3rd Grade - Gateway 3

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 ($$\frac{1}{4}$$ 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.