3rd Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 3 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. For example: 

• In 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 states, “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 then 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?” 
• In 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.” 
• In 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 states, “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. For example: 

• In 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 states, “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.” 
• In 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 states, “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 $$\frac{1}{8}$$.”
• In 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. The materials state, “Write three different equations that we could use to find the area of the following rectangle. Then, find the area of the rectangle.”

The instructional materials for Achievement First Mathematics Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. The instructional 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. For example:

• In Practice Workbook B, Problem 11 states, “Calculate. $$605 - 327 =$$ ; $$708 - 439 =$$ ; $$875 - 218 =$$ ; $$575 + 219 =$$ ; $$238 + 573 =$$ ; $$117 + 582 =$$ .” (3.NBT.2) 
• In Practice Workbook B, Problem 2 states, “$$303 - 165 =$$ .” (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. For example, Problem 6 states, “$$8×$$__$$=56$$.” (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. For example:

• In Unit 5, Cumulative Review 5.2, Problem 2, students practice adding and subtracting within 1000. It states, “Solve. 909 - 690.” (written vertically). (3.NBT.2)
• In Unit 5, Cumulative Review 5.4, Problem 4, students practice solving division problems within 100. It states, “Solve. $$40÷8=$$; $$16÷8=$$; $$7÷7=$$; $$72÷8=$$; $$27÷3=$$.” (3.OA.7) 
• In Unit 7, Cumulative Review 7.2, Problem 2, students practice solving multiplication problems. It states, “Solve. $$6×7$$; $$8×4=$$ ; $$9×3=$$ ; $$3×6=$$ ; 8 x 3 = ; 5 x 5 = .” (3.OA.7)

The instructional materials for Achievement First Mathematics Grade 3 meet expectations for balancing the three aspects of rigor. Overall, within the instructional materials the three aspects of rigor are not always treated together and are not always treated separately.

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

• In 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 states, “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)
• In Practice Workbook B, Problem 29, students demonstrate procedural skill and fluency related to addition and subtraction as they solve problems. It states, “Solve to find the missing numbers. $$142+\_=225$$, $$506-\_=329$$,  $$\_+344=764$$.” (3.NBT.2) 
• In Unit 8, Lesson 10, Independent Practice, students apply their understanding of the four operations as they solve a word problem. Problem 7 states, “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. For example:

• In 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 9 states, “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)
• In 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. The materials state, “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)
• In 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 states, “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 instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The instructional materials provide opportunities for students to engage in constructing viable arguments and analyzing the work and understanding of others. Examples of opportunities for students to construct viable arguments and/or critique the reasoning of others include:

• In Unit 3, Lesson 2, Exit Ticket students critique the reasoning of others based on their knowledge of estimation and rounding. Problem 3 states, “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.” 
• In 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. The materials state, “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.”
• In Unit 9, Lesson 4, Exit Ticket students construct a viable argument based on their understanding of shapes. Problem 3 states, “In the grid below, draw a rhombus that is also a rectangle. Explain how your shape fits in both categories.”

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There are opportunities within the teacher materials that guide teachers in assisting students to construct viable arguments and analyze the arguments of others, through the use of questioning.

Examples of the instructional materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include:

• In Unit 3, Lesson 2, Try One More, students are asked to determine if an answer is reasonable. The materials state, “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?”
• In Unit 4, Lesson 4, Workshop, Problem 7 states, “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.” The teacher's guidance includes, “Is the equation 9 divided by 3 correct? Why or why not? What equation represents the story?
• In Unit 5, Lesson 24, Problem of the Day, students analyze the reasoning of others and use models to explain their reasoning. The materials state, “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.” The teacher is guided to support students as they represent the problem and analyze the reasoning of others. Teacher guidance includes, “How did you represent this problem? First, I represented $$\frac{3}{3}$$ with a visual model by drawing one whole that I partitioned into three parts. I labeled each part with $$\frac{1}{3}$$ because that is the unit fraction since there are three parts in one whole. Then, I shaded in three parts because the numerator is 3 which means that there are 3 parts being referred to. Add to VA. What do you notice about $$\frac{3}{3}$$? What is it equivalent to? $$\frac{3}{3}$$ is equivalent to 1 whole. Hmm. The numerator and the denominator are the same. Why do we have 1 whole when the numerator and the denominator are the same? The total number of parts and the number of parts being referred to are the same. That makes one whole, and we can see it here because the whole shape is shaded in.”

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet  expectations that materials explicitly attend to the specialized language of mathematics. 

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them.  

Examples of explicit instruction on the use of mathematical language include:

• In Unit 1, Lesson 6 provides guidance for teachers in introducing the identified vocabulary word area. State the Aim states, “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.” Students then attempt to fill a shape using pattern blocks to determine which one takes up more space. The materials state, “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.”
• In Unit 4, Lesson 1 provides guidance for teachers in introducing mathematical terminology related to measurement. State The Aim, “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. Students pick up a paperclip in their hand; add benchmark to VA. Place paperclip on the scale. 1 gram The second is called a kilogram and weighs about as much as a textbook. Students lift the textbook; add benchmark to VA. Place the text book on the scale. ____ grams. Note: should be close to 1000 grams. ____ grams is ABOUT 1,000 grams. 1,000 grams is the same as 1 kilogram so let’s say that a textbook is ABOUT 1 kilogram. A textbook can be our benchmark for kilograms. Add to VA.”
• In Unit 9, Lesson 1 identifies new vocabulary used in the lesson and provides specific guidance for teachers in introducing the terminology, including counterexamples. State the Aim states, “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. Reveal KP on VA. Take a look at the top of your page. The first shape is not a polygon because it has an open-spaced 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.”

Examples of the materials using precise and accurate terminology and definitions in student materials: 

• In Unit 4, Lesson 3, Independent Practice, accurate terminology is used as students solve problems related to volume. Problem 2 states, “Mrs. Goldstein pours 3 juice boxes into a bowl to make punch. Each juice box holds 236 milliliters. How much juice does Mrs. Goldstein pour into the bowl?”
• In Unit 7, Lesson 6, Exit Ticket, accurate terminology is used as students are expected to draw an array and write an expression. Problem 1 states, “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.”
• In Unit 8, Lesson 3, Pose the Problem, accurate terminology is used as students work with partners to solve a problem. The materials state, “Khaleel says that whenever you add an odd number plus an odd number, you should get an odd sum because both addends are odd. Is Khaleel correct? Work with your partner to explain why; prove your answer using a visual model or the addition table.”