5th Grade - Gateway 2

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 5 expected fluencies, 5.NBT.2, multi-digit multiplication.

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 5 expected fluencies 5.NBT.5, fluently multiply multi-digit numbers using the standard algorithm. For example, in Module 2, Lessons 2-5 extends knowledge of the standard algorithm for multi-digit multiplication through multi-digit by multi-digit multiplication. Investigations 1, 2, and 3, students build procedural skill and fluency involving multi-digit numbers. Activities 1, 2, and 4, students work with the standard algorithm to multiply multi-digit numbers. In addition, the instructional materials embed opportunities for students to independently practice procedural skills and fluency. Examples include:

• “Fundamentals Games” contains a variety of games that students can play to develop grade level fluency skills. For example, Use a Ten Fact (multiplication with two-digit numbers) develops fluency in multi-digit multiplication.

• 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. In Module 10, Lesson 2,  Decimal fractions: Reinforcing strategies for multiplying by a whole number, provides computation practice with multi-digit dividends and one-digit divisors.

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

• Assessments also give problems that call for fluency and procedural skill. For example, in the Module 2 performance task, students use the standard algorithm to complete multi-digit by multi-digit multiplication.

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 5 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 8, Lesson 3, Common fractions: Finding a fraction of a whole number (unit fractions), Student Journal, Step Up, page page 289, Problem 2a, addresses the standard 5.NF.2, “The cost of a hamburger is one-sixth the price of a family meal. What is the cost of one hamburger if the family meal costs $12?”

• Module 4, Lesson 6, Common fractions: Solving word problems, Student Journal, Step Up, page 135, Problem 2d, addresses the standard 5.NF.1, “Lisa has 2 red apples and 2 green apples. She cuts the red apples into fourths, and the green apples into eighths. She eats 2 pieces of red apple and 3 pieces of green apple. Which color of apple has more left over?”

• Module 9, Lesson 11, Length/mass/capacity: Solving word problems (metric units), Student Journal, Step In, page 350, Problem b, addresses the standard 5.MD.3, “Three packages are each filled with 400 g boxes. Each package weighs 2 kg. How many 400g boxes were used?”

• Module 5, Lesson 7, Decimal fractions: Subtracting tenths (decomposing ones), Teaching the lesson, Problem Solving Activity 2, students work in pairs to discuss the problem and addresses the standard 5.NBT.7, “A weather research center records temperatures to the nearest hundredth of a degree (table provided). On which day was the greatest variation in temperature? Show your thinking.”

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

• Module 3, Lesson 12, Decimal fractions: Interpreting results on a line plot, Teaching the lesson, Thinking Task, Problem 1 asks, “Nancy begins the game by building this tower (students refer to the picture). What is the volume? Show your thinking." This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 6, Lesson 12, Division: Three- and four-digit dividends and any two-digit divisor, Teaching the lesson, Thinking Task, Problem 2 asks students, “The Marathon organizers will purchase energy powder. The Leadership Team will prepare a 10 gallon cooler of energy drink for each water stop. One three-pound tub of energy drink powder makes 24 quarts and costs $8.50. How many tubs will they have to buy and how much will it cost?” Students must use the Course Map provided to solve. This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 9, Lesson 12, Mass/data: Interpreting a line plot to solve problems, Teaching the lesson, Thinking Task, Problem 1 asks, “What is the difference in height between the tree with the greatest growth and the tree with least growth?” Students use a line plot provided to solve the problem. This non-routine question prompts students to apply mathematical knowledge/skills to real-world contexts.

• Module 12, Lesson 12, Division: Calculating unit costs to determine best buys (dollars and cents), Teaching the lesson, Thinking Task, students are given two portable building options with different dimensions (Building A 7m x 5.3m Building B 15m x 9 m). The students must help the school decide which building is the better option for the school to purchase. Problem 1 states, “Calculate the floor area for building A. Show your thinking. Remember to write the unit of measurement.” 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 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)”