3rd Grade - Gateway 2

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

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

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

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

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

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

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

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

The materials for Achievement First Mathematics Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

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

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

• Practice Workbook B, Problem 2, “$$303 - 165 =$$ .” (3.NBT.2) 

• Practice Workbook F contains 25 independent practice problems that allow students to build procedural skill and fluency with multiplication and division within 100. Problem 6, “$$8 × ______ = 56$$.” (3.OA.7)

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications, especially during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding.

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

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

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

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

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. 

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

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

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

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

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

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

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

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

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

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

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

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

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