5th Grade - Gateway 2

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Examples of problems and questions that develop conceptual understanding across the grade level include:

• In 5.MD.C.3a-b Cubicle Dudes, the Teacher Instruction and the Practice Printable include several diagrams and drawings of unit cubes packed into rectangular prisms to determine the volume. 

• In 5.NF.B.3 “In Much Ado About Honey, the three fairies Beeblossom, Coyote, and Columbina are fighting over five jars of honey. King Oberon appears and commands that they stop arguing and share the honey evenly among them. The data provided is an image of the three fairies looking at the five jars of honey.” During the Teacher Instruction, visual models are used to further develop this concept, and the teacher guides students to imagine sharing eight pizzas among five friends and putting four pounds of flour equally into six containers.

• In 5.NF.B.6 Water in the World, an example is “Multiplication of fractions may not be as intuitive for students as multiplication of whole numbers, so using diagrams when discussing the problems in this lesson may be helpful. In Water in the World, Kate is doing a true public service announcement, bringing awareness to the water crisis. She points out that in the United States, very few of us have trouble accessing clean water. However, around the globe there are millions of people who do not have access to clean water, and many of those are children. The data provides the fraction of the world population without access to clean water, as well as the fraction of the population who are children.” During the Teacher Instruction, the teacher works with the students to create a visual model of $$\frac{1}{3}$$ times $$\frac{1}{8}$$.

• In 5.NBT.A.1 The Traveling Suitcase, instruction includes a visual display of movements along a place value line showing how multiplying by a power of 10 results in a different place value. “As a digit shifted spots, it became clear a digit in one place represents 10 times as much as it represents in the place to its rights and $$\frac{1}{10}$$ of what it represents in the place to its left.

Examples where students independently demonstrate conceptual understanding throughout the grade include:

• In 5.NBT.A.6 Hardtack, Practice Printable, Question 7 states, “Jacqueline wrote 1,152 pages during the first 12 months of college. Assuming she wrote the same number of pages each month, how many pages did she write each month? Solve by using equations, rectangular arrays, and/or area models.”

• In 5.NF.B.4a The Horse Doctor, Practice Printable, Question 2 states, “Draw a visual fraction model to represent $$\frac{5}{12}× 3$$, then write the product on the line below.”

• In 5.G.B.3 Squaring Off, Practice Printable, Question 9 states, “Isaiah says that a parallelogram is a square. Dominique says a parallelogram is not a square. Draw and describe a figure Isaiah might use to prove he is correct. Draw and describe a figure Dominique might use to prove she is correct.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for attending to the standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level in the Math Simulator, examples in Teacher Instruction, Cluster Intensives, domain specific Test Trainer Pro and the Clicker Quiz. Examples include:

• In 5.MD.A.1 Treacle Treatment, the teacher demonstrates conversion factors and ratios to change units. The teacher is instructed to say, “Let’s calculate each conversion. We’ll start by finding the conversion information between each pair of units. Now we can start with the given information and then set up the ratios so that the given units cancel out, leaving us with the desired units.”

• In 5.NF.A.1 Hay, the Teacher Instruction includes examples of different ways students could identify common denominators including listing multiples, multiplying the denominators by each other, and using prime factorization to identify the least common multiple.

• In 5.NBT.B.5 O’Hara’s Oversized Order, the Teacher Instruction walks students through examples of multiplying using the standard algorithm: first to solve $$36×720$$, then $$567×54$$, and finally $$3152×251$$. There is also a “worked example” video provided by a “student” to refresh the skills when students work independently.

Examples of students independently demonstrating procedural skills and fluencies include:

• In 5.NBT.B.5 O’Hara’s Oversized Order, the Clicker Quiz includes six questions, including two questions that include error analysis, for multiplication practice. The Practice Printable includes two problems with the directions, “Estimate each product first. Find the actual product by using the standard algorithm. Use your estimate to check the reasonableness of the product.” There are three word problems, and one requires error analysis by stating, “Diana made an error on one of her homework problems. Circle Diana’s error, and redo the problem correctly.”

• In 5.MD.A.1 Treacle Treatment, the Practice Printable has several problems for students to complete independently such as Question 3, “Bruno is 1.75 meters tall. How tall is Bruno in: millimeters? centimeters? kilometers?”

• In 5.NF.A.1 Hay, the Practice Printable provides several opportunities by adding daily feed logs for 4 animals, where amounts have unlike denominators. Questions 1-4 state, “For each set of fractions, write equivalent replacement fractions with a common denominator,” and Questions 5-10 state, “Find the sum or difference.” All problems begin with unlike denominators.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine problems, presented in a context in which mathematics is applied. 

Examples of students engaging in routine application of skills and knowledge include:

• In 5.NF.B.6 Water in the World, Practice Printable, Question 1 states, “Below is a recipe for Nana’s Banana Muffins. How much of each ingredient is needed to make $$\frac{1}{4}$$ of the recipe?” The recipe has five ingredients: three are whole number amounts,  one is a unit fraction amount, and one is a non-unit fraction amount. 

• In 5.MD.A.1 Treacle Treatment, Practice Printable, Question 5 states, “Annette, Adriana and Sasha are all training for an upcoming race. Annette ran 1.5 kilometers; Adriana ran $$\frac{2}{5}$$ of a kilometer; and Sasha ran 1,600 meters. How many total meters did they run?” 

• In 5.NF.C.7c Fairy Fractions, Practice Printable, Question 11 states, “Terri has 18 pounds of dog food for her dog, Migs. One serving for Migs is $$\frac{1}{6}$$ of a pound, and he is supposed to eat 2 servings each day. How many days will the dog food last?”

• In 5.OA.A.2 Liftie Lesson, Practice Printable, Question 5 states, “Carla has 8 baseball cards. She gives 3 to her sister, and then goes to the store and doubles the number of cards she has. Write an expression to represent how many cards she has.”

Examples of students engaging in non-routine application of skills and knowledge include:

• In 5.NF.A S’mores, “Matty, Lucia, and Keshia are buying supplies to make s’mores on an upcoming camping trip. They each have specific recipes they follow to make their perfect s’more. Matty uses $$\frac{1}{2}$$ bar of chocolate, Lucia uses $$\frac{1}{4}$$ bar of chocolate, and Keshia uses $$\frac{1}{3}$$ of a bar of chocolate. They each plan to eat 3 s’mores. a) How many bars of chocolate should they each buy? b) Will there be any chocolate bar left over? If so, how much? c) If they did not want to have any chocolate left over, yet they each want to eat an equal number of s’mores using their own favorite recipe, how many s’mores would they have to eat? How many chocolate bars would they buy? Once you are confident in your solution, draw a picture to show your reasoning. Be ready to present and explain your drawing.” 

• In 5.MD.C.5c Polly Packs, Practice Printable, Question 7 states, “Elijah is building a sandcastle made up of two rectangular prisms stacked atop one another. He has 504 cubic inches of sand. He knows the bottom prism will be 15 inches long, 8 inches wide and 3 inches tall. If he uses all the sand, what could be the dimensions of the top rectangular prism?

• In 5.NF.B.7c Fairy Fractions, Practice Printable, Question 5 states, “A giraffe typically spends $$\frac{4}{5}$$ of a day standing, walking, and eating. After one week, how many “days” has a giraffe spent standing, walking, and eating?”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Examples of the three aspects of rigor being present independently throughout the materials include:

• In 5.NF.B.5a Sweet Success!, students develop conceptual understanding of comparing the size of a product to the size of one factor on the basis of the size of the other factor. In the Practice Printable, Question 1, “Without doing the calculations, circle the greater value. a) $$25$$ or $$\frac{2}{5}×25$$; b) $$3\frac{1}{4}×45 $$ or $$\frac{1}{4}×25$$; c) $$\frac{1}{3}×2$$ or $$1\frac{1}{3}×2$$; d) $$\frac{4}{5}×\frac{1}{2}$$ or $$\frac{1}{2}$$.” In Question 2, “Look at Problem C from Question 1 above. Explain how you knew which was the greater value without doing the calculation.”

• In 5.NBT.A.4 Round and Round, students develop procedural skills related to rounding numbers. During the Immersion Problem students answer, “How is DJ Mastermind doing these calculations in his head?” During the resolution video, DJ Mastermind explains the procedure for rounding numbers. The procedure is developed during the Teacher Instruction, and as students have opportunities to practice during the Simulation Trainer; the Practice Printable, Question 7, students “Round 14.256 to the nearest hundredth.”; and Clicker Quiz, “What is 11.98 rounded to the nearest tenth of a second?”

• In 5.MD.C.4 Shipment Shenanigans, students engage in application problems related to volume. Throughout the lesson students try to answer “What is the volume of the truck?” The lesson narrative explains, “In Shipment Shenanigans, Bud and Lou are loading a truck full of boxes that are cubes, each measuring 1-foot by 1-foot by 1-foot. As soon as the truck is loaded, they get a call from the boss asking for the volume of the truck. Neither Bud nor Lou counted the number of boxes as they loaded the truck, so the only thing left to do? Unload and recount! The data provided is an image of the truck, showing the individual boxes stacked up inside.” Additionally, in the Practice Printable, Question 6 states “Jillian works at a shoe store. The store received a shipment of children’s boots for the upcoming season. The shipment came in one large box full of smaller shoeboxes. How many shoeboxes were inside the shipping box?”

Examples of multiple aspects of rigor being engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In 5.NF.B.3 Much Ado About Honey, students solve application problems using their conceptual understanding about division. In the Practice Printable, Question 7 states “The girls’ swim team is having an end of the year celebration. There are 8 girls on the team and they ordered 6 pizzas to share. How much pizza will each girl get if they split all pizzas evenly? Draw a representation to show your thinking.”

• In 5.NBT.B.6 Hardtack, students use procedural skills of long division while solving application problems. The Resolution video teaches the partial quotients method for long division in the context of solving a real-world problem about how many hardtack biscuits there are for each crew member. Students use the partial quotients method in additional application problems such as the Clicker Quiz, “The Castillo family always goes camping in the summer. The four of them share a tent. The tent floor is 4,508 square inches. How much space does each family member get?”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for supporting the intentional development of MP1 and MP2 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP1 to meet its full intent in connection to grade-level content include:

• In 5.MD.B.2 Anesthesia Outcome, Detailed Lesson Plan, “Anesthesia Outcome provides a good opportunity for students to make sense of and persevere in solving problems in all three phases of The Math Simulator. During the Immersion phase, students begin this practice immediately as they are prompted to make sense of a type of task that is relatively unstructured. Encourage students to take time to develop questions and assumptions and to not feel the need to rush to try to solve the problem. In Data & Computation, students will continue to persevere by developing a plan to approach the problem, create a visual that helps students make sense of the problem, and complete the calculation. In Resolution, students share different approaches to solving the problem and even present mistakes they may have made.”

• In 5.OA.A.2 Liftie Lesson, Lesson Plan Overview, Applying the Standards for Mathematical Practice, “During Data & Computation, students take information gathered by Lou about the number of people the double chairlifts can hold, and use this data to represent it logically as an expression. During Practice Printable and Clicker Quiz, students continue to represent situations symbolically as they create numerical expressions.”

• In 5.NF.B.4a The Horse Doctor, “In Immersion, students create a visual to conceptualize the problem without having all of the information and will begin formulating an approach to solve the problem. In Data & Computation, students calculate an answer that must be checked for reasonableness, and in Resolution, students reflect upon their learning and revise their work. Through each of these phases, students are engaged in the practice of making sense and persevering.”  

Examples of the intentional development of MP2 to meet its full intent in connection to grade-level content include:

• In 5.NF.A.1 Hay, Lesson Plan Overview, “Students will be able to connect a visual model with a numeric representation of equivalent fractions when adding. In Resolution, after students have had an opportunity to apply strategies to the problem, students are simultaneously shown visual diagrams of the hay bales transforming to correspond to the mathematical process of equivalent fractions for adding. In Student Reflection, students use words and numbers, along with a strong visual representation, to build brain pathways that encourage students' ability to decontextualize and contextualize and reason abstractly and quantitatively.”

• In 5.NF.B.6 Water in the World, Detailed Lesson Plan, “As students move away from visuals and use equations to represent and solve real-world problems, they often decontextualize to calculate and then contextualize again to interpret their results. During Practice Printable, students will encounter various contexts in which they have to calculate a product and interpret the results.” Practice Printable Question 1 states, “Below is a recipe for Nana’s Banana Muffins. How much of each ingredient is needed to make of the recipe?” Students are given a recipe with five ingredients and required to calculate $$\frac{1}{4}$$ of the recipe. 

• In 5.OA.A.1 Patrol Schedule, Lesson Plan Overview, Applying Standards for Mathematical Practice, “During Data & Computation, students realize that numerical expressions can represent real-world relationships, and can be evaluated to get a final value, which then must be contextualized to have meaning. Students see an expression which represents the relationship between the various ski lifts, including the type of lift (double, triple, quad), and how many chairs the respective lifts have. Students simplify the expression and use the real-world context to make meaning of the final value, which is the total number of people all of the chair lifts can hold.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for supporting the intentional development of MP3 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials include 10 protocols to support Mathematical Practices. Several of these protocols engage students in constructing arguments and analyzing the arguments of others. When they are included in a lesson, the materials provide directions or prompts for the teacher to support engaging students in MP3. These include: 

• “Lawyer Up! (12-17 min): When a task has the classroom divided between two answers or ideas, divide students into groups of four with two attorneys on each side. Tell each attorney team to prepare a defense for their ‘case’ (≈ 4 min). Instruct students to present their argument. Each attorney is given one minute to present their view, alternating sides (≈ 4 min). Together, the attorneys must decide which case is more defendable (≈ 1 min). Tally results of each group to determine which case wins (≈ 1-2 min). Complete the protocol with a ‘popcorn-style’ case summary (≈ 2-3 min).”

• “Math Circles (15-28 min): Prior to class, create 5 to 7 engaging questions at grade level, place on different table-tops. For example, Why does a circle have 360 degrees and a triangle 180 degrees? Assign groups to take turns at each table to discuss concepts (≈ 3-4 min each table).”

• “Quick Write (8-10 min): After showing an Immersion video, provide students with a unique prompt, such as: ‘I believe that the store owner should…’, or ‘The person on Mars should make the decision to…’ and include the prompt, ‘because…’ with blank space above and below. Quick writes are excellent for new concepts (≈ 8-10 min).”

• “Sketch It! (11-13 min): Tell students to draw a picture that includes both the story and math components that create a visual representation of the math concept (≈ 5-7 min). Choose two students with varying approaches to present their work (≈ 1 min each) to the class (via MidSchoolMath software platform or other method) and prepare the entire class to discuss the advantages of each model (≈ 5 min).”

The materials include examples of prompting students to construct viable arguments and critique the arguments of others.

• In 5.G.B.3 Squaring Off, Practice Printable, Introduction Problem, “As Poseidon and Zeus moved on to another round of ‘Shape Up,’ another disagreement occurred. Zeus said that a parallelogram is a trapezoid, while Poseidon said that a parallelogram is not a trapezoid. Who is correct - Zeus or Poseidon? Explain how you know.” Practice Printable Question 9, “Isaiah says that a parallelogram is a square. Dominique says a parallelogram is not a square. a) Draw and describe a figure Isaiah might use to prove he is correct. b) Draw and describe a figure Dominique might use to prove she is correct.”

• In 5.G.B.4 It’s a Polygon World, Practice Printable, Question 4, “Stacey said a rhombus is always a square. Joel said a square is always a rhombus. Use what you know about the properties of these shapes to explain who is correct.” 

• In 5.NF.B.5a Sweet Success!, Practice Printable, Question 6, “Ryan said that the product of 24 and $$\frac{5}{2}$$ is less than 24 because any whole number multiplied by a fraction will be less than 24. Is he correct? Explain your answer without doing the actual calculation.”

• In 5.NF.B.5b The Beef with Beef, Practice Printable, Introduction Problem, “Marie was grocery shopping. She came to the meat counter and asked Kenneth for $12 worth of ground pork. Ground pork was priced at $8.00 per pound. Marie suggested Kenneth start with measuring two pounds; since $12 was so much more than $8, it would have to be at least 2 pounds of meat. Kenneth told her it would be less than 2 pounds of meat. Which one of them is correct? Explain how you know.”

• In 5.MD.C.3a-b Cubicle Dudes, Practice Printable, Introduction Problem, “The dudes have a new box, the volume of which they need to determine. They have agreed to use a cube with a side length of 1 inch; however, they have gotten different answers. This means that one of the dudes has made a mistake in his calculation. Which dude is right? Explain how you know.” 

The materials provide guidance for teachers on how to engage students with MP3. In several lessons, the Detailed Lesson Plan identifies MP3 and provides prompts that support teachers in engaging students with MP3. Examples include:

• In 5.MD.C.3a-b Cubicle Dudes, “ During Data and Computation, the procedure outlines setting up a mock jury where students will listen to arguments presented by three students with opposing claims to reach a final verdict. This exercise reinforces SMP3 by having students explain and justify the logic of their assumptions.” Teachers are guided to “choose three students to present one of each claim with their supporting evidence. This can be done using a document camera or other technology. Encourage them to make statements using assumptions, data and definitions that support a viable argument in logical statements: My claim is____.; My mathematical evidence is___.; My assumptions are__.; My calculations support ____.  While the three students are prepping, group students to act as a jury to discuss the arguments presented to them. During each presentation, they may ask one clarifying question. At the end they must each agree on a verdict.”

• In 5.NF.B.3 Much Ado About Honey, “Students construct an individual argument to show how much honey each fairy should receive using the ‘Sketch It!’ protocol. Students are tasked with providing visual evidence to accompany computational evidence during the protocol. They share their work in small groups, justifying their conclusions. Students in the group vote on whether they believe each person's sketch and presentation would end the fairies’ argument or if the logic/visual needs to be revised. Teachers are provided with directions to help facilitate this process of analyzing arguments: Have students join in groups of three where each student is given one minute to present their sketch and create a viable argument. Other students in the group vote as to whether each sketch will end the fairies’ argument or if revision is needed to improve the argument. If revisions are needed, group members agree on one piece of feedback for the presenter.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for supporting the intentional development of MP4 and MP5 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP4 to meet its full intent in connection to grade-level content include:

• In 5.OA.B.3 Snowfall, Lesson Plan Overview, “MP4: Model with Mathematics. On Day 1, during the Immersion and Data & Computation phases, students will be given information about the rate of snowfall and asked to model this information on a graph.”

• In 5.MD.A.1 Treacle Treatment, the Detailed Lesson Plan states, “MP4: Model with mathematics. During Immersion, students begin modeling with limited information, primarily just a visual to solve the problem. Because they do not know how much each bottle holds, they must make assumptions and approximations pulling references from their own experiences (milk jugs, water bottles, etc.), which helps build a personal entry point for each student. Students collaborate to create an initial solution pathway prior to having enough information to do so, realizing they may need to make adjustments and revisions later. This pathway may consist of diagrams, estimations and initial calculations.”

• In 5.MD.C.5b Phil & Ned’s Excellent Assignment, Lesson Plan Overview, “MP4: Model with Mathematics. In Immersion, students begin the modeling process as the problem is unstructured during this phase. Students sketch a luxury doghouse which helps them conceptualize what a ‘foundation’ is, and helps them determine what they need to know to solve the problem and what strategies they might use. During Resolution students are encouraged to think about the modeling process, specifically why the process is so important. Additionally, during Clicker Quiz and Practice Printable, students will recognize that the formulas V = Bh and V = lwh can be used to solve real world problems involving volume and that the formulas ‘model’ the filling (or packing) of right rectangular prisms.

Examples of the intentional development of MP5 to meet its full intent in connection to grade-level content include:

• In 5.NF.B.4b Find a Field, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. Students choose tools to develop a strategy for solving the problem. In Immersion, students begin with a brief ‘tool session,’ recommended by Jo Boaler, that reminds them of the various tools at their disposal at any given time. In Data & Computation, students are asked to create a visual, selecting tools of their choice. In Resolution, students complete visuals by writing a statement about how their tools were effective in helping them think about and solve the problem. In the Practice Printable, students will choose tools or representations that help them model multiplication of fractions.”

• In 5.NF.B.7b Alfalfa Amount, the Detailed Lesson Plan states, “MP5: Use appropriate tools strategically. Students will likely use various types of diagrams and drawings to help them think through the math. For example; on Day 3 in the Practice Printable phase, students are asked to use number lines to model division problems, as well as to create their own drawing, such as circle fraction diagrams to represent dividing up pizzas.” In Problem 8, “A softball team is having their end of the year celebration. There are 6 pizzas to share. Each person gets $$\frac{9}{4}$$ of a pizza. How many people can be fed with the pizzas? Draw a visual representation to support your answer.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for supporting the intentional development of MP6 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

The materials use precise and accurate terminology and definitions when describing mathematics, and the materials provide instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include:

• Each Detailed Lesson Plan provides teachers with a list of vocabulary words and definitions that correspond to the language of the standard that is attached to the lesson; usually specific to content, but sometimes more general. For example, 5.G.4 states “Classify two-dimensional figures in a hierarchy based on properties.” The vocabulary provided to the teacher in 5.G.B.4 It’s A Polygon World is, “Hierarchy: A system or organization in which figures (usually polygons) are ranked one above the other according to their properties.” 

• The vocabulary provided for the teacher is highlighted in red in the student materials on the Practice Printable.

• Each Detailed Lesson Plan prompts teachers to “look for opportunities to clarify vocabulary” while students work on the Immersion problem which includes, “As students explain their reasoning to you and to classmates, look for opportunities to clarify their vocabulary. Allow students to ‘get their idea out’ using their own language but when possible, make clarifying statements using precise vocabulary to say the same thing. This allows students to hear the vocabulary in context, which is among the strongest methods for learning vocabulary.” 

• Each Detailed Lesson Plan includes this reminder, “Vocabulary Protocols: In your math classroom, make a Word Wall to hang and refer to vocabulary words throughout the lesson. As a whole-class exercise, create a visual representation and definition once students have had time to use their new words throughout a lesson. In the Practice Printable, remind students that key vocabulary words are highlighted. Definitions are available at the upper right in their student account. In the Student Reflection, the rubric lists the key vocabulary words for the lesson. Students are required to use these vocabulary words to explain, in narrative form, the math experienced in this lesson. During ‘Gallery Walks,’ vocabulary can be a focus of the ‘I Wonder..., I Notice…’ protocol.”

• Each lesson includes student reflection. Students are provided with a list of vocabulary words from the lesson to help them include appropriate mathematics vocabulary in the reflection. The rubric for the reflection includes, “I clearly described how the math is used in the story and used appropriate math vocabulary.” 

• Vocabulary for students is provided in the Glossary in the student workbook. “This glossary contains terms and definitions used in MidSchoolMath Comprehensive Curriculum, including 5th to 8th grades.” 

• The Teacher Instruction portion of each detailed lesson plan begins with, “Here are examples of statements you might make to the class:” which often, though not always, includes the vocabulary with a brief definition or used in context. For example, the vocabulary provided for 5.NF.B.4a The Horse Doctor, is “Partition” and “Product.” The sample statements provided are, “In The Horse Doctor, Dr. Equinas decides to give Bella a larger dose than is typically given to a regular-sized horse; Dr. Equinas interprets $$\frac{4}{3}$$ of 9 grams as 4 parts, when 9 grams is partitioned into 3 parts.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for supporting the intentional development of MP7 and MP8 for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of the intentional development of MP7 to meet its full intent in connection to grade-level content include:

• In 5.NBT.A.1 The Traveling Suitcase, the Lesson Plan Overview includes, “On Day 1, during both the Immersion and Data & Computation phases, students see Sonia change the dial on a suitcase several different times, each time resulting in her moving a certain distance from her original spot. Students are encouraged to see a pattern that emerges in relation to place value, and use that pattern and structure to make a prediction on a similar scenario. This type of place value pattern recognition continues on Day 2, as students work on the Clicker Quiz and Practice Printable.”

• In 5.NF.B.5a Sweet Success!, the Detailed Lesson Plan states, “MP7: Look for and make use of structure. In Immersion, students begin the lesson with a 'Number Talk' designed to help them look for and make use of structure of a multiplication problem, specifically the relationship between the factors and the product. Students are not given the pattern, but rather must discern these patterns themselves. In Resolution, teacher prompts encourage students to look for structures and to reason about how they might be useful.”

• In 5.MD.C.4 Shipment Shenanigans, “On Day 1, during the Immersion and Data & Computation phases, students will look for and make use of structure as they note the pattern of boxes from the data given and then use their knowledge of dimensions and unit cubes to find the volume.” During the Data & Computation phase, students answer, “What is the total number of layers of the boxes in the truck? How many boxes are in each layer? What is the volume of one box? How many total boxes are in the truck? What is the total volume of all of the boxes? Can you see another way that you might determine the volume?” The students apply their knowledge of counting unit squares to counting unit cubes, which is repeated in several problems that the students complete. In the Practice Printable, students answer, “The figures below are made of unit cubes. Identify the dimensions of each figure, and determine the volume.”

Examples of the intentional development of MP8 to meet its full intent in connection to grade-level content include:

• In 5.NBT.A.2 The Power of Ten!, the Detailed Lesson Plan states, “MP8: Look for and express regularity in repeated reasoning. During Resolution, students experience an additional exercise with calculating powers of 10. Students talk with a partner about the patterns and repeated reasoning they notice, focusing on the generalization required for the final value involving 10 to the nth power.”

• In 5.OA.B.3 Snowfall, Lesson Plan Overview, “MP8: Look for and express regularity in repeated reasoning. In Resolution, students experience how to maintain oversight of the process, while attending to details. Students experience repeated reasoning and regularity as they focus on the patterns of snowfall over time. Students (1) describe what patterns they noticed, (2) explain whether a pattern is repeating, and (3) determine if the pattern applies to all situations or not to be able to make a generalization (a rule). This practice is reinforced by having the students watch a complimentary video in which Jo Boaler has students modeling how to look and identify patterns in real-life scenarios.”