The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

• Unit 1, Lesson 13, students develop conceptual understanding of 3.OA.2, interpret whole- number quotients of whole numbers. During the Workshop, students are provided with a variety of sharing situations and representations. In the Workshop, Problem 3, “Mr. Ziegler bought a pack of 18 markers. He wants to split them equally between himself and his niece, Sarah. How many markers will each person get?” Students are shown two picture representations, one showing two groups with nine items each and the other showing nine groups with two items each and asked, “Which drawing represents Mr. Ziegler’s problem? Why?” 

• Unit 5, Lesson 10, students develop conceptual understanding of 3.NF.3, explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. During Pose the Problem students compare fractions using models to support their answer, “Lily and Jasmine each bake a chocolate cake. Lily puts $\frac{3}{8}$ of a cup of sugar in her cake. Jasmine puts $\frac{5}{8}$ of a cup of sugar into her cake. Who uses less sugar? Draw a model to support your answer.” 

• Unit 7, Lesson 6, students develop conceptual understanding of 3.OA.B, as they draw arrays and write equations to model the distributive property of multiplication. In the Independent Practice, Problem 2, “Draw an array to match the equation 6 x 9 then use the distributive property to break apart the array and solve it. Array: Equation: $( )+( )$” 

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

• Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding of 3.MD.C, as they determine the area of shapes and solve problems based on provided grids with unit squares. Problem 6, “Anna’s garden is 7 feet long and 7 feet wide. Noah’s garden is 8 feet long and 6 feet wide. Which garden has a smaller area? ______’s garden is smaller.” 

• Unit 5, Lesson 2, Independent Practice, students demonstrate conceptual understanding of 3.NF.1, as they build a model of a unit fraction. Problem 3, “Build a model of the unit fraction below with your fraction strips. Then, record the shape you made on the rectangle and label one unit fraction $\frac{1}{8}$.”

• Unit 7, Lesson 12, Problem of the Day, Let’s Try One More, students demonstrate conceptual understanding of 3.OA.5, as they create equations based on their knowledge of the distributive model. “Write three different equations that we could use to find the area of the following rectangle. Then, find the area of the rectangle.”

The materials for Achievement First Mathematics Grade 3 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 Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

Math Practice is intended to “build procedural skill and fluency” and occurs four days a week for 10 minutes. There are six Practice Workbooks in Achievement First Mathematics, Grade 3. Two workbooks, B and F, contain resources to support the procedural skill and fluency standards 3.NBT.2: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction; and 3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. In the Guide To Implementing Achievement First Mathematics Grade 3, teachers are provided with guidance for which workbook to use based on the unit of instruction. Examples include:

• Practice Workbook B, Problem 11, “Calculate. $605 - 327 =$ ; $708 - 439 =$ ; $875 - 218 =$ ; $575 + 219 =$ ; $238 + 573 =$ ; $117 + 582 =$ .” (3.NBT.2) 

• Practice Workbook B, Problem 2, “.” (3.NBT.2) 

• Practice Workbook F contains 25 independent practice problems that allow students to build procedural skill and fluency with multiplication and division within 100. Problem 6, “.” (3.OA.7)

Cumulative Reviews are intended to “facilitate the making of connections and build fluency or solidify understandings of the skills and concepts students have acquired throughout the week to strategically revisit concepts, mostly focused on major work of the grade.” Cumulative Reviews occur every Friday for 20 minutes. Examples include:

• Unit 5, Cumulative Review 5.2, Problem 2, students practice adding and subtracting within 1000. “Solve. $909 - 690$.” (written vertically). (3.NBT.2)

• Unit 5, Cumulative Review 5.4, Problem 4, students practice solving division problems within 100. “Solve. $40 ÷ 8 =$; $16 ÷ 8 =$; $7 ÷ 7 =$; $72 ÷ 8 =$; $27 ÷ 3 =$.” (3.OA.7) 

• Unit 7, Cumulative Review 7.2, Problem 2, students practice solving multiplication problems. “Solve. $6 × 7$; $8 × 4=$ ; $9 × 3=$ ; $3 × 6=$ ; $8 × 3 =$ ; $5 × 5 =$ .” (3.OA.7)

The materials reviewed for Achievement First Mathematics Grade 3 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 during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding.

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

• Unit 3, Lesson 11, Problem of the Day, students engage with 3.OA.8 as they solve a two-step word problem in a non-routine format. "Monday morning, Ashley starts a wood pile by stacking 215 pieces of wood. Monday afternoon, Dad takes 17 pieces of wood from the pile to burn in the fireplace. Tuesday morning, Ashley stacks 118 pieces of wood on the pile. Tuesday afternoon, Dad takes 26 pieces of wood to burn in the fireplace. Ashley wants to have exactly 350 pieces of wood on the pile on Wednesday. Does Ashley have to stack more wood on the pile or does Dad have to burn more wood in the fireplace? Show all your mathematical thinking.”

• Unit 4, Guide to Implementing AF, Math Stories, students engage with 3.OA.3 as they use multiplication and division within 100 to solve routine word problems in situations involving equal groups, arrays, and measurement quantities. Sample Problem 8, “Carla has 12 stuffed bears, 18 stuffed rabbits, and 6 stuffed elephants to donate to the local charity shop. The shop wants to arrange them into equal groups. What are two different ways the shop could arrange Carla’s donated stuffed animals?”

• Unit 8, Lesson 13, Math Stories, Perimeter Robot Project, students engage with 3.MD.7 as they apply their knowledge of solving problems involving area in a non-routine format. "You have worked on so many different kinds of word problems in this unit, and the last few days we have been focusing on area and perimeter. Today we will use what we know about the perimeter formula and our addition patterns from our addition table to help us create our own Perimeter Robot! We are going to use this table to help us brainstorm dimensions for the different body parts for our robot. Turn and talk with your partner, what do you notice about the table? We have to find the length and width for each of the perimeters, we have to make dimensions for all the different body parts.”

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

• Unit 2, Lesson 7, Independent Practice, students engage with 3.MD.3 as they draw a scaled picture graph and a scaled bar graph to represent a data set with several categories in a non-routine problem. Question 5, “Mr. Park spent 38 minutes grading exit tickets on Friday night. Ms. Duke spent 44 minutes. Ms. Ervey spent 26 minutes grading exit tickets. Ms. Negron spent 30 minutes. Ms. Max-McCarthy spent the most time grading exit tickets with 46 minutes. Use this data to create a pictograph and bar graph. Write 3 questions for your partner to solve.”

• Unit 7, Lesson 1, Exit Ticket, students demonstrate application of 3.OA.3 as they use multiplication to solve routine word problems. Question 3, “Gretta says there would be 17 hands on 9 people. Use what you know about the patterns for multiples of 2 to explain why you agree or disagree with Gretta.”

• Unit 8, Lesson 8, Independent Practice, students demonstrate application of 3.OA.8 as they solve a routine two step word problem involving the four operations. Question 2, “Marlon buys 9 packs of hot dogs. There are 6 hot dogs in each pack. After the barbeque, 35 hot dogs are left over. How many hot dogs were eaten?”

The materials reviewed for Achievement First Mathematics Grade 3 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 2, Cumulative Review 2.2, students demonstrate conceptual understanding as they interpret products of whole numbers as the total number of objects in groups by comparing two grouping strategies used to evaluate the same expression. Problem 7, “Finn and Sadie are both solving the problems 4 x 5 x 2. Their teacher said they are both correct. Their work is below. Why are they both correct?” (3.OA.1)

• Practice Workbook B, Problem 29, students demonstrate procedural skill and fluency related to addition and subtraction as they solve problems. It, “Solve to find the missing numbers. $142 + \_ = 225$, $506 - \_ = 329$,  $\_ + 344 = 764$.” (3.NBT.2) 

• Unit 8, Lesson 10, Independent Practice, students apply their understanding of the four operations as they solve a word problem. Problem 7, “Tajah washes 4 loads of laundry each week. Each load requires 2 ounces of washing powder. If she washes laundry for 50 weeks this year, how many ounces of washing powder will she use?” (3.OA.8)

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 7, Independent Practice, students demonstrate conceptual understanding and procedural skills as they solve problems using strategies based on place value with multiple addends. Problem 3, “Ryan, Dominic, and Brittney were collecting acorns. Ryan gathered 109 in his bag. Dominic collected 87 in his bag. Brittney picked up 132 acorns. At the end of the day, they put all the acorns into a cardboard box. How many acorns were in the box?” (3.NBT.2)

• Unit 4, Cumulative Review 4.1, students demonstrate conceptual understanding and application of multiplication within 100 as they write an equation and solve problems with an array. “Write a multiplication equation to match the picture below. Use p to represent the unknown number. How many paint cans are there?” (3.OA.7)

• Unit 7, Lesson 6, Independent Practice, students demonstrate conceptual understanding and application of multiplication as they create arrays and apply their knowledge of multiplication to solve problems. Problem 3, “Franklin collects stickers. He organizes his stickers in 5 rows of four. a. Draw an array to represent Franklin’s stickers. Use an x to show each sticker. b. Solve the equation to find Franklin’s total number of stickers. $5 × 4 = \_$.” (3.OA.3)

The materials reviewed for Achievement First Mathematics Grade 3 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 is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 2 Overview outlines the intentional development of MP1. “Students look for entry points and strategies to solve story problems. Students annotate the problem and its graph to identify the information given and the information needed to solve the problem. Students assess the reasonableness of their solutions against their representations and strategies.”

• Unit 3, Lesson 10, How will embedded MPs support and deepen the learning? describes the intentional development of the MP within the lesson. “Students engage with MP 1 as students solve story problems and make sense of what the problem is asking. They preserve when solving problems by annotating, representing the ‘big questions’ and ‘little questions,’ and using calculation strategies to help solve and make sense of the problem. As students solve, they must assess the reasonableness of their solutions against their representations and strategies.”

• Unit 6, Lesson 12, Independent Practice, students access relevant knowledge and work through a task with multiple entry points. “Directions: The zookeepers are designing a habitat for their newest animal, the pandas! They know they need a pen with an area of 24 square meters, but they want to know all of the possible options. Find the four different pens they could make with an area of 24 square meters. Record the details about each shape in the chart below!”

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

• The Unit 4 Overview outlines the intentional development of MP2. “Students decontextualize metric measurements as they solve problems involving addition, subtraction, multiplication, and division. They round to estimate and then precisely solve, evaluating solutions with reference to units and with respect to real world contexts. Students use real-world benchmarks of metric units to estimate measures for weight, determining whether an object is greater than or less than the benchmarks. Students use intervals on scales to determine the most precise measurement. Students represent their final answers using appropriate units from the problem context (e.g., dollars and coins or g versus kg).”

• The Unit 7 Overview describes development of MP2. “Students make sense of quantities and their relationships as they explore the properties of multiplication and division and the relationship between them. Students decontextualize when representing equal group situations as multiplication, and when they represent division as partitioning objects into equal shares or as unknown factor problems. Students contextualize when they consider the value of units and understand the meaning of quantities as they compute. Students will build towards abstraction by composing and decomposing tiled shapes first and then working with just given dimensions or the area of a shape.”

• Unit 8 Lesson 4, Pose the Problem, students engage with MP2 as they predict patterns between factors and products, and explore the strategy of predicting whether a product is odd or even. “Draco says that 4 times a number will always give you an even product. His partner Harry isn’t quite sure. Is he correct? Discuss with your partner, you may use the multiplication table to prove your thinking.”

The materials reviewed for Achievement First Mathematics Grade 3 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:

• Unit 3, Lesson 2, Exit Ticket, students critique the reasoning of others based on their knowledge of estimation and rounding. Problem 3, “There are 525 pages in a book. Julia and Kim round the number of pages to the nearest hundred. Julia says it is 600. Kim says it is 500. Who is correct? Explain your thinking.” 

• Unit 3, Lesson 2, Try One More, students are asked to determine if an answer is reasonable. “Samantha solved this problem: $472 + 371$ She got an answer of 743. Round to the nearest hundred and then solve to determine whether or not her answer is reasonable.” The teacher’s guidance includes, “How did you round 472? How did you round 371? Is Samantha’s answer reasonable? Why or why not?”

• Unit 4, Lesson 4, Workshop, Problem 7, “The capacity of a pitcher is 3 liters. What is the capacity of 9 pitchers? John says that $9 ÷ 3 = P$ represents this story. Do you agree or disagree? Explain.” 

• Unit 5, Lesson 24, Problem of the Day, students analyze the reasoning of others and use models to explain their reasoning. “Treasure and Shianne are having an argument. Shianne thinks that $\frac{3}{3}$ is larger than $\frac{3}{1}$. Treasure disagrees. She thinks that $\frac{3}{3}$ is smaller than $\frac{3}{1}$. Use models to show both numbers and explain which is larger.” 

• Unit 8, Lesson 2, Try One More, students critique the reasoning of others and construct a viable argument based on their knowledge of the properties of addition. “Khallel also says that when you add an even number and an odd number you will get an odd sum. Is Khaleel correct? Work with your partner to explain why; prove your answer using a visual model or the addition table.”

• Unit 9, Lesson 4, Exit Ticket, students construct a viable argument based on their understanding of shapes. Problem 3, “In the grid below, draw a rhombus that is also a rectangle. Explain how your shape fits in both categories.”

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Math Stories Guide, Promoting Reasoning through the Standards for Mathematical Practice, MP4, “Math Stories help elementary students develop the tools that will be essential to modeling with mathematics. In early elementary, students become familiar with how representations like equations, manipulatives, and drawings can represent real-life situations.” Within the K-4 Math Stories Representations and Solutions Agenda, students are given time to represent, retell, and solve the problem on their own.

• Unit 2 Lesson 2, students engage with MP4 as they create and use tape diagrams to collect and organize data. Pose the Problem, “Reisha plays in three basketball games. She scores 12 points in Game 1, 8 points in Game 2, and 16 points in Game 3. Each basket that she made was worth 2 points. Represent each game with a tape diagram.”

• Unit 3, Lesson 10, Narrative assists the teacher with intentional development of MP4 within the lesson. “Students also engage in MP 4 as they solve story problems by using mathematical models and connecting it back to the story problem. Students model the story problem with an appropriate representation (tape diagrams or equations) and use an appropriate strategy (expanded notations, add by place, or number lines) to solve. Students also engage with MP4 as they determine if their answer makes sense and connect their answer back to the story problem by finishing the story.”

• Unit 7, Lesson 11, Exit Ticket Question 1, “students engage with MP 4 as students compose smaller areas of rectangles to determine the area of a larger rectangle, ‘Heather has two rugs. One rug is 5 feet by 6 feet. The other rug is 6 feet by 3 feet. She puts the two rugs next to each other on her floor. a) Draw the rugs in the grid. Then write a number sentence to find the area covered by both rugs on the floor.’”

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

• Unit 2 Overview, “Students use cubes or other items to concretely illustrate the values represented in graphs. Students use the problem context to define how the representation matches the problem situation in creating a key and scale for the representation. Students define each value of a pictograph using multiplication facts.”

• Unit 5 Overview, “Students use pattern blocks, fraction tiles, and circles to illustrate unit fractions and non-unit fractions as copies of unit fractions as the basis for reasoning about comparing fractions with the same denominators or numerators as well as using the foundation of non-unit fractions as a basis for creating equivalent fractions by partitioning part size in models or intervals using number lines.”

• Unit 6, Lesson 4, Narrative assists the teacher with intentional development of MP5 within the lesson. “Students practice SMP 5 as they consider how the title, range, and number of X indicate the unit of measurement, range of the measurements represented, and the number of times each measurement occurred.”

At times, the materials are inconsistent. The Unit and Lesson Overview narratives describe explicit connections between the MPs and content, but lessons do not always align to the stated purpose. 

The materials do not provide students with opportunities or guidance to identify and use relevant external mathematical tools and resources, such as digital content located on a website.

The materials reviewed for Achievement First Mathematics Grade 3 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: 

• Unit 3, Lesson 5, Exit Ticket, Question 3, students accurately calculate and explain how to use expanded notation to find a sum. “Your friend at another school wants to combine 684 and 134. Show and explain how they could add with expanded notation.”

• Unit 7, Lesson 6, Exit Ticket Question 1, precision is used as students are expected to draw an array and write an expression. “Mrs. Stern roasts cloves of garlic. She places 9 rows of 6 cloves on a baking sheet. Part A: Draw an array to show the total number of cloves. Write an expression to describe the number of cloves Mrs. Stern bakes. Part B: Use the distributive property to solve this problem. Draw a model and write an equation to show your thinking.”

• The Unit 9 Overview, “Students use geoboards to model and name polygons (specifically quadrilaterals). Students use the corner of a piece of paper or square tile to compare the sizes of the angles on the page to verify which are and are not square angles; Students measure lengths of sides with rulers; Students extend sides with rulers to check to see if sides are parallel or not; Use geoboards to model and name polygons (specifically quadrilaterals); Students attend to precision when drawing polygons and quadrilaterals (ensuring straight lines, right angles, parallel lines, equal length sides).”

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

• Unit 1, Lesson 6, State the Aim provides guidance for teachers in introducing the identified vocabulary word area. “For the last few days we have been studying multiplication using equal group pictures and arrays. Today we are going to use some of that knowledge to study a new topic called area...How many triangles did it take to fill shape A? What about shape B? Great work! We just found the area of these shapes. Area is the amount of flat space that an object takes up.”

• Unit 4, Lesson 1, State the Aim provides guidance for teachers in introducing mathematical terminology related to measurement. “You’ve worked on measuring in 2nd grade and this year we will continue our growth with measurement. By the end of today you will be able to estimate the weight of objects in grams, kilograms, and measure using scales. This is another day that units will be super important! Today we will be talking about two units of weight. The first is called a gram. A gram is about the weight of 1 paper clip. The second is called a kilogram and weighs about as much as a textbook.”

• Unit 9, Lesson 1, State the Aim identifies new vocabulary used in the lesson and provides specific guidance for teachers in introducing the terminology, including counterexamples. “Today we are starting a new unit on geometry. Geometry is the study of shapes and their attributes, or the way we describe them. We will start our units describing polygons. Polygons are closed shapes made up of line segments or straight lines. Take a look at the top of your page. The first shape is not a polygon because it has an open-space and is not closed, the second shape is not a polygon because the top is curved so it is not made up of line segments or straight lines. The last shape is a polygon because it is a closed shape made up of straight lines. Let’s get started on our study of polygons and how we can talk about them.”

The materials reviewed for Achievement First Mathematics Grade 3 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 3 Lesson 17, Independent Practice, Question 6, students make sense of structure as they solve problems to find start time by counting back on a number line in hour and minute intervals. Problem 6, “Mr. Wellborn arrives at work at 8:30. He leaves for work 50 minutes before. What times does Mr. Wellborn leave for work?”

• Unit 5, Lesson 1, Independent Practice, Questions 1-4, students engage with MP7 as they use unit fractions as the basic building blocks of all fractions on the number line. “DIRECTIONS: Build the unit fraction below with your pattern blocks. Then, record the shape you made on the dot paper space and label one unit fraction. $\frac{1}{2}, \frac{1}{6}, \frac{1}{9}, \frac{1}{5}$.”

• Unit 8, Lesson 5, Independent Practice, students extend a pattern based on a provided pattern. Problem 2, “Marc-Anthony wrote the number pattern below. 15, 19, 23, ____, 31. Part A: What is the missing number in Marc-Anthony’s patterns? e) 24; f) 29; g) 27; h) 30. Part B: What is the rule for this pattern? Part C: What would the next three numbers in his pattern be? __,___,___.”

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

• Unit 1, Lesson 20, Independent Practice, Question 2c, students engage with MP8 as they use the commutative and associative properties of multiplication to solve problems. “Find the products for each. First solve the part in parenthesis and write a new multiplication fact on the first line. Then write the product on the bottom line. $(4\times5)\times2=4\times(5\times2)$.”

• Unit 3, Lesson 5, Independent Practice students add two and three digit numbers using expanded notation. Problem 3, “Juan added 375 and 128 and got 503. Do you agree with his answer? Why or why not? Explain your thinking on the lines below.”

• Unit 5, Lesson 9, Pose the Problem, students reason about the size of fractions and look for strategies they can use when comparing fractions with the same numerator or same denominator. “Genesis cut his hot dog into thirds and ate $\frac{1}{3}$. Ameera cut her hot dog into fourths and ate $\frac{1}{4}$. Who ate more? Use >, < or = to write a comparison statement. Use your fraction strips or a picture to show and explain your thinking.”

The materials reviewed for Achievement First Mathematics Grade 3 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, Math Stories, Math Practice, and Cumulative Review.

• The Math Stories Guide (K-4) provides a framework for problem solving.

• Each Unit Overview includes a section called “Key Strategies” that describes strategies that will be utilized during the unit.

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

• In the narrative information for each lesson, there is information such as “What do students have to get better at today? Where will time be focused/funneled?”

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 5, Fractions, Lesson 25 include: 

• “What do students have to get better at today? Students need to combine all of the skills they have been building over the unit to solve problems today. Scholars will work to place fractions on number lines, as well as compare fractions and find equivalent fractions using number lines. What is new and/or hard about that? This is challenging because scholars may still be struggling to accurately place fractions on number lines, therefore finding equivalent fractions and comparing fractions may be difficult. In addition, number lines require precision, particularly for comparing and finding equivalent fractions. So where will time be focused/funneled? The Intro and Interruption will focus on creating accurate number lines and reiterating the key points of placing, comparing, and finding equivalencies.”

• “Exemplar Student Response: “I can use a number line to solve many different types of fraction problems. To use a number line correctly I need to remember to partition the number line first into the wholes, then partition each whole into the parts shown in the denominator of my fraction. Then I can count by my unit fractions to place my fractions accurately. Once I have done this, I can either place a second fraction to compare by thinking about which fraction is further away from zero and therefore larger, or I can create more smaller partitions or fewer, larger partitions to find an equivalent fraction for the one I have placed.”

• “Mid-Workshop Interruption What is the next level for the skill in the objective? What do you want most of your students to start doing?”

• “Discussion: Discuss relevant student work samples to push thinking to the next level. What questions will you ask to get students to articulate the big ideas before moving on?”

• “Closure: How can we get kids to summarize what was learned today and connect back to the aim? Exit ticket.”

Each lesson includes both “What” and “How” Key Point sections that describe what students should know and be able to do and how they will do it. Examples from Unit 5, Fractions, Lesson 25 include:

• “What Key Points What should students know and be able to do? I can place fractions on a number line. I can compare fractions using a number line. I can compare fractions using reasoning. I can find equivalent fractions using a number line.”

• “How Key Points How will they do it? We can place fractions accurately on a number line by drawing a number line and partitioning it into the parts shown by the denominator, then counting by the fraction until we reach the fraction shown by the numerator and marking that spot. We can compare fractions with the same denominator using a number line by drawing a number line and partitioning it into the number of parts shown by the denominator, then marking both fractions on the line. The fraction closer to 0 is the lesser fraction. We can compare fractions with the same numerator using a number line by drawing two number lines and partitioning them into the parts shown by the two different denominators, then counting tick marks until we get to the amount shown in the numerator. The fraction closest to 0 is the lesser fraction. We can compare fractions by reasoning about their size. When fractions have the same denominator we compare their numerators, the larger numerator is the larger fraction because that means it has more parts or more of the whole would be shaded. When fractions have the same numerator, we compare their denominators. The fraction with the smaller denominator is the greater fraction because that means the whole has been broken up into fewer pieces, and therefore the pieces are larger. We can find equivalent fractions by drawing a number line and then partitioning the intervals further in order to create more smaller parts, or grouping partitions together to create less, larger parts.”

The materials reviewed for Achievement First Mathematics Grade 3 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.

• Guide to Implementing AF Grade 3, 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 5 Overview, “Students begin their introduction to fractions (at least fractional language) in first grade when students partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Students are also primed by being taught to recognize that equal shares of identical wholes need not have the same shape. In fourth grade, scholars extend their understanding of fractions by learning how to explain a/b as being equal to (n x a)/ (n x b). This allows scholars to generate equivalent fractions. Scholars also extend their ability to compare fractions with different numerators and denominators. Scholars also learn to use the identity property to make either the numerators or denominators the same for the purpose of comparing. Scholars also learn how to add and subtract, and multiply fractions by thinking about joining and separating parts of the same whole. Furthermore, scholars applying these concepts for the first time in the form of story problems. Lastly, fourth grade introduces decimal notation for fractions with denominators 10 or 100. Scholars then compare these decimals to hundredths.”

• Unit 9 Overview, “In second grade, scholars learn to recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Specifically, they are taught to identify triangles, quadrilaterals, pentagons, hexagons, and cubes, many of which they have been able to recognize for years. In this unit, scholars use their same understanding of shape characteristics to classify different kinds of quadrilaterals. By the end of elementary school, scholars continue their work with classifying shapes based on their attributes. Specifically, scholars learn to classify shapes based on the presence or absence of parallel or perpendicular lines. They also learn to identify types of right triangles.”

The materials reviewed for Achievement First Mathematics Grade 3 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 at equal intensities weekly with all 3 tenets, we structured our program into three main daily components Monday-Thursday: Math Lesson, Math Stories and Math Practice. Additionally, students engage in Math Cumulative Review each Friday in order for scholars to achieve the fluencies and procedural skills required."

• The Implementation Guide includes descriptions of “Math Lesson Types.” Descriptions are included for Game Introduction Lesson, Task Based Lesson, Math Stories, and Math Practice. Each description includes a purpose and a table that includes the lesson components, purpose, and timing. 

Research-based strategies are cited and described within the Program Overview, Guide to Implementing AF Math: Grade K-4, Instructional Approach and Research Background. 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.”

• Principles to Actions: Ensuring Mathematical Success for All, 2014, “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.”

• Problem-solving as a basis for reform in curriculum and instruction: the case of mathematics by Heibert et. al., “Students learn mathematics as a result of solving problems,” and that “mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving.”

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Each lesson includes a list of materials specific to the lesson. Examples include:

• Unit 3, Lesson 12, Lesson Overview: “Materials: student work packets, stopwatches for partners, VA.”

Unit 4, Lesson 3, Lesson Overview: “Materials: Student work packet, VA (stands for visual aid), Objects to model with: eye dropper, 1 liter water bottle, 500 mL water bottle, 1 liter beaker with milliliter intervals, Containers as models for Workshop, such as spoon, soda can, milk, or other containers to estimate and measure capacity.”

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for having assessment information included in the materials to indicate which standards are assessed. There are connections identified for standards, but not the mathematical practices. 

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: Multiplication, Division, & Area Pt. 1, denotes the aligned grade-level standards and mathematical practices. Question 2, “Nia has 45 pieces of chocolate. She split up the pieces equally into bags. Each bag had 5 pieces of chocolate. How many bags of chocolate did Nia have? Show your work.” (3.OA.2, MP1, MP2, MP4, MP6)

• Unit 4 Overview, Unit 4 Assessment: Measurement, denotes the aligned grade-level standards and mathematical practices. Question 6, “Ophelia had 64 ounces of milk. She wants to pour an equal amount of milk into 8 glasses for her children. How many ounces will Ophelia pour into each glass?” (3.MD.2, MP5, MP7, MP8)

• Unit 8 Overview, Unit 8 Assessment: Story Problems, denotes the aligned grade-level standards and mathematical practices. Question 10, “Cary and Nick wanted to go for a run around the park. The park has a perimeter of 36 miles, and is in the shape of a square. Nick and Cary got tired and decided to run only two sides of the park. How far did they run? Draw a picture to explain your answer.” (3.OA.8, MP1, MP2, MP4, MP6)

The materials reviewed for Achievement First Mathematics Grade 3 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 3, 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 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: Estimation, Addition, Subtraction, & Time, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 3.NBT.2, “If students subtract smaller digits from larger digits no matter their position: Refer to lessons 5 - 7. Using the problems from the assessment, have students represent each number using base ten blocks or stick model with the minuend above and the subtrahend below. Then prompt students to identify why they can't ‘take’ the smaller number from the larger number just because it is larger. Prompt students to think about how they can regroup from the left in order to have enough above to take away the count from the bottom.”

• Unit 5 Overview, Unit 5 Assessment: Fractions, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 3.NF.2, “If students don't recognize that a fraction with the same numerator and denominator represents one whole or don't recognize the part-to-whole relationship of numerators and denominators: Refer to lessons 8 & 15. Using the assessment item, ask students to draw the representation and label with information from the problem. Prompt students to identify and label which parts of their representation based on the problem information, writing a fraction for each. Then prompt students to identify the question the problem is asking and have them consider which one of their written fractions answers that question.”

• Unit 8 Overview, Unit 8 Assessment: Story Problems, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 3.OA.8, “If students attempt to visualize but is unable to make sense of the story (often due to language or vocabulary gaps), visualize accurately but do not understand what actions/ situations pair with what operations, or expect the problem to have only a single operation that needs to occur multiple times to solve a two-step problem: Refer to lessons 7-12. Provide students with a blank copy of the assessment question and prompt them to create a representation for all of the parts of the problem. Prompt students to consider what part(s) of the problem ask them to Add To/Take From, Put Together/Take Apart, or Compare. Prompt students to relate an operation or action to each of those kinds of problems and write an expression or equation that reflects that operation or action, ultimately calculating in order to be able to answer the big question of the problem.”

The materials reviewed for Achievement First Mathematics Grade 3 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 3 Assessment contributes to the full intent of 3.NBT.1(use place value understanding to round whole numbers to the nearest 10 or 100) and 3.NBT.2 (fluently add and subtract within 1000). Items 1-6, “1) Round 452 to the nearest ten. 2) Round 452 to the nearest hundred. 3) When rounding to the nearest ten, what is the least whole number that will round to 50? Explain how you know. 4) Solve 205 - 186 = . 5) Compute 98 + 427 = . 6) Part A. First grade eats 242 bananas and second grade eats 183 bananas. Find the sum of the bananas they eat. Show your strategy to solve and explain your thinking. Part B. Mr. Teague estimates the sum of the bananas by rounding each amount to the nearest ten before adding. He decides to order the estimated number of bananas for tomorrow’s lunch. Will Mr. Teague have ordered enough bananas? Why or why not?”

• Unit 5, Lesson 25, Exit Ticket, Problem 1 contributes to the full intent of 3.NF.3 (Explain equivalence of fractions in special cases), “Martin says that if a fraction has an even denominator, its equivalent fractions will also always have an even denominator. Is Martin correct? Explain why or why not on the lines below.”

• The Unit 9 Assessment contributes to the full intent of 3.G.1 (recognize rhombuses, rectangles, and squares as examples of quadrilaterals). Item 4, “Frederick draws a quadrilateral with two sets of parallel sides. Draw what his shape could be on the grid below. Then, name the shape you drew.”

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 look for entry points and information needed to solve a word problem. “Baseball practice starts at 6:30 in the evening. Nelson has a list of activities to do before going to baseball practice: 15 minutes to feed and walk the dog; half an hour to eat dinner; twenty-five minutes to do homework. It takes fifteen minutes to get to baseball practice. What time does Nelson need to start his activities in order to get to baseball practice on time? Show all your mathematical thinking.” 

• Unit 7 Assessment, Item 9, supports the full development of MP3 as students justify their answer. “Mr. Yearwood is arranging desks in rows with a front section and a back section. The array below represents his desks. Fill in the blanks below with the numbers that make the equation true: (__ x __) + (__ x __) = 6 x 6.  Explain why his number sentence will work to find the total.”

• Unit 8 Assessment, Item 1, supports the full development of MP8 as students look for and explain patterns of odd and even factors. “The first row in a pattern of tiles had 5 tiles. Each row after the first had 2 more tiles than the row before it, as shown below. Which statement is true about the number of tiles in any row?  a. It is divisible by 10; b. It is an even number; c. It is a multiple of 3; d. It is an odd number.”

The materials reviewed for Achievement First Mathematics Grade 3 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 3, Differentiation and Working with Special Populations, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning mathematics and feel unsuccessful. Therefore, it is critical that strong curricular materials are designed to provide support for all student learners, especially those with diagnosed disabilities (Hott et al., 2014). The Achievement First Mathematics Program was designed with this in mind and is based on several bodies of research about best practices for the instruction of students with math disabilities, including the work of the What Works Clearinghouse (an investment of the Institute of Education Sciences within the U.S. Department of Education) and the Council for Learning Disabilities (an international organization composed of professionals who represent diverse disciplines). 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. Teachers should reference these materials in conjunction with the information that follows in this Implementation Guide when planning instruction in order to best support all students.” Within Daily Lesson Plans there are two versions of Independent Practice Problem Sets, “one set is more scaffolded and can be used for all students and in combination with intervention as needed; the other is less scaffolded.”

Examples of supports for special populations include: 

• Unit 1, Lesson 6, Workshop, Suggested intervention(s), “Have paper square units so students can glue them on.”

• Unit 4 Overview, Measurement, Differentiating for Learning Needs, “In this unit, students are learning to make sense of measuring in standard units, building on the skills of measurement from 1st and 2nd grade. This includes extending their understanding of measuring and estimating measurements as well as applying the skills of adding and subtracting within the base ten system. Whatever your students’ comfort level with and understanding of the base ten system, we can use that to help them conceptualize and apply multiplication and division. Teachers will need to know their students’ data and use that to differentiate both up and down while ensuring that students are all engaging in solving the same grade-level problems, no more and no less.” Suggested Interventions, “Provide students with physical manipulatives to help them ‘feel’ the difference in weight. Prompt students to act problems out before they attempt to represent with visual models or concrete manipulatives, identifying the first and second step to solve.”

• Unit 8 Overview, Story Problems, Differentiating for Learning Needs, “In this unit, students continue to build their conceptual and practical understanding of addition and multiplication patterns, extending their flexibility using properties of operations. Students also begin to represent and solve two step word problems with all operations and in the context of perimeter and area problems. This unit is designed to build fluency with these concepts in multiple contexts. As the unit concludes, students use these skills to build a robot! Whatever your students’ comfort level with and understanding of addition and multiplication patterns, we can use their understanding of factors and multiples to help them conceptualize and apply properties of operations to find more efficient ways of solving. Teachers will need to know their students’ data and use that to differentiate both up and down while ensuring that students are all engaging in solving the same grade-level problems, no more and no less.” Suggested Interventions, “Prompt students to consider multiple ways to represent the problem in order to provide the opportunity to determine if they understand the little question and big question in the context of the problem. Provide addition and multiplication tables to students as they solve problems, prompting them to color or label the patterns they see in these tables as they work.”

• K-4 Math Stories Guide, Differentiating Math Stories Instruction, “As noted in the Implementation Guides for each grade level, supporting all learners, including those with disabilities and special needs, English and Multilingual learners and advanced students, is a commitment of the Achievement First program, and Math Stories, like the other program components, is designed to meet all students where they are and to move them to grade level proficiency and deeper understanding of the Common Core Math standards through research-based best practices for differentiation.”

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

According to the Guide to Implementing AF Math: Grade 3, 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.” Daily lessons provide “suggested extension activities for students in the Workshop Section of the lesson plan so that teachers can encourage students to engage with the content at a higher level of complexity if they are not doing so naturally but are ready to. These extension suggestions include variations of the game that encourage more sophisticated strategies in Game Intro Lessons (K-2) and variations of the tasks or suggested strategies or tools students may use in Exercise Based Lesson (2-4). The independent practice for grades Exercise Based Lessons also includes 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. Additionally, the Discussion section of the daily lesson plans always include a potential whole class extension/ application problem. These are often additional problems or tasks that ask students to apply the mathematical concepts taught that day, and like the focal problem of the day, students should be encouraged to use the strategy that makes sense to them in order to solve, once again allowing students to engage with the grade level content at a level that is appropriate to them.” Examples Include:

• Unit 3, Lesson 12, Workshop, Extension, “Encourage students to write their own story problems about time. Then allow them to try and solve them.”

• Unit 5, Lesson 4, Workshop, Extension. “Have students create a number bond that represents the whole, the fraction shaded, and fraction not shaded.”

• K-4 Math Stories Guide, Differentiating Math Stories Instruction, “In the Math Stories block, heterogeneous groups of students are expected to work with a variety of tools and strategies as they work through the same set of problems; this ensures that all students access the content and build conceptual understanding while allowing advanced students to engage

with the content at higher levels of complexity.”

The materials reviewed for Achievement First Mathematics Grade 3 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.

According to the Program Overview, Guide to Implementing AF Math: Grade K, Differentiation, Supporting Multilingual and English Language Learners, “Both the Game Introduction Lessons in lower elementary and the Exercise Based Lessons in upper elementary along with the Math Stories Protocols used in Math Stories at all grade levels 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 (Zwiers et al., 2017), Understanding Language/SCALE and recommended by the English Language Success Forum…” The series provides the following design principles that promote mathematical language use and development: 

• “Design Principle 1: Support sense-making, Principle in Action - Daily lesson plan scripts and the math stories protocols intentionally amplify rather than simplify student language by anticipating where students may have difficulty accessing the concepts and language and providing multiple ways for them to come to understanding. Every lesson includes multiple opportunities for students to engage in discussion with one another, often through turn and talks, as they make sense of the content, and this sense-making is also supported through the use of concrete and pictorial models and a lesson visual anchor that captures student thinking and mathematical concepts and key vocabulary… 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, Principle in Action - Lessons and the math stories protocols are strategically built to focus on student thinking. Students engage in each new task individually or with partners, have opportunities to discuss with one another, and then analyze student work samples as a whole class…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, Principle in Action - A key element of all lesson types is student discussion. Daily lesson plans and the math stories protocol rely heavily on the use of individual or partner think 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, Principle in Action - Every daily lesson and math stories lesson is structured so that students have many opportunities to get ‘meta’ about the mathematical processes they engage in. Students explain how they model and solve problems to the teacher and one another throughout the lesson, often through turn and talks in which they also evaluate their peers’ strategies and thinking. Lesson scripts also encourage students to draw connections between new content and previous learning as well as between different strategies....”

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 3, 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. We have strategically included several mathematical language routines (MLRs) to support the language and content development of MLLs/ELLs in all lesson plans and called them out explicitly for teachers in a third of lesson plans.” The Mathematical Language Routines, Vocabulary, and Sentence Frames are present throughout the materials. Examples include:

• Unit 6 Overview, Length, Perimeter, and Line Plots, Differentiating for Learning Needs, Supporting MLLs/ELLs, Mathematical Language Routines, “8 mathematical language routines are outlined in detail in the Implementation Guide for Grade 3. These routines are worked into the lesson plans throughout the unit and explicitly highlighted for teachers in lessons 3, 6, 9, & 11. Teachers should use these lessons as a model for recognizing when routines occur in all the remaining lessons and thinking about how they might incorporate additional routines into the remaining lessons if they feel their students need more language development support. A brief overview of each of the math language routines along with general guidance about how to implement them in the context of this unit are outlined below:

  • MLR 1 Stronger and Clearer Each Time: Teachers provide students with multiple opportunities to articulate their mathematical thinking, with the opportunity to refine their language with each successive share. Students have several opportunities to articulate their understanding of data sets and their representations as well as how to use a ruler. For example, when describing the features of and data points in a data set, students engaged in this routine are prompted to discuss these aspects using more precise mathematical language first in pairs and then in a whole group share. Students have additional opportunities to refine their mathematical thinking and language as they consolidate the learning. Teachers should consider making all of these questions turn and talks to maximize practice for MLLs/ELLs, and they may modify by breaking questions into smaller parts or by having students engage in successive shares.

  • MLR 2 Collect and Display: The teacher captures student thinking and/or strategies visually and leads the class in a discussion. In all lessons, teachers co-create a visual anchor with students. This visual anchor should include illustrations of the strategies at work, and teachers should reference them and encourage students to reference them in whole group discussion.

  • MLR 4 Info Gap: Students are put into pairs; each student in the pair is given partial information that when combined with their partner’s information provides the full context needed to solve the problem. Students must communicate effectively in order to solve the problem. Teachers should use info gap routines by pairing students with different information needed to represent and solve the problem together. The recommendation is that teachers prompt for this if they wish to add more opportunities for practice with student input or output.

  • MLR 6 Three Reads: Teachers support students in making sense of a situation or problem by reading three times, each time with a particular focus. This routine is recommended for use during the math stories blocks for classrooms with MLLs/ ELLs who need additional support, especially for the POD and TOM problems. Teachers can modify the protocol to read the story aloud 3 times instead of two. Teachers may want to have students visualize twice and then represent, or for students who are stronger with listening comprehension, they may visualize for the first read, represent for the second, and check their representation for the third. (Another option would be to visualize for the first read, represent what they know from the story for the second, and represent what they need to figure out for the third.)

  • MLR 7 Compare and Connect: Teachers prompt students to understand one another’s strategies by comparing and connecting other students’ approaches to their own. Students engage in this routine multiple times in each lesson as they connect the different focal strategies of the lesson. Several questions are scripted into each lesson’s introduction and then again in the second bullet of the MWI and Discussion that ask students to consider how strategies relate to one another. These questions should be posed as turn and talks with think time to best support language development.

  • MLR 8: Discussion Supports: Teachers use a number of moves to help facilitate student discussion including revoicing, encouraging students to agree, disagree, build on, or ask questions of their peers, incorporating choral response to build vocabulary, showing concepts multi-modally, and modeling clear explanations/ think alouds. Teachers should continue to prompt students to use habits of discussion like agreeing and disagreeing with one another, building off one another's thinking, and asking clarifying questions. Teachers should prompt for these discussion habits any time students are engaged in discussion with one another and support them with visual anchors and sentence frames. Additionally, teachers can consider incorporating nonverbal symbols for students to use when they wish to agree/disagree/ build off/ ask a question. As students familiarize themselves with these discussion habits, teachers will need to support less as they will become automatic.”

• Vocabulary: “When introducing new vocabulary, words and their meanings should be explicitly taught with the use of concrete objects and/or visual models. Kinesthetic motions and choral response also are helpful for introducing new vocabulary, and when it is possible, it is often useful to pre-teach vocabulary for MLLs/ ELLs. To support sense-making, make sure that vocabulary is posted throughout the unit with visual illustrations of meaning.” Examples include: “data (a group of information we can count); inch (in., a unit of measure for length); quarter inch ( of an inch); perimeter (the units around the shape).”

• Unit Sentence Frames/ Starters: “Providing sentence frames and starters is helpful for cultivating conversation, particularly in lower elementary. Teachers should have these sentence frames posted in the classroom to assist students in engaging in discourse.  Additionally, teachers can provide sentence starters at the start of each turn and talk by posing the question and then providing the starter. For example, if the turn and talk is ‘Turn and tell your partner how you solved 4+4,’ the teacher would give the cue for students to turn and then say, ‘I solved 4+4 by…’ before students begin talking.” Examples include: “Sentences Frames for the Unit: I believe this is the correct answer because _____. My strategy uses ______ and shows ______. I can verify my final answer by _____.”

The materials reviewed for Achievement First Mathematics Grade 3 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 most commonly found in the Intervention suggestion at the end of Workshop time. However, there is little teacher guidance to explain how and when the intervention is intended to be used. Examples include:

• Unit 1, Lesson 2, “students interpret arrays as multiplication and use arrays to represent multiplication.” The Intervention includes the use of manipulatives, “Allow students to use counters to create their groups and then their arrays.” 

• Unit 5 Overview, Lesson Sequence, Lesson 2, “Scholars will fold rectangular strips to create their own fraction strips. Scholars will continue their work with naming fractional parts and identifying where a whole (strip) is split equally.” In Lesson 2, Problem of the Day,  “Maya baked a rectangular cake. She wants to break it into fourths so that she can share it with her mom, dad and sister. Use your fraction strips to represent $\frac{1}{4}$ and show your work below.”

• Unit 7, Lesson 20, Narrative, “students need to find the area of shapes by partitioning the shape into smaller rectangles.” The Intervention includes the use of manipulatives,“Transfer shapes onto grid paper. Have students cut shapes in order to see that the area of the shape remains the same even if you cut the shape into 2 or more rectangles. Guide students to break down shape into rectangles and squares. Have them write in the dimensions for the small shapes. Prompt them to use what they know about the irregular shape’s sides to help them figure out the side lengths for the small shapes.”

There are a few instances where manipulatives are not connected to written methods. Examples Include:

• Unit 2 Overview, “Students use cubes or other items to concretely illustrate the values represented in graphs. Students use the problem context to define how the representation matches the problem situation in creating a key and scale for the representation.”  However there is no use of cubes in any of the lessons of Unit 2.