3rd Grade - Gateway 2

The instructional materials for Math Expressions 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 identify Five Core Structures: Helping Community, Building Concepts, Math Talk, Quick Practice, and Student Leaders as the organizational structures of the program. “Building Concepts in the classroom experiences in which students use objects, drawings, conceptual language, and real-world situations - all of which help students build mathematical ideas that make sense to them.”

The instructional materials present opportunities for students to develop conceptual understanding. For example:

• Unit 1, Lesson 2, students write multiplication equations for equal groups from pictures and tables to find the total number. “How many bananas?” requires students to write “4 x 3 = 12” from the figure. In the following activity, students make a drawing for each problem, label the drawing with a multiplication equation, and write the answer to the problem.
• Unit 2, Lesson 9, Activity 3, students interpret products and quotients of whole numbers. “Louis put 72 marbles in 8 bags. He put the same number of marbles in each bag.” The directions state, “Write a question for the given information and solve.” Students determine which operation to use and formulate a question that requires that operation.
• Unit 4, Lesson 1, Understand Fractions, Activity 1, "Students use what they have learned about decomposing shapes as a foundation for understanding unit fractions as equal parts of a whole." Activity 2, "Students use fraction bars to visualize and represent unit fractions as the elements for building other fractions."

The instructional materials present opportunities for students to independently demonstrate conceptual understanding. For example,

• Unit 1, Lesson 16, students determine the type of multiplication or division problem represented in word problems, and solve the word problems. For example, Problem 3, “Zamir bought 21 treats to the dog park. He divided the treats equally among the 7 dogs that were there. How many treats did each dog get?” Students need to determine this is an Equal Groups Division with an Unknown Multiplier (number of groups) problem, and write and solve the equation.
• In Unit 5, Lesson 2, students find the area and perimeter of rectangles. Check Understanding problem, “Draw a rectangle with an area of 36 centimeters square and one side length of 4 centimeters. Find the unknown side length. Then find the perimeter." 
• Unit 6, Lesson 4, students engage with comparison problems. Students represent comparison problems in two ways: using a drawing and using a bar diagram and completing statements using more and fewer. For example, “Claire has 8 marbles. Sasha had 15 marbles.” Problem 17, “How many more marbles does Sasha have than Claire? Problem 18, “How many fewer marbles does Claire have than Sasha?”

The instructional materials for Math Expressions Grade 3 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials provide regular opportunities for students to attend to the standards. For example, 3.OA.7, fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division or properties of operation; and 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.

The instructional materials develop procedural skill and fluency throughout the grade-level. Each lesson includes a “Quick Practice” described as “routines [that] focus on vitally important skills and concepts that can be practiced in a whole-class activity with immediate feedback”. Quick Practice can be found at the beginning of every unit on the pages beginning with the letters QP. Student materials and instructions are also found in the Teacher Resource Book on pages beginning with Q. Examples include:

• The Introduction includes a chart “Path to Fluency: Kindergarten through Grade 6” which provides a pathway for fluency and memorization of basic facts, and operations with multi-digit operations. Also included are Reteach and Practice Sheets, Quick Practices, and Daily Routines supporting fluency.
• Unit 1, Teacher Resource Book, Multiplication Equations as Groups of 9, “The Student Leader points at the equation in the 9s column of the Multiplication Table Poster in order: 1x9=9, 2x9=18, 3x9=27, and so on. Class says: 9 is 1 group of 9 and raises 1 finger. 18 is 2 groups of 9 and raises 2 fingers. 27 is 3 groups of 9 and raises 3 fingers, and so on.”
• Unit 5, Teacher Resource Book, “Display these fractions bars. Student Leader 1 says two unit fractions with different denominators (such as 1/3 and 1/5 ) and asks which is greater and why. Class: 1/3 is greater because it has fewer unit fractions to make the same whole.” Students practice fraction fluency with students leading the activity.

The instructional materials provide opportunities for student to independently demonstrate procedural skill and fluency throughout the grade-level. These include: Path to Fluency Practice, and Fluency Checks. For example:

• Unit 1, Lesson 14, Activity 2, students review strategies for multiplying and dividing. The Student Activity Book includes questions that “focus on using strategies students were introduced to in previous lessons.” “Emily knows that 4 x 10 = 40.  How can she use subtraction and multiples of 9 to find 4 x 9?” (3.OA.7)
• Unit 1, Lesson 3, Path to Fluency, students practice “count bys”, mixed up multiplication, and mixed up division facts for 5s, 2s, 10s, and 9s.
• Unit 3, Fluency Check 3, students practice single digit by single digit multiplication. For example, Problem 10, “9 x 3 = __.”  
• Student Activity Book, Unit 3, Lesson 8, students practice writing addition problems vertically by lining up the place values correctly before adding. (3.NBT.2)

In addition, Homework and Remembering activity pages found at the end of each lesson provide additional practice to build procedural skill and fluency.

The instructional materials for Math Expressions Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

Students engage with application problems in many lessons for the standards that address application in solving real-word problems. For example, in the Student Activity Book, Unit 4, Lesson 9, students solve application problems involving elapsed time. “Berto spent from 3:45 p.m. to 4:15 p.m. doing math homework and from 4:30 p.m. to 5:10 p.m. doing social studies homework. How much time did he spend on his math and social studies homework?”

Each lesson includes an Anytime Problem listed in the lesson at a glance, and Anytime Problems include both routine and non-routine application problems. For example:

• In Unit 4, Lesson 12, Anytime Problem, “Sarah spent $5 from her savings to buy a book. Then, she spent half of her remaining savings when she paid $9 for a shirt. How much money did Sarah have in savings before she bought the book and the shirt?”
• Unit 1, Lesson 2, Anytime Problem, “Dulal, Lien, and Kate have either the bird, the cat, or the fish. Dulal does not have the bird. Lien does not have the cat. Kate does not have the fish or the cat. Which pet does each person have?”

The instructional materials present opportunities for students to engage in routine application throughout the grade-level. Examples of routine applications of grade-level mathematics are found in the Student Activity Book. For example:

• Unit 1, Lesson 15, “There are 3 shelves in a bookshelf. Each shelf has 2 piles of books on it. If there are 3 books in each pile, how many books are in the bookshelf?”
• Unit 2, Lesson 2, Student Activity Book, “Ana has a ribbon that is 18 inches long. She cut the ribbon into 3 equal pieces. Then she cut each of those pieces in half. How many small pieces of ribbon are there? How long is each piece?”
• Unit 3, Lesson 3, students use place value drawings to assist in solving problems. “Scott made a batch of rolls. He gave a bag of 10 rolls to each of 7 friends. He kept 1 bag for himself. How many rolls did he bake in all?”
• Unit 6, Lesson 6, Problem 2, “Mark has 6 shirts and 5 pairs of pants. Today his aunt gave him 4 more shirts and another pair of pants. How many shirts does he have now?” 

Remembering pages at the end of each lesson are designed for Spiral Review anytime after the lesson occurs. One feature of the Remembering problems are those titled Stretch Your Thinking, which often present opportunities for students to engage with non-routine problems. For example:

• Unit 2, Lesson 15, Remembering, Stretch Your Thinking, Exercise 14, “Matt runs four days a week. On the first day he runs 30 minutes. On the second day he runs 5 minutes more than the first day. On the third day, he runs the same number of minutes as on the second day. On the fourth day, he runs 10 minutes more than the previous day. After Matt runds on the fourth day, how many minutes in all has he run?”
• Unit 3, Lesson 5, Remembering, Stretch Your Thinking, Exercise 17, “Adult and student tickets were sold for a concert. When the numbers of adult tickets and students tickets are rounded, the total number of tickets sold was about 1,200. List four different combinations of adult and students tickets that might have been sold.”
• Unit 7, Lesson 4, Remembering, Stretch Your Thinking, Exercise 7, “Jake has 12 liters of water. Name four different ways he can divide the water into buckets so each bucket has the same number of liters.”

The instructional materials for Math Expressions Grade 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are represented in the materials, for example:

• Each lesson has a 5-minute Quick Practice providing practice with skills that should be mastered throughout the year.
• There are Performance Tasks throughout the series, where students use conceptual understanding to perform a mathematical task. For example, Unit 3, Problem 4, “June, Ella, and Joshua are collecting pennies for the service project. June collected 324 pennies. Ella collected 442 pennies, and Joshua collected 248 pennies… Part C: How many more pennies do the students need to collect to buy a second flat of flowers? Show your work.”
• Fluency Checks are included throughout the series, where students practice procedural skills and fluency. For example, Student Activity Book, Lesson 19, Fluency Check, Problem 3, “Add: 8+5=_.”
• Application problems are embedded into practice in the Student Activity Book. For example, Unit 5, Lesson 5, Solve Perimeter and Area Problems, Problem 1, “The dimensions of a rectangular picture frame are 9 inches and 6 inches. What is the greatest size picture that would fit in the frame?”

Examples where student engage in multiple aspects of rigor:

• Unit 3, Lesson 7, students are introduced to proof drawing addition. This lesson builds students conceptual understanding of addition by asking students to visualize and model regrouping by making proof drawings to illustrate adding two 3-digit numbers. Students are introduced to four methods for solving addition problems: Show All Totals Method, New Groups Below Method, New Groups Above Method, and Proof Drawing. Students use each method to solve: “Tonya and Mark collect seashells. Tonya has 249 shells and Mark has 386 shells. How many do they have in all?”
• Unit 3, Lesson 16, students practice and discuss addition and subtraction methods. In Activity 1, students create an addition word problem using the given numbers 672 and 228. Students then create a subtraction problem using the given numbers 814 and 439. Students create a Math Mountain for those numbers, and then find addition problems that match the mountain. Students apply learned skills by creating word problems. They use procedures to solve the equations in the problems.

The instructional materials reviewed for Math Expressions Grade 3 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Mathematical Practice Standards are clearly identified in a variety of places throughout the materials. For example:

• The Mathematical Practices are identified in both volumes of the Teacher’s Edition. Within the introduction, on page I13 in the section titled The Problem Solving Process, the publisher groups the Mathematical Practices into four categories according to how students will use the practices in the problem solving process. Mathematical Practices are also identified within each lesson.
• Each time a Mathematical Practice is referenced it is listed in red with a brief description of the practice.
• At the beginning of each Unit is a section devoted to the Mathematical Practices titled "Using the Common Core Standards for Mathematical Practices". Within this section, each Mathematical Practice is defined in detail. In addition, an example from the Unit is provided for each practice. For example, Unit 1, “Using the Common Core Standards for Mathematical Practices” illustrates how MP2 is used in Lesson 1-1 and Lesson 1-3.  
• The Mathematical Practices align and connect with the content of daily lessons, rather than being included as stand-alone topics.

Examples of Mathematical Practices that are identified, and enrich the mathematical content include:

• Unit 1, Lesson 4, MP7 - Look for Structure | Identify Relationships, “Emphasize that division undoes multiplication. In multiplication, start with the factors and then find the product. In division, start with the product and one of the factors and then find the other factor." Student rewrite division equations as an unknown multiplication equation.
• Unit 1, Lesson 10, Teaching the Lesson, MP8 - Use Repeated Reasoning/Identify a Pattern, “Ask what patterns students see in the count-bys and equation. Two common patterns are: The sums of the digits of the count-bys follow the pattern 3, 6, 9, 3, 6, 9; and the products follow the pattern odd even, odd, even,...”
• Unit 2, Lesson 15, Teaching the Lesson, MP1 - Make Sense of Problems, teachers are given no guidance to support students as they “have Student Pairs work together to make sense of Problems 1 and 2 and to decide what operations to use.”
• Unit 3, Lesson 4, Teaching the Lesson, MP7 - Use Structure, “Allow time for students to build the number using their Secret Code Cards. What number did you build? [1,278] How is this exercise different from the first one? [The place values are given out of order.]”
• Unit 5, Lesson 9, Teaching the Lesson, MP2 - Reason Abstractly and Quantitatively, “Why can you use different fractions to name 1/2?”

It should be noted that while the Mathematical Practices are clearly identified in the teacher materials, they appear to be over identified. Many lessons have multiple Mathematical Practices listed.

The instructional materials reviewed for Math Expressions Grade 3 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Math Expressions includes a Focus on Mathematical Practices lesson as the last lesson within each unit. Activity 3 of each of these lessons prompts students to determine whether a mathematical statement is true or false or to establish an arguable position surrounding a mathematical statement. These activities provide students opportunities to construct an argument and critique the reasoning of others. Student volunteers ask questions of other students to verify or correct their reasoning. Examples of Focus on Mathematical Practices lessons include, but are not limited to:

• In Unit 2, Lesson 15, students defend their position of the following statement, “You can always break an 8s multiplication into two equal addends.” Students are asked to “Establish an arguable position by writing or stating sentences that support a specific point of view.” Student work should include examples or counterexamples to justify their position. Volunteers are invited to share their positions with the class. Classmates are encouraged to ask questions to verify reasoning or correct reasoning errors. 
• In Unit 3, Lesson 18, students establish an arguable position by writing or stating sentences or equations supporting a specific point of view. “If you add two 3-digit numbers, the sum will always be a 3-digit number.” Students share their positions and explanations with the class and verify their reasoning.
• In Unit 5, Lesson 10, students determine a position for the following statement, “When comparing two fractions with the same numerators and different denominators, the fraction with the smaller denominator is greater than the fraction with the larger denominator.” Classmates are encouraged to ask questions to verify reasoning or correct reasoning errors. 

Puzzled Penguin problems are found throughout the materials and provide students an opportunity to correct errors in the penguin’s work. These tasks focus on error analysis, and many of the errors presented are procedural. Examples of Puzzled Penguin problems include:

• In Unit 1, Lesson 4, Puzzled Penguin problem, students identify that the penguin made a computation error when relating a division fact with the inverse multiplication fact and correct it.
• In Unit 2, Lesson 4, Puzzled Penguin problem, students identify that the penguin added instead of multiplying.
• In Unit 4, Lesson 2, Puzzled Penguin problem, students identify the error to 8 x 6 = 14.

In addition, Remembering pages at the end of each lesson often present opportunities for students to construct arguments and/or critique the reasoning of others. For example:

• Unit 2, Lesson 6, Remembering, Stretch Your Thinking, Exercise 8, “Explain two different squares that can be made using the number 9.” 
• Unit 3, Lesson 2, Remembering, Stretch Your Thinking, Exercise 5, “Anton says 2,000 + 300 + 70 + 5 is the same as 23 hundreds + 7 tens + 5 ones. Is he correct? Explain.”
• Unit 4, Lesson 3, Remembering, Stretch Your Thinking, Exercise 6, “Use the numbers 3 and 4 to make a fraction that is greater than 1 and a fraction that is less than 1. Explain how you made your fractions without using a number line.”
  

The instructional materials reviewed for Math Expressions Grade 3 meet expectations that materials use accurate mathematical terminology.

• New vocabulary is introduced at the beginning of a Lesson or Activity.
• The Teacher Edition provides instruction for teachers on how to develop the vocabulary, with guidance for teachers to discuss and use of the vocabulary.
• The student materials include Unit Vocabulary Cards that students can cut out and use in school or at home to review vocabulary terms.
• The Student Activity resource contains activities that students can do with the vocabulary cards; however, the teacher materials do not provide guidance as to when students should engage in these activities to support learning the vocabulary.  
• There is an eGlossary providing audio, graphics, and animations in both English and Spanish of the vocabulary needed in the lessons.  
• Study POP! is an interactive digital charades app that includes Math Expressions vocabulary to help students practice and develop mathematical vocabulary. Study POP! is listed at the beginning of many lessons, but is not referenced during the lesson.

Examples of how vocabulary is incorporated within lessons include:

• Unit 1, Lesson 1, the terms equation, multiplication, factor, and product are listed in the student materials. The Teacher Edition suggests teachers may want to add these words to a vocabulary list on chart paper.     
• Unit 5, Lesson 1, the terms area, perimeter, and unit square are listed in the student materials. Students use perimeter and area as they work on their Math Boards with rectangles. Students use unit squares to find the area of the rectangles. Students use 1-inch unit squares and a ruler to determine the area and perimeter of a rectangle.
• Unit 7, Lesson 5, the terms ray, angle, right angle, triangle, quadrilateral, polygon, concave, convex, pentagon, hexagon, decagon, and octagon are listed in the student materials.  Students work with the vocabulary terms throughout the lesson in multiple activities. Definitions and visual examples are provided for the vocabulary words.

In addition, there are instances where teachers are told to look for precise use of words, facts, and symbols. For example:

• Unit 2, Lesson 15, “MP6-Attend to Precision: The sentences must include precise mathematical words, facts, and symbols.” Students use precise mathematical language to defend their position on the statement, “You can always break an 8s multiplication into two equal addends.”  
• Unit 5, Lesson 10, “MP6-Attend to Precision: The sentences must include precise mathematical words, facts, and symbols.” Students use precise mathematical language to defend their position on the statement, “When comparing two fractions with the same numerators and different denominators, the fraction with the smaller denominator is greater than the fraction with the larger denominator.”