5th Grade - Gateway 2

The instructional materials for Math Expressions Grade 5 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 five crucial components that are 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 3, Lesson 4, students use diagrams and number lines to represent multiplication by a unit fraction. In the teaching notes, “Making Connections. The fraction-bar model has number-line-diagram labels to connect these two length models, fraction bars and number line diagrams.” Students add units of 1/4 to find 2/4, 3/4, etc., in both the diagram and on the number line.
• Unit 6, Lesson 7, Model a fraction problem. “Trey spent 3/4 of an hour doing homework. Kylie spent 1/2 hour. How much less time did Kylie spend doing homework than Try?” Teachers are given the following guidance, “Be sure students understand that comparison bars can be used to model and solve additive problems involving, whole numbers, decimals, or fractions.”

The instructional materials include opportunities in the Student Activity Book for students to independently demonstrate conceptual understanding. For example:

• Unit 3, Lesson 1, Solve Comparison Problems, Problem 19, “Fred has 24 model cars. Scott has 1/6 as many. How many models cars does Scott have?" Students need to understand the whole - 24 cars - and that 1/6 represents 24/6 as the number of cars that Scott has.
• Unit 6, Lesson 2, Reasonable Answers, Problem 6, “Suppose you were asked to multiply the numbers at the right (2,500 x 0.6). Without actually multiplying, give a reason why an answer of 15,000 is not reasonable.”
• Unit 7, Lesson 4, students explore patterns and relationships. Problem 6a, “Write the first five terms of two different patterns.” Problem 6b, “If possible, describe two different relationships that the corresponding terms of your patterns share.”

The instructional materials for Math Expressions Grade 5 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 standard 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

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:

• Unit 1, Teacher Resource Book, Find a Common Denominator: Denominators Have No Common Factors, “Student Leader 1 points to the problem and asks: How can we do this? The class responds: Find a common denominator. Student Leader 1: How? Class: The denominators have no common factors. We can multiply the numerator and denominator of each fraction by the other denominator.”
• Unit 7, Teacher Resource Book, "The Student Leader uses a similar routine for each expression in the problem set. Class responses for the order of operations will vary for the different expressions in the set.” For example, $$(20-10)\div2$$.

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 2, Lesson 4, students build computational fluency with addition and subtraction of decimals. For example: 0.9 + 0.06; 0.47 + 0.25. (5.NBT.7)
• Unit 4, Lesson 4, Pathway to Fluency, students practice multiplying multi-digit numbers using any method. Problem types include: multiplying two-digit by one-digit numbers, three-digit by one-digit numbers, and two-digit by two-digit numbers. Students are also presented opportunities to determine the error in a completed problem using the standard algorithm.
• Fluency Checks are provided throughout the materials. For example, fluency standard 5.NBT.5 is assessed in Fluency Check for Unit 4, Lesson 12, with 15 multiplication questions.

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 5 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. In Unit 2, Lesson 4, Student Activity Book, students solve contextual problems involving measurement with metric lengths. “Matt is competing in the long jump event. His first jump was 3.56m. So far, the longest jump in the event is 4.02m. How much farther than his first jump must Matt jump to be in first place?” Students are applying mathematics with metric conversions and decimal place value within the context of a real-world situation. 

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, Unit 5, Lesson 6, Anytime Problem, “To ride his bike to school, Emilio rides 2 blocks west, 5 blocks north, and then 3 blocks west. To get home, he rides 5 blocks south, and then he rides east. How many blocks does he ride east to get back home?”

The instructional materials present opportunities for students to engage routine applications of grade-level mathematics. Examples include:

• Unit 1, Lesson 12, Student Activity Book, students write equations to solve word problems involving addition and subtraction of mixed numbers. “Ariel ran 6 1/4 miles on Saturday. This is 1 3/4 miles more than Harry ran. How far did Harry run? Why is the answer reasonable?” 
• Unit 3, Lesson 3, Student Activity Book, students apply their understanding of fractions to model multiplication with fractions. “Farmer Smith has 4 acres of land. She plows 1/3 of her land. Divide and shade the drawing at the right to show the part of the land she plows.” In Item 2, students “Express 1/3 x 4 as a sum of unit fractions. 1/3 x 4 =___.”
• Unit 6, Lesson 2, Teacher Edition, the teacher presents the following problem: “A pastry chef divided a 1/2 pound block of cream cheese into 4 identical pieces. What was the weight of each piece?” Students are asked, “In this problem, what is the total amount of cream cheese? We don’t know the weight of each of the 4 pieces. But we do know that if we multiply the weight of one piece by 4, the result should be 1/2, the total amount of cream cheese in pounds. What situation equation represents this problem?”
• Unit 6, Lesson 10, Student Activity Book, Question 2, students apply mathematics to solve a multi-step story problem involving multiplication, addition, and subtraction. “Sasha earns $8 per hour working at her grandparents’ farm. During July she worked 39 1/2 hours at the farm and earned $47 babysitting. How much more money does Sasha need (d) to buy a gadget that costs $399?”

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 3, Lesson 11, Remembering, Stretch Your Thinking, Exercise 19, “Harrison is playing a board game that has a path of 100 spaces. After his first turn, he is 1/5 of the way along the spaces. On the second term, he moves 1/4 fewer spaces than he moved on his first turn. On his third turn, he moves 1 1/4 times as many spaces than he moved on his first turn. What space is he on after three turns?”
• Unit 6, Lesson 1, Remembering, Stretch Your Thinking, Exercise 16, “Garrett wants to buy a soccer ball, a pair of shorts, and a pair of soccer shoes. The ball costs $12.55, the shorts cost $22.98, and the shoes cost $54.35. Garrett has $85.00. How much more money does Garrett need? Write an equation to solve the problem.”
• Unit 8, Lesson 9, Remembering, Stretch Your Thinking, Exercise 6, “Shannon pours four different liquid ingredients into a bowl. The sum of the liquid ingredients is 8.53 liters. Two of her measurements are in liters and two of her measurements are in millimeters. Give an examples of possible measurements for Shannon’s four liquids.”

The instructional materials for Math Expressions Grade 5 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 4, “How Much Does it Cost: Jon is a travel agent. This week he is planning tours for groups of 6, 8, 20, and 40 people. The table shows the cost of each tour, per person. Problem 1: A group of 6 people wants to take either the Caribbean cruise or the U.S. train tour. How much more money would it cost to take the Caribbean cruise?”
• Fluency Checks are included throughout the series, where students practice procedural skills and fluency. Unit 4, Fluency Check 1, students solve 15 multiplication problems involving multi-digit by single digit, and two-digit by two-digit. For example, Problem 9, “3,070 x 7.” (Problems are presented vertically.)
• Application problems are embedded into practice in the Student Activity book. For example, Unit 6, Lesson 2, Problem 1, “How many individual pieces of cheese, each weighing 1/4 lb. can be cut from a block of cheese weighing 5 pounds?”

Examples where student engage in multiple aspects of rigor:

• Unit 2, Lesson 4, students write equations and solve routine word problems involving given measurements. Problem 14, “Tori had fabric that was 6.2 meters long. She used some and now has 1.45 meters. How much did she use?” Problem 17, “Sarita has some ribbon. After she used 23.8 cm of it, she had 50 cm left. How much ribbon did Sarita start with?”
• Unit 6, Lesson 2, Activity 1, students write equations for multiplication and division situations. For example, “In a school gymnasium, 375 students have gathered for an assembly. The students are seated in 15 equal rows. How many students are seated in each row?” and “A pastry chef divided a 1/2 pound block of cream cheese into 4 identical pieces. What was the weight of each piece?”

The instructional materials reviewed for Math Expressions Grade 5 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, in Unit 6, page MB25-U6 illustrates how MP2 is used in Lesson 6-1 and Lesson 6-3.  
• 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 2, Lesson 3, MP5, Use Appropriate Tools | MathBoard. Students use their MathBoards to develop an understanding of hundredths and tenths related to a whole.
• Unit 4, Lesson 2, MP7, Look for Structure | Identify Relationships. Teachers write multiplication problems on the board and have students give the answers quickly (5x1, 5x2, 5x3, etc.). Students will be reminded that every other answer ends in zero because it is a multiple of 10.
• Unit 6, Lesson 4, MP8, Use Repeated Reasoning | Generalize. Students use rounding to assess the reasonableness of answers in a variety of story problems. Groups are asked to share their explanations with the class. Students demonstrate other ways those same strategies can be used to decide reasonableness.  
• Unit 8, Lesson 9, MP1 Make Sense of Problems is identified. Student Activity Book, Problems 13-17, students solve multistep problems that require one or more metric unit conversions. Teachers are instructed to remind students that in order to perform addition or subtraction on numbers with units, the units must be the same. Students may choose to convert to different units.

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 5 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:

• Unit 1, Lesson 13, students determine a position for the following statement: “If you add two fractions less than 12, the sum will always be less than 12.” Students establish an arguable position in writing and include examples or counterexamples. Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
• Unit 6, Lesson 11, students determine a position for the following statement: “The Commutative Property can be applied to subtraction, so the following equation is true: 5.55 - 3.25 = 3.25 - 5.55.” Students establish an arguable position in writing and include examples or counterexamples. Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
• Unit 7, Lesson 7, students determine a position on the following statement: “The expression 5 + 3(4 ÷ 2) − 1 simplifies to 15.” Students establish an arguable position in writing and include examples or counterexamples. Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects 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:

• Unit 3, Lesson 5, Puzzled Penguin problem, students find a calculation error in a perimeter problem.
• Unit 5, Lesson 8, Puzzled Penguin problem, students identify the error the penguin made in a division problem.
• Unit 7, Lesson 5, Puzzled Penguin problem, students determine whether the coordinates of a point are accurate.

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 3, Lesson 14, Remembering, Stretch Your Thinking, Exercise 10, “If you start with 1 and repeatedly multiply by 1/2 will you reach 0? Explain why or why not.”
• Unit 4, Lesson 10, Remembering, Stretch Your Thinking, Exercise 15, “Taylor estimated the music departments would raise $1,100 by selling tickets to a performance next week. Each ticket will be $12,75. About how many tickets does the music department need to sell for Taylor’s estimate to be reasonable?”
• Unit 6, Lesson 4, Remembering, Stretch Your Thinking, Exercise 8, “Kaley has 2 3/8 yards of fabric. She cuts and uses 1 1/16 yards from the fabric. She estimates that less than one yard of fabric is left over. Is her estimate reasonable? Explain.”

The instructional materials reviewed for Math Expressions Grade 5 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 3, Lesson 12, lists dividend, divisor, and quotient as vocabulary at the beginning of the lesson. The terms are reviewed in the Student Activity Book in Activity 2, but are not used anywhere else in the lesson or Student Activity Book directions.
• Unit 6, Lesson 4, lists benchmark as vocabulary at the beginning of the lesson. The term is used in the title and directions for Activity 2. At the beginning of the Activity there is a teacher-led whole group discussion introducing the use of benchmark fractions. The teacher prompts provide very few opportunities to make meaning of benchmark numbers; therefore, students only have a limited opportunity to make sense of the new vocabulary.  
• Unit 8, Lesson 3, lists perimeter, area, square centimeter, and square unit as new vocabulary. The Teacher’s Edition suggestions include a discussion on perimeter and area. The teacher asks students what they remember about perimeter and area. Five specific points are listed for teachers to elicit from students through discussion. In this teacher-led discussion, once students have made the five points listed in the Teacher’s Edition, the discussion concludes. This type of discussion does not ensure all students have a full understanding or memory of area and perimeter. In addition, the terms square centimeter and square unit are not written anywhere within the Student Activity pages for this lesson, with the exception of the word square unit being included in the the definition of area at the beginning of the student page.

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

• Unit 1, Lesson 13, “MP6 - Attend to Precision: The sentences must include precise mathematical terms and any examples or counterexamples must include precise facts and symbols.” Students must use precise mathematical language to develop an argument of true or false regarding the following statement: “If you add two fractions less than 1/2, the sum will always be less than 1/2.”
• Unit 5, Lesson 11, “MP6 - Attend to Precision: The sentences must include precise mathematical words, and any examples or counterexamples must include precise facts and symbols.” Students must decide whether the inequality statement, $$20\div0.95>20$$, is true or false and develop an argument that supports their position.