The instructional materials for Achievement First Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. Examples include: 

• Unit 3, Lesson 4, students develop conceptual understanding of 5.NBT.5, as they calculate products of two- and three-digit numbers by one-digit factors using area models. Partner Practice, Problem 1, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each. How much money does her brother raise? a) Find the product using the distributive property and an area model.” (a partially filled out area model is provided) “b) Use the standard algorithm to solve the multiplication problem. c) Describe each of the partial products you calculated, in order, when using the standard algorithm.”

• Unit 6, Lesson 4, students develop conceptual understanding of 5.MD.5, as they use visual models of shapes to write expressions related to volume. In the Independent Practice, Bachelor Level, Problem 1, provides students with a $4×4×5$ rectangular prism. “The same prism is shown below three times. Each cube represents one cubic meter. On each prism, use the lines to show you how you can deconstruct it into layers in a different way. Then, below each prism, write an expression to find the volume of each prism and solve.” 

• Unit 8, Lesson 2, students develop conceptual understanding of 5.NF.3, as they use tape diagrams to solve division problems. In Think About It, students are introduced to tape models to solve, “8 ÷ 4 = and 3 ÷ 4 = . The models below are called tape diagrams. Part A. Use the models provided to determine each quotient. Circle the quotation in your model. Part B. In the space below each model, show a check step to prove that each quotient is correct.”

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: 

• Unit 2, Mixed Practice 2.1, students demonstrate conceptual understanding of 5.NBT.A, as they explain patterns in products when multiplying by powers of ten. Problem 2, “Matthew multiplied $1.5×10^3$ and said that the answer was 1.5000. Which statement, if any, explains Matthew’s error? a. Matthew multiplied 10 by the exponent 3 b. Matthew multiplied 1.5 by the exponent 3 c. Matthew added 3 zeroes to the end of 1.5 d. Matthew’s statement is correct and contains no errors.”

• Unit 7, Lesson 1, students demonstrate conceptual understanding of 5.NBT.7, as they use a decimal grid to solve a subtraction problem involving decimals. Independent Practice, Bachelor Level, Problem 2, “Use the decimal grid below to solve: $0.81-0.16=$?” 

• Unit 8, Lesson 7, students develop conceptual understanding of 5.NF.4, as they create area models to multiply unit fractions. Independent Practice, Bachelor Level, Problem 3, “What is the area of a rectangle that is $\frac{1}{2}$ yard long and $\frac{3}{8}$ yard wide? A 1 by 1 yard rectangle has been started for you below.”

The materials for Achievement First Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both Skill Fluency and Cumulative Review (Mixed Practice) components. 

The publisher states that the Skill Fluency component of the curriculum “addresses the skill, procedures and concepts that students must perform quickly and accurately in order to master a standard or a skill imbedded within a standard. Skill Fluency is delivered during a 10-minutes segment of a 90-minute period.” The Skill Fluency and Cumulative Review (Mixed Practice) components contain resources to support the procedural skill and fluency standard 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• Unit 3, Lesson 4, Independent Practice, Bachelor Level, students estimate and connect partial products to the standard algorithm as they multiply a one-digit number by a three-digit number. Problem 3, “For each problem, make an estimate first. Then calculate the product using the standard algorithm and show your work. For number 3, list each of the partial products being calculated in order as shown in number 1. Use estimation to check the reasonableness of your product: $464 × 5 =$ ____.” (5.NBT.5) 

• Unit 3, Lesson 8, students reflect upon and choose an appropriate strategy for multiplication. Think About It, “We’ve studied several methods for multiplying in this unit and in previous grades, including mental math, the distributive property (with an area model or expression) and the standard algorithm. Look at each problem below and decide which of these strategies makes the most sense to use.” Students solve, “, $85 × 10$, $5 × 17$, and $422 × 329$” (5.NBT.5) 

• Unit 5, Mixed Practice 5.1, students develop procedural skill and fluency related to multiplication as they solve a word problem. Problem 3, “Over the course of fifteen days, a museum counts the number of guests that enter. They count an average of 2,362 people on each of the days. How many guests visited the museum altogether. Show your work. Answer ________.” (5.NBT.5)

The materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. Examples include: 

• Unit 3, Mixed Practice, 3.2, Day 2, students demonstrate procedural skill and fluency as they multiply multi-digit factors while solving a problem with a provided chart. “Rory, Elaina and Yashika are all on a reading marathon team. The time each girl reads each day is shown in the chart below. If each girl reads for 36 days, how many total minutes will they have read?” (5.NBT.5)

• Unit 4, Skill Fluency 4.2, Day 2, students demonstrate fluency in multiplying multi-digit whole numbers using the standard algorithm. Problem 1, “Find the product of 736 and 92.” (5.NBT.5) 

• Unit 7, Skill Fluency 7.1, Day 3, students demonstrate procedural skill and fluency with multiplication. Problem 3, “. Find the value of ?.” (5.NBT.5)

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications especially within exercise based lessons as well as the problem of the day in each cumulative review. 

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 1, Mixed Practice Day 2, Problem 5, students solve a real-world, non-routine problem by comparing two decimals to thousandths based on meanings of the digits in each place. (5.NBT.3) "Noah threw a Frisbee 4.89 yards. a) Noah threw the Frisbee farther than Lin. How far could Lin have thrown the Frisbee? b) Andre threw the Frisbee farther than Noah but less than 4.9 yards. How far could Andre have thrown the Frisbee? Explain your reasoning.”

• Unit 3, Lesson 3, Problem of the Day, Day 1, students write and interpret an expression then solve a routine real-world problem involving multiplying multi-digit whole numbers using the standard algorithm (5.NBT.5). "Use the chart to solve. (note: the chart shows minimum and maximum length and width of High School and FIFA Regulation Soccer Field Dimensions) a. Write an expression to find the difference in the maximum area and minimum area of NYS high school soccer fields. Then, evaluate your expression. b. Would a field with a width of 75 yards and an area of 7,500 square yards be within FIFA regulation? Explain why or why not.”

• Unit 7, Lesson 11, Interaction with New Material, students engage in a routine problem with 5.NF.1 as they add and subtract fractions with unlike denominators. "Victor is making a special enchilada dish for the Latin Heritage festival at his school. To make the dish, he needs a lot of fresh tomatillos. To make enough for 60 servings he needs $12\frac{1}{2}$ pounds of tomatillos. He finds $5\frac{1}{4}$ pounds at King’s Grocery and $3\frac{3}{5}$ pounds at Metropolitan Grocers. He decides to call a third store to see if they’ll have enough in stock. How much should he ask for?”

• Unit 9, Lesson 3, Think About It, students engage with 5.NF.7 as they apply and extend previous understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions in a non-routine problem. Think About It, “Carmine and Miguel are working together on the following problem: Mrs. Silverstein is having a college graduation party for her son. She buys enough cake so that each guest at the party can have up to $\frac{1}{6}$ of a cake. She buys 3 cakes. How many guests is she expecting? Carmine writes the equation $\frac{1}{6} ÷ 3 = \frac{1}{18}$. Miguel writes the equation $3 ÷ \frac{1}{6} = 18$. Is either student correct? Create a model to prove your thinking. Then explain your reasoning on the lines below.”

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include: 

• Unit 6, Lesson 10, Independent Practice Question 1 (Bachelor Level), students apply the volume formula (5.MD.5) and convert among different-sized standard measurement units within a given measurement system (5.MD.1) in the context of solving a non-routine real-world problem. "At the flea market, a shopper asks Geoffrey if it is possible to use his 3 foot by 2 foot by 2-foot large planter as a bookcase or storage instead. Geoffrey considers this and estimates that a typical book has a volume of about 40 cubic inches. How many books would a large planter hold if filled with as many books as possible?”

• Unit 9, Lesson 3, Independent Practice Question 2 (Bachelor Level), students engage with 5.NF.7 as they solve a routine real-world problem involving division of unit fractions. "Virgil has $\frac{1}{6}$ of a birthday cake left over. He wants to share the leftover cake with 3 friends. What fraction of the original cake will each of the 3 people receive? Draw a picture to support your response.”

• Unit 11, Lesson 5, Real World Problems, students represent routine real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation (5.G.2). "There is a $25 annual fee for membership at the gym. It also costs $5 per visit to use the gym. Fill in the table to show the total cost of x visits to the gym. a) Write the ordered pairs, and graph the data on the coordinate graph. b) Write the ordered pair that represents 6 visits to the gym. Explain what the ordered pair means. c) If Amaya can only spend up to $50 in one month, how many times can she visit the gym? Explain.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

The instructional materials include opportunities for students to independently demonstrate the three aspects of rigor. Examples include:

• Unit 4, Mixed Practice 4.1, students develop procedural skill and fluency as they solve problems involving multi-digit multiplication. Problem 3, “Find a 3-digit number and a 1-digit number that when multiplied together will result in a product between 3,000 and 4,000. Show your work.” (5.NBT.5) 

• Unit 5, Cumulative Review, Problem of the Day, Day 2, students apply skills related to measurement conversions as they solve a routine problem. “A city wants to install fencing around two new playgrounds. Playground A is 5 yards long and 25 feet wide. Playground B is 3 yards long and 27 feet wide. A) Which playground will require more fencing, and by how much? B) Fencing costs $15 per two feet. How much will it cost to put up fencing around both playgrounds?” (5.MD.1) 

• Unit 7, Lesson 2, Independent Practice, Bachelor Level students develop conceptual understanding of adding and subtracting decimals to the hundredths as they use a hundreds grid to solve a problem. Problem 3, students are shown a 100 grid with two rows of 10 filled in. “Jonah added 0.36 to the value below and got 2.36. Is his answer reasonable? Why or why not? (Use the space to the right to explain.)” (5.NBT.7)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Unit 3, Lesson 4, Partner Practice, students develop conceptual understanding of place value and procedural skills and fluency as they solve a problem involving the standard algorithm, to find a product in a real world context. Problem 1, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each. How much money does her brother raise? a) Find the product using the distributive property and an area model. b) Use the standard algorithm to solve the multiplication problem.” (5.NBT.5)

• Unit 7, Lesson 12. Independent Practice, Master Level, students develop conceptual understanding of fractions and apply skills related to addition and subtraction of fractions as they solve a problem and develop a model. Problem 1, “Directions: Create a model of both scenarios. Write an equation that could be used to find a solution in each scenario. Explain how the scenarios are similar and how they are different. Problem A: Jennah has one piece of string that is $3\frac{1}{8}$ meters long, and another that is $3\frac{5}{10}$ meter. How much longer is the longer string? Model: ____, Equation: ____ . Problem B: Jennah had a piece of string that was $3\frac{5}{10}$ meters long. She used $3\frac{1}{8}$ meters. How much string was left? Model: ____ Equation: ___. How are the problem scenarios mathematically similar? What is one important difference in the problem scenarios?” (5.NF.1, 5.NF.2) 

• Unit 8, Lesson 18, Independent Practice, Masters Level, students apply their understanding of fractions as they solve problems involving multiplication of fractions and mixed numbers, and demonstrate procedural skill to add and subtract decimals to hundredths. Problem 1, “Oliver came home from the store with .250 L of heavy cream only to find that he needed $1\frac{1}{3}$ times that much for his recipe. How much more heavy cream does he need when he goes back to the store? Represent the problem with a model and an expression or equation. Then solve.” (5.NF.6, 5.NBT.7)

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit, but the lessons do not cite the same MPs.

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

• The Unit 6 Overview outlines the intentional development of MP1. “In lesson 6, students work to make sense of problems by identifying unknowns in various volume-related contexts. In lessons 9 and 10, students make sense of complex volume problems in various contexts, persevering to properly formulate solution pathways and solutions.”

• Unit 7, Lesson 12, Partner Practice Question 1 (Bachelor Level), students make sense of how equations connect to verbal descriptions. “Which equation or equations can be used to represent the following: Team A built a tower that was $1\frac{1}{2}$ feet taller than Team B’s. Team B’s tower was $3\frac{1}{5}$ tall. How tall is Team A’s tower? Create a model, and then circle all equations that apply. a) $1\frac{1}{2} - \frac{1}{5} =$?; b) $1\frac{1}{2} + ? = 3\frac{1}{5}$; c) $3\frac{1}{5} - 1\frac{1}{2} =$?; d) $3\frac{1}{5} + 1\frac{1}{2} =$ ?”

• The Unit 9 Overview describes development of MP1. “In both lessons 5 and 12, students extend their understandings of division by making sense of and persevering in solving multi-step problems in real world contexts. In lesson 5 students do this with the division of fractions, in lesson 12 with all operations of decimals.”

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

• Unit 4, Lesson 8, Interaction With New Material, students reason abstractly and quantitatively when working with dimensions of a room. “The owner of the art gallery knows that his rectangular space is 1500 square feet. The width of the space is 60 feet. In order to plan for a new exhibit, he needs to know the full perimeter of the gallery. Help him find it in the space below.”

• Unit 7, Lesson 13 Check for Understanding, students engage with MP2 as they solve a real world problem involving addition and subtraction of fractions with different denominators. “John got directions to his new high school for orientation day. He knows the school is 9 miles away. When he pulls the directions from his pocket, some of the last step was rubbed off. ~Take Atlantic Ave. $3\frac{1}{4}$ miles. Turn right on 17th. ~Go $2\frac{1}{8}$ miles. The road becomes Jefferson Ave. ~Take Jefferson Ave .... If the school is on Jefferson Ave., how many miles should John be on Jefferson Ave.? Draw a model and write an equation to represent the scenario.”

• The Unit 8 Overview outlines the intentional development of MP2. “In lesson 1-3, students relate fractions to division in different contexts. By de- and re-contextualizing fractions in these contexts students reason abstractly and quantitatively about the situations. In lessons 16 and 17, students reason quantitatively about the placement of their decimal in a product based on their estimates.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. “Purpose: Through the use of error analysis, guided questioning and discussion students will identify and fix a common misconception related to a skill they learned the previous day. These are sequenced so that after a particularly complex conceptual lesson or a lesson involving a skill that surfaces a common misconception, students get another focused at bat to either fix their misunderstanding or deepen their reasoning around key mathematical concepts and viable strategies to guide them away from making the same error. These lessons start with analyzing fictional student work and are structurally based off of the Standards for Mathematical Practice 3.”

• Unit 4, Lesson 7, Partner Practice, Masters Level, students critique the reasoning of others and construct an argument based on their knowledge of division. Problem 1, “Paul divided 8,280 by 36 and got 23. Do you agree or disagree? Prove your thinking and explain in the space below.”

• Unit 6, Lesson 2, Independent Practice, Bachelor Level, students critique the reasoning of others and construct an argument based on their knowledge of shapes. Problem 7, “Tyler builds the shape below and then turns it on its side. He says that the figure takes up less space now because it is shorter. Do you agree or disagree with his claim and why?” 

• Unit 7, Lesson 11, Day 2, Partner Practice, Bachelor Level, students construct an argument based on their knowledge of fractions. Problem 1, “Which of the following differences will require regrouping to solve? $1\frac{1}{3} -\frac{1}{2}$ OR $1\frac{1}{2} -\frac{1}{3}$ Explain how you know without doing any calculations.” 

• Unit 8, Lesson 17, Day 2, Exit Ticket, students critique the reasoning of others as they use estimation to assess the reasonableness of an answer. Problem 2, “Tyler multiplies 3.1 and 4.2. He gets a product of 130.2. Using estimation as your evidence explain if his product is reasonable or unreasonable and what his mistake might have been.” 

• Unit 10, Lesson 9, Interaction With New Material, students critique the reasoning of others as they classify triangles. “Ms. Cox’s class is analyzing the two figures below. Mya says that they can be given the same name. Justin says the shapes have different names. Ms. Cox says that both students are correct. Part A. How is it possible that both students are correct? Explain your reasoning. Part B. What is the most specific name that can be given to each triangle? Justify your response.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• The Unit 4 Overview, "In lessons 3-7 students attend to precision when they are transitioning their preferred division strategy to the written methods. This requires students to make connections to more concrete representations and to keep place values of quotients in order. MP 6 is a major focus of unit 4 as students must precisely identify place values when dividing throughout the unit."

• The Unit 6 Overview, “In lesson 1, students attend to precision by intentionally learning the proper expression of cubic units in volume and the why behind these three dimensions. This understanding is expanded upon in lesson 4 when students learn a different way of breaking down a 3D structure into layers. Finally in lesson 7, students learn of the additive nature of volume and how to define their units when adding multiple structures together. Students often attend to precision in lessons, however, in Unit 6, MP6 is specifically emphasized with a focus on the identification and tracking of proper units in volume contexts.”

• Unit 10, Lesson 8 provides strategies for teachers to use in guiding students to use precise vocabulary when classifying triangles. The debrief, “Using the precise words for angles less than, equal to, or greater than 90, what name could we give each group, and why?” Students might say, “Acute, Right, and Obtuse, because group 1 has only acute angles, group 2 has a right angle, and group 3 has an obtuse angle.”

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. Unit 2, Lesson 2, Key Vocabulary,

• “Denominator – The bottom number in a fraction; shows the number of parts in the whole

• Numerator – The top number in a fraction; shows how many parts of the whole are being described

• Equivalent fraction – A fraction with the same value as another fraction but with different numerators and denominators.”

• Unit 3, Lesson 2, Independent Practice, Question 2 (Bachelors Level), accurate terminology is used as students identify expressions. “Which expression represents twice the product of 15 and 4? Circle all that apply. a. $2 + (15 × 4)$ b. $2 × (15 × 4)$ c. $2 × (15 + 4)$ d. 62 e. 120.”

Unit 5, Lesson 4, Independent Practice, Question 5 (Bachelors Level), students are expected to understand and use accurate terminology as they solve a division problem and explain their answer. “Myra converted 5,300 feet into miles using the correct expression $5,300 ÷ 5,280$. She got a correct answer of 1 R20. What does the 1 in her quotient represent? What does the 20 represent? Explain.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 4, Independent Practice, Question 1 (Bachelor Level), students engage with MP7 as they compare and order fractions. “Review: Compare the pairs of fractions by reasoning about the size of the units. Use >, <, or =. a) 1 fourth _____ 1 fifth; b) 3 fourths _____ 3 fifths; c) 1 tenth _____ 1 twelfth; d) 7 tenths _____ 7 twelfths.”

• Unit 10, Lesson 6, Think About It, students look for structure as they classify quadrilaterals. “Below each shape, list as many names as you can for the shape. Then, circle every name that they have in common.”

• Unit 11, Lesson 1, Exit Ticket, students interpret the structure of the coordinate plane as they construct a coordinate plane and use it to name the location of points. “Use a ruler on the grid below to construct the axes for a coordinate plane. The x-axis should intersect points L and M. Construct the y-axis so that it contains points K and L. Label each axis. a) Place a hash mark on each grid line on the x- and y-axis. b) Label each hash mark so that A is located at (1, 1). c) What are the coordinates of point M? d) What is point L called?”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 1, Independent Practice, Question 2 (Bachelor Level), students repeatedly create equivalent fractions and make connections to multiplying and dividing by fractions equal to one. “Generate four fractions that have the same value as the fraction $\frac{2}{5}$ . Show your work and record your answers on the line below.”

• Unit 7, Lesson 9, Day 2, Independent Practice, Question 3 (Bachelor Level), students find the least common denominator as an efficient shortcut or additional subtraction strategy with fractions. “Madame Curie made some radium in her lab. She used $\frac{15}{36}$ kg of the radium in an experiment and had $1\frac{1}{18}$ kg left. Part A. How much radium did she have at first?”

Unit 10, Lesson 3, Test the Conjecture, Question 2, students use repeated reasoning to make sense of polygons by classifying quadrilaterals based on the presence of parallel sides. “True or false, a quadrilateral is always a trapezoid.”

The materials reviewed for Achievement First Mathematics 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. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

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

• The Guide to Implementing AF Math provides a Program Overview for the teacher with information on the program components and scope and sequence. This includes descriptions of the types of lessons, Skill Fluency, Mixed Practice, and Problem of the Day.

• The Teacher’s Guide supports whole group/partner discussion, ask/listen fors, common misconceptions and errors, etc. 

• Each lesson includes a table identifying the steps and actions for the teacher which helps in planning the lesson and is intended to be reviewed with a coach.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 6, Measurement and Data-Volume, Lesson 5 include:

• “What do we want every student to take away or do as a result of this lesson? How will a teacher know if students have met this goal? As a result of this lesson, every student can determine the volume of a prism using one of two formulas and given dimensions.”

• “Anticipated Misconceptions: Scholars may struggle to identify the given dimensions as a base area, length, width, or height. This will be particularly true if the area of the “base” is not given as the bottom or top layer, but rather the side. Students will need to rotate their perspective (but practiced this in L4). If students do not have a solid understanding of how the base area is derived, they may struggle to make connections between the formulas and understand why B and lw are interchangeable.”

• “Debrief: Show student work that created 3 layers of 8 cubes. How are the students' methods similar? Student might say, We think Jade’s method is correct because she found that there were 8 cubes in one layer and multiplied by 3 layers. Michael did the same thing, but he multiplied the length and width to find the number of cubes in one layer and then multiplied by 3 layers [Planner’s note: Scholars must be able to articulate a clear connection between the strategies].”

• Test the Conjecture provides multiple prompts to help teachers guide students through the problem. “What is the question asking us to do? How can we apply our conjecture to solve this problem? Is the height given, and if so, what is it, how do you know? We can use your conjecture to write a formula and evaluate. [Write v = Bh]. What do we substitute for B and h? Talk to your partner, and solve. How can we prove that our conjecture worked?”

Each lesson includes a “How” section that lists the key strategies of the lesson and delineates what “top quality” work should include. Examples from Unit 6, Measurement and Data-Volume, Lesson 5 include:

• “Key Strategy: To apply the volume formulas: Identify whether the base area or the length and width are given. Determine the height. Select a formula and substitute known dimensions in.”

• “CFS (Criteria for Success) for top quality work (generating equivalent fractions): Annotate and label given dimensions using their variables. The formula is written before substitutions are made. Work is shown for substitution and computation. The answer is recorded with the correct units.”

The materials reviewed for Achievement First Mathematics 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. 

Materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. Examples include:

• Unit Overviews provide thorough information about the content of the unit which often includes definitions of terminology, explanations of strategies, and the rationale about incorporating a process. Unit 8 Overview, “In lessons 12 and 13 students explain multiplication as scaling (5.NF.5). They rationalize: 1) Why multiplying by a second factor greater than 1 results in a product that is greater than the first factor; and 2) why multiplying a second factor less than 1 results in a product that is less than the first factor. This is the middle school mathematician’s first introduction to the concept of a fractional scale factor, which is an important concept throughout middle school math, and reappears later in this unit as ‘scaling factors’ are used to convert between units of measure. In lesson 12, students develop this concept with fractions. They see leverage patterns in products as well as number sense from earlier in the unit to explain that multiplication by a number less than one can never result in a number bigger than the second factor (i.e. $\frac{3}{4}$ of 6 can never be more than 6, because it is just 3 of the 4 parts of the whole). At the same time, multiplying by a number greater than 1 will always result in a product bigger than the second factor. In lesson 13 students explain and prove that the same is true with decimal fractions. For now, the concept of scaling will be key to developing number sense for multiplying and dividing by rational numbers, estimating products containing fractions and decimal fractions, and solving problems involving ‘x times as many.’”

• Some Unit Overviews include background knowledge for the teacher at the end of the file. The sources include Envision by Pearson, EngageNY, the progression documents, and several listed as “author and source unknown.” The Unit 4 Overview includes, “Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as $4\times8+3$. In multi-digit division students will need to find the greatest multiple less than a given number.  For example, when dividing by 6, the greatest multiple of 6 less than 50 is $6\times8=48$.  Students can think of these ‘greatest multiples’ in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put into each group is 8 and there are 2 objects left over.” 

• The Unit Overview includes an Appendix titled “Teacher Background Knowledge” which includes a copy of the relevant pages from the Common Core Math Progression documents which includes on grade-level information.

Materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. Examples include:

• Unit 1 Overview, Numbers in Base Ten – Place Value Understanding, Identify the Narrative, “Following this unit on place value understanding, students will begin a mini-unit on fraction understandings. They will make connections to the work done with decimal fractions to review, reinforce, and develop foundational fraction concepts and build fluency and number sense before learning to add and subtract fractions with unlike denominators. Gaining strong understanding and fluency with the core concepts in Unit 1 is essential to students’ success with fraction and decimals operations, as well as to multiplying with multi-digit numbers in units 2 and 3. In later grades, students continue to leverage this work when extending the number line to include negative rational numbers (grade 6), operating with all forms of rational numbers (grades 6 and 7), understanding ratios and rates of change (grade 6-8), creating probability models (grade 7), working on a coordinate grid (grades 5-8), and creating graphs to represent data (grades 5-8). In High School, these foundational skills continue to be paramount given that a deeply internalized number sense that includes fractions and decimals is critical to understanding concepts such as Exponential growth and decay (what it means to have a base of 1.25 v. 0.75). Limits: what would happen eventually if the denominator of a rational function was increasing faster than the numerator?”

• Unit 6 Overview, Measurement and Data – Volume, Identify the Narrative, “Looking ahead to the remainder of 5th grade, students will continue to apply their understanding of volume as they write numerical and algebraic expressions to represent real world and mathematical contexts (5.OA.A), perform operations with multi-digit whole numbers (5.NBT.B), and convert units of measurement (5.MD.A). In future grades, students will continue to develop their understanding of 3-Dimensional figures using the figures’ properties as they extend their work with volume to include fractional edge lengths (6.G.2), learn to create nets as a strategy to calculate surface area (6.G.4), describe two-dimensional figures resulting from a slicing three-dimensional figures (7.G.3), and apply formulas to find surface area and volume of other solids that are not rectangular prisms (7.G.6, 8.G.9). In High School, students continue to build on the foundation they developed in Middle School by learning how to find the volume of additional solids (i.e. pyramids), to find the volume of non-right solids using slant heights, to apply algebraic principals to explore the relationship between Algebra and Geometry, to model real-world and mathematical contexts, and to recognize and understand the relationships between volume, perimeter and surface area.”

• Unit 8 Overview, Multiplying Fractions and Decimals, Identify the Narrative, “Immediately following Unit 6, scholars move into Unit 7 on dividing fractions, decimals, and mixed numbers. By 6th grade, students are expected to have a deep conceptual understanding of multiplying fractions and decimals as well as fluency with multiplying fractions. Once in 6th grade, students extend their understanding of multiplying fractions to working with division of fractions and mixed numbers, and they extend their understanding of decimals to fluently multiply decimal numbers. Additionally, scholars apply their knowledge of fractions to ratios and rates, fluently compute multi-digit multiplication and division with decimals, and begin working with percents. In 7th grade, students’ understanding of and ability to perform calculations culminates with performing all operations with rational numbers. This unit is pivotal in students’ continued understanding of and ability to fluently calculate with rational numbers. For High School, fluency with rational numbers sets students up to focus on learning new algebraic material in high school that incorporates the use of these numbers and assumes knowledge of them. An understanding of rational number operations also facilitates the understanding of rational functions and how to work with them appropriately.”

The materials reviewed for Achievement First Mathematics 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. Examples include:

• Guide to Implementing AF Grade 5, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

• The Program Overview informs teachers “about how to ensure scholars have sufficient practice with all of the Common Core State Standards. Standards or parts thereof that are bolded are addressed within a lesson but with limited exposure. It is recommended that teachers supplement the lessons addressing these standards by using the AF Practice Workbooks to ensure mastery for all students. Recommendations for when to revisit these standards during Math Practice and Friday Cumulative Review are noted in the Practice section of each unit.”

• The Unit Overview includes a section called Identify Desired Results: Identify the Standards which lists the standards addressed within the unit and previously addressed standards that relate to the content of the unit.

• In the Unit Overview, the Identify The Narrative provides rationale about the unit connections to previous standards for each of the lessons. Future grade-level content is also identified.

• The Unit Overview provides a table listing Mathematical Practices connected to the lessons and identifies whether the MP is a major focus of the unit.

• At the beginning of each lesson, each standard is identified. 

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work. 

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

In the Unit Overview, the Identify the Narrative section provides the teacher with information to unpack the learning progressions and make connections between key concepts. Lesson Support includes information about connections to previous lessons and identifies the important concepts within those lessons. Examples include:

• Unit 4 Overview, “Unit 4 content is integral to students’ success in the rest of 5th grade as well as future grade levels. Later in 5th grade, students apply strategies for division of whole numbers to decimals (5.NBT.7) and general principals of division to division of fractions (5.NF.7). In 6th grade, students solidify their division of whole number skills as they learn to fluently apply the division algorithm (6.NS.2). Later, they develop fluency with decimal division as well (6.NS.3). They are expected to use this fluency in applications of division across a number of topics in 6th grade and in 7th grade when they learn to operate with rational numbers (7.NS. A). Students continue to calculate quotients throughout the rest of middle school. Therefore, it’s imperative that the foundation built in 5th grade is strong in order to set them up for success in future grades.”

• Unit 8 Overview, “Immediately following Unit 6, scholars move into Unit 7 on dividing fractions, decimals, and mixed numbers. By 6th grade, students are expected to have a deep conceptual understanding of multiplying fractions and decimals as well as fluency with multiplying fractions. Once in 6th grade, students extend their understanding of multiplying fractions to working with division of fractions and mixed numbers, and they extend their understanding of decimals to fluently multiply decimal numbers. Additionally, scholars apply their knowledge of fractions to ratios and rates, fluently compute multi-digit multiplication and division with decimals, and begin working with percents. In 7th grade, students’ understanding of and ability to perform calculations culminates with performing all operations with rational numbers. This unit is pivotal in students’ continued understanding of and ability to fluently calculate with rational numbers. For High School, fluency with rational numbers sets students up to focus on learning new algebraic material in high school that incorporates the use of these numbers and assumes knowledge of them. An understanding of rational number operations also facilitates the understanding of rational functions and how to work with them appropriately.”

The materials reviewed for Achievement First Mathematics 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. Examples include:

• The Implementation Guide states, "Our program aims to see the mathematical practices come to life through the shifts (focus, coherence, rigor) called for by the standards. For students to engage daily with all 3 tenets, we structure our program into two main daily components: math lesson and math cumulative review. The math lessons are divided into three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson. On a given day students will be engaging in EITHER a conjecture-based, exercise-based lesson or less often an error analysis lesson. The math cumulative review component has three sub-components: skill fluency, mixed practice, and problem of the day. Three of the five school days students engage with all three sub-components of the math cumulative review. The last two days of the week have time reserved for lessons, reteach lessons, and assessments. See the diagram below followed by each category overview for more information.”

Research-based strategies are cited and described within the Program Overview, Guide to Implementing AF Math: Grade 5-8, Instructional Approach and Research Background and References. Examples of research-based strategies include:

• Concrete-Representational-Abstract Instructional Approach, Access Center: Improving Outcomes for All Students K-8, OESP, “Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.”

• Introduction to the Math Shifts, by Achieve the Core, 2013, “According to the National Council of Teachers of Mathematics, Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.”

• Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell by Common Core Tools, “According to the National Mathematics Advisory Panel (2008), explicit instruction includes ‘teachers providing clear models for solving a particular problem type using an array of examples,’ students receiving extensive practice, including many opportunities to think aloud or verbalize their strategies as they work, and students being provided with extensive feedback.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Program Overview, Guide to Implementing AF Math: Grade 5, Scope and Sequence Detail, Supplies List includes a breakdown of materials needed for each Achievement First Mathematics Program. Examples include:

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2mind Rainbow Fraction Tiles Set, Class set.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2mind Classroom Number Line (-20 to 100), 1 for each math classroom.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2mind Unit Cubes, Set of 1000.”

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

Unit Assessments consistently and accurately identify grade-level content standards along with the mathematical practices within each unit. Examples from unit assessments include:

• Unit 1 Overview, Unit 1 Assessment: Understanding Place Value, denotes the aligned grade- level standards and mathematical practices. Question 9, “Kareen has $20 and his brother, Hakeem, has$200. Which statement describes the relationship between the amount of money each brother has? Hakeem has ten times as much money as Kareen. Kareen has $\frac{1}{10}$ as much money as Hakeem. Explain your thinking on the lines below.” (5.NBT.1, MP2, MP3)

• Unit 5 Overview, Unit 5 Assessment: Measurement Conversions, denotes the aligned grade- level standards and mathematical practices. Question 4, “In the Middle School Basketball finals, the Eagles faced off against the Wolves. During the game the Eagles drank 45 pints of water and the Wolves drank 6 gallons of water. Which team drank more water during the game?” (5.MD.2, MP1, MP4, MP6)

• Unit 7 Overview, Unit 7 Assessment: Adding & Subtracting Decimal – fractions, denotes the aligned grade-level standards and mathematical practices. Question 4, “Ned caught $\frac{1}{5}$ pound of fish. Sarah caught $\frac{5}{12}$ pound of fish. Jessa caught $\frac{1}{6}$ pound of fish. Their goal was to catch at least 1 pound of fish to cook dinner. Did they meet their goal? Show your work.” (5.NF.2, MP1, MP2, MP4)

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

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

• Assessments include an informal Exit Ticket in each lesson and a formal Unit Assessment for every unit. 

• There is guidance, or “look-fors,” to teachers about what the student should be able to do on the assessments.

• All Unit Assessments include an answer key with exemplar student responses.

• The is a rubric for exit tickets that indicates, “You mastered the learning objective today; You are almost there; You need more practice and feedback.” 

Program Overview, Guide to Implementing AF Math: Grade 5, Differentiation, Unit-Level Errors, Misconceptions, and Response, “Every unit plan includes an ‘Evaluating and Responding to Student Learning Outcomes’ section after the post-unit assessment. The purpose of this section is to provide teachers with the most common 1-2 errors as observed on the questions related to each standard, the anticipated misconceptions associated with those errors, and a variety of possible responses that could be taken to address those misconceptions as outlined with possible critical thinking, strategic practice problems, or additional resources.” Examples include: 

• Unit 3 Overview, Unit 3 Assessment: Whole Number Multiplication, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 5.OA.2, “Suggested re-teach activities by question group: Q2, 4 – Students will struggle to abstractly imagine scenarios, especially if they are unfamiliar with them. Model using a Think Aloud when creating your expression with the support of students. Record key questions to ask yourself when translating. Ground the thinking in the construction of a model (i.e. bar model) to aid in students visualizing both the scenario and its order. Below are some suggestion questions to guide student thinking: What do each of these values represent? How can we annotate them? Can we model this scenario? What order do these events happen? Can we include that in our model? How can we represent the specified order in the expression? How can we check if our expression matches our scenario/makes sense? Lessons for possible re-teach focus: Lesson 2 – This lesson has aligned practice. Differentiate the packet based on scholar needs, having scholars who need practice of the skill focus on the Bachelor level problems. Scholars who are demonstrating proficiency should focus on the Master level and PhD level problems.”

• Unit 6 Overview, Unit 6 Assessment: Volume, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 5.MD.5, “Suggested re-teach activities by question group: Q6 – 16: Utilizing a guided model to illicit a Criteria for Success (CFS) when finding composite volume. Leverage student voice to guide you through each step, pausing to have students surmise what they are doing using transferable language. Scholars will likely struggle to decompose and accurately label the figure’s dimensions and will be fluent in using the volume formulas. Ask students to demonstrate the various ways in which you can decompose, stamping that they are accurate. Next, label the dimensions of each and prompt for student observations of how the dimensions’ change based on the decomposition. Below is a suggested Criteria for Success: Decompose and label figures. Create table of dimensions for each decomposed figure. Label dimensions & organize in table. Find volume of Figure A. Find volume of Figure B. Find total volume. Answer w/ units. Lessons for possible re-teach focus: Lesson 7 – This lesson focuses on the concept that volume is additive and included targeted practice where students must find volume using decomposition. Strategically focus feedback laps on each part of the CFS, ensuring that all students are successfully implementing CFS 1.”

• Unit 11 Overview, Unit 11 Assessment: Coordinate Geometry, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 5.G.1, “Suggested re-teach activities by question group: Q2 – 4: Students who struggle to accurately label points may benefit from practice using criteria for success. Chart for sophistication as a class identifying the key elements of creating a graph/plotting points. Below is a suggested CFS: Label x and y axis. Label all units/titles. Label scale using same sized intervals. All coordinate pairs are labeled and written in the format (x, y). Lessons for possible re-teach focus: Lessons 1 & 2: These lessons focus on students being able to identify and plot points on a coordinate plane. Students should have multiple at-bats at both with feedback focused on the accuracy of coordinate pairs.”

The materials reviewed for Achievement First Mathematics 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. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems. 

Assessments include opportunities for students to demonstrate the full intent of grade-level standards across the series. Examples include:

• The Unit 4 Assessment contributes to the full intent of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors). Item 5, “Tom’s teacher wrote the following word problem on the board. Elvis wrote 155 songs during his career and shared them on 9 albums without repeating any songs on multiple albums. About how many tracks were on each of his albums? What is a reasonable estimate for the number of tracks on each album? a) 6 b) 12 c) 16 d) 1,600.”

• Unit 6, Lesson 4, Exit Ticket, Problem 2 contributes to the full intent of 5.MD.5 (relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). “The prism to the right measures 18 cubic units. Which of the following statements may correctly describe the prism? Select 2:  a) 3 layers, 2 cubes in each layer. b) 6 layers, 3 cubes in each layer. c) 3 cubes wide, 2 cubes high, 6 cubes long. d) 2 cubes wide, 1 cube high, and 9 cubes long.”

• The Unit 8 Assessment contributes to the full intent of 5.NF.4 (apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). “Anthony has 12 marbles. If $\frac{3}{4}$ of the marbles are clear, how many clear marbles does Anthony have. Draw a model to show your answer. ( 2 points)”

Assessments include opportunities for students to demonstrate the full intent of grade-level practices across the series. Examples include:

• Unit 3 Assessment, Item 10, supports the full development of MP1 as students make sense of a complex problem. “Layla starts her own slime business. She orders 63 boxes of glue and 45 boxes of baking soda and 10 bottles of different colors of dye. Every box of glue has 24 bottles of glue in it. Each bottle contains 16 ounces of glue. Every bottle of dye has 36 ounces of dye in it. How many more ounces of glue does Layla have than ounces of dye?”

• Unit 10 Assessment, Item 8, supports the full development of MP3 as students construct a viable argument and explain their reasoning. “What is the best, or most specific name of the quadrilateral shown below? (3 points) Can you use any other classifications to describe the quadrilateral above? If so, what are they, and how do you know? If not, why not?”

• Unit 5 Assessment, Item 10, supports the full development of MP6 as students attend to precision in place value as they convert units. “Brandon and Kwame both buy sodas at the Nets concession stand during the game. Brandon looks at the label for his soda and sees that his soda contains 1,260 mL. Kwame looks at his bottle and sees that his soda contains 1.4 liters. Kwame insists that he has more soda than Brandon. Is he correct? Explain your thinking on the lines below. ”

The materials reviewed for Achievement First Mathematics 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 as suggestions are outlined within each unit overview. According to the Program Overview, Guide to Implementing AF Math: Grade 5, Differentiation, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning and not feel successful. Therefore, it is critical that strong curricular materials provide support for all student learners, but are created in a way to support students who have diagnosed disabilities. Our program has been designed to align to the elements identified by the Council for Learning Disabilities that should be used in successful curriculum and instruction: Specific and clear teacher models, Examples that are sequenced in level of difficulty, Scaffolding, Consistent Feedback, and Frequent opportunity for cumulative review. Unit Overviews and lesson level materials include guidance around working with students with disabilities, including daily suggested interventions in the Workshop Section of the daily lesson plan.” 

Examples of supports for special populations include: 

• Unit 1 Overview, Numbers in Base Ten- Place Value Understanding, Differentiating for Learning Needs, “Previous Grade Content: Students have worked with place value in Grade 2’s Place Value, where they focus on representing whole numbers in different forms based on their place value. The following lessons may be useful for differentiation of pre-skill content: Grade 2, Unit 6: Lessons 3 – 8. Furthermore, in Grade 3’s Unit 3 Estimation, Addition, Subtraction, and Time, students round numbers to various place values using a number line. Lessons 1 – 3 can be useful for differentiating the pre-skills related to estimation. Grade 4 Unit 7, Decimals, focuses on both numbers in different forms and comparing and ordering decimals. Lessons 1 – 8 can also be useful in differentiating pre- skill content.” Error Analysis, “This unit includes multiple error analysis lessons. The first is Lesson 4, Day 2 which focuses on addressing the error of students counting the base as a power of ten. The next is Lesson 6, Day 2 which addresses students forgetting place holders when writing numbers in standard form from word form. Lesson 8 Day 2 addresses the mistake of forgetting place holders when shifting digits in decimal place values. Lesson 9, Day 2 addresses the error of students comparing the number of digits instead of the value of place values. Lastly, Lesson 10, Day 2 addresses the mistake of students not regrouping to the next highest place value when rounding up a 9. Each of these lessons are an extension of the previous lessons. While these errors are common, it is recommended that teachers look at the ‘Day 1’ Exit Tickets of each to address the needs of each group. Based on data, the teacher may opt to have students who have demonstrated mastery work independently, focusing on practice and feedback, while students who are approaching/below mastery focus on the error analysis and feedback with more teacher support.” Responding to Student Learning Outcomes, “See the Unit Assessment ‘Evaluating and Responding to Student Learning Outcomes’ at the end of the Overview for suggestions on unit-level common errors, misconceptions, and suggestions on how to respond. These can be useful for supporting struggling learners proactively throughout the unit. Identify specific problems to sequence through cumulative review and create groupings of students for small group instruction during that period. Consider changing student seating so that students who struggled with the pre-test are seated next to students who had higher mastery for support throughout the unit or strategically group students who struggle together for teacher support or small group instruction.” Student Grouping Suggestions, “Pre-Test: Use the 5th Grade, Unit 1 Pre-test and Key to identify student strengths and weaknesses when it comes to comparing the value of digits/numbers, writing numbers in different forms, and estimation. Exit Tickets: Closely analyze student mastery of the Exit Tickets for Lessons 2, 3, 4, and 5 as these are critical for understanding why and how we shift digits when multiplying or dividing by powers of 10. Students who have struggled with these should be prioritized for small group instruction and/or more support during independent practice prior to the end of the unit where students will need to extend their understanding of place value and powers of 10 to decimals.”

• Unit 4 Overview, Number and Operations in Base Ten: Whole Number Division, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to gain fluency in dividing 2 by 2 digit and 3 by 2 digit dividends and divisors, respectively. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example, in Lesson 5, to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: PP Bachelor 2 and IP Master 2. These problems would ensure students have had practice with the isolated skill of division in addition to determining whether statements are true or false based on a quotient.”

• Unit 7 Overview, Number and Operations: Division of Fractions and Decimals, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to create various models and expressions to represent fraction division and decimal scenarios. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example, in Lesson 2, to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: PP Bachelor 1 and 2 and Master 1. These problems would ensure students have had practice with using a tape diagram or model as a well as an equation to solve in addition to a word problem application.”

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

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. According to the Program Overview, Guide to Implementing AF Math: Grade 5, Differentiation, Supporting Advanced Students, “Part of supporting all learners is ensuring that advanced students also have opportunities to learn and grow by engaging with the grade level content at higher levels of complexity. A problem-based approach is naturally differentiated as students choose the strategies they use to model and solve the problem. Teachers highlight particular strategies for the class, but they always affirm any strategy that works, regardless of its level of complexity. In a classroom implementing the Achievement First Mathematics program, students are expected to work with a variety of tools and strategies even as they work through the same set of problems; this allows advanced students to engage with the content at higher levels of complexity. Daily lessons resources (DLRs) also provide differentiated problems labeled by difficulty. Teachers should differentiate for student needs by assigning the most challenging problems to advanced students while allowing them to skip some of the simpler ones, so that they can engage with the same number of problems, but at the appropriate difficulty level.” Independent Practice in each lesson provides three levels of rigor in the lesson for student work: Bachelor, Master, and PhD work. Examples include:

• Unit 8, Lesson 18, Independent Practice Bachelor Level, “A container holds 0.7 liters of oil and vinegar. $\frac{3}{4}$ of the mixture is vinegar. How many liters of vinegar are in the container? Express your answer as both a fraction and a decimal.”

• Unit 8, Lesson 18, Independent Practice Master Level, “An artist builds a sculpture out of metal and wood that weighs 14.9 kilograms. $\frac{3}{4}$ of this weight is metal, and the rest is wood. How much does the wood part of the sculpture weigh?”

• Unit 8, Lesson 18, Independent Practice PhD Level, “Jacob buys a project board that is 2.5 feet long and 1.75 feet wide. Before he gets started, he needs to get the length and width down to $\frac{3}{4}$ of its original size. Part A. What will the new dimensions of his project board be, in inches? Show your work. Part B. What is the area of his new project board, in square feet? Show your work.”

The materials reviewed for Achievement First Mathematics 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.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Program Overview, Guide to Implementing AF Math: Grade 5, Differentiation, Supporting Multilingual and English Language Learners, “The Achievement First Mathematics Program appreciates the importance of creating a classroom environment in which Multilingual and English language learners (MLLs/ ELLs) can thrive socially, emotionally, and academically. MLLs/ ELLs have the double-task of learning mathematics while continuing to build their language mastery. Therefore, additional support and thoughtful curriculum is often needed to ensure their mastery and support in learning. Our materials are designed to help teachers recognize and serve the unique educational needs of MLLs/ ELLs while also celebrating the assets they bring to the learning environment, both culturally and linguistically. Our three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson all build on the four design principles for promoting mathematical language use and development in curriculum and instructions outlined by Stanford’s Graduate School of Education, Understanding Language/SCALE.” The series provides the following principles that promote mathematical language use and development: 

• “Design Principle 1: Support sense-making - Daily lesson resources (DLRs) are designed to promote student sense-making with an initial ‘Think About It’ task that engages students with a meaningful task upon which they can build connections. Students have time to read and understand the problem individually and the debrief of these tasks include clear definitions of new terminology and/or key ideas or conjectures…Additionally, teachers are provided with student-friendly vocabulary definitions for all new vocabulary terms in the unit plan that can support MLLs/ELLs further.”

• “Design Principle 2: Optimize output - Lessons are strategically built to focus on student thinking. Students engage in each new task individually, have opportunities to discuss with partners, and then analyze student work samples during a whole class debrief…All students benefit from the focus on the mathematical discourse and revising their own thinking, but this is especially true of MLLs/ELLs who will benefit from hearing other students thinking and reasoning on the concepts and/or different methods of solving.”

• “Design Principle 3: Cultivate conversation - A key element of all lesson types is student discussion. Daily lesson resources (DLRs) rely heavily on the use of individual think/write time, turn-and-talks with partners, and whole class discussion to answer key questions throughout the lesson script. The rationale for this is that all learners, but especially MLLs/ELLs benefit from multiple opportunities to engage with the content. Students that are building their mastery of the language may struggle more with following a whole-class discussion; however, having an opportunity to ask questions and discuss with a strategic partner beforehand can help deepen their understanding and empower them to engage further in the class discussion….”

• “Design Principle 4: Maximize linguistic and cognitive meta-awareness - The curriculum is strategically designed to build on previous lesson mastery. Students are given opportunities to discuss different methods to solve similar problems and/or how these concepts build on each other. The focus of the ‘Think About It’ portion of the Exercise-Based lesson is to help students build on their current understanding of mathematics in order to make a new key point for the day’s lesson. The entire focus of the Test the Conjecture lesson is for students to create their own conjecture about the new learning and then to test this by applying it to an additional problem(s). Students focus on building their own mathematical claims and conjectures and see mathematics as a subject that involves active participation of all learners. By ending each lesson type with this meta-awareness, all learners, but especially MLLs/ELLs benefit by building deeper connections.”

The series also provides Mathematical Language Routines in each unit. According to the Program Overview, Guide to Implementing AF Math: Grade 5, Differentiation, Supporting Multilingual and English Language Learners, “Beyond these design principles, our program outlines for teachers in every unit plan the most appropriate mathematical language routines (MLRs) to support language and content development of MLLs/ELLs with their learning within the specific unit.” Examples include:

• Unit 6 Overview, Measurement and Data – Volume, Differentiating for Learning Needs, Supporting MLLs/ELLs: 

  • Vocabulary: “MLLs/ELLs should be provided with a student-friendly vocabulary handout throughout the unit that is either completed for them and/or that they add to each day. All terms included in the ‘Vocabulary’ section below should be included. This scaffold can be incredibly helpful for other learners to help them see a verbal and visual definition for each term. Each of the terms, definitions, and examples should be translated into the students preferred language using Google Translate or a translator (Spanish in the example provided).”

  • Sentence Frames: “MLLs/ELLs and all students can greatly benefit from specific guidance around sentence frames for standard justifications or explanation within the unit. For this unit, Lessons 2 – 7 include practice problems which ask students to explain their thinking or justify their reasoning. Teachers can provide students with the following sentence frames to use throughout these problems: ‘The units we should use is _____ because _____.’ ‘“I agree/disagree with this claim because _____.’ ‘_____ is correct/incorrect because _____.’”

  • Language Development Routines: “Throughout the unit, teachers should focus on student discussion and use of critical thinking when analyzing student work samples. See the ‘Implementing Language Routines’ of the Implementation Guide for the course for further detail of how these routines live within all lessons. Within this unit, students should specifically focus on the following Mathematical Routines.

    • MLR1: Stronger and Clearer Each Time - Students will focus in ALL lessons on analyzing student work and revising their thinking either during the Think About It, Interaction with New Material, or Test the Conjecture portion of each lesson.  

    • MLR 2: Collect and Display: For lessons 1 and 2 students will focus on learning, capturing, and applying new vocabulary terms for volume. Students will be exposed to key vocabulary of the unit and will need to be able to flexibly use their understanding to apply various volume concepts. 

    • MLR 6: Three Reads – In Lessons 10, students will be using volume in the context of real word problems. The 3 reads strategy will be helpful in unpacking the problem. The first read, students read with the goal of comprehending the general context. The second read, students are analyzing the language that is used and annotating with mathematical vocabulary. The third read, students are brainstorming strategies to approach the problem.

    • MLR7: Compare and Contrast – During independent and partner practice sections of ALL lessons, collect student exemplars and most common errors to perform an error analysis of the most common misconception. Allow students to critique the exemplar and non-exemplar, stamp a take away, and implement it in their own work if appropriate.”

The materials reviewed for Achievement First Mathematics 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.

Manipulatives are described as accurate representations of mathematical objects in the narrative of the Unit Overviews, and although there is little guidance for teachers or students about the use of manipulatives in the lessons, the use of manipulatives can be connected to written methods. Examples include:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             

For example, in Unit 6 Overview, “In lesson 2, students use cubic units (unit cube manipulatives) to build figures to a specified volume. Through this, they see that the units may be rearranged to create a different looking figure which has the same volume. More importantly, in this lesson students get an introduction to the process of finding volume as they count the number of cubic units making up a solid figure to determine its volume.”