The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 3, Lesson 9, Decimal fractions: Comparing and ordering with unequal places, Step 2 Starting the Lesson, students make sense of problems that involve comparing decimal fractions and then model their strategies. “Project slide 1. Read the problem aloud and ask, Is it possible for a decimal fraction that has only one decimal place to be greater than a decimal fraction with two decimal places? Encourage students to explain their thinking. Project slide 2 and invite students to shade the squares to support their arguments (MP1). Project slide 3 to repeat the activity. In this problem, the two decimal fractions have an unequal number of places, yet they represent the same number. Again, have the students demonstrate their solutions. However, this time project slide 4 so students can model their solutions on a thousandths square.”

• Module 4, Lesson 11, Mass/capacity: Solving word problems (customary units), Step 3 Teaching the Lesson, students analyze multi-step word problems that involve converting customary units of mass and capacity and persevere in solving them. “Project slide 2, as shown. Slowly read the problem aloud, twice through. Slide 2: A bottle of juice holds 64 fl. oz. Naomi pours juice into a pitcher. The pitcher holds $1\frac{1}{2}$ quarts and is now full. How much juice is left in the bottle? Organize students into pairs then discuss the points below (MP1): What is the problem asking you to do? What information do you know? What information do you need to find out? How will you solve the problem? What steps will you follow? Allow time for the students to solve the problem. Invite volunteers to share their strategies on the board. Emphasize the importance of working with only one measurement unit. Have the students identify the measurement unit that they decided to convert.”

• Module 7, More Math, Problem Solving Activity 4, Word Problems, students make sense and persevere in solving word problems involving operations with mixed numbers. “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 operation will we use? What will we do first? What will we do next? How could you show your thinking? Slide 1: Deon made $3\frac{1}{4}$ qt of vegetable soup and $2\frac{2}{3}$ qt of chicken soup. Megan made $3\frac{1}{2}$ qt of lentil soup and $2\frac{3}{8}$ qt of potato and leek soup. Who made the greater amount of soup?”

• Module 12, Lesson 5, Division: Working with the standard algorithm (with remainders), Step 3 Teaching the Lesson, students make sense and persevere in solving division word problems. “Project side 3 and discuss the points below (MP1): What is the problem asking us to do? What do you estimate the answer will be? Who can solve the problem using the standard algorithm? Slide 3: The path at the golf club is 10,935 yards long. If I have gone half-way around the course, how far have I traveled?”

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 10, Algebra: Working with expressions (without parentheses), Step 2 Starting the Lesson, students reason abstractly and quantitatively as they represent a word problem as an expression or equation and think of a word problem to match given conditions. “Organize students into pairs and have them work together to write a word problem that involves adding three numbers (MP2). Encourage them to choose numbers that are mentally manageable. Afterward, invite three or four students to share their word problems. As each problem is read, write key points on the board, and then discuss the points below: What expression would you write to match the problem? Is it possible to write more than one expression? When it is written can we arrange the parts to make it easier to calculate an answer? Do we need parentheses? Why not? Repeat for a word problem that involves multiplication and three factors.”

• Module 5, Lesson 7, Decimal fractions: Subtracting tenths (decomposing ones), Step 3 Teaching the Lesson, students reason abstractly and quantitatively as they represent a problem symbolically and explain the solution in context of the original problem. “Refer to the tide heights for the first high tide and the first low tide on Sunday. Have pairs calculate the difference between these tides using the standard subtraction algorithm (MP2). Remind them to first estimate the difference and also to align the place values and decimal point. Afterward, project slide 3, as shown. Invite a volunteer to verbalize the procedural steps they followed to calculate the difference.”

• Module 6, Lesson 10, Division: Three- and four-digit dividends and one-digit divisors (with remainders), Step 3 Teaching the Lesson, students reason abstractly and quantitatively as they write a word problem to show how remainders can be interpreted differently depending on context, and then interpret the remainders in the context of a word problem. “Review the role that context plays in interpreting remainders. For example, depending upon the context the remainder might be broken up and shared, it might represent the actual answer, or it might be irrelevant to the answer and excluded. Encourage students to provide an example for each situation. Organize students into pairs, then move around the room asking each pair of students to roll the cube. The number that they roll will tell the remainder in the word problem that they must write (MP2). For example, if (Hailey) rolls 2, she will write a word problem that leaves a remainder of 2. Select several word problems and read them aloud. Encourage the students to discuss the problem in their pairs to make sense of what is happening and how the quantities are related (MP1). Then ask students to interpret the remainder in the context of the problem (MP2).”

• Module 10, Lesson 12, Decimal fractions: Solving multiplication and division word problems, Step 3 Teaching the Lesson, students decontextualize and contextualize word problems that involve dividing or multiplying decimal fractions. “Slide 2: A gardening machine needs 0.1 gallons of oil for each gallon of gas. How much oil is needed for the gas in this container?” An image shows a 5 gallon container. “A remote-controlled plane uses 0.1 gallons of gas for each half-hour of flight. What is the greatest amount of flying time possible with 5 gallons of gas?  Encourage them to use a letter or symbol to represent the unknown amount in each equation. For example, students could write T =  0.1 x 5 for the first word problem, or T = $5\div 0.1\div 2$ for the second word problem (MP2). The latter requires two steps, so students should justify why they decided to divide the total number of half-hours of flight (50) by two. Make sure students explain what the solutions represent in the context of each problem (MP2).”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 2, Student Journal, page 81, Convince a friend, students justify their reasoning about the standard algorithm for multiplication and then critique the reasoning of a classmate, “Ashley says when completing the standard algorithm for multiplication the greater factor must be recorded first or the product will be incorrect. Gabriel disagrees with Ashley. Who do you agree with? Explain your reasoning. Share your thinking with another student. They can write their feedback below.” On the journal page, students complete, “I agree/ disagree with your thinking because … ”

• Module 4, Lesson 5, Common fractions: Converting mixed numbers to improper fractions, Step 3 Teaching the lesson, students construct arguments and critique the reasoning of others as they develop a rule for rewriting mixed numbers as improper fractions. “Encourage students to develop a general rule that would work to convert mixed numbers to improper fractions (MP8). Allow time for them to test their rule with a few examples. Then invite students to share their rule with the class and justify how it works and why it makes sense conceptually (MP3). Encourage students to use diagrams in their justifications. (Note: Students who cannot describe why the rule works conceptually are not ready to use the rule. Premature use of rules puts students at a disadvantage in later learning.) Prompt students to critique the reasoning of their peers (MP3) by providing sentence stems such as: I agree/disagree with you because …  I don’t understand …  My rule is different because … So, what you are saying is …”

• Module 6, Thinking Tasks, Question 5, students construct viable arguments and critique the reasoning of others as they solve a multi-step real-word problem involving operations with fractions and then determine whose cost plan is the best option. “After the marathon $\frac{1}{4}$ of the Apple Pie Granola Bars and the Fig and Walnut Bars were left. Students calculated how many total bars were sold and how much money they made. They charged $0.50 for each bar. Cody predicted that after buying the fruits and nuts, they lost money and should have charged $1.00 for each bar instead. Nancy argued that they did make money and might have sold fewer bars if they were more than $0.50 for each bar. Do you agree with Cody or Nancy? Write a letter to the Leadership Team explaining why.”

• Module 8, Lesson 5, Common fractions: Finding a fraction of a whole number symbolically (non-unit fractions), Step 3 Teaching the Lesson and Student Journal, page 294, Step In, students critique the reasoning of others as they find a fraction of a whole number. Student Journal, “Andre races cars. He has 25 miles of the race left to complete. His pit stop crew tell him he has enough fuel to travel $\frac{3}{4}$ of that distance. How many miles can he travel before refueling? Katherine showed her thinking like this. What steps does she follow? Why does she rewrite 25 as $\frac{25}{1}$? Why does she decide to convert the improper fraction to a mixed number? Cole and Sara showed their thinking like this (image is shown of their work). What do all the methods have in common? How are they different? Why did Cole and Sara change the order of the factors?“

• Module 12, Student Journal, page 470, Convince a friend, students construct viable arguments and critique the reasoning of others as they solve world problems using the standard division algorithm. “Awon says that buy 2 get 1 free is the best deal. Cathay says that it was better to buy 5 packs because you get 2 free packs. Who do you agree with? Explain your reasoning. 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.” An image is shown, “Baseball Card Bargains: 1 pack of 10 cards $4.92, Buy 2 packs get 1 pack free, Buy 5 packs and get 2 packs free.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 2, More math, Problem solving activity 2, students model a real world problem, for the best chair arrangement for graduation, using multiplication strategies. “Between 1,200 and 1,300 people will attend the Freemont High School graduation. The chairs need to be arranged in a rectangular array and 18 to 23 chairs can fit in a row. How many rows and how many chairs in a row are needed to make sure they have enough chairs for all the people?” Directions for the teacher state, “Allow time for them to work independently or in pairs to experiment with different arrangements. Notice how the students work with the numbers. (Are they estimating? What multiplication strategies are they using? Do they use the standard multiplication algorithm?) Invite volunteers to share how the chairs could be arranged and what strategies they used to figure out the answer. Emphasize the range of dimensions that work.”

• Module 4, Lesson 1, Common fractions: Reviewing equivalent fractions (related denominators), Step 3 Teaching the lesson, students model with math as they determine which area models can be used to represent equivalent common fractions. “Encourage pairs to explain how they knew an equivalent fraction could be shown on another rectangle and how they decided which rectangle to use and how many parts to shade. (MP4) Record the fractions and encourage them to describe how the numerators and denominators of the fractions are related, for example, "Multiply both the numerator and denominator by 3 and the fraction is equivalent."

• Module 7, Student Journal, page 280, Mathematical modeling task, students model a real- world problem involving operations with mixed numbers. “Nancy found this old recipe for fruit punch. Unfortunately, some of the measurements of ingredients are missing. Nancy knows that the recipe makes a little less than $8\frac{1}{2}$ cups. She also knows there is more ginger ale than apple juice in the punch and that the quantity of each is a mixed number. Write the amount of apple juice and ginger ale that is missing. Explain your thinking.” The recipe shows ___ cups of apple juice, ___ cups ginger ale, $3\frac{1}{5}$ cups of cranberry juice, $1\frac{1}{4}$ cups orange juice, $\frac{1}{2}$ cup cold raspberry.

• Module 12, More Math, Thinking Task, Classroom Conundrum, students model with math as they graph and represent patterns on a coordinate grid. Question 3, “Use the information from Classroom Conundrum to solve. a. Complete the tables in two different colors to show the number of students (y) that can be seated in any number of classrooms (x). b. Plot each set of data on the coordinate plane. c. The school can purchase 7 of Building A for the same price as 2 of Building B. Which is the better option? Consider student numbers in the explanation of your answer.” Two tables show Classroom seating capacities for building A and building B.

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 10, Decimal fractions: Rounding thousandths, Step 3 Teaching the lesson, students choose strategies they have learned as tools to round numbers to thousandths. “Organize students into pairs and distribute the support page. Project the numbers 1.829, 1.170, and 1.493 (slide 4). Then have the students work together to round each number to the nearest whole number, tenth, and hundredth. Remind them that they can use the thousandths squares on the support page or draw number lines to help their thinking. Some students may prefer to use place-value strategies (MP5).”

• Module 5, Lesson 6, Decimal fractions: Using the standard algorithm to subtract, Step 3 Teaching the lesson, students choose an appropriate strategy as a tool to subtract decimals. “Project the picture of two dogs (slide 2) and say, The mass of each dog is measured in kilograms. What is your estimate of the difference in mass? What digits did you look at to make your estimate? How could you calculate the exact difference? Allow time for the students to use a preferred method to calculate the exact difference (MP5). Then project an empty number line (slide 3) and invite volunteers to draw jumps on the number line to show their thinking. Other students may have written equations or used a method, such as recognizing that 16.94 – 13.4 gives the same difference as 16.54 – 13.”

• Module 9, Student Journal, page 356, Mathematical modeling task, students choose strategies they have learned as tools to solve a real-world problem. “1.25 meters of ribbon is used for this gift. The bow is made with 15 cm of the ribbon. Write two different sets of possible dimensions the box could be. Show your thinking.” 

• Module 12, More Math, Investigation 3, Using partial quotients, students choose strategies they have learned as tools to solve a real-world problem using partial quotients. “How many different ways can you split $74.80 into 2 parts to help divide by 8?”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 2, Lesson 8, Volume: Analyzing unit cubes and measuring volume, Step 4 Reflecting on the work, students use precision when reasoning about volume using unit cubes. “Discuss the students’ answers to Student Journal 2.8. Have the students share their rule for determining the total number of cubes when they know the dimensions of the base and the number of layers. Refer to Step Ahead and ask, Do you think Carlos’s calculation is accurate? Encourage students to cite the gaps and overlaps that are left by the cubes in the container (MP6). Discuss the difficulties of using cubes to measure the volume of containers that have curved sides. Ask, How else could you determine the volume? Encourage suggestions such as, filling the container with sand or rice then pouring the rice into a rectangular-based prism.”

• Module 7, Lesson 10: Number: Working with exponents, Student Journal, page 271, Step Up, students attend to precision as they use exponents to accurately represent powers of 10. Question 3, “Write each number using exponents. a. one hundred, b. one million, c. ten, d. ten million, e. one, f. ten billion.” Step 4 Reflecting on the work, “Have students explain how they identified the correct exponent to use (MP6).” 

• Module 9, More math, Thinking tasks, The Ecology Club, Question 1, students attend to precision as they read a line plot and accurately calculate operations with fractions. “The Grade 5 Ecology Club stays after school once a week to maintain the garden and the trees that surround their school. One year ago they hosted a big tree-planting event. Every year they will measure the trees in the grove to see how they are growing and buy new seedlings to replace those that have died. This year, the club measured all of the young trees to compare the heights of all the trees. They measured the growth of the new trees and recorded it on this line plot. The trees that died are not represented. What is the difference in height between the tree with the greatest growth and the tree with the least growth?” 

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. Associative property of multiplication, calculate, centimeter (cm), compare, cubic centimeters (cm³), cubic inches (in³), cubic units, decimal fraction, dimensions, equivalent fractions, estimate, expanded form, feet (ft), fraction, height, inch (in), length, meter (m), multiplication algorithm, multiplication, multiply, number name, one whole, partial products, prism, product, volume, width.” 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 5, Lesson 11, 2D shapes: Exploring categories of quadrilaterals, Step 3 Teaching the lesson, students use clear and precise language as they examine relationships between different types of quadrilaterals. “Project the Step In discussion from Student Journal 5.11. Review what the students know about tree diagrams. Specifically, highlight how they show the relationship between different categories (MP6). Then discuss the points below: What does the tree diagram show? What does it tell you about the relationship between some of these shapes? What does it tell you about rectangles? What does it tell you about squares? Discuss what the students know about shapes based on their position in the tree diagram. For example, squares have four sides (because they are quadrilaterals), two pairs of parallel sides (because they are parallelograms), four equal sides (because they are rhombuses), and four equal angles (because they are rectangles). Invite students to come to the front and draw examples of the shapes that match the label for each white box. As they draw examples, ask individuals to justify why the shapes they draw belong in a particular part of the tree diagram (MP3). Encourage others to confirm or challenge the shapes drawn by their peers. Protractors and rulers can be used to measure the angles and side lengths of contentious shapes (MP6).”

• Module 11, Lesson 12, Volume: Solving word problems, Step 2 Starting the lesson, students use the specialized language of mathematics in order to communicate ideas about volume. “Project slide 1, as shown, and discuss the points below (MP6): How do we calculate the area of a rectangle? Who can use this image to explain the difference between area and volume? What is the definition of volume? How do we calculate volume? How do we record volume? Why is the record for volume written as cubed?”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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, Multiplication: Extending the standard algorithm, Step 3 Teaching the lesson, “students relate the procedural step of the standard multiplication algorithm back to the partial-products strategy.” “Project slide 4 and have the students carry out the procedural steps of the standard algorithm. Discuss the points below: What numbers do we multiply in the first row of calculations? What numbers do we multiply in the second row of calculations? What about the third row?  Does any regrouping take place? How should we record the regrouping digit? Why do we write a zero in the ones place of the second row, and then again in the tens place and ones place in the third row? How do we calculate the final total? What numbers do we add? Relate each of the procedural steps back to the partial products in the rectangle (MP7).”

• Module 3, Lesson 10, Decimal fractions: Rounding thousandths, Step 3 Teaching the lesson, “students use benchmarks between two marked numbers on a number line to help them round.” “Project the number line partitioned into thousandths (slide 3) and have the students identify the position of 1.371. If necessary, remind the students that 1.37 is equivalent to 1.370. Ask, How does this number line help us identify the hundredth that is nearest? Again, the students should describe how the halfway point between 1.37 and 1.38 is used as a benchmark to help them decide (MP7).”

• Module 7, Lesson 2, Common fractions: Subtracting (related denominators), Student Journal, page 246, Step Up, Question 1, students make use of structure as they create equivalent fractions. “Rewrite each fraction to calculate the difference. Use the diagram to help. Then write the difference. a. $\frac{4}{5}$ - $\frac{7}{10}$ = , b. $\frac{2}{3}$ - $\frac{2}{9}$ =.”

• Module 11, Lesson 2, Algebra: Examining relationships between two numerical patterns, Step 2 Starting the lesson and Student Journal, page 398, students make use of structure as they work with numerical patterns. “Project the table (slide 1) and have the students identify the two variables being compared. Ask, What are some missing numbers we can write? How many candles are in each box? How do you know? How can we calculate the total number of candles for any number of boxes? What rule could we write? Invite volunteers to write the missing numbers in the table (MP7).” Student Journal, Step Up, Question 1, “Emilio has a favorite fruit punch recipe. To make one glass of punch he uses 2 fl oz of pineapple juice and 6 fl oz of orange juice. Write in this table to help you answer the questions below. a. How much pineapple juice will be used for 5 glasses of punch? b. If 12 fl oz of orange juice is used, how many glasses of punch can be made? c. How much orange juice will be used if 20 fl oz of pineapple juice is used? d. How much pineapple juice will be used if 90 fl oz of orange juice is used?” A table is provided that shows Number of glasses- 1 and then six empty boxes, Pineapple juice (fl oz)- 2, and then six empty boxes, Orange juice (fl oz) - 6, and then six empty boxes. 

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 1, Lesson 5, Number: Reading and writing eight- and nine-digit numbers, Step 2 Starting the lesson, “students apply what they know about the multiply by 10 relationship of the base-ten number system, to work the other way and divide by 10.” “Project slide 1 and ask, What do you notice about this table? What patterns do you see? Highlight that the places are grouped into threes (millions, thousands, and ones) and that the places within each group repeat in the pattern H, T, O. Project slide 2, and refer to the arrows below the table. Say, Ten is 10 times as many as 1, and one hundred is 10 times as many as 10. What other relationships can you share? Volunteers can draw more arrows to show the relationship between different place values. Say, The size of each place value is ten times as much as the place value just before when we move from right to left in this table. What happens if we move from left to right? Draw arrows in the opposite direction above the table to show that the size of each place value is one-tenth of the size of the place value just before when we move from left to right (MP8).”

• Module 6, Lesson 1, Common fractions: Making comparisons and estimates, Step 4 Reflecting on the work, “students make a generalization about how the difference between the numerator and the denominator of a fraction affects the size of the fraction.” “Project slide 4. Discuss the common fractions that the students made. Explain how the difference in value between the numerator and denominator affects the size of the fraction. Students can check this relationship and investigate whether it applies with other collections of four digits. (MP8)”

• Module 8, More math, Investigation 1, Fraction Patterns, students use repeated reasoning as they look for shortcuts for operations with fractions. “What numbers will make these true? $\frac{1}{4}\times$__ = 15, $\frac{2}{4}\times$__ = 15, $\frac{3}{4}\times$__ = 15, $\frac{4}{4}\times$__ = 15. What other patterns involving fractions and multiplication can you make? Then challenge students to look for patterns in the equations and write other groups of equations that show the same patterns.” 

• Module 11, Lesson 6, Algebra: Interpreting coordinate grids, Student Journal, page 411, Step Ahead, students identify and write rules from patterns on coordinate planes. “This rule describes the relationship between the number of cups of water (C) and number of scouts (S) on page 410: $S\times6$ = C. Write rules to describe the relationship between the items in Questions 3 and 4. a. Distance (D) and Time (T): b. Green beads (G) and Red beads (R):”  Step 4 Reflecting on the work, “They can then share the relationship rules they wrote in Step Ahead. (MP8)”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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. “The students have been introduced to six-digit whole numbers, and have been introduced to the idea of one million and a special place name for this place. The millions place is important because it is the start for the next three places that are described using the word millions. This module begins with a review of six-digit numbers. In Lesson 2, students focus on reading and writing seven-digit numbers to help them see that they now have three main groups of three when saying number names: the millions, the thousands, and the ones. A place-value expander helps to clearly show these groups. The number line is used to provide a picture of seven-digit numbers to help students when they work with these greater numbers. The number line representation stresses the fact that the greatest distance from zero is determined by looking at the place with greatest value, and then the digit in that place. The idea of relative position — what the number line tool displays — is useful when students compare, order, or round numbers. Comparing and ordering involve the position of numbers relative to 0 (the origin.) For rounding, the number line indicates the position of a number relative to nearby multiples of 10, 100, 1,000, and so on. In Lesson 5, the work with seven-digit numbers is extended to eight- and nine-digit numbers, so students become confident with reading, writing, comparing, ordering, and rounding these whole numbers.”

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 6, Number: Working with millions expressed as fractions, Step 2 Starting the lesson, teachers provide context about equal parts on a number line. “Project the number line (slide 1) and say, Draw marks to divide this number line into four equal parts. What number will we write at each of the marks? How do you know? Have a student draw the marks and then encourage other students to describe how they would figure out the value of each number. Erase the numbers and the marks and repeat to divide the number line into five and then eight equal parts.”

• Module 5, Lesson 1, Decimal fractions: Reviewing addition strategies (without composing), Step 3 Teaching the lesson, provides teachers guidance about how to perform operations and work with decimals to hundredths. “Project the empty number line below the table (slide 3). Invite volunteers who used a number line to model their thinking and demonstrate their strategy on the board. Students who used a place-value strategy or worked with common fractions should also demonstrate their steps. Refer to the three methods on the board. Ask students to compare and contrast the models and describe the appropriateness of each (SMP4). Repeat the discussion by asking the students to calculate the sum of polyunsaturated and monounsaturated fats. Again, have the students model their strategy on the number line or on the board. Look for students who convert the 2 tenths to 20 hundredths to think 5.45 + 2.20. This can be a useful strategy because it allows students to work with an equal number of decimal places (SMP7). Relate the addition of decimal fractions to money. Then project the price tags (slide 4). Ask, When might you add prices like these? What is the total cost? Students can then share other amounts of money that they may add on a daily basis. Project the expressions (slide 5). Students can then work independently or in pairs to calculate each sum. Afterward, select a few students to share their strategies and answers with the class. Project the Step In discussion from Student Journal 5.1 and work through the questions with the whole class. Read the Step Up and Step Ahead instructions with the students. Make sure they know what to do, then have them work independently to complete the tasks.”

• Module 9 Lesson 9, Mass: Converting metric units,  Lesson overview and focus,  Misconceptions, include guidance to address common misconceptions as students work with converting metric units. “For students who struggle with metric conversions, work to separate struggles with decimal fractions from challenges with the units of measure. Provide experience with benchmarks and measuring tools to support those who are not familiar with the units of measure. Provide fraction and place-value support for those who struggle with decimal fractions.”

The materials reviewed for Origo Stepping Stones 2.0 Grade 5 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, Research into Practice, Multiplication, supports teachers with concepts for work beyond the grade. “Work in this module on multiplicative comparison sets the stage for future work with unit conversion in Grade 5 and equivalent ratios in Grade 6. Unit conversion is an instance of multiplicative comparison (how many times as many inches as feet in a given length, for example) while equivalent ratios are a sequence of multiplicative comparisons. Read more about unit conversions in the Research into Practice sections for Grade 5, Modules 4 and 9. Read more about the ways students understand equivalent ratios in the Research into Practice section for Grade 6 Module 3.”

• Module 4, Preparing for the module, Research into practice, Common fractions, includes explanations and examples connected to fraction equivalence and fraction notation. To learn more includes additional adult-level explanations for teachers. “Work with fractions in this module incorporates two understandings of fraction notation. When students work with equivalent fractions, they are building on their understanding of multiplication and building their skills towards proportional reasoning. By looking at fractions in the context of the multiplication table rows, students see that the numerator and denominator in equivalent fractions vary together. The skill of converting improper fractions to mixed numbers begins to suggest the division interpretation of a fraction where 15⁄4 is the numerical value of 15 ÷ 4. Students may come to this understanding as they create groups of four fourths (each one whole) by repeated subtraction and find that 15 fourths can be described as three wholes and 3 more fourths. It is not necessary to teach students this strategy for finding equivalence. Allow them to use their knowledge of numbers and operations to realize what is happening. It may be helpful to some students to relate this to the idea of regrouping used in base-ten place value. When working with fractions, we do not regroup for ten of a given item; we regroup when we reach one whole, whether four fourths, five fifths, or any way the whole is partitioned.” To learn more, “Neumer, Chris. 2007. “Mixed Numbers Made Easy: Building and Converting Mixed Numbers and Improper Fractions.” Teaching Children Mathematics 13 (9): 488–92.”

• Module 5, Preparing for the module, Research in practice, Decimal fractions, support teachers with concepts for work beyond the grade. “As the Mathematics Focus suggests, this work prepares students for the multiplication and division of decimal fractions (Module 10), then for the use of the standard algorithms for addition, subtraction, and multiplication in Grade 6 (Module 2) and for division (Module 3) with decimal fractions. In preparation for this, use every opportunity for students to work with the place value structure of the base-ten system and to internalize the multiplicative relationships between different place values. For example, have them represent 0.527 and 527 with a numeral expander or place-value slider then explain how each number changes as it is multiplied and divided by 10, 100, and 1,000. Be sure students describe how the digits shift, not the decimal point, as its position remains constant. Read more about operating with decimal fractions in the Research into Practice section of Module 10, and in Grade 6 Modules 2 and 3.”

• Module 9, Preparing for the module, Research into practice, Measurement, includes explanations and examples connected to measurements within the metric system. To learn more includes additional adult-level explanations for teachers. “Measurement provides a powerful context for continuing to think about fractions and the relationships they represent. Students must continue to develop a benchmark sense of the measures they know (for example, “How much is 2 liters of soda?” of “How many laps around the track is 1 kilometer?”) They then use this number sense to reason about measuring the same quantity using different units. Because the metric system is driven by base-ten relationships, this reasoning strengthens student understanding of decimal fractions as well.” To learn more, “Lehrer, Richard, and Hannah Slovin. 2014. Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics in Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 5 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 5 and the Common Core Standards, includes all Grade 5 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 3, 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, Decimal fractions: Reviewing addition strategies (without composing), the Core Standard is identified as 5.NBT.B.7. The Prior Learning Standards are identified 5.OA.A.2, 5.NF.B.7, 5.NF.B.7c. 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 4, Mathematics Overview, Measurement and Data, includes an overview of how the math of this module builds from previous work in math. “In Grade 4, students learned the relationships between customary units of length, capacity, and mass, and converted between common units. All answers were whole numbers. This module extends that to converting between units, and includes examples that involve familiar fractions. For example, students convert 8 fluid ounces to one-fourth, one-half, and then three-fourths of one quart. The module concludes with solving word problems involving capacity/mass and, in a separate lesson, data collection activities in which students work with ounces to construct and answer questions related to line plots.”

• Module 10, Mathematics Overview, Coherence, includes an overview of how the content in fifth grade connects to mathematics students will learn in sixth grade. “Lessons 5.10.1–5.10.12 focus on multiplication and division with decimals, utilizing the partial-products strategy and the area model for multiplication, and using models, the relationship between multiplication and division, and place-value strategies for division. This extends the previous work with multiplication and division (4.1.2, 4.5.2–4.5.4) to examples involving decimals with tenths or hundredths and connects to work using the standard algorithm to multiply and divide decimals (6.2.10–6.2.12).”

• Module 4, Lesson 6, Common fractions: Solving word problems, Topic progression, “Prior learning: In Lesson 5.4.5, students use fraction strips and related parts to explain how they can rewrite a mixed number as an improper fraction. They consider general rules that can be applied. 5.NF.A.1; Current focus: In this lesson, students are encouraged to use a range of strategies to solve multi-step word problems involving the comparison of two or more common fractions. 5.NF.A.2; Future learning: In Lesson 5.8.2, students relate finding a unit fraction of a quantity to division. 5.NF.B.3” 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 5 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, “Multiplication: Students in Grade 5 extend their knowledge of the standard algorithm for multiplication from specific cases to multiplying any pair of whole numbers. Their understanding of place value allows them to extend their models for multiplication to multi-digit numbers, continuing the learning progression begun in Grade 3. Students have a wide range of strategies and methods, including area diagrams and partial products, available to them. Extending area diagrams to three-digit numbers helps students connect their work with specific cases to more general cases with multi-digit numbers. Connecting area diagrams to written records, students see that the same strategies and patterns extend into additional columns of the place-value chart. This supports their work to generalize the algorithm. Students review multiplicative comparison word problems in preparation for work this year with scale factor. These problems help students transition their thinking from an additive model to multiplicative reasoning. When comparing two values, multiplicative thinking emphasizes how many times greater (or lesser) one value is compared to the other. This reasoning lays the foundation for ratio and proportion, a key element of middle school mathematics. Volume: Grade 5 students work with volume as a measurement concept as well as an opportunity to apply and extend their knowledge of multiplication. Students reason about building structures and filling shapes with unit cubes to explore ideas of volume and develop a formula for calculating volume. Students need many concrete experiences filling space with unit cubes before they can visualize the elements of volume and use the formula fluently. To learn more: Farmer, Sherri A., Kristina M. Tank, and Tamara J. Moore. 2013. “Using STEM to Reinforce Measurement Skills.” Teaching Children Mathematics 22(3): 196-199.; Fuson, Karen C. 2003. “Toward computational fluency in multi-digit multiplication and division.” Teaching Children Mathematics 9(6): 300-305. Wallace, Ann H. and Susan P. Gurganus. 2005. “Teaching for Mastery of Multiplication.” Teaching Children Mathematics 12(1): 26-33. References: Battista, Michael T., and Douglas H. Clements. 1996. “Students’ understanding of three- dimensional rectangular arrays of cubes.” Journal for Research in Mathematics Education 27(3): 258-292. Fuson, Karen C. and Sybilla Beckmann 2012. “Standard algorithms in the Common Core State Standards.” National Council of Supervisors of Mathematics Journal of Mathematics Education Leadership 14(2); 14-20. McCallum, William, Phil Daro, and Jason Zimba. “K, Counting and Cardinality; K–5, Operations and Algebraic Thinking” in Progressions for the Common Core State Standards in Mathematics. The University of Arizona. https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf (retrieved 21 September 2016)”

• Module 8, Preparing for the module, Research into practice, “Common fractions: Students in Grade 5 extend their understanding of fraction multiplication to include multiplying a fraction by a fraction. In many cases, an area model serves as a more accessible representation than an equal groups model for this operation. As with all operations, it is essential that students see the operations in context and reason about their solution process in the context of specific problem situations. Students may have the misconception that multiplication makes things greater, not realizing that multiplication by a fraction less than one results in a product of lesser value. Reasoning in the context of problem situations helps students build confidence in their solutions. When students use an area model to multiply fractions, there are several important ideas to develop. First, the whole changes from a length (units along each side of the rectangle) to an area (square unit area of the rectangle itself). Multiplying the denominators tells how many pieces the new square unit area is divided into. Multiplying the numerators tells how many of those pieces are part of the solution to the problem. These basic principles extend to multiplying mixed numbers as well. It is also important to ensure that the visual model being used is appropriate to the situation and mathematics being represented. Students often require more time to see the relationships between multiplication and division than they needed for addition and subtraction. As students gain experience with fractions, they begin to see that sharing between two people (dividing by two) gives the same result as halving (multiplication by $\frac{1}{2}$). This shows the inverse relationship between multiplication and division and helps students develop their understanding of this relationship more deeply. To learn more: Webel, Corey, Erin Krupa, and Jason McManus. 2016. “Using Representations of Fraction Multiplication.” Teaching Children Mathematics 22 (6): 366–73. References: Robinson, Katherine M., and Jo-Anne LeFevre. 2012. “The Inverse Relation between Multiplication and Division: Concepts, Procedures, and a Cognitive Framework.” Educational Studies in Mathematics 79 (3): 409–28. Van de Walle, John, Karen Karp, and Jenny Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally 8th ed. Boston: Pearson.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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, “cube labeled: 17, 19, 24, 26, 31, 35 and a cube labeled 15, 20, 25, 30, 35, 40 for lesson 3, small to medium-sized rectangular prisms (such as a toothpaste box, matchbox, soap box, etc.) for lessons 7 and 8. Each group of students need base-10 blocks (ones) for lessons 7, 8 and 11, inch cubes, marbles, counters and coins for lesson 8, small to medium-sized rectangular prisms (such as a toothpaste box, matchbox, soap box, etc.) for lesson 8, and Support 14 for lesson 7. Each pair of students need base 10-blocks (ones) for lessons 7 and 10. Each individual student needs base-10 blocks (ones) for lessons 7 and 9, and Student Journal for each lesson.” 

• Module 2, Lesson 9, Volume: Developing a formula, Lesson notes, Step 1 Preparing the lesson, “Each student will need: base-10 blocks (ones) and Student Journal 2.9.” Step 2 Starting the lesson, “Say, Imagine you have 20 cubes. What are some different rectangular- based prisms you could make using all the cubes? What would be the dimensions of the three faces? How would you check that you have used all the cubes in each prism? If necessary, provide cubes so the students can construct the prisms to confirm their thinking.”

• Module 5, Preparing for the module, According to the Resource overview, teachers need, “3.5 meter length of string, 4.25 meter length of string and metric measure tape for lesson 4, a cube labeled: 1, 1, 2, 2, 3, 3, and another cube labeled: O, O, T, T, H, H for lesson 5, a protractor and ruler for lesson 11. Each group of students need crayons or markers and Support 15 in lesson 1 and geostrips and fasteners, paper, and a protractor in lesson 12. Each pair of students needs a protractor in lesson 10, scissors in lesson 7, Support 16 in lesson 3, Support 19 in lessons 7 and 8, and Support 20 in lesson 9. Each individual student needs scissors and glue and Support 21 in lesson 10, and the Student Journal in each lesson.”

• Module 6, Lesson 3, Common fractions: Adding (related denominators), Lesson notes, Step 1 Preparing the lesson, “Each student will need: 1 copy of Support 11, scissors, Student Journal 6.3”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 6, Decimal fractions: Using the standard algorithm to subtract, include:

• Extra help, “Activity: Organize students into pairs and distribute one card to each student. Have pairs compare their mass pieces, identify the mass that is heavier, and explain their thinking. Encourage each student to then find another student who has a card showing a similar mass, for example, 6.4 kg and 6.09 kg. Establish that 6.4 kg is equivalent to 6.40 kg. The students use this understanding to help calculate the difference. Have students form new pairs and repeat the activity as time allows.”

• Extra practice, “Activity: Write the equations 5.42 – 3.2 = ____ and 3.98 – 2.45 = ____ on the board. Have the students work independently to solve each equation using both the standard subtraction algorithm and another method of their choice. Afterward, they should share their alternative methods and place a check mark beside those methods that were more efficient to use than the standard algorithm.”

• Extra challenge, “Activity: Organize students into groups and distribute the resources. Have students cut out the cards from the support page and place them facedown on the floor. Ask each student to select two cards. They then calculate the difference between the masses indicated on their cards. One point is awarded to the student who records a difference that is nearest (2). Repeat the activity: give a different whole number and have students select two new cards to form a difference that is nearest the new target number.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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: Comparing and ordering seven-digit numbers, Differentiation, Extra Challenge, “Organize students into pairs to play Guess My Number. Ask one student to discreetly write a seven-digit number (where zero is not permitted in the millions place). The other student then asks a series of yes/no questions to figure out the number. For example, “Is your number greater than 5,000,000?” or “Is your number within 100,000 of 3,450,000?” Students keep a tally showing how many questions the student asks before they guess the number (up to 20 questions). When either the number has been determined or 20 questions have been asked, students alternate roles and repeat the game. For each round, the student with the fewest tallies wins. The game is repeated as time allows. Rules may be altered so students are to guess within 10,000 of the secret number.”

• Module 4, Lesson 2, Common fractions: Reviewing equivalent fractions (related and unrelated denominators), Differentiation, Extra Challenge, “Organize students into pairs and distribute the strips. Ask each pair to place their multiplication strips facedown on the floor. They take turns to choose two strips to compose several equivalent fractions. The student who records the greater fraction scores one point. A bonus point is awarded if they can identify equivalent fractions that share a common denominator. The first student to record six points wins. For example, in the image below, (Lomasi) scores 1 point because $\frac{7}{10}$ is greater than $\frac{3}{6}$. She then identifies two equivalent fractions that share a common denominator ( and $\frac{30}{60}$) to score a bonus point. (Note: Retain the multiplication strips for Lesson 6.4).”

• Module 10, Lesson 4, Decimal fractions: Multiplying with whole numbers using partial products, Differentiation, Extra Challenge, “Organize students into pairs. Have students take turns to write any three single digits. They then use their digits to create three different multiplication expressions. Each expression has one factor that is a whole number and one factor that is a decimal fraction recorded in hundredths. For example, a student writes 3, 9, and 5, and uses these digits to write 3 × 0.95, 5 × 0.39, and 9 × 0.35. Students then calculate each product. One point is awarded to students who form a product nearest a whole number. The student with the most points after five rounds wins.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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: Comparing and ordering seven-digit numbers, Lesson notes, Step 2 Starting the lesson, “ELL: Allow students to discuss the meaning of three- and four-digit numbers before moving on with the activity. Demonstrate an example of the word halfway before moving on with the activity.” Step 3 Teaching the lesson, “ELL: Use hand gestures to show left to right as you say the phrase. Ensure students understand the difference between the words right as in direction and write as in transcribing text, and order as in sequence and order as in placing a request for food at a restaurant. Allow students to discuss the words nearest, between, and order before moving on with the activity. Pair the students with fluent English-speaking students. During the activity, have students discuss the concepts in their pairs, as well as repeat the other student’s thinking. Allow the students to work in their pairs to complete the Student Journal, if necessary.” Step 4 Reflecting on the work, “ELL: Allow pairs of students to discuss their thoughts on the question, What rolls will you need to get the closest number? Then, invite the student to explain their thoughts to the class.”

• Module 4, Lesson 1, Common fractions: Reviewing equivalent fractions (related denominators), Lesson notes, Step 3 Teaching the lesson, “ELL: Pair students with a fluent English-speaking partner. During the activity, have students discuss the concepts in their pairs, as well as repeat the other thinking. Encourage them to explain what they are learning in their own words to check for understanding of the concepts. Create an anchor chart about common fractions and display in the classroom for students to reference when necessary. Encourage them to use the words numerator and denominator in a sentence to show understanding of the language. Allow students to work in their pairs to complete the Student Journal, if necessary.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 7, Volume: Developing the concept, Step 2 Starting the lesson, identifies cubes to support student understanding of volume. “Organize students into small groups. Distribute base-10 ones blocks and the support page to each group. Project slide 1, as shown. Refer to the first base picture and select one individual in each group to form one base then build up five layers of the same size. Guide the other members of the group to record the number of cubes in the first table on the support page as each layer is added. Repeat for the other two bases.”

• Module 4, Lesson 3, Common fractions: Reviewing the relationship with mixed numbers, Step 2 Starting the lesson, describes the use of number lines in order to represent fractions. “Open the Flare Number Line online tool and locate these amounts on a number line. Emphasize the distance from 0 to 1 represents 1 whole for each case, but the whole changes depending on the context. In the discussion, establish that these are called mixed numbers.”

• Module 10, Lesson 2, Decimal fractions: Reinforcing strategies for multiplying by a whole number, Step 3 Teaching the lesson, outlines the use of tenths and hundredths squares to support multiplication of whole numbers by decimal fractions. “Project slide 2 as shown. Read the word problem aloud and clarify that there are two questions attached to the problem. Organize students into pairs and distribute the resources. Likewise, students who used the tenths and hundredths squares can compare the areas that they shaded.”