The instructional materials for Achievement First Mathematics Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

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

• Unit 4, Lesson 8, students develop conceptual understanding of 4.NBT.5, as they use place value blocks to help them solve multi-digit multiplication problems. In Problem of the Day, “Problem: A video store display shelf has DVDs stacked in 3 rows. There are 246 videos in each row. How many videos can the shelves hold? TT: How can we represent this problem with an equation? We can write 246 videos x 3 rows = K total videos. Add to VA. Why does that work? It works because this is a problem about equal groups. In this problem, we have 3 groups—the rows—with 246 DVDs in each. We need to figure out the total number of DVDs. We can do this with multiplication. Today we’re going to solve 2, 3, and 4-digit multiplication with place value blocks. Work with your partner to solve this equation with place value blocks.”

• Unit 6, Lesson 6, students develop conceptual understanding of 4.NF.1, as they use tape diagrams and number lines to find equivalent fractions. In Workshop, Problem 2, “Markette is using a number line to figure out how many sixths are equal to $\frac{2}{3}$. She tells her partner, “We should partition each interval on our number line into 3 new parts, because $3+3$ is $6$. Is Markette’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.”

• Unit 7, Lesson 6, students develop conceptual understanding of 4.NF.7, compare two decimals to the hundredths by reasoning about their size. In the Workshop, Problem 1, students use visual models and number lines to support their reasoning about comparisons. “For each problem, shade each decimal amount on the given grids and plot them on the number line. Then use those models to compare the decimals using <, > or $=0.2$__.” 

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

• Unit 2, Lesson 4, students demonstrate conceptual understanding of 4.NBT.2, as they use a provided place value chart and their knowledge of place value to determine the reasonableness of a provided answer. Independent Practice, Problem 2, “Kate used the place value chart to write the number below in standard form. Is Kate’s work correct? Explain why or why not on the lines below.”

• Unit 6, Lesson 1, students demonstrate conceptual understanding of 4.NF.3, as they draw a visual model and write an equation to solve a problem. Independent Practice, Problem 1, “Terrell is keeping track of his running for the week. Draw a visual model and write an addition equation to model Terrell’s running plan. How far will he have to run at the end of the week?” 

• Unit 10, Lesson 3, students demonstrate conceptual understanding of 4.MD.5, as they use manipulatives to find the measure of a given angle. In the Exit Ticket, Problem 3, “Using pattern blocks, how can you find the measure of the angle below? Use pictures, words and numbers to show how you found your answer.”

The materials for Achievement First Mathematics Grade 4 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 eight Practice Workbooks in Achievement First Mathematics, Grade 3. One workbook, C, contains resources to support the procedural skill and fluency standard 4:NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm. In the Guide To Implementing Achievement First Mathematics Grade 4, teachers are provided with guidance for which workbook to use based on the unit of instruction. Examples include:

• Practice Workbook C, Problem 1, students solve subtraction problems. “Find the difference. 51,348 and 22,122. Use the standard algorithm to solve.” (4.NBT.4)

• Practice Workbook C, Problem 6, students solve subtraction problems. “Use a strategy that makes sense to you to solve. 59,637 – 34,721 = .” (4.NBT.4)

• Practice Workbook C, Problem 9, students practice subtraction. “ _____.“ (4.NBT.4)

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 Review occurs every Friday for 20 minutes. Examples include:

• Unit 4, Cumulative Review 4.5, Problem 4, students solve subtraction problems. “Find the difference. Show your work. $6,241 - 1,366 =$ _____.” (4.NBT.4) 

• Unit 4, Cumulative Review 4.5, Problem 2, students solve addition and subtraction problems using the standard algorithm. “Camden Yard sold 5,864 tickets and 2,549 students tickets to last Friday's Baltimore Orioles baseball game. How many total tickets were sold for last Friday’s game?”(4.NBT.4)

• Unit 6, Cumulative Review 6.4, Problem 5, students solve a subtraction problem. “Find the difference. $2,301 - 1,976 =$ ___” (4.NBT.4)

The materials reviewed for Achievement First Mathematics Grade 4 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 2, Guide to Implementing AF, Math Stories, October, students engage with 4.OA.3 as they solve a multi-step word problem posed with whole numbers and having whole-number answers using the four operations in a non-routine format. Sample problem 13, “Jamal and Sarah are playing a game with 5 counters. On each person’s turn they can take either 1or 2 counters from the pile. The player with the last turn loses. If Jamal starts the game and takes 1 counter away, what are two possible outcomes for the game?”

• Unit 7, Lesson 11, Problem of the Day, students engage in a routine problem with 4.NF.5 as they apply their understanding of fractions. "Victoria finds a multicolored quilt that exactly matches the colors in her bedroom. Victoria is so excited that she phones her mom to tell her about the quilt. This is what Victoria tells her mom: The quilt is a rectangle with one hundred squares, $\frac{36}{100}$ of the quilt is made of red and yellow squares $\frac{5}{10}$ of the quilt is blue squares, $\frac{14}{100}$ of the quilt is green squares. Victoria's mom is very excited about the new quilt. She asks Victoria what total fraction of the quilt is made of blue and green squares. What fraction should Victoria tell her mom is the total fraction of the quilt made of blue and green squares? Show all your mathematical thinking.”

• Unit 9, Guide to Implementing AF, Math Stories, May, students engage with 4.NF.1 as they apply the use of a visual fraction model to generate equivalent fractions and solve a non-routine problem. "Akilah draws two rectangles of the same size, and divides them into a different number of total parts. In the first rectangle, she colors in 4 parts. In the second rectangle, she colors in 5 parts. The colored areas are equal. What fractions could she have divided her rectangles up into?”

• Unit 10, Guide to Implementing AF, Math Stories, June, students engage with 4.MD.2 as they use the four operations to solve a routine word problem involving money. Sample Problem 1, “There are 3,418 students at Brookside Elementary and 2,192 students at La PLaya Elementary. All of the students are going on a field trip to the Natural history museum, where tickets for children are $3 each. The schools have a budget of $20,000 to spend on field trips. How much money will they have left over?”

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

• Unit 1, Lesson 11, Independent Practice, students engage with 4.OA.2 as they solve a routine word problem involving multiplicative comparisons. Problem 2, “Kenny is 56 years old. His sister is 7 years old. How many times younger is Kenny’s sister than him?”

• Unit 2, Cumulative Review 2.2, Problem 4, students engage in non-routine application of 4.OA.4 as they use their knowledge of factor pairs to solve a problem in more than one way. "Yvette is making bracelets for her friends. Each bracelet will have an equal number of charms. She has 24 charms and she wants each bracelet to have at least 2 charms, but no more than 8 charms. Part A: Which is NOT a way that Yvette can make her bracelets? a) 8 bracelets with 3 charms on each bracelet. b) 6 bracelets with 4 charms on each bracelet. c) 4 bracelets with 6 charms on each bracelet. d) 4 bracelets with 8 charms on each bracelet. Part B: Are there any other ways that work for Yvette to make her bracelets? Show your work below.”

• Unit 6, Lesson 22, Independent Practice, students engage with 4.NF.3 as they solve routine word problems involving addition and subtraction of fractions. Problem 1, “A cabinet has shelves that are $11\frac{1}{4}$ inches tall. Mike stacked a speaker that is $4\frac{3}{4}$ inches tall on top of a DVD player that is $3\frac{2}{4}$ inches tall. How much space is left between the objects and the top of the shelf?”

The materials reviewed for Achievement First Mathematics Grade 4 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 of factors by determining whether there are additional factors for a number within 100. Problem 7, “Marco and Desiree made 56 cookies for a bake sale. They will put an equal amount of cookies into bags. Marco and Desiree want to put more than 2 cookies but fewer than 10 cookies into each bag. Desiree says that they can only put 7 cookies into 8 bags or 8 cookies into 7 bags. Marco thinks there are more ways to put an equal number of cookies into bags. Who is right? Why are they right?” (4.OA.4)

• Practice Workbook D, students develop procedural skill and fluency as they multiply whole numbers. Problem 14, “Calculate the product of $64 × 35$.”  (4.NBT.5) 

• Unit 6, Lesson 23, Exit Ticket, students apply their understanding of fraction multiplication as they solve word problems. “Edwin uses $\frac{3}{4}$ of a teaspoon of baking powder for each batch of muffins he makes. He needs to make 3 batches for his Cub Scout meeting and 4 batches for his study group. How many teaspoons of baking powder will Edwin need?” (4.NF.4) 

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 7, Lesson 11, Independent Practice, students apply their conceptual understanding of adding decimals to solve a real-world problem. Part 1, “Mrs. Evans, the physical education teacher, is forming relay teams to help raise money for cancer research. There must be two students on each relay team. To determine the teams, Ms. Evans uses the students’ practice times from the last physical education class. Ms. Evans wants the teams to be as evenly matched as possible so they have a fair chance to win the race. What would the best combination of students be for each of the relay teams? Show all your mathematical thinking.” (4.NF.5, 4.NF.7) 

• Unit 5, Lesson 7, Exit Ticket, students apply their conceptual understanding of multiplication to solve a two-step word problem using tape diagrams and equations. Problem 1, “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. $(A × 2) + 4,892 = 6,392$.” (4.OA.3) 

• Unit 8, Lesson 4, Independent Practices, students apply their conceptual understanding of place value to solve a problem involving the value of coins. Problem 6, “Which is more, 68 dimes or 679 pennies? Prove with a place value chart and then explain on the lines below.” (4.MD.2)

The materials reviewed for Achievement First Mathematics Grade 4 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:

• Unit 3, Lesson 3, Independent Work Question 8, students engage with MP1 as they make sense of a problem involving multi-digit addition of whole numbers. “Milos’s family is keeping track of their steps each week. So far his sister has walked 15,678 steps, his father has walked 123,098 steps, and his mother has walked 435,607 steps. If Milos has walked twice as many steps as his sister, how many total steps has the family walked altogether?”

• The Unit 6 Overview outlines the intentional development of MP1. “Students apply the meaning of area and perimeter in order to interpret word problems in which area and perimeter are implicitly stated. Students practice division and multiplication strategies in the context of word problems. Calculations can be tedious and long, and students must continue to persevere through many steps in order to solve problems. Students employ a variety of problem solving skills in order to solve conversion problems. They must use ratios and calculations, but also determine which operation to use to convert.”

• Unit 5, Lesson 4, “How will embedded MPs support and deepen the learning?”, teachers are provided with explanations of connections between content and practices, “Students continue to practice SMP 1 as they plan to represent and solve multi-step problems by identifying all of the values and relationships between the values in the word problem, paying close attention to the questions asked in the word problem.”

• Unit 8, Lesson 6, Workshop Problem 3, students engage with MP1 as they work through multi-step word problems that require them to apply the concept of elapsed time. “Ms. Johnson has 20 minute meetings with students during the school day. She has a five-minute break between meetings. She does not have a break before her first meeting or after her last meeting. If she starts meetings at 7:45am, and has 4 meetings scheduled, what time will she be finished?”

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

• The Unit 2 Overview outlines the intentional development of MP2. “When students start to work with numbers in greater place values such as the hundred thousands and ten thousands, they use abstract reasoning to understand quantitative meanings. Since there are no place value blocks big enough to show a hundred thousand and they often can’t draw quantities this large, they must apply patterns of the place value system to logically understand the magnitude of larger numbers. SMP 2 is developed in lessons 2, 3, 5, & 9-11.”

• Unit 2, Lesson 3, Independent Practice Question 5, students engage with MP2 as they read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. “Lee and Gary visited South Korea. They exchanged their dollars for South Korean bills. Lee received 5 thousand dollar bills, 6 hundred dollar bills, 9 ten dollar bills, and 5 one dollar bills. What was Lee’s total amount of money in standard, written, and expanded form?”

• The Unit 6 Overview describes development of MP2. “The true understanding of the fundamental meaning of a fraction is abstract reasoning. Students must learn to take an abstract representation of two numbers (a numerator and a denominator) and give it a new meaning referring to a part of a whole – a value less than one. Through visual models and many examples, students should begin to understand that a fraction is a quantity in itself that has a position on a number line. This is extremely abstract quantitative reasoning. Students reason abstractly when they compare fractions of different wholes. The idea that a fraction can have a different value based on the size of its whole, but the same whole is implied when it is not specified, is a challenging abstract concept for students to grasp.”

The materials reviewed for Achievement First Mathematics Grade 4 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 2, Lesson 16, Workshop, students critique the reasoning of others and construct a viable argument as they evaluate the estimation strategies used by other students to determine who is correct. Problem 2, “Patricia said the best way to estimate the solution to $8,421 - 462$ is to round each number to the nearest hundred. Matthew said the best way to estimate is to round each number to the nearest thousand. Who is correct? Explain your answer.” 

• Unit 7, Lesson 4, Exit Ticket, Problem 3, “Patrice is measuring the rainfall for December. On Monday there was 0.09 of an inch of rainfall. On Tuesday there was 0.9 of an inch of rainfall. Patrice tells his sister that it rained the same amount on Monday and Tuesday. Tell whether or not Patrice is correct on the lines below.”

• Unit 7, Lesson 8, Discussion, teachers are provided with guidance and questions to engage students in critiquing the reasoning of others. “Last year I had a scholar who told me that when you compare decimals it's like the opposite of comparing whole numbers. What do you think they meant by that? How are comparing whole numbers and comparing decimals similar?”

• Implementation Guide, Unit 7, Math Stories, February/March, students are provided with an opportunity to share their math thinking as they solve problems involving fractions. Problem 12, “Yanira ran $\frac{3}{4}$ miles each day for 6 days. How many miles did she run over the course of 6 days?” Teachers are provided with three specific protocols to assist them in helping students represent and/or solve the problem, including sentence stems, for example: “First I put ____ because the story ____. Then I put ____ because in the story ____. Finally, I put ____ because in the story/we need to figure out ____.”

• Unit 10, Lesson 4, Exit Ticket, students construct an argument and critique the reasoning of others based on their knowledge of shapes. Problem 3, “Carlos is helping his brother with homework. He tells his brother that if you want to draw an obtuse angle, you should always use the bottom set of degrees on the protractor arc, and if you want to draw an acute angle you should always use the top set. Is Carlos’s reasoning accurate? Explain why or why not on the lines below.”

The materials reviewed for Achievement First Mathematics Grade 4 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.

• The Unit 1 Overview describes the intentional development of MP4. “Students model real-world mathematical situations using equations and tables with patterns. Students use tables, pictures, and mathematical formulas to solve problems involving patterns, multiplicative comparisons, and to determine factors, and classify numbers as prime or composite. SMP4 is developed in lessons 2 and 5 - 9.”

• The Unit 4 Overview provides guidance for connecting MP4 with area, perimeter, and solving conversion problems. “Students interpret word problems referring to area and perimeter and represent the information. When students solve conversion problems, they use many different types of models to determine how to convert, or show how they converted. They use the ratio to draw appropriate pictures, create tables, and write equations that use mathematics to model how to convert from one unit of measurement to another. Students also use benchmarks to understand units of measurement, which is a way of modeling a mathematical concept with real-world objects.”

• Unit 6, Lesson 18, Pose the Problem, students subtract mixed numbers by using fraction tiles and drawings. “Moira ordered 6 pizzas for the Student Council meeting. At the end of the meeting there were $2\frac{3}{4}$ pizzas left. How many pizzas did the student council eat during the meeting?”

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

• The Unit 2 Overview describes the intentional development of MP5. “Students choose between many methods when working with place value. In almost every aspect of this unit (place value relationships, expanding, reading, writing, comparing and rounding numbers, and non-standard partitioning) students have a variety of tools that could assist them. They can use place value charts, place value blocks, pictures of dots in each place value, pictures of place value blocks, organized lists, etc. to solve these types of problems. They must determine which tools are most effective for certain tasks.”

• Unit 6, Lesson 6, Exit Ticket Question 2, students use various tools, models and representations to show the meaning of fractions and different ways of showing fractional quantities. “Delilah is using a number line to figure out how many eighths are equal to $\frac{3}{4}$. She tells her partner, ‘We should partition each interval on our number line into 4 new parts, because 4 + 4 is 8.’ Is Delilah’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.”

• Unit 9, Lesson 4, Independent Practice Question 6, students use square corners and rulers to determine types of lines and angles. “Can a triangle have two right angles? Explain and draw an example to prove your thinking.”

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 4 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 2, Lesson 1, Exit Ticket, students solve problems requiring them to demonstrate an understanding of the term expanded form. Problem 3, “What two hundreds is seven hundred twelve between? Write seven hundred twelve in standard form. Write seven hundred twelve in expanded form.” 

• Unit 8, Lesson 2, Independent Practice Question 1 (Bachelor Level), students use precision to solve problems related to volume and capacity. “The capacity of each pitcher in the teacher work room is 3 quarts. Right now, each pitcher contains 1 quart 3 cups of liquid. If there are 3 pitchers in the room, how much more total liquid can the pitchers hold?”

• In the Unit 9 Overview, “Students attend to precision when naming and identifying lines, angles and triangles based on names using points.”

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

• Unit 4, Lesson 15, Try One More, teachers are provided with instructions to explicitly teach the term remainder. “We do have 1 leftover in this problem. This is called our remainder. A remainder is the amount leftover after dividing a number when one number does not divide evenly into another number. What was our answer before the remainder and why?” Students might say, “114 because we have 1 hundred + 1 ten + 4 ones.” The teacher replies, “Yes. Now our answer becomes 114 R1 because our answer is 114 with a remainder of 1.”

• Unit 6 Lesson 1, teachers are provided with guidance in reviewing vocabulary related to fractions. The introduction, “Before you get started, let’s review some key fraction vocabulary. In a fraction, what does the denominator tell us? The denominator tells us the total number of parts in a whole. In a fraction, what does the numerator tell us? The numerator tells us the total number of parts being referred to.”

• Unit 9, Lesson 4, the introduction provides teachers with guidance in introducing terminology related to triangles through a series of questions. “We have learned about different angle types, which will help us in our work today. What are the different types of angles? Today you will use the different types of angles to help you classify triangles!” Students work to observe and note information about triangles. The teacher prompts, “What did you observe about triangles? Triangles are classified by two names, kind of like how you have a first name and a last name. One name tells us about their angles, and one name tells us about their angles, and one name tells us about their sides.”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

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

• Unit 2, Lesson 5, Independent Practice, students solve problems by looking for structures based on place value. Problem 4, “Tiana drew 12 hundreds blocks on her paper. How many tens is that equal to? a. 1,200  b. 120 c. 12 d. 12,000.” 

• Unit 5, Lesson 4, Independent Practice, Question 2, students look for structure as they solve problems involving parts adding up to a whole by using tape diagrams to represent the situations. “Malia is keeping track of the subway riders on Saturday. At the first stop, some people got on the train. At the second stop, three times more people got on the train than at the first stop. At the last stop some people got off the train. How many people are on the train now?”

• Unit 8, Lesson 5, Exit Ticket, Question 2, students solve word problems involving adding enough of a smaller unit in order to regroup in the context of money amounts and determining change. “Meiling needed $5.35 to buy a ticket to a show. In her wallet, she found 2 dollar bills, 11 dimes, and 5 pennies. How much more money does Meiling need to buy the ticket?”

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

• Unit 5, Lesson 8, Exit Ticket, Question 1, students look for regularity in repeated reasoning as they solve problems by interpreting and labelling a representation such as a tape diagram. “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. $(A\times2)+4,892=6,392$.”

• Unit 6, Lesson 16, Independent Practice, Question 2, students see regularity in regrouping as they add and subtract mixed numbers. “Khalia and Jermaine are in a pie-eating contest. After 5 minutes, Khalia ate $3\frac{3}{4}$ pies and Jermaine at $4\frac{3}{4}$ . How much total pie did they consume altogether?”

• Unit 10, Lesson 2, Independent Practice, students look for repeated calculations as they solve a problem involving a circle divided into angles with given measurements. Problem 3, “Joanne cut a round pizza into equal wedges with angles measuring 30 degrees. How many pieces of pizza does she have?”

The materials reviewed for Achievement First Mathematics Grade 4 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, etc. 

• 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 7, Decimals, Lesson 10 include:

• “What do students have to get better at today? Students add fractions with denominators of 10 and 100 by changing both fractions to have like denominators or by relating them to decimals in expanded form. Students may still be relying on visual models today, though some may be able to ‘just know’ or use place value knowledge to add.”

• “What is new and/or hard about that? This is challenging because scholars may not have solidified adding or subtracting fractions from the previous unit. Scholars may struggle to regroup accurately, and when converting from mixed numbers to decimals may struggle to accurately change from fractions to decimals, particularly when needing to account for whole numbers. Lastly, scholars must be able to convert fractional tenths to hundredths (or understand how to accurately combine with tenths and hundredths) which could be another area for misunderstanding, as scholars can forget to multiply or divide the numerator by ten.”

• “Exemplar Student Response: ‘When I added $\frac{9}{10}$ and $\frac{18}{100}$, I converted $\frac{9}{10}$ to hundredths because we can only add fractions if they have the same denominator. I converted $\frac{9}{10}$ to hundredths by multiplying the top and the bottom of the fraction by 10; I got $\frac{90}{100}$. Then, I added the numerators together and got $\frac{108}{100}$. I regrouped because $\frac{108}{100}$ is a fraction greater than 1. I thought about how many wholes and how many fractional parts I have; I know that $\frac{100}{100}$ makes 1 whole which leaves $\frac{8}{100}$ leftover. I wrote my answer first as a fraction - and then as a decimal -1.08.’”

• “Introduction State the aim: Connect it to their lives and prior knowledge. Discuss how they will be working on it today. Plan a problem and questions to uncover key points and address common errors and misconceptions.”

• “Mid-Workshop Interruption: Share a strategy you’d like more students to use OR clarify a major misconception.”

• “Discussion: Discuss a major misconception OR have students share their work in CPA order OR ask students to apply their learning in a new way OR direct students to complete a pre-planned written response followed by a share from 1-2 students.”

• “Closing & Exit Ticket A quick debrief to clear up confusion OR cement a key point or big idea from the lesson.”

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 7, Decimals, Lesson 10 include:

• “What Key Points What should students know and be able to do? I can accurately add tenths and hundredths as fractions. When adding fractions, they must have the same denominator because we can only combine fractions with the same-sized pieces. I can accurately convert between fractions and decimals.”

• “How Key Points How will they do it? I can accurately add tenths and hundredths by converting tenths to hundredths (if needed) and then adding the numerators (number of pieces) of each fraction and writing my sum over the given denominator.”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Guide to Implementing AF Grade 4, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

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

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

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

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

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

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

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

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

• Unit 4 Overview, “In 5th grade, students must explain patterns in the number of zeroes of the product when multiplying by multiples of 10. This directly relates to strategies and understanding students use in 4th grade to multiply multiples of 10. Students learn rules with zeroes based on place value. These rules also extend to multiplying and dividing decimals by multiples of 10 in 5th grade. They must use this same pattern of zeroes and changes in place value to explain why decimal points move certain amounts of places in certain directions when multiplying and dividing by multiples of 10.”

• Unit 10 Overview, “Angle measurements will not come up as a formal part of the math curriculum again until seventh grade. Although there is a large gap in time between this unit and seventh grade, the seventh grade standards rely heavily on students’ skills and knowledge from this unit in fourth grade. Understanding of the concept of angles, the additive property of angles, and measurements of benchmark angles are foundational for seventh grade angle content. In 7th grade, students use supplementary angles, complementary angles and adjacent angles in problems and write and solve equations to find measurements of unknown angles. They also start to work with vertical angles and use them to find angle measurements. The strong ties between the 4th grade and 7th grade angle measurement standards, and the fact that students will not revisit this content for three years makes it crucial for students to deeply understand the fourth grade angle measurement material.”

The materials reviewed for Achievement First Mathematics Grade 4 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 4 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 2, Lesson 2, Lesson Overview: “Materials: all pages of this packet, 10 hundreds chart pags per student or per partner pair, colored pencils, scissors, staples, VA.”

• Unit 9 Lesson 1, Lesson Overview: “Materials: All pages of this packet, VA (stands for visual aid).”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

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

• Unit 1 Overview, Unit 1 Assessment: Multiplication and Division 1, denotes the aligned grade- level standards and mathematical practices. Question 6, “The length of the kitchen counter at Sal’s house is 9 times the length of a book. The length of a book is 8 inches. What is the length of the kitchen counter? Write an equation and solve.” (4.OA.2, MP1, MP2, MP6, MP7, MP8)

• Unit 4 Overview, Unit 4 Assessment: Multiplication and Division 2, denotes the aligned grade- level standards and mathematical practices. Question 21, “Ribbon that surrounds the border of a small bulletin board is 500 inches. The board is 150 inches wide. How tall is the bulletin board?” (4.MD.3, MP1, MP2, MP4, MD6, MP7)

• Unit 9 Overview, Unit 9 Assessment: Geometry, denotes the aligned grade-level standards and mathematical practices. Question 10, “Draw a quadrilateral that is not a parallelogram or a trapezoid.” (4.G.2, MP4, MP5, MP6, MP7)

The materials reviewed for Achievement First Mathematics Grade 4 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 4, 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: Addition and Subtraction, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 4.NBT.4, “If students ignore the places between the place they needed to regroup from to the place they needed to regroup to when subtracting across zeros: Model with place value blocks or pictures and ask students what is equivalent to 1 of the larger unit they are regrouping from. Remind students that when we regroup, we’re making exchanges of 1 of a larger unit for 10 of the next smaller unit such that we are creating an equivalent amount. If we take 10 hundred thousands, as an example, and exchange it for 10 ones, we’ve now changed the value of the original amount and this changes the problem completely. As you model with place value blocks or pictures, record the process with the standard algorithm to allow students to connect what is happening visually with the abstract method.”

• Unit 5 Overview, Unit 5 Assessment: Story Problems 1, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 4.OA.3, “If students do not interpret the remainder correctly: Revisit Lesson 1 from this unit and discuss the various ways we might need to deal with the remainder in a problem - ignore it, add another group or the remainder is the answer. Allow students to solve simple problems, like those found in Lesson 1. When they end up with a remainder, encourage students to label what the remainder is from the context of the story. For example, in item #6 on the assessment, the remainder represents a student. Ask students to consider, within the context of the story, the three different options for how to deal with the remainder and which makes sense given what the remainder represents from the story. Encourage students to always label what the remainder is when they’ve solved before deciding what to do with it. If students are struggling to identify what the remainder is, it can be useful to have them draw a labeled picture of the story so they can concretely see the remainder and what it represents.”

• Unit 9 Overview, Unit 9 Assessment: Geometry, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 4.G.2, “If students do not understand how a square can be classified as a rectangle and a rhombus: Remind students that some shapes can fall into more than one category based on their attributes. Discuss various quadrilaterals and sort them into categories using a venn diagram like the one shown earlier in the narrative of the unit. Play games like ‘guess my shape’ where students give hints about the shape and others have to try and identify it by name.”

The materials reviewed for Achievement First Mathematics Grade 4 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:

• Unit 1, Lesson 13, Exit Ticket contributes to the full intent of 4.OA.5 (Generate a number or shape pattern that follows a given rule). “1) Look at the towers and table below and complete the table. 2) Explain how you figured out how many blocks are in tower 4.”

• The Unit 7 Assessment contributes to the full intent of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100). Item 10, “Jenny split one cake into 10 equal pieces and ate 3 of the pieces. Then she split another cake into 100 equal pieces and ate 13 of the pieces. Write a sentence telling the total amount of cake(s) Jenny ate as a decimal.”

• The Unit 10 Assessment contributes to the full intent of 4.MD.6. Item 4 includes an image of a kite, “Use your protractor to answer the question below. Which measure is closest to the measure of angle H? A) 30° B) 35° C) 150° D) 155°” Item 7, “Which angle has a measure of 65°?” Item 11, “Draw an angle with a measure of exactly 147°.”

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

• Unit 4 Assessment, Item 15, supports the full development of MP2 as students convert between units of measurement, using abstract reasoning to determine which operations to use. “A race is 5 km long. How many meters long is the race? How many centimeters long is the race? a. 500 meters; 5000 centimeters; b. 5,000 meters; 50,000 centimeters; c. 5,000 meters;  500,000 centimeters; d. 500 meters; 50,000 centimeters.”

• Unit 1 Assessment, Item 6, supports the full development of MP4 as students write and solve an equation to represent a situation. “The length of the kitchen counter at Sal’s house is 9 times the length of a book. The length of a book is 8 inches. What is the length of the kitchen counter? Write an equation and solve.”

• Unit 10 Assessment, Item 2, supports the full development of MP7 as students look for and make sense of structure. “Which statement about angles is true? A) An angle is formed by two rays that do not have the same endpoint; B) An angle that turns through 1/360 of a circle has a measure of 360 degrees; C) An angle that turns through 5- 1 degree angles has a measure of 5 degrees D) An angle measure is equal to the total length of the two rays for the angle.”

The materials reviewed for Achievement First Mathematics Grade 4 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 4, 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 7, Workshop, Suggested intervention(s), “Have students show each clue with manipulatives or pictures, have students act out problems/have students focus on fact families with which they are most comfortable.”

• Unit 5 Overview, Story Problems 1, Differentiating for Learning Needs, “As children work to interpret complex story problems and apply addition, subtraction, multiplication, and division strategies to solve, it is likely that they will bring a variety of experiences and levels of mastery from the previous units on addition, subtraction, multiplication and division strategies and from story problem units in previous grade levels. All students should be familiar with the story problem protocol as a tool for making sense of story problems and should have a bank of strategies for modeling problems and solving problems involving all 4 operations, but their levels of success and comfort with these models and strategies may vary. Regardless of the experiences with which children enter Unit 5, teachers must meet their students where they are and ensure that all students are learning and deepening their understanding of the math concepts introduced in this unit. 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, “Scaffold by beginning with one-step problems including problems in which remainders must be interpreted first. Once students can interpret remainders in this context, move to two-step problems and then to multi-step problems. 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. Move back on the CPA continuum to help students move toward success with representing equations. Prompt students to represent with concrete/ pictorial models or less abstract models (ie - tape diagrams). Then, explicitly model using these less abstract models to generate equations, thinking aloud and visually capturing your work and thinking steps as you go. Then take students through these same thinking steps in the form of guided practice, providing them with many opportunities to articulate their own thinking steps as they work before releasing to independent practice with ample feedback.”

• Unit 9 Overview, Geometry, Differentiating for Learning Needs, “As children work to develop a flexible understanding of basic geometric attributes and how those attributes can be put together and analyzed to categorize shapes and form complex figures, it is likely that they will bring a variety of experiences and levels of mastery from geometry units in previous grade levels and home. All students should be familiar with categorizing shapes from third grade and should be fluent with the vocabulary used to describe many but not all quadrilaterals. Students will also likely have some familiarity with describing angles and lines from previous grade levels, though the vocabulary they use may vary. Regardless of the experiences with which children enter Unit 9, teachers must meet their students where they are and ensure that all students are learning and deepening their understanding of the math concepts and geometric vocabulary introduced in this unit. 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, “Scaffold tasks by having students work on classifying based on one attribute at a time. For example, have them focus first on classifying by angle type. Provide students with cut-outs of shapes that they can physically fold in order to develop conceptual understanding of symmetry.”

• 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 4 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 4, 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 5, Lesson 3, Workshop, Extension, “Have scholars begin to attempt multi-step problems where they must interpret remainders.”

• Unit 10, Lesson 4, Workshop, Extension, “Have students come up with their own rows for the table – their own circle with a certain fraction represented, and then have them determine the degrees of the angle. They will have to realize that only factors of 360 will work as denominators in order to get whole numbers of degrees.”

• 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 4 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 4, 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 2 Overview, Place Value, Differentiating for Learning Needs, Supporting MLLs/ELLs, Mathematical Language Routines, “8 mathematical language routines are outlined in detail in the Implementation Guide for Grade 4. These routines are worked into the lesson plans throughout the unit and explicitly highlighted for teachers in lessons 5, 8, 9, 11, 13, and 16. Teachers should use these lessons as a model for recognizing when routines occur in 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. This routine is often incorporated into lessons as students have multiple opportunities to articulate the key understanding/ key points of the lesson through turn and talks in the intro, workshop, and discussion. Over the course of the lesson, students refine their understanding of the concepts and the language they use to articulate that understanding as they engage in these successive turn and talks. Exemplar responses to turn and talks in the introduction of most lessons are often broad and mutl-part. The expectation is that students answer the question at hand, explain how they came to it, and why that works. Students who are not in the habit of giving such thorough answers or who are struggling with oral language may need more support than is scripted into the plans in order to give complete, exemplar responses. Teachers can and should feel comfortable modifying these turn and talks questions to align with the Stronger and Clearer Routine by breaking the question into 2-3 separate turn and talks, using scaffolds to break the question into more manageable parts without reducing the rigor of the question. For example, when asking students to discuss how they solved a problem, teachers may ask first what the solution is, then how they figured that out, and finally why that works. All turn and talks can also be posed as successive questions in which students engage in the same turn and talk several times in a row with different partners. As students practice articulating their ideas multiple times and hear different peers explain the concepts using different language and vocabulary, they will refine their language each time.

  • 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 3 Critique, Correct, and Clarify: Teachers present students with a statement, an argument, an explanation, or a solution, and prompt them to analyze and discuss. Nearly all lessons include an error analysis option as a potential focus either of the introduction or discussion. When following a misconception protocol, teachers should give students plenty of think time and allow them time to discuss the error and misconception with partners.

  • 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 may wish to work this routine into the math stories block by providing pairs of students with opposite parts of the story problem; the pairs will have to work together to communicate the important information needed to solve. This routine can also be worked into any lessons involving story problems to support MLLs/ELLs with language development. We recommend that this is incorporated as students master the mathematical concepts so that students are not struggling to grapple with language and mathematical thinking simultaneously to the point of frustration.

  • MLR 5 Co-Craft Questions and Problems: Teachers guide students to work with one another to create questions or situations for math problems or to create entire problems and then solve them. Teachers may wish to incorporate this routine into Math Stories by having students work in pairs to create story problems to exchange with one another, particularly on days when the class finishes the protocol early. This routine can also be used as an extension in any of the daily lessons involving story problems.

  • 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. Teachers should work this routine into the math stories block and any other time MLLs/ ELLs work with story problems, including when the problem of the day or try one more problem are contextual. When reading a story problem, prompt students to do a particular task for each read. For example, for the first read, teachers might direct students to focus on visualizing only. Then they might prompt students to represent during the second read and to check their representation against the story during the third read.

  • 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 most lessons as they connect the different focal strategies of the lesson. Several questions are scripted into each lesson’s introduction and often in the 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 strategically build new vocabulary throughout the unit through the use of visuals, repetition, and challenging students to explain the meaning in their own words. Teachers continue to build habits of discussion in this unit. Continue to prompt for students to engage in discourse by agreeing/disagreeing with one another, building on one another’s thinking, and asking clarifying questions.”

• 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: “Rounding: approximating the value of a given number; Place: the location of a digit within a number; Place value: the value of a digit based on its location.” 

• Unit Sentence Frames/ Starters: “Providing sentence frames and starters is helpful for cultivating conversation, particularly for students who are developing oral langauge skills in new or multiple languages. 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 42x40,’ the teacher would give the cue for students to turn and then say, ‘I solved 42x40 by…’ before students begin talking.” Examples include: “Sentence Frames for Lessons 8-10: When I multiply/divide by 10, the new number will be _______ because _______. All of the digits will change places to the _______ because _______. _______ is 10/100/1000 times greater/ less than _______. I know because _______.”

The materials reviewed for Achievement First Mathematics Grade 4 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 4, Lesson 4, Narrative, “students will find the area of rectilinear figures by decomposing them into rectangles.” The Intervention includes the use of manipulatives, “Have scholars use tiles or counters to fill the shape, counting as they go to find area. Have scholars count squares inside and around shapes to determine area.”

• Unit 6, Lesson 1, Narrative, students will use “fraction tiles and visual models to break fractions down into their unit fractions.” Students use manipulatives to solve the Problem of the Day, “Park Ranger O’Hara is putting trail markers down on Mount Bonnell. On the Mockingbird trail, he puts down a trail marker at every $\frac{1}{8}$ of a mile. If the trail is $\frac{7}{8}$ of a mile long, how many trail markers will he use? Use your fraction tiles to help you.”

• The Unit 8 Overview, “Students use many different strategies to solve different types of measurement problems such as timelines (or clock manipulatives) to solve elapsed time problems.” The Lesson 6 materials list includes clocks and the Intervention suggests “students just use clocks to solve” but there is no additional guidance. 

Manipulatives are not always connected to written methods. There are several instances where manipulatives are listed as materials but not incorporated into the lesson. Examples Include:

• Unit 8, Lesson 4, Materials list, “Coin/dollar manipulatives for intervention.” “Use a place value chart to represent each money/decimal/coin amount. In each column on the place value chart, write the decimal place value and the coin associated with that place value. Then have students put each problem from workshop into the place value chart.” However the Intervention does not include the use of such manipulatives.