4th Grade - Gateway 3

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 work with place value involving whole numbers is extended to six-digit numbers. Students learn to read and write these greater numbers by using an expander and representations of the numbers on an abacus. When asked to write numbers with zeros, students often insert too many zeros. For example, to write the number three hundred four, they might incorrectly write 3004. To clarify these potentially challenging numbers, this module discusses numbers with zeros and teens separately. By using the expander to show six-digit numbers, students can see the group of three places that are collectively called the thousands. They observe that this group of three has similar place names to the group of three places to the right — the hundreds, tens, and ones. The expander is used to help the students write numbers in expanded form. The students locate greater numbers on a number line to reinforce the role of place value to help position the numbers on the line. For example, the digit in the hundred thousands place tells how many steps of one hundred thousand to move from 0, the digit in the ten thousands place tells how many ten thousands farther along the number line to move, and so on. Students focus on place value to identify the value of digits in different places, and increase or decrease digits by 10, 100, 1,000, or 10,000.”

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 7, Number: Working with place value, Step 2 Starting the lesson, teachers provide context about equal parts on a number line. “Project the abacus (slide 1). Choose one student to roll the place-value cube. They draw a bead on the abacus to match their roll. Another volunteer rolls the same cube to record a different place value and draws a bead on the abacus to show their value. The values of the two beads are then compared. Ask, What number does each bead represent? What is the relationship between (Patricia’s) number and (Ben’s) number? If necessary, draw (×10) relationship arrows between each rod on the abacus, as shown. Establish that each rod represents a value that is 10 times as much as the value of the rod to the immediate right.”

• Module 5, Lesson 5, Multiplication: Using tape diagrams to solve word problems, Step 3 Teaching the lesson, provides teachers guidance about how to solve problems using the four operations. “Organize students into pairs to solve the problem from Step 2. Make sure they represent the relationships between the ages with tape diagrams and write equations to match. Afterward, invite pairs to share their models and equations on the board. Make sure they explain the connections between the two representations. Project the Step In discussion from Student Journal 5.5 and read aloud to the end of the clues. Say, This is a photo of Brady and his family when he was 8 years old. You will need to use the clues below the photo to figure out the age of each person in the photo. Refer to the clues and highlight how each clue describes a relationship between the ages of two people in the photo. Have the students work in their pairs to first create a solution plan, writing the clues in the order that they will need to use them. Listen to their discussions, assisting where needed with questions such as: Whose age do you know? Which clue refers to Brady’s age? What model will you use to show the relationship between Brady’s age and his little brother’s age? Why is that a suitable model? When students have completed their plan, have them work through it to find the solutions. As they do so, remind them to continually ask themselves, “Does this make sense?” and to adjust their plan if needed. Continue to observe and listen to the pairs as they work. Afterward, invite pairs to present their solutions and methods to the class. Compare and contrast the different methods and models, highlighting the connections between them. Work with the students to analyze conflicting answers to determine how and where the mistakes were made (SMP3). Discuss whether the students’ original solution plans worked or had to be adjusted, and if they were adjusted, discuss how. Read the Step Up and Step Ahead instructions with the students. Encourage them to draw tape diagrams to help model each problem. In doing so, they should consider whether the problem involves part-total thinking or equal-groups thinking. Make sure they know what to do, then have them work independently to complete the tasks.”

• Module 9 Lesson 9, Mass: Reviewing pounds and introducing ounces, Lesson overview and focus, Misconceptions, include guidance to address common misconceptions as students work with units of measure. “If students are struggling with units of measure, provide objects and experiences (pouring water, stacking mass pieces in a pan balance) which will give them concrete experiences with benchmark measurements. "How long does it take to fill a 1 quart container at the water fountain?" "How much does a baseball weigh?" This enables number sense about measurements which will help students solve problems accurately.”

The materials reviewed for Origo Stepping Stones 2.0 Grade 4 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 5, 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 6, Preparing for the module, Research into practice, Angles, includes explanations and examples connected to concepts of angles. To learn more includes additional adult-level explanations for teachers. “An angle can be understood in two different ways. In the first way, an angle measures the amount of space based on how far the angle arms are apart from each other. This space is understood as a static shape, and can be visualized as a wedge. In the second conception, an angle measures the amount of turn, or rotation that occurs as the rays move apart from each other. This conception is dynamic in that it focuses on the action of the angle arms moving apart. In this module, students focus on the dynamic, or turning, interpretation of angle, learning to focus their attention on how the opening changes and measuring it using familiar concepts such as quarter and half turns.” To learn more, “Browning, Christine, Gina Garza-Kling, and Elizabeth Hill Sundling. 2007. “What's Your Angle on Angles?” Teaching Children Mathematics 14 (5): 283–87.”

• Module 7, Preparing for the module, Research in practice, Division, supports teachers with concepts for work beyond the grade. “Learning about reminders prepares students to expand the range of values they consider in division and allows them to explore situations where the solution is not a whole number value. This prepares students for dividing fractions and decimals in Grades 5 and 6. It is important to emphasize the meaning of the remainder, particularly when it represents a partial group, to prepare students for dividing fractions in Grade 6. Read more about the development of fraction division in the Research into Practice sections for Grade 5 Module 6 and Grade 6 Module 5.”

• Module 9, Preparing for the module, Research into practice, Measurement, includes explanations and examples connected to the magnitude of various measurement units. To learn more includes additional adult-level explanations for teachers. “It is essential that students develop a sense of the magnitude of various units of measure. Students should be comfortable with the fact that a typical slice of bread weighs about 1 ounce while a baseball weighs about 1 pound. As students think about measuring the same object with different units, they use their multiplicative reasoning skills to understand that the baseball would weigh about 16 ounces since it weighs about 1 pound. The slice of bread would weigh a small fraction of a pound. Working with different units of measure is another example of composing and decomposing. When counting in 1 ounce increments, every 16 counts represents another pound of mass. As students work with measurement units, they also develop a deeper understanding of the operations. Addition and subtraction become tools for comparing measurements (how much more or how much less). Multiplication (how many times as many or as much) becomes the beginning of proportional reasoning, an essential concept developed in the middle grades. Multiplication and fractions both support reasoning about the relative size of different units.” 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 4 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 4 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 4 and the Common Core Standards, includes all Grade 4 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 10, 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 4, Lesson 1, Subtraction: Making estimates, the Core Standards are identified as 4.OA.A.3 and 4.NBT.A.3. The Prior Learning Standard is identified 3.NBT.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 4, Mathematics Overview, Number and Operations in Base Ten, includes an overview of how the math of this module builds from previous work in math. “The first section of this module focuses on using the standard subtraction algorithm with multi-digit numbers. This includes reinforcing the importance of estimation when subtracting multi-digit numbers. Lesson 2 reviews the standard algorithm as studied in Grade 3, decomposing tens or hundreds. The standard algorithm is developed for three-digit numbers with decomposing in any place (regrouping does not extend across a place with a zero). The algorithm is then extended to multi-digit numbers, but does not involve regrouping across a place with a zero, although zeros are included in some numbers.”

• Module 10, Mathematics Overview, Coherence, includes an overview of how the content in fourth grade connects to mathematics students will learn in fifth grade. “Lessons 10.1–10.12 focus on work with common fractions with denominators that are 10 or 100, and reading, writing, comparing, and ordering decimal fractions, as well as, relating common fractions and decimal fractions to carry out addition. This work builds on experiences with identifying common and equivalent fractions less than and greater than one (3.8.6–3.8.9) and serves as a foundation for using addition and subtraction with decimal fractions (5.5.1–5.5.9).”

• Module 5, Lesson 7, Length: Introducing millimeters, Topic progression, “Prior learning: In Lesson 4.5.6, students consider what the numbers and markings represent on tools that measure unit of length. They express measured lengths in both meters and centimeters, and use abbreviations. 4.MD.A.1; Current focus: In this lesson, students learn about the need for a unit of length that is shorter than a centimeter. They use a metric ruler to measure short lengths in centimeters and millimeters. Students then use the relationship between millimeters and centimeters to convert units. 4.MD.A.1; Future learning: In Lesson 4.5.8, students explore the relationship between meters, decimeters, centimeters, and millimeters. Students use the times 10, and divide by 10 pattern to convert between metric units of length. 4.MD.A.1, 4.MD.A.2.” 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 4 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, “Addition: In Grade 4, students work to master a standard algorithm for addition. This builds from work in previous grades around modeling addition on the number line or with base-10 blocks and now extends to multi-digit numbers and strings of multiple numbers where these physical and visual representations are less practical. Computational estimation is an important part of this process because it provides students with an effective way to check their work. Students use rounding as one method of estimating computational results. Supporting students to see what happens in the standard algorithm, with base-10 blocks or other representations, builds conceptual understanding of the algorithm. Allowing alternate notations can also build confidence and competence as the students see the connections between the symbolic representation and the models they have built in the past. Multiplication: Grade 4 students extend their knowledge of multiplication facts to use the same principles with multi-digit numbers. This is an important structural element of the base-ten place-value system. Problem solving situations allow students to generalize their understanding to work with multi-digit numbers in a practical way. Strategies, such as multiplying by ten and then halving the solution to find the product when multiplied by five, take on new life when students see that they also work with multi-digit numbers. To learn more: Kling, Gina and Jennifer M. Bay-Williams. 2015. “Three Steps to Mastering Multiplication Facts.” Teaching Children Mathematics 21(9): 548-559. O’Connell, Susan and John SanGiovanni. 2014. Mastering the Basic Math Facts in Multiplication and Division. Portsmouth, NH: Heinemann. Problem Solvers: Problem 2015. “Eggsactly How Many?” Teaching Children Mathematics 21 (9): 521-523. References: 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. Lefevre, Jo-Anne, Stephanie L Greenham, and Nausheen Waheed. 1993. “The Development of Procedural and Conceptual Knowledge in Computational Estimation.” Cognition and Instruction 11(2): 95-132.”

• Module 9, Preparing for the module, Research into practice, “Common fractions: As students consolidate their understanding of fraction comparison, they bring together a number of strategies. Common numerators, common denominators, and referencing benchmark fractions are all effective strategies for comparing fractions. The number line is an effective tool for illustrating fraction comparisons, in part because it provides a strong foundation for understanding the magnitude of fractions and their relative value when based on the same whole. Strategies for finding common denominators (or common numerators) become important as students compare fractions when a known strategy will not work at first sight. Measurement: It is essential that students develop a sense of the magnitude of various units of measure. Students should be comfortable with the fact that a typical slice of bread weighs about 1 ounce while a baseball weighs about 1 pound. As students think about measuring the same object with different units, they use their multiplicative reasoning skills to understand that the baseball would weigh about 16 ounces since it weighs about 1 pound. The slice of bread would weigh a small fraction of a pound. Working with different units of measure is another example of composing and decomposing. When counting in 1 ounce increments, every 16 counts represents another pound of mass. As students work with measurement units, they also develop deeper understanding of the operations. Addition and subtraction become tools for comparing measurements (how much more or how much less). Multiplication (how many times as many or as much) becomes the beginning of proportional reasoning, an essential concept developed in the middle grades. Multiplication and fractions both support reasoning about the relative size of different units. To learn more: Bray, Wendy S., and Laura Abreu-Sanchez. 2010. “Using Number Sense to Compare Fractions: Reflect and Discuss.” Teaching Children Mathematics 17 (2): 90–97. Freeman, Daniel W. and Theresa A. Jorgensen. 2015. “Moving Beyond Brownies and Pizza.” Teaching Children Mathematics 21 (7): 412–20. 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. References: National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Jeremy Kilpatrick, Jane Swafford, and Bradford Findell (eds.). Washington, D.C.: National Academy Press.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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, “base-10 blocks (hundreds, tens, and ones) in lessons 2 and 3, base-10 blocks (thousands, hundreds, tens, and ones) in lessons 4, 5, and 6, scissors and Support 16 in lesson 11. Each pair of students needs base-10 blocks (hundreds, tens, and ones) in lessons 3 and 8, cube labeled: 15, 18, 24, 28, 34, 35 in lesson 10, a cube labeled: 90, 150, 160, 180, 210, 240 in lesson 8, a cube labeled: 5, 5, 5, 10, 10, 10 in lesson 10, a cube labeled: 1, 2, 3, 4, 5, 6 and a cube labeled: 7, 7, 8, 8, 9, 9 in lesson 9, a cube labeled: halve, halve halve, double, double, double in lesson 8, paper and Support 13 in lesson 7, scissors in lesson 11, Support 14 in lesson 9, and Support 15 in lesson 10. Each individual student needs counters in lessons 9 and 10, scissors in lessons 11 and 12, the Student Journal in each lesson, and Support 17 in lessons 11 and 12.”

• Module 2, Lesson 9, Multiplication: Reviewing the fives strategy, Lesson notes, Step 1 Preparing the lesson, “Each pair of students will need: 1 copy of Support 14, 2 cubes labeled: (Note: Retain for Extra Practice.) cube A: 1, 2, 3, 4, 5, 6, cube B: 7, 7, 8, 8, 9, 9. Each student will need: 10 counters (a different color for each student) and Student Journal 2.9.” Step 3 Teaching the lesson, “Organize students into pairs and distribute the resources. Students take turns to roll the cubes, then choose one of the numbers to multiply by 5. A counter is then placed on that square.”

• Module 4, Preparing for the module, According to the Resource overview, teachers need, “base-10 blocks (thousands, hundreds, tens, ones) in lessons 4 and 6, cube labeled: 1, 2, 3, 4, 5, 6 and cube labeled: 4, 5, 6, 7, 8, 9 in lesson 6, and The Number Case in lesson 9. Each group of students needs paper in lesson 6. Each pair of students needs base-10 blocks (hundreds, tens, and ones) in lessons 2 and 3, blank sheet of paper in lesson 7, non- permanent markers in lesson 6, paper in lesson 5, Support 33 in lesson 8, and The Number Case in lesson 6. Each individual student needs adhesive tape or glue, a ruler, several strips of paper and construction paper in lesson 9, paper in lessons 2, the Student Journal in each lesson, and Support 32 in lesson 5.”

• Module 4, Lesson 4, Subtraction: Reviewing the standard algorithm (decomposing tens or hundreds), Lesson notes, Step 1 Preparing the lesson, “Each pair of students will need: base-10 blocks (hundreds, tens, and ones); Each student will need: Student Journal 4.2.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

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

• Module 2, Check-up 1 and Performance task and Module 4, Performance task, develop the full intent of standard 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm. Check-up, Question 2, “Use the standard algorithm to calculate the totals. a. 25714 + 3205; b. 57023 + 27894.” Performance task, Question 2, ”A student made a mistake when using the standard addition algorithm. Describe the error. 3621 + 2835 = [5] [14] [5] [6]” Module 4, Performance task, Question 1, “Use the standard subtraction algorithm to calculate the difference between each pair of numbers. a. 8.756 and 3,621; b. 67,384 and 4,313.” Question 2, “A student made a mistake when using the standard subtraction algorithm. Describe what they did wrong. 47602 - 31281 = 16421.” 

• Module 6, Quarterly test questions support the full intent of MP4, model with mathematics, as students use a tape diagram to represent multiplication and division. Question 1, “Nicole has 42 game tokens. She has 6 times as many as Dixon (D). How many tokens does Dixon have? Choose the equation that would not help you solve this problem. A. D \times 6 = 42, B. 42 \div 6 = D, C. 6 \times D = 42 D. D \div 6 = 42.”

• Module 10, Interview 1 and Check-up 2, develop the full intent of 4.NF.6, use decimal notation for fractions with denominators 10 or 100. Interview 1, “Steps: Display the number line. Draw an arrow toward one of the increments between 3 and 4. Ask, What number would be here? How did you figure that out? Repeat with an arrow at a position between an adjacent pair of increments. Make sure the arrows are not close to one another. Repeat with one other arrow. Indicate two of the arrows on the number line and say, Write a number that is between these two numbers. Show me where it is on the number line. Repeat with another pair of arrows. Ensure the student does not choose a number that only involves ones and tenths. Draw a ✔ beside the learning the student has successfully demonstrated.” Check-up 2, Question 1, “Write the common friction. a. 0.8 = ___   b. 0.50 = ___.”

• Module 12, Quarterly test B questions support the full intent of MP6, attend to precision, as students add numbers written in decimal form. Question 12, “Complete the equation. Show your thinking. 0.31 + 0.5 = ___.” Question 13, “Complete the equation. Show your thinking. 1.08 + 2.49 = ___.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 9, Length: Introduction kilometers, include:

• Extra help, “Activity: Invite a student to write digits in each place on the expander and share their length with the class. Repeat as time allows. As students gain confidence have them close the expander and read the length.”

• Extra practice, “Activity: Organize students into pairs and distribute the cards. Have one student write a length in kilometers (for example, 3 km) on a card. The other student then writes the equivalent length in meters (for example, 3,000 m) on another card. After five rounds, students alternate roles and repeat the activity until all their cards are used. The cards can then be used to play mix-and-match games. (Note: Collect and retain the cards for use in the differentiation activity for Lesson 5.12.)”

• Extra challenge, “Activity: Organize students into pairs and distribute the cards. Have one student write a length including tenths of a kilometer (for example, 3\frac{6}{10} km) on a card. Another student then writes the equivalent length in meters (for example, 3,600 m) on another card. After five rounds, students alternate roles and repeat the activity until all their cards are used. The cards can then be used to play mix-and-match games. (Note: Collect and retain the cards for use in the differentiation activity for Lesson 5.12.)”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 2, Lesson 1, Addition: Making estimates, Differentiation, Extra Challenge, “Organize students into pairs. Project the four cars (slide 1). Inform the students that you will write a number above each car that tells the number of cars of that color in the sales lot and they will have 15 seconds to consider the numbers, then write their estimate of the total number of cars. (Note: This amount of time can be increased or decreased to suit the students.) Write three three-digit numbers and one four-digit number. Students then show their estimates to their partner and work together to calculate the exact total. The difference between each student's estimate and the exact total is the number of points that they receive. The student with fewer points after five rounds wins.”

• Module 5, Lesson 4, Multiplication: Making comparisons involving division and subtraction (tape diagram), Differentiation, Extra Challenge, “Have students write comparison problems that involve division or subtraction. The examples should be similar to those on page 167 of Student Journal 5.4. The problems can then be exchanged and completed.”

• Module 10, Lesson 10, Decimal fractions: Adding hundredths, Differentiation, Extra Challenge, “Distribute the resources. Have students cut out various food items showing prices from the food catalog and place them faceup on the table. Then have them cut out the wallets and place those facedown in a pile. Students then turn over one wallet. They race to see who can collect food items to meet the approximate amount the wallet allows without going over the amount. After students have selected items, they add up the amounts. The student whose total is closest to the wallet amount will win the wallet and receive a point. If a student goes over the wallet value, they lose a turn in the next round. Students repeat the activity until all the wallets have been collected. The student with the most points at the end wins.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 3, Lesson 4, Number: Rounding four-, five-, and six-digit numbers, Lesson notes, Step 2 Starting the lesson, “ELL: Pair the students with fluent English-speaking students. Encourage them to think aloud as they find the missing answers. Encourage the students to listen to the answers given and then say if they agree or disagree and why.” Step 3 Teaching the lesson, “ELL: Provide time for the students to process the questions, formulate an answer, then speak about their thoughts to another student before presenting their ideas to the class. 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 thinking for the question, Can you think of other occasions when rounding to estimate may be useful? Then, give them an opportunity to explain their thinking to the class.”

• Module 9, Lesson 9, Mass: Reviewing pounds and introducing ounces, Lesson notes, Step 2 Starting the lesson, “ELL: Say the words mass, weight, and pound slowly and clearly. Invite the students to read each word with you one or two times. Ensure students understand the difference between the words weight as in mass, and wait as in being patient; and the difference between mass as in weight, and mass as in a religious event.” Step 3 Teaching the lesson, “ELL: Pair the students with fluent English-speaking students to discuss the concepts, as well as repeating the other student’s thinking. Say the word ounce slowly and clearly. Invite the students to read each word with you one or two times. Encourage them to explain what they are learning to check that they understand the concept. When the students are holding the mass pieces, ensure they make the connection between the physical objects and the word weight associated with the mass.” Step 4 Reflecting on the work, “ELL: Allow pairs to discuss their response to the questions, What is heavier, a pound or a kilogram? An ounce or a gram? Invite them to explain their thoughts to the class.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 3, Addition: Reviewing the standard algorithm (composing hundreds), Step 2 Starting the lesson, references an online game to support two and three digit addition strategies. “To review two- and three-digit addition strategies, organize the class into two groups and project the online Fundamentals game Jump On for students to play as a class. Discuss mental strategies with students to compose tens as required while they play (SMP8).”

• Module 6, Lesson 9, Angles: Identifying fractions of a full turn, Step 3 Teaching the lesson, references a “tester” as a strategy to support students when measuring angles. “Distribute copies of the support page and have the students cut out the top tester. Work through the instructions beside the tester on the support page to demonstrate how to fold it into a fan-like object. Demonstrate how the tester can be used to measure a wide range of angles and emphasize that the unit of measure is based on one-eighth of a full turn.”

• Module 10, Lesson 8, Decimal fractions: Comparing and ordering, Step 3 Teaching the lesson, references a handout, colored pencils and scissors to support comparing and ordering decimal fractions. “Organize students into five small groups. Distribute the resources and project slide 4. Assign each group one number and have them work together to represent their number on the page. Afterward, the page is cut into strips along the dotted lines.”