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.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials.

• A Curriculum Overview provides a chart of the components and description for the lessons, assessments, and Domain Review. The curriculum components are described briefly in the Overview section. 

• A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines helps the teacher understand rubrics for the assessments.

• In the Teacher Guide, there is instruction on planning a lesson with a sample sequence for lessons and assessments. The materials provide pacing for the year.

• In the Teacher Guide, the instructional protocols used throughout the series are described and connected to the Mathematical Practices they support. 

• In the Detailed Lesson Plan, there is a section to help support Diverse Learners with a chart of Accommodations, Modifications, and Extensions, as well as Language Routines. 

• Common Misconceptions are listed in each Detailed Lesson Plan.

• In each Detailed Lesson Plan, teachers are given suggestions for vocabulary incorporation such as, “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.”

• Guidance is given to teachers for applying and reinforcing math practices in the Teacher Guide and in Detailed Lesson Plans. For example, MP8: “This practice is reinforced by having the students watch a complimentary video in which Jo Boaler has students modeling how to look for and identify patterns in real-life scenarios.” Guidance shared directly from Jo Boaler states, “Students need time and space to develop their capacity to ‘look for and express regularity in repeated reasoning.’ When you provide tasks that are specific to supporting MP8, explicitly tell students that it’s ok to slow down, and to think deeply.” Several “tips” to address the MP are also shared.

• “Detailed Lesson Plans provide a step-by-step guide with specific learning objectives for the math standard, lesson summary, prerequisite standards, vocabulary and vocabulary protocols, applying Standards for Mathematical Practice, Jo Boaler's SMP Tips, cluster connection, common misconceptions, instruction at a glance, and day-by-day teaching instructions with time allotments. Also included are suggestions for differentiation, and instructional moves as well as tips for the English Language Learner student.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Throughout each lesson’s Detailed Lesson Plan, there is narrative information to assist the teacher in presenting student material throughout all phases. Examples include:

• In 5.G.B.3 Squaring Off, Teacher Instruction: “Here are some examples you might make to the class. Zeus and Poseidon were arguing about whether or not a square is a rectangle: We saw that even though a square is a square, it’s also a special rectangle; Polygon hierarchies can help us understand the relationship among different polygons, especially quadrilaterals.”

• 5.NF.A.2 Acre Acquisition, Common Misconceptions: “Students may forget that when adding or subtracting fractions, the fractions must refer to the same whole. Often when students use fraction models to represent addition or subtraction, they use mixed models (rectangles and circles, for example), which makes it difficult to compare the fractional segments. Tell students to use one type of model for each problem.”

• 5.NF.B.7a Pirate Pay, Cluster Connection: “Direct Connection: In Pirate Pay, students extend their previous understanding that $$a+b$$ is the same as $$a×\frac{1}{b}$$ to fractions. Namely, that $$\frac{1}{a}÷b$$ is the same as $$\frac{1}{a}×\frac{1}{b}$$. Students will divide $$\frac{1}{3}$$ pound of gold by 4 galley crew members, $$\frac{1}{3}÷4=\frac{1}{3}×\frac{1}{4}$$.”

• 5.OA.A.2 Liftie Lesson, Part 3 Resolution: “1. Play Resolution video to the whole class, and have the students compare their solutions as they watch. 2. After the video, prompt students with the following questions: What did you do that was the same? What was different? What strategy do you think was more efficient to find the equation? Why? Students may respond aloud or in a journal.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Under the Resources Tab, there is a section dedicated to Adult-Level Resources. These contain adult-level explanations including examples of the more complex grade-level concepts so that teachers can improve their own knowledge of the subject. There are also professional articles provided on topics such as mathematical growth mindset, cultural diversity in math, and mathematical language routines.

The Teacher Guide contains a page at the beginning of each cluster section titled, “Cluster Refresher for the Teacher - Adult Level Explanation”. This provides a page of basic background information for the teacher including strategies to develop understanding. For example,

• “5.NBT.A works to extend the understanding of place value, the magnitude of digits in a number, and the relationship between adjacent places. By emphasizing the meaning behind our base-ten system, the mathematical relationship can be extended in both directions, relating places that are ten times or one tenth the size of any given place. Understanding place value and the basic relationships between places not only establishes a foundation for understanding the magnitude of numbers, but also reveals how numbers can be broken down, expanded, or compared. ... The key is the order of the digits and how they are in relation to the decimal point. Once digits are in order, the magnitude of each can be shifted by powers of ten in either direction. The shift along the place value spectrum can show how adjacent places are ten times or one-tenth the size depending on which direction we move.”

The Adult-Level Explanations booklet under the Resources tab includes a progression through each domain from Grade 5 through High School. The last section is Beyond Grade 8, which explains how concepts from the middle grades connect to high school standards. For example: Beyond Grade 8: Number and Quantity:

• “Understanding the relationship between numbers, the various ways to represent them, their place on the number line, and the familiarity of the properties that they possess describes the concept that is the number system. The middle grades allow for opportunities to discover and conceptualize different types of numbers and their place on the number line. Starting with counting numbers in early years eventually leads to integers, and then to all rational numbers. Rational numbers are all numbers that can be written as a fraction p/q, when p is an integer and q is a nonzero. Studying the placement of these numbers on the number line is of equal importance as understanding different representations of the same number. Once one has a firm understanding of all rational numbers, their relationships and different representations, irrational numbers are introduced. ... This notion and its properties extends to sets of polynomial and rational functions in later courses. The rules that govern these are as follows: The sum or product of two rational numbers is rational. The sum of a rational number and an irrational number is irrational. The product of a non-zero rational number and an irrational number is irrational. Understanding the relationships between rational and irrational numbers and how they relate to form real numbers leads to future work in understanding complex numbers and their role in the number system. These are not rules and procedures, but numerical relationships that provide the foundation to these rules.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series.

• Each course in this series includes a document called Planning the Year that provides the standards and pacing for each lesson. 

• There are standards correlations in the Scope and Sequence Chart that lists each Lesson, Domain Review, and Major Cluster Lessons throughout a year.

• Each lesson is designed to address a single standard.

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series.

• The Teacher Guide contains a page at the beginning of each cluster section titled “Role of Mathematics” which clearly identifies the grade-level clusters and standards within a domain and describes the intent of the cluster. The Cluster Role Across Grade Levels describes the grade-level content in context of the domain progression from when the initial related skills were introduced to how the skills progress through high school. For example, “The 5.NBT.A cluster involves place value and using it to represent, round and compare decimal numbers to the thousandths. The development of these skills begins in Grade 2 where place value is introduced for three-digit numbers (2.NBT.A.1). In Grade 4, throughout Number and Operations - Fractions, students use place value to understand fractions with denominators of 10 and 100, relating them to their decimal equivalents (4.NF.C.6) as well as adding them (4.NF.C.5). Within the Numbers and Operations in Base Ten domain, students continue using their place value knowledge in the 5.NBT.B cluster as they learn operations with multi-digit whole numbers and decimals. Students expand their knowledge of powers to bases other than 10 as they write and evaluate numerical expressions involving whole number exponents in Grade 6 (6.EE.A.1) and work with radicals and integer exponents in Grade 8 (8.EE.A.3).” 

• The Detailed Lesson Plan for each lesson lists the Prerequisite Standards required for students to be successful in the lesson. For example, in 5.G.B.3 Squaring Off, the Prerequisite Standards listed are 3.G.A.1 and 4.G.A.2.

• The Detailed Lesson Plan for each lesson includes Cluster Connections that identify connections between clusters and coherence across grade levels. For example, in 5.G.B.4 It’s a Polygon World, Cross-Cluster Connection, “This activity connects 5.G to 6.G, 7.G, 8.G as students will continually use their knowledge of figures and their attributes to determine missing dimensions and to calculate area and volume, including figures in 3-dimensions. It also connects to High School Geometry, as students will know what assumptions can be made about figures, and they will be able to write better proofs.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program.

• The Curriculum Overview in the Teacher's Guide states that the curriculum is designed to “STOP THE DROP.” The materials state, “Core Curriculum by MidSchoolMath is developed to fix this problem through a fundamentally different approach... MidSchoolMath emphasizes structured, conceptual learning to prepare students for Algebra I... MidSchoolMath is specifically designed to address the ‘The Mid School Math Cliff’.”

• In the Teacher Guide, the overview on Scoring Guidelines states, “In coordination with Dr. Jo Boaler, MidSchoolMath has developed an approach to using rubrics and scoring with an emphasis on making them useful and practical for helping teachers support student learning. This is in contrast to the use of scoring guidelines for the primary purpose of giving grades.” 

• In the Letter to the Parent, the instructional approaches are summarized, “MidSchoolMath strives to help students see that math is relevant and holds value and meaning in the world. The curriculum is designed not only to enhance student engagement, but also to provide stronger visual representation of concepts with focus on logic structures and mathematical thinking for long-term comprehension. ... Peer Teaching: Students learning from other students is a powerful mechanism, wherein both the ‘teachers’ and the ‘learners’' receive learning benefits.”

Materials reference relevant research sources:

• “Hattie, J. (2017) Visible Learning

• Cooney, J.B., Laidlaw, J. (2019) A curriculum structure with potential for higher than average gains in middle school math

• Tomlinson (2003) Differentiated Instruction

• Dweck (2016) Growth Mindset

• Carrier & Pashler (1992) The influence of retrieval on retention: the testing effect

• Boaler, J. (2016) Mathematical MindSets

• Rohrer, D., & Pashler, H. (2007) Increasing retention without increasing study time

• Kibble, J (2017) Best practices in summative assessment

• Laidlaw, J. (2019) Ongoing research in simulators and contextualized math

• Lave, J. (1988) Cognition in practice: Mind, mathematics and culture in everyday life

• Schmidt and Houang (2005) Lack of focus in mathematics curriculum: symptom or cause.”

Materials include research-based strategies. Examples include:

• “Detailed Lesson Plans (Research Indicator: Teacher pedagogy and efficacy remains the highest overall factor impacting student achievement. Multiple instructional models show greater gains than ‘stand and deliver’.)

• The Math Simulator (Research Indicator: On randomized controlled trials, The Math SimulatorTM elicited high effect sizes for achievement gains across educational interventions. Contextual learning and Productive Failure are likely influences contributing to the large achievement gains.)

• Teacher Instruction (Research Indicator: Clarity of teacher instruction shows a large effect on student achievement.)

• Practice Printable (Research Indicator: Differentiation of instruction leads to higher effect sizes compared to full-time ‘whole-group’ instruction. Varied instructional approaches support a growth mindset, an indicator for student success.)”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Teacher Guide includes Planning the Year, Comprehensive Supply List which provides a supply list of both required and recommended supplies for the grade. For example: “Required: Markers, Chart, Paper, Colored Pencils, Dry-erase Markers, Graph Paper, Ruler, Protractor; Recommended: Individual white boards/laminated alternative, Calculator, Base-10 Blocks, Fraction Tiles, Multiplication Chart, Fake Money, Number magnets, Unit cubes."

Each Detailed Lesson Plan includes a Materials List for each component of the lesson. For example, in 5.OA.B.3 Snowfall:

• “Immersion: Materials -  Snowfall Immersion video; Chart paper/Interactive whiteboard

• Data & Computation: Materials - Copies of Snowfall Data Artifact, one per student 

• Resolution: Materials - Snowfall Resolution video

• Math Simulator: Materials - Snowfall Simulation Trainer; Student Devices; Paper and Pencil; Student Headphones

• Practice Printable: Materials - Copies of Snowfall Practice Printable, 1 per student

• Student Reflection: Materials - Copies of Student Reflection rubric, 1 per student; White Paper; Colored Pencils; Sticky Notes

• Clicker Quiz: Materials - Snowfall Clicker Quiz; Student Devices; Paper and Pencil”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the standards and Mathematical Practices addressed by formal assessments.

• In the Teacher Guide, Curriculum Components lists several assessments: Clicker Quiz, Test Trainer Pro, and the summative Milestone Assessment. Each cluster has a Pre-assessment and a Post-assessment (Milestone Assessment) which clearly identifies the standard(s) being assessed. The standard is part of the title, for example, “Milestone Post-Assessment 5.MD.A.”; individual tasks and items are not identified on the actual assessment. However, each problem is identified with the standard being assessed in the teacher answer key. 

• Standards are identified accurately and are from the appropriate grade level. 

• Assessment problems are presented in the same order as the lessons. They are sequential according to Domain and Cluster headings.

• The Milestone Assessments include a chart that aligns Mathematical Practices to each question on the assessment, including identifying if the assessment is online, print, or both.

• The end of each lesson includes a student self-assessment rubric that has students evaluate their understanding of the content standard and the mathematical practices that align with the lesson.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. There is guidance provided to help interpret student performance and specific suggestions for following-up. 

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance.

• In the Teacher Guide, the Domain Curriculum Components lists two assessments: Test Trainer Pro (formative) and Milestone Assessment (summative). 

  • “Milestone Assessment is a summative evaluation following each cluster per grade. They are automatically graded, yielding the percentage of items answered correctly. The math items are crafted to include items of varying difficulty.” “Please note: Milestone Assessments should not be used to determine student growth. As summative assessments, they are not as sensitive nor as accurate as the adaptive tool, Test Trainer Pro, for providing individual student data for achievement gains over time.”

  • “Test Trainer Pro acts as a low-stakes, formative learning tool for students to practice testing under more relaxed and stress-free conditions. It is an adaptive tool and is designed to elicit the largest gains in student achievement possible in the shortest period of time.”

• The Teacher Guide contains a section titled “A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines.” This section provides several assessment rubrics: 

  • The MidSchoolMath Rubric and Scoring Framework aligns a percentage “raw score” with a 4-point rubric and proficiency levels.

  • The Milestone Assessment Rubric aligns a percentage “raw score” with a 4-point rubric and has suggestions for follow-up.

  • The Student Self Assessment has students reflect and identify understanding for each lesson component. 

• An article by Jo Boaler, “Assessing Students in a Growth Mindset Paradigm with Jo Boaler” provides “recommendations for assessment and grading practices to encourage growth mindsets.”

• Each Curricular cluster contains a tab for Assessments which has a Milestone Assessment Overview & Rubric. There is a rubric from 0 to 3 provided for the open response section of the assessment. To earn all 3 points, students must demonstrate accuracy, show work, and may only have minor mistakes.

  • “Recommended Scoring for Milestone Assessments: A 3-point response includes the correct solution(s) to the question and demonstrates a thorough understanding of the mathematical concepts and/or procedures in the task. This response: Indicates that the student has completed the task correctly, using mathematically sound procedures; Contains sufficient work to demonstrate a thorough understanding of the mathematical concepts and/or procedures; May contain inconsequential errors that to not detract from the correct solution(s) and the demonstration of a thorough understanding.”

• The Overview states, “All items in Milestone Assessments are at grade level and evaluate student understanding of the content at the ‘cluster’ level. Milestone Assessments should only be administered to students after all lessons are completed within the cluster, following recommended sequence and pacing.”

• The answer key for each Milestone Assessment provides examples of correct responses for each problem. There is a sample response for the open-ended questions. 

• Several of the other lesson components could be used as formative assessments or for progress monitoring such as the Clicker Quiz.

The assessment system provides task-specific suggestions for following-up with students. There are suggestions for follow-up that are generic strategies, and there are some that direct students to review specific content.

• The Milestone Assessment Rubric includes Recommendations for Follow Up. These are found in the front matter of the Teacher Guide. They are generic to all assessments and align with the 4 points of the rubric: 

  • “Review and correct any mistakes that were made. Participate in reteaching session led by teacher.

  • Review and correct any mistakes that were made. Identify common mistakes and create a ‘Top-3 Tips’ sheet for classmates. 

  • Review and correct any mistakes that were made. Participate in the tutorial session. 

  • Review and correct any mistakes that were made. Plan and host a tutorial session for the Nearing Proficient group.”

• The Milestone Assessment also includes suggestions based on which problems are missed. The guidance directs students to review the worked example and Clicker Quiz in the lessons that align to the missed problems and then revise the problems they missed in the assessment. This provides specific feedback to review the content of the lesson. 

• The Student Self-Assessment provides a generic strategy for follow-up: “Recommended follow-up: When students self-identify as ‘Don’t get it!’ Or ‘Getting there!’ on an assignment, is it essential for teachers to attempt to provide support for these students as soon as possible. Additionally, it is helpful for teachers to use scoring on Practice Printables and Clicker Quizzes to gauge student comprehension. Use the general scoring guidelines to determine approximate proficiency. It is highly recommended that all assignments may be revised by students, even those which are scored.”

• The Student Self-Assessment provides suggestions based on where the students rate themselves. Students are directed to review specific parts of the lesson to reinforce the parts they do not feel successful with. There are also more generic strategies suggested that go across lessons and grade levels.

• The materials state that “Test Trainer Pro automates assessment and recommendations for follow-up under the score. As an assessment, Test Trainer Pro is the most specific, and most accurate measure available in MidSchoolMath to determine how students are performing in terms of grade and domain level performance.” A teacher can view the Test Trainer Pro question bank; however, there is no way to review the specific follow-up recommendations provided since they are adapted to each student. 

• Exit Ticket results are sometimes used to suggest grouping for instructional activities the following day.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. Assessments include opportunities for students to demonstrate the full intent of grade-level standards and the mathematical practices across the series.

• Assessments are specific to each standard, so there is opportunity for students to demonstrate to the full intent of grade-level standards. 

• Considering both formative and summative assessments, there are a variety of item types offered including Exit Tickets, Clicker Quizzes, Test Trainer Pro, Lesson Reflection, Self-Reflection, and Milestone Assessments.

• Most assessments are online and multiple choice in format, though there is a print option for milestone assessments that includes open response. 

• Students have the opportunity to demonstrate the full intent of the practices in assessments; practices are aligned in Milestone assessments and addressed in the student self-assessments for each lesson.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In each Detailed Lesson Plan, Supporting Diverse Learners, there is a chart titled Accommodations, Modifications, and Extensions for English Learners (EL) and Special Populations that provides accommodations for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 5.NBT.A.4, Round and Round, include:

• The Math Simulator Data & Computation: “Provide students with a place value chart. Provide students with manipulatives such as base ten blocks and fake money. Provide a rounding rules chart: (two rounding rules included).”

• Simulation Trainer: “Pair students to allow for peer teaching and support.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Make tracing paper available. Consider allowing students to answer questions verbally to a scribe. Students may benefit from doing fewer problems or receiving extended time to complete this assignment. Provide students with a place value chart, manipulatives chart, a number line, and a rounding rules chart. Underline the place that each question is asking the students to round to, and draw an arrow above every digit to the right of the underlined number.” The Practice Printable also has interactive buttons that allow students to complete work online through draw and text tools as well as a work pad that includes an opportunity to chat with the teacher. 

• Clicker Quiz: “Provide students with a  place value chart, manipulatives chart, a number line, and a rounding rules chart.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity. The Exit Ticket in each lesson provides a differentiation plan that includes extension. While some strategies are the same across lessons, there are a variety of tasks offered. Examples include:

• 5.MD.C.5 Polly Packs, “Distribute this figure to students. Challenge them to find as many ways as possible to find the total volume. This may include enclosing it in an even larger prism and subtracting out the missing volume.”

• 5.NBT.B.7 Bread & Butter, “Write and solve four word problems involving decimals, one for each operation (add, subtract, multiply and divide). Use diagrams or other strategies to illustrate your reasoning for each problem.”

• 5.OA.B.3 Snowfall, “Consider having students play act their own math problem with the rest of the group tasked to solve it (≈ 2-3 min each) if time allows. Have students create and graph rules that have more than one step. For example, the rule is multiply by 2, and then subtract 4. Start with 1.”

In each Detailed Lesson Plan under Supporting Diverse Learners, there is a chart titled Accommodations, Modifications and Extensions for English Learners (EL) and Special Populations that provides extensions for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 5.NBT.A.4 Round and Round include:

• The Math Simulator Resolution: “Task the students with describing another real-life situation when rounding to the nearest whole would not work to estimate.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Give the students multi-operational problems involving decimals and estimation. Ex: Estimate each number to the nearest whole and solve using order of operations. 67.8 + 9.34 + 12.3.”

• Clicker Quiz: “Task students with writing and solving their own ‘clicker quiz’ question.”

• Extensions are optional; there are no instances of advanced students doing more assignments than their classmates.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Materials consistently provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics. Examples include:

• In the Detailed Lesson Plan for every lesson, the same two strategies are suggested: “Access Closed Caption and Spanish Subtitles within the video.” and “Pair students to allow for peer teaching and support. Consider allowing EL students to write the narrative in their native language, then use a digital translator to help them transcribe it into English.” 

• Each Detailed Lesson Plan makes a connection with one of the eight identified Math Language Routines (MLR), listed and described in the Teacher Guide. The MLRs include: Stronger and Clearer Each Time, Collect and Display, Critique, Correct, and Clarify, Information Gap, Co-Craft Questions and Problems, Three Reads, Compare and Connect, Discussion Supports.

• All materials are available in Spanish. 

• The use of protocols such as Think-Pair-Share, Quick Write, and I Wonder I Notice provides opportunities for developing skills with speaking, reading, and writing.

• Vocabulary is provided at the beginning of each lesson and reinforced during practice and lesson reflection, “In the Practice Printable, remind students that key vocabulary words are highlighted.” 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 the lesson. 

• There is teacher guidance under the Resources tab - Math Language Routines. “Principles for the Design of Mathematics Curricula: Promoting Language and Content Development”, from the Stanford University Graduate School of Education, provides background information, philosophy, four design principles, and eight math language routines with examples. 

• There are no strategies provided to differentiate the levels of student progress in language development.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. 

Examples where manipulatives are accurate representations of mathematical objects include:

• The students have access to virtual manipulatives on the Work Pad which is available online in their Simulator Trainer, Practice Printable, Assessments, and Clicker Quiz. These include shapes, 2-color counters, base 10 blocks, algebra tiles, protractor, and ruler. In addition, there are different styles of digital graph paper and dot paper on the digital whiteboard.

• Throughout the materials, there are visual models with number lines, graphs, or bars, though these cannot be manipulated. 

• During the Immersion and Resolution videos, items from the real world are used to represent mathematical concepts. 

• The Teacher Guide has a section titled “Guidance on the Use of Virtual Manipulatives.” This section includes sub-sections titled: Overview, General Guidance, During Lessons, Manipulative Tools, and Examples of their Use & Connecting to Written Methods. The “Examples of their Use & Connecting to Written Methods” provides teachers with guidance about how to use and make connections with the manipulatives.

• In the Detailed Lesson Plan, Practice Printable, there is a “Manipulative Task!” where students use the virtual tools in the Work Pad and specifically connect manipulatives to written methods. For example, in 5.G.4 It’s a Polygon World, Manipulative Task, “Create a polygon! It's a Polygon World provides a good opportunity for students to use the WorkPad manipulatives. Have students create a polygon of their choice (with 5 or fewer vertices), name the polygon, and provide a definition.”