4th Grade - Gateway 2

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 4 expected fluencies, 4.NBT.2, add/subtract within 1,000,000.

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

The materials attend to the Grade 4 expected fluencies: 4.NBT.4 fluently add and subtract multi-digit whole numbers using the standard algorithm. For example, in Module 2, Lesson 3,  Addition: Reviewing the standard algorithm (composing hundreds) addresses the addition component of the standard. Investigations 2 and 3 provide opportunities to build students’ procedural fluency. In Activity 1, students choose three items that add together to get close to $875 without going over using the standard algorithm. In Activity 2, students examine the work of their peers to determine who used the standard algorithm correctly and identify possible mistakes. In addition, the instructional materials embed opportunities for students to independently practice procedural skills and fluency. Examples include:

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

• Each module contains a summative assessment called Interviews. According to the program, “There are certain concepts and skills , such as the ability to route count fluently, that are best assessed by interviewing students.”  For example, in Module 5’s Interview, students must demonstrate fluency of subtracting decimals.

• The “Fundamentals Games” contains a variety of games that students can play to develop grade level fluency skills. For example, in Jump On, students add multi-digit numbers.

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

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

Examples of routine application problems include:

• Module 2, Lesson 7, Addition: Solving word problems, Student Journal, Step Up, page 63, Problem 2b, addresses the standard 4.OA.3, “The record number of cell phones sold over a 12 month period is 1,089. Gloria beat the record by 94 sales. How many cell phones did she sell?”

• Module 7, Lesson 4, Division: Solving word problems with remainders, Student Journal, Step Up, page 253, Problem 2b, addresses the standard 4.OA.3, “A roll of plastic wrap is 70 meters long. Thomas cuts the plastic wrap into lengths of 9 meters. How many of these lengths can he cut?”

• Module 9, Lesson 12, Capacity/mass: Solving word problems involving customary units, Student Journal, page 353, Problem 2b, addresses the standard 4.OA.3, “Deon buys 3 16 oz boxes of raisins. He shares the raisins equally among 4 bowls. What is the mass of raisins in each bowl?”

• Module 9, Lesson 8, Common fractions: Consolidating comparison strategies, addresses the standard 4.NF.1, Student Journal,Maintaining Concepts and Skills, Ongoing Practice, page 343, Problem 1, includes “Imagine you wanted to lay turf in this barnyard. Calculate the area. Show your thinking.” The materials present a rectangle labeled “barnyard” with perimeter measurements given.

• Module 3, Lesson 12, Perimeter/area: Solving word problems, Problem Solving Activity 4, Problem e, addresses the standard 4.NBT.6, “One yard is 25 ft long and 10 ft wide. Another yard is 22 ft long and 10 ft wide. What is the difference in area?”.

• Module 10, Lesson 12, Decimal fractions: Solving word problems, Student Journal, Step Up, page 390, addresses the standard 4.MD.1, students solve a word problem and show their thinking. For example, Problem 1b states, “Vishaya rides 4.6 miles. Jacob rides 8.3 miles more than Vishaya. How far does Jacob ride?”

• Module 7, Enrichment Activity 1 includes routine one-step problems and addresses standard 4.NBT.2). For example, ‘There are 43 strawberries to place onto toothpicks. Each toothpick can hold 2 strawberries. How many toothpicks will be needed to hold all the strawberries?”

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

• Module 3, Lesson 12: Perimeter/area: Solving word problems, Teaching the lesson, Thinking Task, Problem 1 states, “Abigail buys one roll of chicken wire. She runs the chicken wire around the outside of the posts to make four walls. How much chicken wire will she use?” Students use information provided to answer. This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 6, Lesson 12: Angles: Estimating and calculating, Teaching the lesson, Thinking Task, Problem 1 states, “Compare the amount that the adults and students pay to go on the field trip. Describe the relationship between the two amounts. Show your thinking.” The problem states that students pay $4 and adults pay $12. This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 9, Lesson 12, Capacity/mass: Solving word problems involving customary units, Teaching the lesson, Thinking Task, Problem 2 states, “In the Fresh-Fruit Punch, how much more cranberry juice is there than lemon juice? Show your work.” Students must use the information for the Drink Recipes to solve. This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 12, Lesson 12, Money: Solving word problems, Teaching the lesson, Thinking Task, Problem 1, students use time, money, fractions/decimals, and information from School Fun Run to solve. The question states, “The students in Ms. Yorba’s class want to each raise $5.00. They are trying to calculate the number of laps needed to reach $5.00, and what distance that will be. For this item: Show how many laps each student must run to raise $5.00. Show the distance in miles.” This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 1, Lesson 3, Number: Reading and writing six-digit numbers, Student Journal, page 13 and Step 4 Reflecting on the work, students make sense of problems and persevere in solving them as they reason about place value. Student Journal, Step Ahead, page 13, “Figure out the total number shown by each set of cards. Write the numbers on the expanders below.” Step 4 Reflecting on the work, “Discuss the students’ answers to Student Journal 1.3. Refer to Step Ahead and discuss the points below: How did you decide which digits should be written on the expanders? How did you know where to write each digit? How does the number of groups show on the place-value cards? (MP1 and MP7)” 

• Module 4, Lesson 5, Subtraction: Analyzing decomposition across places involving zero (three-digit numbers), Step 3 Teaching the Lesson, students make sense of subtraction problems and persevere in solving them. “Project slide 6, as shown, and discuss the points below: What problem does this algorithm represent? (307 − 118.) (MP2) What do you estimate the difference to be? Will you need to regroup more than once to find the difference? (Yes.) How would you use blocks to show the regrouping? Ask the students to use a strategy of their choice to calculate the difference (MP1 and MP5).”

• Module 6, Lesson 8, Length: Exploring the relationship between miles, yards, and feet, Step 3 Teaching the Lesson, students make sense of equivalence statements involving length, connecting problems to those solved in the previous lessons as they persevere in solving them. “Project slide 2, as shown. Organize students into pairs to work together to find the solutions. Slide: 1760 yards is equal to 1 mile. ___ yards is equal to 2 miles. 1 mile is 1760 times longer than 1 yard. 2 miles is 1760 times as long as ___ yards. 1760 yards is equal to 1 mile. ___ yards is equal to 3 miles. 3 miles is 1760 times as long as ___ yards. If students have difficulty, help them persevere (MP1) by reminding them how similar statements were solved when working with just yards, feet, and inches in the previous lesson. Point to the first relationship and ask, How has the number of miles changed? (It was doubled.) What do we need to do to the number of yards to keep the same relationship? If students are still confused, ask them to write the equation 12 inches = 1 foot and ___ inches = 2 feet, then repeat the questions with these numbers.”

• Module 8, More Math, Problem Solving Activity 4, Word Problems, students make sense and persevere in solving multi-digit division word problems. “Project slide 1 and read the word problem with the students. Ask, What information do we need to solve this problem? What operations will we use? What will we do first? What will we do next? How could you show your thinking? Slide 1: Robert bought a used car for $5,995 and a GPS for $109. He is paying the total in equal monthly amounts for 8 months. How much does he pay each month? Allow time for students to find a solution. Then invite students to share their solution ($763) and explain their thinking.”

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 2, Lesson 7, Addition: Solving word problems, Step 3 Teaching the Lesson, students decontextualize addition word problems to find answers and contextualize numbers to create addition word problems. “Distribute a copy of the support page to each pair of students. Project the two tables of sales information (slide 4) and ask the students to work together to write an addition word problem in the top left corner of the support page using the information on the tables as a guide. Specify that students can only write a problem where a maximum of two addition equations are required to find the solution. Move around the room to ensure that each word problem is suitable and that only addition is used. The word problems can be exchanged among the students and the support page completed. Suggest that the students describe what the word problem is asking them to do (top right), before estimating the total by rounding and adding (as they would do mentally — bottom left), then finding the exact total using a written method of their choice (bottom right). The problems are then returned to the original authors, who check the solutions for accuracy. Select various pairs to present their word problems and solutions (MP1, MP2, and MP4).”

• Module 3, Lesson 11, Perimeter: Working with rules to calculate the perimeter of rectangles, Step 2 Starting the Lesson, students reason abstractly and quantitatively as they explain the meaning of the rules to find perimeter. “Project slide 1, as shown, then discuss the points below: P = 2 x L + 2 x W; P = 2 x (L + W)  How do these rules describe how to calculate the perimeter of this rectangle? What does L (W) mean? Where is L (W) on the rectangle? Why are the length and width both doubled in the first rule? Why can the length and width be added first then the total doubled in the second rule? Invite volunteers to use the numbers from the diagram and mathematical understanding to explain why each rule makes sense (MP2).”

• Module 5, Lesson 8, Length: Exploring the relationship between meters, centimeters, and millimeters, Step 4 Reflecting on the work, students decontextualize word problems involving various lengths, make sense of the relationships between units, and work with symbols to solve the problem. “Discuss the students’ answers to Student Journal 5.8 and the points below: Why is the number of meters less than the number of millimeters? How did you decide which lengths would be equivalent?  What did you have to do first to solve the problems in Question 4? (Convert the lengths to the same unit of measurement.) (MP1 and MP2).” 

• Module 9, Lesson 12, Capacity/Mass: Solving word problems involving customary units, Student Journal, Step Ahead, page 353 and Step 4 Reflecting on the work, students reason abstractly and quantivity as they solve word problems with customary units. “Write numbers to complete the story. Make sure it makes sense. Andrea buys a carton of juice. The curtain holds __ fluid ounces. She fills __ glasses with juice from the carton. Each glass holds __ fluid ounces. There are __ fl oz left in the carton.” Step 4 Reflecting on the work, “Ask, What equations did you use? Is there more than one way the story will work? Compare two cases of the same story where both of the numbers work and ask, “What relationships make both of these stories work? (MP2)”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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, Lesson 1, Addition: Making estimates, Step 3 Teaching the Lesson, students construct viable arguments and critique the reasoning of others when they round multi-digit numbers and estimate sums. “How can we use compatible numbers, or rounding, to estimate the total of different combinations of items so that we do not have to find exact totals each time? Are there enough cars for charity? What combination of items will be the closest to 500 without going over? Which two classes collected the most toy cars? Invite groups of students to share their thinking. Highlight the different strategies they used and the estimations they made. Encourage respectful critique by asking questions such as, Do you agree with this group’s estimate? Why/why not? Who used a different strategy? Did your strategy result in a more accurate estimate? How could you explain that strategy in a different way?“

• Module 3, Student Journal, page 119, Convince a friend, students justify their reasoning about prime and composite numbers and then critique the reasoning of a classmate. “Which number does not belong? 9, 27, 36, 17, 51. Show your thinking. The number does not belong. I think this because …  Share your thinking with another student. They can write their feedback below. I agree/disagree with your thinking because. . .”

• Module 6, Thinking Tasks, Question 5, students construct viable arguments using their understanding of fourths and halves as they design a travel plan and share evidence for why they think their plan will work. “Ms. Lesh’s class will try to earn over 500 points on the field trip day all at once. Is it possible to create a plan for each group that will: Not skip lunch, AND Earn the class 500 total points or more? For this item you need to: Fill in a Walking Plan for each group on the table below. Fill in Total Miles Walked on the table below. Fill in the Total Mileage Club Points on the table below. Write a letter to the class explaining how it is or is not possible to make a plan that will earn 500 points or more?”

• Module 10, Lesson 9, Decimal fractions: Adding tenths, Student Journal, page 383, Step Ahead, students construct viable arguments as they reason about adding tenths, “Maka and Lillian ran a relay. Maka ran the first 3.1 kilometers, then Lillian ran the last 3.3 kilometers. a. Did they run greater than or less than 6.05 kilometers in total? b. Write how you know.”

• Module 11, Student Journal, page 433, Convince a friend, students construct viable arguments as they draw lines of symmetry and make predictions in real world problems. “Draw all the lines of symmetry in each shape. What do you notice? Use what you know to predict how many lines of symmetry will be in a shape that has 10 sides equal in length. Share your thinking with another student. They can write their feedback below. Discuss how you could prove your prediction.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 1, Lesson 8, Multiplication: Reviewing the doubles strategy, Step 3 Teaching the lesson, students model with math as they draw pictures or write multiplication equations to justify their thinking. “Invite volunteers to select a price, explain why they chose it, and describe the thinking they would use. For example, “I know double 23 is 46. I doubled 20 and then doubled 3.” Encourage the students to draw pictures on the board or write equations to help explain their thinking to others (MP4). Explore all of the strategies for each price tag. Emphasize strategies such as: Use a known fact: Double 7 is 14 so double 70 is 140. Use a whole quantity: Two 45s is 90 so two 46s is two more, or two 20s is 40 so two 18s is four less. Split into parts: Double 20 is 40 and double 3 is 6.”

• Module 2, More math, Problem Solving 4, students model real word multi-step problems and explain the problem and strategy they chose to find a solution. “Project slide 1 and read the word problem with the students. Slide 1: The school sports department purchased 5 stopwatches for $19 each and a set of hurdles for $1,100. How much did they spend in total? Ask, What information do we need to solve this problem? What operations will we use? What will we do first? What will we do next? How could you show your thinking? (For example, use the standard algorithm, draw a diagram, or write an equation.) Allow time for the students to find a solution. Then invite students to share their solution ($1,195) and explain their thinking. Distribute the support page to each student and have them work independently to solve each problem. Remind them to show their thinking.”

• Module 7, Student Journal, page 280, Mathematical modeling task, students model a real- world multi-step problem involving operations with common fractions. “Jessica is sending three packages to her family. The combined mass of the three packages is just less than 2 kg. One package has a mass of \frac{7}{10} kg, another package has a mass of \frac{1}{4} kg. What could the mass of the third package be? Show your thinking.”

• Module 12, More Math, Thinking Task, School Fun Run, students model with math as they debate and justify their ideas about whether more time elapses while students are running or doing other things on the Fun Run day. Question 4, “Some students in Ms. Flanagan’s class are debating the time that is spent running versus the time that is spent at lunch and stretching. Sharon says they already spend twice as long stretching as they spend running. Sandra says they actually spend twice as long running as they spend stretching. Emilio says that if they include the time spent at lunch, each grade level actually spends twice as long eating lunch and stretching as they do running. For this item, write true or false beside each statement. Explain/show your thinking for each using words, numbers, and/or models.”

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 2, More Math, Investigation 1, students show all the different strategies they can use as tools to solve a problem. “Project slide 1 and read the investigation question. (Slide 1, What different written methods can be used to calculate 537 + 374?) Organize students into small groups and have them brainstorm all of the different ways that students may choose to solve the problem. Make sure they record the results using different pieces of paper to show each different method. After, have each group share one method at a time until no other methods are offered. Have students post the written methods on the board. Ask questions such as, How are the methods the same (different)? What are the advantages (disadvantages) of the methods? Are there times when you would use one method over the other? Why?”

• Module 4, Lesson 7, Addition/subtraction: Solving word problems, Student Journal, page 139, Step Up, students choose a suitable method or strategy to solve real world problems. They have been learning about how estimation can help determine a reasonable answer and guide the choice of solution strategies. Question 2, “Solve each problem. Show your thinking. a. A club store reported sales of $12, 550 for shirts, $6805 for sweaters, and $2090 for caps. What were the total sales for shirts and caps?” Step 4 Reflecting on the work, “Discuss the answers to Student Journal 4.7. Ask, Did anyone use the standard algorithm to solve the problems? What other strategies did you use? Allow time for students to share those strategies that did not involve the standard algorithm. For each strategy, ask, Why did you decide to use that strategy? Highlight those occasions where the student first considered the given numbers before deciding on a strategy.”

• Module 7, Lesson 8, Common fractions: Subtracting with same denominators, Step 4 Reflecting on the work, students choose a calculation method as a tool to solve problems with fractions. “Ask, When subtracting the fractions, was it easier to think of taking some away, or was it easier to think of how much space was between them on the number line? Lead a discussion that highlights the benefits and drawbacks of each way of thinking and the contexts that promote each method (MP5).”

• Module 9, Student Journal, page 356, Mathematical modeling task, students choose from tools or strategies they have learned to solve a real-world problem involving operations with fractions. “Abey, Max, and Janice are driving to visit their cousin. The trip is long, so they want to make sure they each drive about the same distance. Abey drives about \frac{1}{5} of the distance, Janice drives about \frac{2}{4} of the distance, and Max drives \frac{3}{10} of the distance. What fraction of the return journey should each person drive? Show how you decided.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 2, Addition: Reviewing the standard algorithm (composing tens), Step 3 Teaching the lesson, students use precision when using the standard addition algorithm to calculate sums with regrouping. “Project the Step In discussion from Student Journal 2.2 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, remind students to work methodically and carefully, checking their work before they move on (MP6), then have them work independently to complete the tasks. (Note: Some students may need the additional support of base-10 blocks as they make sense of the standard addition algorithm.)

• Module 9, More math, Thinking tasks, Question 3, students attend to precision as they convert between measurements. “Students in Mr. Pham’s class will put the drinks in ten- gallon coolers. They will make enough for every student in Grade 4 to have 2 servings. Use this conversion chart to figure out how many servings will fit into a ten-gallon cooler. 1 serving is 8 fluid ounces, 1 cup is equivalent to 8 fluid ounces, 1 quart is equivalent to 4 cups, 1 gallon is equivalent to 4 quarts. How many servings will fit into one ten-gallon cooler? How many ten-gallon coolers will they need for each student to get at least 2 servings? Show your thinking.”  

• Module 10, Lesson 1, Decimal fractions: Introducing decimal fractions, Student Journal, page 359, Step Up, students attend to precision as they interpret and write decimal fractions. Question 2, ”Read the fraction name. Write the amount as a common fraction or mixed number. Then write the matching decimal fraction on the expander. a. four and two-tenths, b. sixty-three tenths, c. five and eight-tenths.”

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 3, Module overview, Vocabulary development, students can attend to the specialized language of math as teachers are provided a list of vocabulary terms. “The bolded vocabulary below will be introduced and developed in this module. These words are also defined in the student glossary at the end of each Student Journal. A support page accompanies each module where students create their own definition for each of the newly introduced vocabulary terms. The unbolded vocabulary terms below were introduced and defined in previous lessons and grades. Addition algorithm, area, array, compare, composite number, difference, divide, estimate, digit, dimensions, factor, greater than, greatest, least, length, less than, measure, millions, multiple, number line, number, order, perimeter, place value, prime number, round, skip count, square centimeter (cm²), square units (sq units), square yard (yd²), subtraction algorithm, 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 4, Lesson 8, Common fractions: Reviewing concepts, Step 2 Starting the lesson, students use precise language when describing common fractions and making equivalent fractions. “Project slide 1 and ask, Which number is the numerator (denominator)? What does the numerator (denominator) tell us? (MP6)”

• Module 8, Lesson 8, Division: Solving word problems, Student Journal, page 304, Words at Work, students use the specialized language of mathematics as they reason about concepts connected to multiplication and division. “These math terms are related to multiplication and division. Write the meaning of each. a. partition a number, b. dividend, c. quotient.”

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 4 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 11, Multiplication: Reviewing the nines strategy, Step 2 Starting the lesson, “students decompose a known tens fact to help calculate an unknown nines fact, for example, 9 × 4 = 10 × 4 subtract 1 × 4”. Lesson notes state, “Project the 10-by-6 array (slide 1). Have students describe the array, referring to the number of columns and rows, and the number of dots in each column or row. Then ask What multiplication fact could you write to match this array? Encourage suggestions, then write 10 × 6 = 60 on the board. Say, This array shows 10 rows of 6. How could we use the same array to show 9 rows of 6? Suggestions might include crossing out, or circling one row of six. Cross out the bottom row of dots and ask, How can we figure out 9 rows of 6? What product should we write? Encourage students to explain that 9 × 6 is equal to 10 × 6 subtract 1 × 6 (MP7). Write the equation 9 × 6 = 54 on the board.”

• Module 4, Lesson 3, Subtraction: Using the standard algorithm (decomposing in any place), Step 3 Teaching the lesson, students use repeated regrouping to solve complex subtraction problems. “Project the table showing various representations of 326 – 157 (slide 2) and ask students to explain how this table shows the same subtraction just conducted (MP7).”

• Module 7, Lesson 7, Common fractions: Adding mixed numbers (composing whole numbers), Student Journal, page 263, Step Up, Question 2, students make use of structure as they use benchmark fractions to add mixed numbers. “Split each mixed number into whole numbers and fractions before adding. Then write the total. Show your thinking. a. 5\frac{3}{4} + 2\frac{2}{4} = .” There are parts a to f for students to practice.

• Module 9, Lesson 4, Common fractions: Calculating equivalent fractions, Step 3 Teaching the lesson, students make use of structure as they recognize that equivalent fractions can be made by multiplying both the numerator and denominator by the same factor. “Organize students into pairs and distribute the support page. Direct students to use the support page to compare \frac{3}{6} and \frac{5}{12}, then \frac{4}{6} and \frac{9}{12} (MP4). After sufficient time, ask, What stayed the same? What changed? What is the relationship between the numerators and denominators? (MP7)” Step 4 Reflecting on the work, “Discuss the students’ answers to Student Journal 9.4. For each question, ask the students to describe how they calculated the factor they used to change the first fraction to find the equivalent second fraction (MP7).”

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 10, Multiplication: Extending the fours and eights facts, Step 4 Reflecting on the work, “students describe how their doubling strategy can be extended to multiply a factor by 16.” “​​Discuss the students’ answers to Student Journal 1.10. Focus students’ attention on Step Ahead and have the students describe how the same strategy might be extended to multiply by 16 (MP8). Project the relationship sentences (slide 6) and say, 15 × 8 is the same value as 15 × 2 × 2 × 2. How can we use the same thinking to calculate 15 × 16? Encourage students to describe how 16 can be decomposed into groups of 2 (MP7). On the board, complete the sentence: 15 × 16 is the same value as 15 × 2 × 2 × 2 × 2. Say, We can double once to multiply by 2, double twice to multiply by 4, double three times to multiply by 8, and then double four times to multiply by 16.”

• Module 3, Lesson 8, Multiplication: Identifying prime and composite numbers, Step 2 Starting the lesson, “students look for patterns to identify a mystery number.” “Invite students to identify (or exclude) possible starting numbers (2, 3, 6, or 9) by shading or writing numbers on the empty hundred chart. Make sure they share their thinking as they look for solutions, for example, “It cannot be 5 because 54 is not a multiple of 5.” (MP8) The investigation can be repeated by using other starting numbers with similar clues.”

• Module 7, Lesson 5, Common fractions: Adding with same denominators, Student Journal, page 257, Step Ahead, students write a rule they can use to add common fractions with the same denominator. “Write a rule that you could use to add two common fractions with the same denominator.” Step 4 Reflecting on the work states, “Refer to Question 4 and ask, “How did you decide which totals were greater than 3? For Step Ahead, invite students to share the rule they wrote (MP8).”

• Module 12, Lesson 1, Patterns: Working with multiplication and addition patterns, Student Journal, page 435, Step Ahead, students use repeated reasoning as they generalize a pattern rule. “Yuma hires a private room at a restaurant for a party. The food for each guest costs the same amount. The room costs $20 no matter how many people are there. Look at the total costs she figured out. Write a rule to calculate the total cost for nine guests.” A table chart shows Number of Guests 1, 2, 3, 4 and Total Price ($), 29, 38, 47, 56.