The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meets expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. 

Cluster 6.RP.A addresses understanding ratio concepts and using ratio reasoning to solve problems. Multiple modules explore a variety of real-world applications using a few mathematical representations. Some opportunities exist for students to work with ratios that call for conceptual understanding and include the use of some visual representations and different strategies. Examples include:

• Module 6, Lesson 8, Ratio: Developing the concept of rate, Step 3 Teaching the lesson, “A car travels 200 miles in 4 hours. How far does it travel in 1 hour? Project slide 3 and read the problem aloud. Ask, What does this problem want you to find? How would you solve it? What strategy or model could you use? Allow pairs to try different strategies to solve the problem (MP4). They may choose to draw a diagram or use a table to model their results. Invite volunteers to share their thinking, using the board if needed. If a table isn’t suggested, project the table (slide 4) and ask, How could this help us solve the problem? Confirm that a table can be used to organize rates to calculate the answer. Invite volunteers to share their ideas on how to complete the table. The image below shows two strategies for using the table. Ideas could include repeating the pattern, or finding out how long it takes Amy to run 10 laps. Confirm that the answer is 30 minutes.” Module 3, Lesson 2, Ratio: Building equivalent ratios pictorially, Step 3 Teaching the lesson, “Allow students time to work with a partner and create a representation for the scenario using a tape diagram for one sandwich first, then for two. Invite volunteers to share their responses with the class. Ask, What do you notice about the relationship between the amounts? Does the ratio change? Establish that there are more slices of ham in total, but because there are more sandwiches, the ratio stays the same. Say, Ratios describe a constant relationship between quantities. This relationship stays constant as the quantities change. This means equivalent ratios have the same relationship between each quantity.” (6.RP.1).

• Module 3, Lesson 1, Ratio: Introducing ratio, Step 3 Teaching the lesson, “Ask, When might you use a ratio? What are some real-world examples of ratios? Allow students time to discuss and invite volunteers to share their thinking. If students have difficulty thinking of a context for this, provide examples such as birthday parties (guests, drinks, food), sports teams (scores, players) or school (classes, lessons) to assist with this. Have students work in pairs to create their own example of a scenario describing a relationship between two quantities (for example, for every x girls in the class, there are y boys). Have them write their example on a blank card and collect them to be used later in the lesson. Shuffle and distribute the cards to pairs of students (ensure that all students have a card that they did not create). Have the students work with the relationship of the quantities written on the cards, creating different representations of the ratio. This could include drawing tape diagrams, writing sentences describing the ratio, or writing the ratio as x:y. Allow time for students to work on their cards, moving around the room to ensure all students know what to do. Bring students back together and invite volunteers to share their ratio and related representations.

Invite students to critique the reasoning of others.” Students relate abstract to concrete examples (6.RP.1).

• Module 8, Lesson 1, Ratio: Linking part-whole ratios to fractions, Step 3 Teaching the lesson, “Distribute the cubes. and say, You will be building a character with some or all of your cubes. But first you need to count how many of each color cube you have. Ask students to build a character with their cubes. Afterward, have them write as many different ratios as they can about the cubes used to build the character. Encourage them to include part-part and part-whole ratios. Project the statement and table (slide 3), and read the scenario aloud. Analyze the fraction of profits () being donated. Then ask, What part of the fraction represents the total? What part represents the amount donated? How much is remaining?  Project slide 4 to show these numbers in the table. Point out that if only $4 profit is made, then only $1 is donated to charity. Then ask, What numbers can we write to complete the table? Work with the students to write the missing values. For each value, ask, How did you figure out that amount? Look for students who have identified the multiplicative relationships in each column of the table.” (6.RP.1)

Cluster 6.EE addresses the need to be able to apply and extend previous understandings of arithmetic to algebraic expressions, reason about and solve one-variable equations and inequalities and represent and analyze quantitative relationships between dependent and independent variables. Multiple Modules explore a variety of real-world applications using a few mathematical representations. Some opportunities exist for students to work with expressions and equations that call for conceptual understanding and include the use of some visual representations and different strategies. Examples include:

• Module 2, Lesson 1, Algebra: Reviewing language and conventions, Step 3 Teaching the lesson, “Invite groups to share their working definitions from Step 2. Then discuss the points below: What is the difference between an expression and an equation? What is the difference between a product and a sum? What examples can you give for each?  Establish that an expression can be a number by itself, or a combination of numbers and operations that do not have a relationship. Whereas an equation shows that two expressions are equal.” (6.EE.1)

• Module 7, Lesson 1, Algebra: Simplifying expressions, Step 2 Starting the lesson, students are presented with a visual representation of 16 blocks representing 4 squared. “To review square numbers, project slide 1 and discuss the points below: What number does this picture represent? What expression can we write to match? (4 • 4, 16, or 4².) What type of number is 16? (Square number.) What other square numbers do you know?” Step 3 Teaching the lesson, “Distribute the resources and say, These are called algebra tiles. They are used to represent expressions. Ask students to examine the tiles and look for relationships between the different sizes (MP7). Display a small square tile. Say, The length of each side of this square is 1. What do we know about the area of this tile? (1 by 1.) 1 by 1 is 1 so this tile is called 1. Display the long tile end-to-end with the small square tile. Ask, What is the width of this rectangle? (1.) We could try to determine an exact length of this tile but we want this tile to represent a variable so we will think of it as a tile that can change length according to the unknown amount it is representing. Since we do not know the exact length of the tile, we will use the variable x to label, so the area is 1x or just x. Display the large square tile aligned to the length of the long tile. Say, We know the length of the long tile is x, so the side length of the large square tile is also x. What can we write to represent the area of the large square tile? (x².) Emphasize that each tile should be treated as a single quantity.” (6.EE.3)

• Module 7, Lesson 4, Algebra: Simplifying expressions with more than one variable, Step 3 Teaching the lesson, “ Organize students into groups of three or four and distribute the cubes to each group. In turn, each student rolls the cubes and uses the result to form a term, such as 2a for a roll of 2 and a, or 6 or 9 for a roll of 3 and 3. A roll of a and b would result in the two terms a + b. After there are at least two terms with differing variables, have the students work independently to write one expression to represent the total of all the individual terms, for example, for the terms 2a, 6, 4b, and 8, they could write 2a + 4b + 14. Encourage them to use factors to write the expression a different way where possible, for example, 2(a + 2b + 7), and to check the equivalence by substitution. Observe and listen to the students as they work, to identify and support those using incorrect thinking. Afterward, have the students compare their expressions and explain the steps they used. Encourage them to compare and contrast the different methods, if any.” (6.EE.3)

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skills and fluencies throughout the grade-level. Opportunities to formally practice procedural skills are found throughout practice problem sets that follow the units. Practice problem sets also include opportunities to use and practice emerging fluencies in the context of solving problems. Ongoing practice is also found in Assessment Interviews, Games, and Maintaining Concepts and Skills.

The materials attend to the Grade 6 expected fluencies: 6.NS.2, multi-digit division and multi-digit decimal operations; 6.NS.3, add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation; and 6.EE.A, apply and extend previous understandings of arithmetic to algebraic expressions. Examples include:

• In Module 3, Lessons 8-12, students use multi-digit division of whole numbers and decimals (6.NS.2,3).

• In Module 5 interview, students calculate quotients in expressions that include fractions and whole numbers (6.NS.2,3).

• Module 2, Lesson 8, Algebra: Using the distributive property, includes using <, >, or = to compare expressions that include skills such as subtracting or multiplying decimal numbers. 

• In Module 2, Lessons 9-12, students solve problems using the standard algorithm for performing mathematical operations with decimals (6.NS.3). 

• Module 3, Lesson 10, Division: Terminating and repeating decimal fractions, students add and subtract decimals, and multiply or divide numbers as they convert units. In Module 2, Lessons 9-12 provide practice for solving problems using the standard algorithm for performing mathematical operations with decimals (6.NS.3).

• Module 4, Lesson 2, Algebra: Writing equations to match word problems, provides practice in writing equations to match word problems. For example, “What variable can you use to represent the value?” and “What operations will we use to calculate that value?” (6.EE.A)

In addition, the instructional materials embed opportunities for students to independently practice procedural skills and fluencies. Examples include:

• The Stepping Stones 2.0 overview explains that every even numbered lesson includes a section called “Maintaining concepts and skills” that incorporates practice of previously learned skills from the prior grade level. 

• Each module contains a summative assessment called Interviews. According to the program, “There are certain concepts and skills , such as the ability to route count fluently, that are best assessed by interviewing students.”  For example, in Module 9 Interview, students must demonstrate fluency of finding the mean, median, and mode of a data set.

• Some lessons provide opportunities for students to practice procedural skills during  the “Step Up” section of the student journal.

• “Fundamentals Games” contains a variety of games that students can play to develop grade level fluency skills.

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Engaging applications include single and multi-step word problems presented in contexts in which mathematics is applied. There are routine problems, and students also have opportunities to engage with non-routine application problems. Thinking Tasks found at the end of Modules 3, 6, 9, and 12, provide students with problem-solving opportunities that are complex and non-routine with multiple entry points.

Examples of routine application problems include:

• Module 6, Lesson 8, Ratio: Developing the concept of rate, Student Journal, Step In, page 216, addresses the standard 6.RP.3, “Antonio mixes teaspoons of yellow and red paint in the ratio of 12:4. How much yellow paint will be used for 1 teaspoon of red?”

• Module 5, Lesson 2, Division: Common fractions (same denominators), Student Journal, Step Up, page 161, Problem 2b, addresses the standard 6.NS.1, “A straw is ten-twelfths of a foot long. Nicole cuts the straw into shorter pieces that are each two-twelfths of a foot long. How many pieces did she cut?”

• Module 7, Lesson 10, Algebra: Solving word problems, Student Journal, page 271, Problem 2b, address the standard 6.EE.7, “Three friends enter a 15-mile fun run. Dwane runs the first 4 1/2 miles. Natalie runs the next $3\frac{1}{4}$ miles. Reece runs the rest of the distance. How far did Reece run? Let r represent the unknown distance.”

• Module 4, Lesson 11, Algebra: Identifying independent and dependent variables, Student Journal, Step Up, page 151, Problem 2c, addresses the standard 6.EE.9, “In a math test, a student scores 5 points for each correct answer. What is a student’s score if they get 15 correct answers?” (identifying dependent and independent variables).

• Module 9, Lesson 12, Area: Solving word problems, Student Journal, Step Up, page 353, Problem 2a, address the standard 6.G.4, “The roof of a cottage needs refurbishing. One side of the roof is 30 ft long by 12.5 ft wide. If a bundle of shingles can be purchased to cover 24 ft², how many bundles are needed?”

• Module 12, Lesson 10, Algebra: Generating and graphing variables (non-equivalent ratios), Student Journal, Maintaining Concepts and Skills, Ongoing Practice, page 463, Problem 2a, “Andrea wants to buy a guitar that costs $150. The music store has a 25% off sale. When she buys the guitar she is given an extra 5% off. What amount does she pay for the guitar?”

• Module 3, Lesson 7, Algebra: Generating and graphing variables (non-equivalent ratios), Teaching the lesson, Investigation 2, students work in pairs to answer, “Emily has beetles to feed her lizards. Altogether she has a total of 52 legs. How many lizards could Emily have? What is the ratio of lizards to beetles?” (6.RP.1).

Examples of non-routine application problems with connections to real-world contexts include:

• Module 3, Lesson 12, Division: Adjusting to divide with decimal fractions, Teaching the lesson, Thinking Task, Problem 1, students read points on a coordinate grid to fill in a table showing the cost of a phone bill per month. Question 3 asks, “Riku investigates the call rate for Option B. It will cost $0.30 for each minute that she spends on the phone. She decides to calculate the total cost of a six-minute call. Which of these displays is most likely to show the total cost? Explain your thinking.” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 6, Lesson 12, Ratio: Comparing rates, Teaching the lesson, Thinking Task, Problem 2 inquires, “An architect hired by the school district draws up these plans (design shown). She decides to split the hall into three areas: stage, auditorium, and entrance. The stage needs wooden flooring and the auditorium will be carpeted. The school has a budget of $900 to spend on the wooden stage flooring. One dimension of the stage is 8m. They would like the other dimension to be somewhere between 3 and 5 meters. Given their budget, write the dimensions for the largest stage the school can afford to build. Show your thinking.” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 9, Lesson 12, Area: Solving word problems, Teaching the lesson, Thinking Task, students find the surface area of a greenhouse with dimensions given and identify the net that matches the greenhouse design. Problem 3 asks, “Archie plans to build the garden bed in this picture (3ft x 6ft x 6in deep). The measurements are taken from inside the garden bed. He will need to buy the wood to build it and the soil to fill it. He has all the other tools and materials that are necessary. What is the cost of building the garden bed?” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 12, Lesson 12, Algebra: Generating and graphing variables (non-linear), Teaching the lesson, Thinking Task, students are provided a collection of data in two tables. Number of vehicles that drive past the school at certain times during the day, the other set of data is a record of each vehicle’s speed. Problem 1 asks, “How many vehicles traveled at a speed that is equal to or greater than 25 miles per hour?” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

The materials reviewed for ORIGO Stepping Stones 2.0 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. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers in several places:  Mathematical Practice Overview, Module Mathematical Practice documents and within specific lessons, alongside the learning targets or embedded within lesson notes.

MP1 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Lesson 6, Number: Finding the least common multiple, Step 3 Teaching the Lesson, students make sense of problems involving multiples and look for efficient ways to represent and solve them. “Project slide 2 and deconstruct the problem by asking questions such as, What do you know? What do you need to find out? How will you solve the problem? (MP1): A school cafeteria serves burgers every third school day and burritos every fourth school day. Both meals were served today. How many days until both meals are again served on the same day?” The teacher asks, “What is this problem asking you to do? Have you solved a problem like this before? What steps have you used to this point? What other strategies could you try? What other math tools could you use?”

• Module 3, Lesson 7, Ratio: Solving word problems with part-part and part-whole situations, Step 2 Starting the Lesson, students make sense of problems and persevere in solving them as they draw upon their understanding of equivalent ratios. “Project slide 1, as shown, and read the word problem aloud (MP1). Ask, What steps do you need to take to solve the problem? Is the question referring to a part-part situation or part-whole? How do you know? Review the language used in the problem to determine that it is a part-whole situation, and the steps students will need to take to solve it. Allow students time to work with a partner to solve the word problem. Invite volunteers to share their solutions and confirm the ratio is 32:48, making the answer to the problem 80 club members.”

• Module 5, Lesson 5, Division: Whole numbers by common fractions (with remainders), Step 2 Starting the Lesson, students analyze given word problems with division of whole numbers and fractions and check that their solution makes sense in the situation. “Project slide 1 as shown, and discuss the point below (MP1): A snail can travel at $\frac{7}{10}$ of a mile each day. How many days would it take to travel 14 miles? What do you know about this problem? (A snail can travel $\frac{7}{10}$ of a mile each day.) What do you need to find out? (How far can it travel in 14 days?)  What expression can you write to match the problem? (14 ÷ .)  What picture could you draw to help solve the problem? (14 rectangles split into tens.) How can you use the diagram to help you change the expression to whole numbers? (Multiply the whole number (dividend) and the fraction (divisor) by the same amount.) What amount would you multiply the numbers by? Why? (The value of the denominator because that will create a numerator that is a multiple of the denominator. The resulting fraction will be equivalent to a whole number.) Invite a student to complete the diagram to show the multiplication. What is the equivalent whole number expression? (140 ÷ 7.) What is the solution to the problem? (20 days.)”

• Module 11, More Math, Problem Solving Activity 4, Word Problems, students make sense and persevere in solving multi-step ratio word problems. “Project slide 1 and read the word problem with the students. Ask questions such as, “What information do we need to solve this problem? What will we do first? What will we do next? How could you show your thinking? Allow time for the students to find a solution. Then invite students to share their solution (1:2:2) and explain their thinking. Slide 1: There are 200 cars parked in a parking lot. 20% are blue, $\frac{2}{5}$ are red, and the rest are white. What is the ratio of blue, red, and white cars in the parking lot?” 

MP2 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Lesson 3, Number: Reviewing abbreviations for numbers greater than one million, Step 2 Starting the Lesson, students reason abstractly and quantitatively as they contextualize numbers, identifying real-world contexts for millions, billions, and trillions. “Project slide 1. Have the students list everyday examples for millions, billions, and trillions. Encourage debate around each suggestion (MP2). For example, a country’s population may be recorded with billions, millions, or even thousands depending upon its size and density. Then ask, How do we record these numbers? Do we write the whole number or do we just abbreviate the number? Bring out that numbers greater than one million are often abbreviated, especially when being reported to the public. For instance, a large company may have a reported net worth of 7.2 billion.”

• Module 4, Lesson 5, Algebra: Order of operations involving variables, Step 3 Teaching the Lesson, students reason abstractly and quantitatively as they write an expression to match a given word problem and write a word problem to match a given expression. “Project slide 2  and read the problem aloud. Slide: Vishaya runs the same distance Monday through Friday. She then runs 12 miles on Saturday.” Notes include, “What expression could you write to represent the total number of miles that Vishaya runs each week? Organize the students into pairs to brainstorm ideas. Then have pairs write their expression (for example, 5m + 12) on the board (MP2) and explain how it matches the problem. Point to one expression and discuss the points below: What part of the expression will you calculate first? How do you know? What is a reasonable value we could substitute for the variable?”

• Module 6, Lesson 1, Area: Exploring parallelograms, Step 2 Starting the Lesson, students reason abstractly and quantitatively as they contextualize a given scenario into a word problem. “Project slide 1 as shown and ask, What do you know about calculating the area of a rectangle? Organize the students into pairs to discuss their ideas. Then invite responses. Emphasize the understanding that the area of a rectangle is calculated by multiplying the length and width together. Project the grid overlaid on the rectangle (slide 2) and ask students to calculate the area of the shape. Invite responses to confirm the answer is 40 ft². Ask pairs to work together again to create a word problem based on calculating the area of the shape shown on the slide (MP2). Bring students back together and invite volunteers to share their word problems with the class.”

• Module 7, Lesson 3, Algebra: Simplifying expressions using distributive property, Student Journal, Step Up, page 251, Question 2, and Step 4 Reflecting on the work, students reason abstractly and quantitatively as they simplify expressions using the distributive property. “Write an expression to match each problem. a. There are 5 equal rows of fruit trees. In each row, there are 15 apple trees and some plum trees, with the same number of plum trees in each row. Let p represent the number of plum trees in each row. How many fruit trees are there in total? b. Each concert ticket costs $49 plus a $5 service fee. Ethan buys a ticket for himself and a group of friends. Len nrepresents the number of tickets that he bought. What is the total cost of the tickets? c. A garden path is $1\frac{1}{2}$ yards wide. The first 10 yards of the path are concrete. The rest of the path is gravel. Let prepresent the length in yards of the path that is gravel. What is the total area of the path?” Step 4, “Discuss the students’ answers to Student Journal 7.3. For Question 2, have students write their expressions on the board and relate each term back to the problem. (MP2)”

The materials reviewed for ORIGO Stepping Stones 2.0 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. 

Students have opportunities to meet the full intent of MP3 over the course of the year as it is explicitly identified for teachers in several places: Mathematical Practice Overview, Module Mathematical Practice documents and within specific lessons, and alongside the learning targets or embedded within lesson notes.

Teacher guidance, questions, and sentence stems for MP3 are found in the Steps portion of lessons. In some lessons, teachers are given questions that prompt mathematical discussions and engage students to construct viable arguments. In some lessons, teachers are provided questions and sentence stems to help students critique  the reasoning of others and justify their thinking. Convince a friend, found in the Student Journal at the end of each module and Thinking Tasks in modules 3, 6, 9, and 12, provide additional opportunities for students to engage in MP3. 

Students engage with MP3 in connection to grade level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Lesson 7, Number: Comparing and ordering positive and negative numbers, Step 3 Teaching the lesson, students critique the reasoning of others as they compare and order positive and negative numbers on a number line. “Make sure students justify the position of each number on the number line. In doing so, they  should justify which side of the 0 to which the number is attached, the distance of the number to 0 and the relative position of the number in relation to other numbers. Encourage the remaining students to critique their reasoning using the sentence stems shown below (MP3). I have a different opinion, I think, I agree (disagree) because, and That makes sense, but...”.

• Module 3, Thinking Tasks, Question 4, students construct viable arguments as they analyze linear relationships in order to determine the best cell phone plan. “Riku asks her friend Thomas about his cell phone usage. He says that he rarely makes any phone calls, but he sends a lot of texts. He estimates that he sends at least 300 text messages each month! If Riku uses her phone in much the same way Thomas does, which cell phone option should she choose? Show your thinking in the space below. Then justify your decision.”

• Module 4, Student Journal, page 157, Convince a friend, students create expressions and equations to solve problems and then critique the reasoning of classmates. “Deana is 3 years older than her brother Marcos, and 2 years older than her pet dog, Hailey. Marcos is 8 years old. Deana reads that one human year is equivalent to 7 dog years. Marcos says that Hailey is over 70 years old in dog years. Do you agree or disagree with Marcos? Explain why. I agree/disagree with Marcos because ...Share your thinking with another student. They can write their feedback below. I agree/disagree with your thinking because … Feedback from:”

• Module 7, Student Journal, page 281, Convince a friend, students construct a viable argument and critique the reasoning of others as they determine equivalent expressions in real-word and mathematical problems. “James and Gemma are talking about expressions. Gemma says that 7a + 10b can be rewritten as 17ab in the same way that 10 + 7 = 17. James agrees that 10 + 7 = 17 but he says it is more accurate to write 7a + 10b as 17 + ab. Who do you agree with? Explain your thinking. Share your thinking with another student. They can write their feedback below. Discuss how the feedback you received will help you give better feedback to others.” 

• Module 9, Student Journal, page 357, Convince a friend, students construct an argument and critique the reasoning of others while interpreting measures of center in real-world and mathematical problems. “Noah asks his classmates to contribute to his math project. He creates a dot plot showing the age of one parent of each in his class so he can try to calculate the overall age. Mary says that calculating the median gives the most accurate measure of center. Do you agree or disagree with Mary? Share your thinking with another student. They can write their feedback below. Discuss how the feedback you received will help you give better feedback to others.”

The materials reviewed for ORIGO Stepping Stones 2.0 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. 

Students have opportunities to engage with the Math Practices throughout the year. The MPs are often explicitly identified for teachers in several places: Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within lesson notes.

MP4 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 5, Lesson 9, Ratio: Comparing ratios in tables and graphs, “students use a ratio table or coordinate plane to model a real world problem”. Step 3 Teaching the lesson, “Organize the students into pairs and distribute the resources. Project slide 3. Pairs then solve the problems (90 dokkins from Store A or 72 from Store B for $48; 60 dokkins for $36 at Store A; 60 dokkins from Store B will cost $40). Organize pairs into groups to share answers and analyze methods used. Ask, Who created a ratio table to find equivalent values? Who created a coordinate plane and plotted ordered pairs to compare the prices? Who worked with the numbers in another way to solve the problems? (MP4) Make sure students justify their choices.”

• Module 7, Student Journal, page 280, Mathematical modeling task, students model a real- world situation with equations. “Ten year old Ricardo and his family are going to a show at the theater. His mom, dad, and five-year-old sister are all attending. They have $130 to spend on the family night out. How would you spend the money to ensure everyone has a ticket, a treat, and a drink? Explain your thinking.” An image shows: Theater tickets: Adults $23.50, child (8-17 years) $19.50, child (2-7 years) $16.50, Booking fee $3.50/ticket, Family ticket (includes booking free) $95.00. Another image shows Refreshment prices: Popcorn $5.50, Chocolate $2.50, Small drink $3.00, Large drink $4.50.

• Module 8, Lesson 9, Ratio: Finding the whole given a part and the percentage, students use a double number line or relationship diagram to represent a problem and explain the connections between the two models. Step 3 Teaching the lesson, “Organize students into pairs. Project slide 6 and read each problem aloud. One student in each pair uses a double number line to solve the first problem, then draws a relationship diagram to solve the second problem. The other student uses a relationship diagram to solve the first problem, and a double number line for the second problem. Listen and observe as they work. Ask questions to gauge their understanding of the models and calculations. Afterward, have the pairs debate the efficiency and effectiveness of each diagram and their methods (MP3). Then ask the class, What is the same about the two models? (MP4) What is different? Which do you prefer? Why?”

• Module 9, Student Journal, page 356, Mathematical modeling task, students model a real-world surface area problem. “Pamela wants to paint the outside of her dog’s kennel. The kennel is shaped like a rectangular-based prism and she would like to cover the sides only with one coat of paint. The paint cans sold at her local store contain enough paint to cover about 2,500 square inches per can. a. Pamela buys two cans of paint from the store. Does she have enough paint for her project? b. If the kennel was twice as long, would Pamela need exactly twice as much paint? c. Explain your thinking.” 

• Module 12, More Math, Thinking Task, School Crossing, students model with math as they graph ratio percentages. “The principal of Sunnydale Elementary is concerned about the increased traffic around the school. She has asked the city council to install a set of traffic lights to help. The council explains they must first study the traffic flow around the school before they can agree. The following Monday, two traffic inspectors arrive. One of the inspectors records the number of vehicles that drive past the school. The other inspector records the speed that each vehicle travels. The results are shown below. Show the number of vehicles that drove past the school throughout the morning in this graph. You will need to add the following information that is missing: title labels for the x- and y-axis values along each scale.”  

MP5 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the modules to support their understanding of grade level math. Examples include:

• Module 3, Lesson 7, Ratio: Solving word problems with part-part and part-whole situations, students choose between a model or diagram as a tool to represent ratio and solve ratio word problems. Step 3 Teaching the lesson, “Project slide 2, as shown, and read the word problem aloud. Cathy makes bouquets with 23 flowers in each to sell at a local market. She can make 6 bouquets in 4 hours. If she needs to make 54 bouquets by the weekend, how many hours will she need to work? Encourage students to solve the problem, choosing a model or diagram to show their thinking (MP5).”

• Module 7, Student Journal, page 194, Mathematical modeling task, students consider and choose from the available tools to solve a real world problem. “Samuel is playing an online game where he can trade the goods he collects for other items he needs. He has some oranges, but he would like to trade them for fish. He knows that: one loaf of bread is worth the same as one apple and two fish, five oranges are worth the same as two loaves of bread, four loaves of bread are worth the same as fourteen fish, and six fish are worth the same as two cantaloupes. How many fish can Samuel trade for five oranges? Explain your thinking. Discuss the different tools, representations, and/or strategies students used to solve the problem. For example, some students may have used tables to represent each relevant clue as a ratio then used equivalent ratios to allow comparison, as shown, and concluded that Samuel can trade 5 oranges for 7 fish. Other students may have simply started with the second clue, because Samuel has 5 oranges, then analyzed the third clue and determined that Samuel can trade 5 oranges for 7 fish.”

• Module 7, Lesson 2, Algebra: Simplifying expressions using the commutative and associative properties, Slide 1, students simplify algebraic expressions and choose a tool to represent the problem. “Victoria is decorating her backyard for a party. She arranges balloons individually or in bunches of the same number. Afterward, there are three bunches of silver balloons and six individual silver balloons. There are also four bunches of purple balloons and three individual purple balloons. How many balloons has she arranged in total?” Step 3 Teaching the Lesson, “What might help you model this problem? Have students work individually to model the problem using any method of their choice. (MP5)”

• Module 12, More Math, Investigation 1, students choose an appropriate strategy as a tool to solve real-world ratio problems. “When would 20% off be a better deal than $20 off?”

The materials reviewed for ORIGO Stepping Stones 2.0 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. MP6 is explicitly identified for teachers in several places: Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within whole class lesson notes. 

Students have many opportunities to attend to precision in connection to grade level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Lesson 9, Number: Using the 1st and 2nd quadrants of the coordinate plane, Step 3 Teaching the lesson, students use accuracy and precision when naming points on the coordinate grid. “Have the students interpret the second quadrant. They should write the coordinates for each color star on the board. Reinforce that the first number still tells the distance to move from the origin along the x-axis, while the second coordinate still tells the distance to move up the y-axis (MP6). The only difference is that it is now possible to move in a negative direction along the x-axis.”

• Module 3, Lesson 8, Division: Reviewing the standard algorithm, Student Journal, page 103, Step Up, Question 2d, students use precision when estimating an answer and finding an accurate quotient using the standard algorithm. “Estimate each quotient first. Then use the standard algorithm to calculate. Use your estimate to help you place the decimal point if needed. $340.2\div 14$”

• Module 11, More math, Investigation 2, Area change, students attend to precision as they measure area and calculate percent change. “Can you calculate the percentage change from the area of the gym to the area of the classroom?” 

Students have frequent opportunities to attend to the specialized language of math in connection to grade level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Module overview, Vocabulary development, students can attend to the specialized language of math as teachers are provided a list of vocabulary terms. “The bolded vocabulary below will be introduced and developed in this module. These words are also defined in the student glossary at the end of each Student Journal. A support page accompanies each module where students create their own definition for each of the newly introduced vocabulary terms. The unbolded vocabulary terms below were introduced and defined in previous lessons and grades. Addition, algorithm, balance, calculate, common factor, composite number, decimal fraction, decimal point, denominator, difference, distributive property of multiplication, dividend, division, divisor, equation, equivalent fractions, estimate, exponent, expression, factor, greatest common factor (GCF), halve, innermost grouping symbols, least common multiple (LCM), multiple, multiplication, multiply, one whole, operation, order of operations, parentheses, partial products, pattern, prime number, problem, product, relationship, remainder, rule, standard algorithm for multiplication, subtraction.” Students are provided with a Building Vocabulary support page. The page includes: Vocabulary term (the bolded terms), Write it in your own words, and Show what it means.

• Module 4, Lesson 8, Algebra: Interpreting tables, Student Journal, page 142, Words at work, students use precise language as they describe words connected to expressions and equations. “Imagine your friend was away from school when you learned about these terms. Write in words how you would explain them. Include examples, variable, constant, coefficient.”

• Module 7, Lesson 5, Algebra: Introducing balancing to solve addition equations, Step 2 Starting the lesson, students use the specialized language of mathematics as they solve and describe simple algebraic equations and reinforce equivalence. “Project slide 2, as shown. Ask, What do you know about the masses of these two boxes? (They are equal.) What is the value of y? What equation would we write to match? (y = 12.) What does the equal symbol mean? (MP6). Some students may still see the equal symbol as an operator meaning the answer is. Students with this misconception need support to recognize that the symbol shows a relationship of equivalence (e.g. y is equivalent to 12).”

The materials reviewed for ORIGO Stepping Stones 2.0 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. Students have opportunities to engage with the Math Practices throughout the year and they are often explicitly identified for teachers in several places:  Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within whole class lesson notes.

MP7 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities throughout the modules to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• Module 2, Lesson 5, Number: Reviewing factors and multiples, Step 3 Teaching the lesson, “students explore the relationship between multiples and factors, and find pairs of numbers that have a multiple-factor relationship.” “Project slide 1 as shown, and discuss the points below (MP7): What pairs of numbers could you write in this diagram? How did you decide? Can you write more than two possible factors for the number 73? How do you know? What do we call these types of numbers? (Prime numbers.) What other prime numbers do you know?”

• Module 4, Lesson 3, ​​Algebra: Writing equations with two variables, Step 4 Reflecting on the work, “students identify the relationships between the equivalent equations and the equivalent ratios.” “Discuss the students’ answers to Student Journal 4.3. For the equations in Question 2 and Question 3, ask, How are these equivalent equations similar to equivalent ratios? Organize the students into pairs to discuss the question. Invite them to share and explain their ideas to the class (MP7). Establish that in this context, the equivalent equations could be written as equivalent ratios. For example, the equation in Question 2a (5b = 3d), and the equation in Question 3a (15b = 9d) could be written as 5:3 and 15:9 respectively. Discuss how the students solved the equation in Step Ahead.”

• Module 7, Lesson 3, Algebra: Simplifying expressions using the distributive property, Student Journal, page 251, Step Up, Question 3, students make use of structure as they analyze and apply the partial-products strategy to solve algebraic expressions. “Simplify each expression. a. 3(m + 4), b. 7(5 + d), c. h(8 + 2), d. 4(1 + 2b), e. 15(a + 7), f. j(1 + 6).”

• Module 9, More math, Investigation 3, Unchanged surface area, students make use of structure as they analyze a problem and look for more than one approach. “Can you change the volume of a rectangular prism without changing the surface area? If students are having difficulty answering this question prompt them to build a prism with the dimensions 4 x 3 x 3. Have them calculate the surface area and volume of the prism they have built. Encourage them to experiment with ways to solve the investigation.” 

MP8 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities throughout the materials, with support of the teacher or during independent practice, to use repeated reasoning in order to make generalizations and build a deeper understanding of grade level math concepts. Examples include:

• Module 3, Lesson 8, Division: Reviewing the standard algorithm, Step 3 Teaching the lesson, “students see the overall process of division of dividends which incorporate decimals and still attend to the processes of the standard algorithm to solve.” “Project the next word problem (slide 4) and read it aloud. Ask, How is this problem different? What do you estimate the answer to be? How will you solve this problem? Encourage students to work in pairs to calculate the quotient, then invite volunteers to share their strategies with the class (MP8).”

• Module 7, More math, Investigation 1, Simplifying expressions, students use repeated reasoning and make generalizations on how to simplify an expression. “How many different equivalent expressions can you write to match: 4 + 4 + 4 + 4 + 4 + x + x + x + x + x + y + y.” 

• Module 8, Lesson 12, Division: Consolidating strategies (common fractions), Student Journal, page 314, Step Up, Question 1, students use repeated reasoning as they look for shortcuts to solve problems. “Complete each equation. Show your thinking on page 318. Then explain in words the strategy you used to find the quotient. a. $4\div \frac{1}{3}$= , b. $\frac{3}{7} \div \frac{9}{5}$ = , c. $\frac{24}{30} \div \frac{8}{30}$, d. $\frac{9}{6} \div \frac{3}{5}$=.”  

• Module 12, Lesson 7, Algebra: Solving equations with percentages and variables, Student Journal, page 453, Step Up, Question 3, students identify and apply a rule to calculate the unknown values in equations. “Check each answer and cross out each incorrect answer. Show your thinking. a. 30%$\cdot$p = $6, p = $21, b. x = 30 + (20% $\cdot$ 30), x = $36 c. 25% $\cdot$ z = 9, z = 13 d. 28 = 50% $\cdot$ y, y = 13.” Step 3 Teaching the lesson states, “Project the equations (slide 5) to discuss two alternate strategies. Encourage students to explain why each method results in the same solution. (MP8). Ask, How can you check your answer?”

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 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.”