to investigate the grade-level content at a higher level of complexity. The Lesson Differentiation in each lesson includes a differentiation plan with an extra challenge. Each extra challenge is unique to an activity completed in class. Examples include:

• Module 1, Lesson 4, Number: Using exponents greater than 2, Differentiation, Extra Challenge, “Review what the students already know about square numbers and list some of these examples on the board (e.g. 1, 4, 9, 16). Say, Square numbers are generated with exponents of two. Cubic numbers are generated with exponents of three. Ask, What are some cubic numbers that you know? Have the students list the first few cubic numbers on the board (e.g. 1, 8, and 27). Once familiar, groups of 3 students can be challenged to a) find the cubic numbers less than 250, and b) find the cubic number that is nearest 500.”

• Module 5, Lesson 2, Division: Common fractions (same denominators), Differentiation, Extra Challenge, “Organize the students into pairs. Have each student write a quotient in the form of a proper fraction or whole number (less than 10). They then exchange cards and write a matching expression involving division. For example, one student writes the quotient \frac{1}{12} and the other student writes the expression \frac{5}{12}\div5. Encourage them to write expressions in the different formats discussed in the lesson, for example, \frac{4}{8}\div4, \frac{4}{8}\div\frac{2}{8}, and \frac{4}{8}\div\frac{1}{8}. The activity continues until all the cards are used. Retain the cards for use in the later differentiation activities.”

• Module 10, Lesson 4, Statistics: Consolidating box plots, Differentiation, Extra Challenge, “Organize students into pairs. Explain that students are to select an international city to research temperature data for the 1st to the 15th of the previous month, then complete a five-number summary, box plot, and report to describe the variation in temperature.”

The materials reviewed for ORIGO Stepping Stones 2.0 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.

Although strategies are not provided to differentiate for the levels of student language development, all materials are available in Spanish. Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards.  According to the Mathematics Overview, English Language Learners, “The Stepping Stones program provides a language-rich curriculum where English Language Learners (ELL) can acquire mathematics in a natural second-language progression by listening, speaking, reading, and writing. Each lesson includes accommodations to be aware of when teaching the lesson to ensure scaffolding of content and misconceptions of language are addressed. Since there may be several stages of language development in your classroom, you will need to use your professional judgement to select which accommodations are best suited to each learner.” Examples include:

• Module 1, Lesson 4, Number: Using exponents greater than 2, Lesson notes, Step 2 Starting the lesson, “ELL: Pair the students with fluent English-speaking students. Allow them to discuss the question, How could you use exponents to represent the number one hundred? Create an anchor chart representing Exponents and display it in the classroom.” Step 3 Teaching the lesson, “ELL: Allow the students to think about the questions being asked. Then give them the opportunity to tell their partner their answer before presenting to the group. Allow the students to use hand gestures (such as thumbs up or down) to show they understand, or are confused by, the language being used. Allow pairs to complete the Student Journal, if necessary.” Step 4 Reflecting on the work, “ELL: Encourage the students to listen to other students’ answers and tell their partner why they agree or disagree.”

• Module 12, Lesson 3, Ratio: Using complementary percentages, Lesson notes, Step 3 Teaching the lesson, “ELL: Encourage the students to think about the questions being asked, then share their answers with their partner before presenting to the group. Allow pairs to complete the Student Journal, if necessary.” Step 4 Reflecting on the work, “ELL: Encourage the students to listen to other studnet’s answers, then rephrase them.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for providing  manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade level math concepts. Examples include: 

• Module 2, Lesson 5, Number: Reviewing factors and multiples, Step 2 Starting the lesson, references the Flare Number Board to review the relationship between factors and multiples. “Open the Flare Number Board online tool. Invite students to shade all multiples of 6. Choose a multiple of 6 and ask, What are some other factors of (36)? Repeat for other multiples of 6 on the chart.”

• Module 4, Lesson 3, Algebra: Writing equations with two variables, Step 3 Teaching the lesson, describes the use of physical items as a way for students to explore the relationship between variables and their equations. “Organize the students into pairs or groups of three. Distribute one balance card to each pair or group. Have them work together to explore the relationship between the mass of the items in each bag, and to write an equation to show that relationship (SMP2). Then have them figure out which bag holds the lightest single item. Afterward, invite pairs or groups to share their equation and explain how they figured out the lightest single item.”

• Module 11, Lesson 1, Ratio: Introducing ratios with three parts, Step 3 Teaching the lesson, identifies the use of connecting cubes to create patterns and explore ratios. “Organize students into pairs and distribute the connecting cubes. Say, I want you to use the cubes to create a color pattern that has three parts (colors) and that has at least eight cubes in the repeating element. As the students are working, distribute the table cards and markers. When their patterns are complete, have the students create a ratio table to show how their pattern can be extended.”