4th Grade - Gateway 2

The instructional materials for Math Expressions Grade 4 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 provide opportunities for students to develop conceptual understanding. For example:

• Unit 2, Lesson 2 addresses 4.NBT.A, generalize place value understanding for multi-digit whole numbers. Students are presented with the following equation: 2 x 10 = 20. The teacher is directed to, “elicit from students that 2 groups of 10 is 20, or 2 tens.” The discussion leads students to see the relationship between place value and multiplication. Students complete exercise 1 to demonstrate this understanding, “Ten times any number of tens gives you that number of hundreds.”
• Unit 3, Lesson 5, Math Talk in Action, student review types of division problems, “Let’s make up some word problems that could be solved by dividing 350 by 5.” Students find examples of equal-group problems, array problems, area problems, and a comparison problem.
• Unit 6, Lesson 3, Real World Problems, students solve fraction addition and subtraction problems using models. “Draw a model. Then solve.” Problem 31, “Reese had 2/4 cup of orange juice. She added pineapple juice to make a total of 3/4 cup of juice. How much pineapple juice did she add?”

Students have opportunities to independently demonstrate conceptual understanding. For example:

• Unit 3, Lesson 9, Activity 1, students make sense of remainders. Students solve problems where the remainder serves different roles in the solutions. Problem D, “Raul bought 4 toy cars for $9.00. Each car cost the same amount. How much did each car cost?” “In this case, the remainder is a decimal part of the answer.”
• Student Activity Book, Unit 2, Lesson 6, students solve two-digit by one-digit multiplication connecting Use Place Value Section Methods, to the area model, and expanded notation. Problem 5, “A marina needs to replace the boards on their pier. The pier is 7 feet by 39 feet. What is the area of the pier. Students complete an area model, and solve the problem using expanded notation. Students show the relationship between the place value of the two digit number in the area model, and in expanded notation.
• Student Activity Book, Unit 8, Lesson 4, Problem 33, students classify and sort triangles based on the characteristics of triangles. For example, isosceles, equilateral, acute, obtuse, etc.

The instructional materials for Math Expressions Grade 4 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 4.NBT.4, fluently add and subtract 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 2, Teacher Resource Book, Zero Patterns, “The Student Leader uses the pointer to point to each of the three multiplications in the first list. The class responds with the product of the non-zero digits and the place-value name of this product (ones, tens, or hundreds) and then gives the answer.”
• Unit 6, Teacher Resource Book, Build a Fraction, "The Student Leader hands out six Class Fraction Strips that say 1/6 to six different students and writes 1/6 on the board. Four students holding a 1/6 strip come to the front. The Student Leader writes 4/6=1/6+1/6+1/6 +1/6 and the class reads this equation.”

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 7, includes a Path to Fluency,  students rewrite horizontally written multi-digit addition problems to line up the place values vertically before adding.
• Unit 3, Fluency Check 7, students develop fluency with multi-digit addition and subtraction.
• Unit 6, Lesson 5, students “Practice Addition and Subtraction with Fractions Greater Than 1.” In Problems 1-14, students add and subtract fractions presented both horizontally and vertically.

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 4 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 the Student Activity Book, Unit 5, Lesson 3, students solve contextual problems regarding elapsed time. “Kevin baked a cake. He started making the cake at 8:03 p.m. It took him 1 hour and 17 minutes to finish making the cake. What time did Kevin finish making the cake?”

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 3, Lesson 2, Anytime Problem, “Jana buys a pack of stickers that she wants to share with her friends. She divides the stickers into 5 equal groups for herself and 4 friends, and gives herself any remaining stickers from the pack. If each friend gets 22 stickers and Jana ends up with 25 stickers, how many stickers were in the pack?” Students are applying mathematics by using the four operations to solve a multi-step word problems with whole numbers. 

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

• Unit 3, Lesson 10, Student Activity Book, “The parents ordered pizzas to serve at the carnival. Each pizza was cut into 8 slices. How many pizzas had to be ordered so that 1,319 people could have one slice?”
• Unit 4, Lesson 4, Student Activity Book, students are instructed to write an equation and draw a model if needed to solve each problem. “Audrey has 1,263 centimeters of fabric, and that is 3 times as much fabric that she needs to make some curtains. How many centimeters of fabric does Audrey need to make the curtains?” 
• Unit 6, Lesson 3, Student Activity Book, “A puppy is now 5 weeks old. It has gained 8/16 pound since it was born. The puppy weighs 11/16 pound now. How much did the puppy weigh when it was born?”
• In Unit 6, Lesson 6, Student Activity Book, students write equations and solve to answer story problems. “The width of a rectangle is 3 5/6 inches. The length is 1 4/6 inches longer than the width. What is the length of the rectangle?”
• In Unit 6, Lesson 10, students use fractions and mixed numbers to solve word problems. “What fraction of the farm is not made up of wheat?” Students use their answer and compare it with another fraction related to the farm using an explanation to describe which fraction is bigger.

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 14, Remembering, Stretch Your Thinking, Exercise 8, “Kia is printing packets of information. There are 23 pages in a packet, and she needs enough copies for 52 people. Each package of paper contains 200 sheets. She estimates she needs 5 packages of paper to print the packets. Will she have enough paper? Explain.”
• Unit 4, Lesson 12, Remembering, Stretch Your Thinking, Exercise 10, “For a cookie exchange, Kaiya bakes 2 pans of 12 chocolate chip cookies each, 3 pans of lemon drops each, and 4 pans of 10 peanut butter cookies each. She is dividing the cookies into 8 tins, with an equal number of each type of cookie in each tin. How many of each type of cookie will be in each tin? How many cookies in all will be in each tin? Explain.”
• Unit 7, Lesson 3, Remembering, Stretch Your Thinking, Exercise 6, “Raylene made a bracelet with 28 beads. She also made a necklace with twice the number of beads as the bracelet. If 1/2 of the beads on the bracelet are green and 1/4 of the beads on the necklace are green, does the bracelet, the necklace, or neither have more green beads? Explain.” 

The instructional materials for Math Expressions Grade 4 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 7, Performance Task, Problem 2, “Jenny thinks that on Tuesday about 1/2 of the bicycle rack was used. Jacob thinks about 1/2 of the rack was used on Wednesday. Who is correct? Use a diagram or number to justify your answer.” The table shows 7 bicycles on Tuesday and 4 bicycles on Wednesday.
• Fluency Checks are included throughout the series, where students practice procedural skills and fluency. For example, Unit 2, Fluency Check 3, students add or subtract problems presented in vertical form. Problem 4, “97,532 + 55,722;” and Problem 6, “88,526 - 79,613.”
• Application problems are embedded into practice in the Student Activity Book. For example, Unit 2, Lesson 15, Problem 11, “Brian is buying T-shirts for the marching band. He knows that at parades the band forms 24 rows. Each row has 13 students. If T-shirts come in boxes of 100, how many boxes of T-shirts should Brian buy?”

Examples where student engage in multiple aspects of rigor:

• Unit 3, students work with multi-digit division. In the Student Activity book, Unit 3, Lesson 2, “The area of the new rectangular sidewalk at the mall will be 3,915 square feet. It will be 9 feet wide, how long will it be?” Students are told to practice the place value section method. During Problem Solving with Three-Digit Quotients, students solve application problems using the expanded notation method for division. Problem 9, “The convention center is expecting 1,434 people for an event. Since each table can seat 6 people, how many tables will the convention center need to set up?”
• Unit 5, students explore the system of metric units of length. In Lesson 2, students engage in procedural skill and fluency as they apply their understanding of metric units of liquid volume and mass to solve real world problems. For example, Student Activity Book, Problem 20, “A race is 5 miles long. Complete the table. How many feet are equal to 5 miles?”
• Unit 6, Lesson 6, Activity 1, students add and subtract mixed numbers with fractions using procedures. Question 31, “A pitcher contains 4 3/8 cups of juice. Antonio pours 5/8 cups into a glass. How much juice is in the pitcher?”

The instructional materials reviewed for Math Expressions Grade 4 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 2, “Using the Common Core Standards for Mathematical Practices” illustrates how MP3 is used in Lesson 2-9 and Lesson 2-14.  
• 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 5, MP1 - Make Sense of Problems I Act It Out, The class discusses the problem and students work together in groups using Math Boards to compare the numbers 101,538 and 101,835 using inequality symbols. Students share their work as classmates compare the numbers using place value.
• Unit 2, Lesson 7, MP4 - Model with Mathematics I Write an Expression, Exercise 6, “There are 9 members on the school’s golf team. Each golfer hit a bucket of 68 golf balls at the driving range. How many golf balls did the entire team hit?” Students are directed to “draw an area model and use the Algebraic Notation Method to solve the problem.”
• Unit 3, Lesson 5, MP8 Use Repeated Reasoning | Generalize, “Ask students to write an equation for checking a quotient that includes the variables q for quotient, d for divisor, p for dividend, and r for the remainder."
• Unit 5, Lesson 5, MP6 - Attend to Precision/ Explain a Solution, Students work in small groups to complete exercises 3-5. Students “discuss how they completed a table,” and “then apply their reasoning to describe the pattern” found in the table.
• Unit 7, Lesson 2, MP7 Look for Structure | Identify Relationships. “Assign students to work in Student Pairs to complete Exercises 12 and 13. Again, emphasize that the number for a point is the total distance from 0 to that point. In Exercise 12, students should notice that the fractions or mixed numbers in each column name the same point on the number line. Some students may also notice that for each fraction in the second row, the numerator and denominator are three times the fraction in the first row. Some may even recognize that they could use this pattern to create additional fractions that name the same number.”

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 4 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 14, students have to determine a position for the following statement: “When you add two whole numbers, the sum will always be greater than each of the two addends.” 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 1, Lesson 19, students determine a position on the following statement: “The sum of two mixed numbers can be a mixed number or a whole number, but not a fraction less than 1.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
• Unit 5, Lesson 8, students decide if the following statement is true or false and develop an argument supporting their position. “In the conversion 1 L = ____ mL, the number of milliliters will always be less than the number of L.” 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 long division problem.
• Unit 4, Lesson 5, Puzzled Penguin problem, students identify the error the penguin made in a multiplicative comparison problem.
• Unit 8, Lesson 2, Puzzled Penguin problem, students determine the error in an angle measurement.

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 4, Remembering, Stretch Your Thinking, Exercise 5, “Jenna divides 2,506 by 4. Explain the error in Jenna’s solution. Then show the correct solution.”
• Unit 5, Lesson 1, Remembering, Stretch Your Thinking, Exercise 13, “Kyle says the number is greater when an object is measured in centimeters than in millimeters. Is Kyle correct? Explain.”
• Unit 7, Lesson 4, Remembering, Stretch Your Thinking, Exercise 5, “Omar cuts a pizza into 4 slices and takes 3 of the slices. He says that he would have the same amount of pizza if he cut the pizza into 8 slices and takes 6 of the slices. Paul says he can cut the pizza into 16 slices and take 12 slices and have the same amount. Who is correct? Explain.”

The instructional materials reviewed for Math Expressions Grade 4 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 2, Lesson 1, the terms area, array, area model, and square unit are the identified vocabulary terms. In the Math Talk for this lesson students have the opportunity to use the vocabulary terms. In the following lesson there are no additional vocabulary terms identified. Within Lesson 2, students are encouraged to use the term square units; however, this is the only term specifically carried forward from Lesson 1.
• Unit 2, Lesson 3, students use the identified vocabulary of factor and product within the Math Talk portion of the lesson.   
• Unit 3, Lesson 1, divisor, quotient, dividend, and remainder are the identified vocabulary terms. Students use remainder, quotient, and divisor, but the term dividend is rarely used. In Lesson 2, the materials continue to build student understanding of multiplication and division, but students are not provided opportunities to engage with the relevant division vocabulary introduced in Lesson 1. In Lesson 3 teachers are informed, “Students should be able to verbalize and define the terms divisor, dividend, and quotient. Students should also be able to identify these as parts of a division problem.” While this is noted in the materials, opportunities to reinforce students use and understanding of the vocabulary is not specifically called for in the Teacher’s Edition.

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

• Unit 6, 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, “The sum of two mixed numbers can be a mixed number or a whole number, but not a fraction less than 1.”
• Unit 7 Lesson 13, “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, “Any fraction with a denominator of 100 can be rewritten as an equivalent fraction with a denominator of 10.”