Alignment: Overall Summary

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for alignment to the CCSSM. The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and the materials connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

See Rating Scale
Understanding Gateways

Alignment

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Meets Expectations

Gateway 1:

Focus & Coherence

0
7
12
14
13
12-14
Meets Expectations
8-11
Partially Meets Expectations
0-7
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
10
16
18
16
16-18
Meets Expectations
11-15
Partially Meets Expectations
0-10
Does Not Meet Expectations

Usability

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Meets Expectations

Not Rated

Gateway 3:

Usability

0
22
31
38
38
31-38
Meets Expectations
23-30
Partially Meets Expectations
0-22
Does Not Meet Expectations

Gateway One

Focus & Coherence

Meets Expectations

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Gateway One Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

Criterion 1a

Materials do not assess topics before the grade level in which the topic should be introduced.
2/2
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for not assessing topics before the grade level in which the topic should be introduced. The materials assess grade-level content and, if applicable, content from earlier grades.

Indicator 1a

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.
2/2
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Indicator Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations that they assess grade-level content.

The assessments are aligned to grade-level standards and do not assess content from future grades. The Grade 4 Assessment Guide includes a Beginning of Year Test, Middle of Year Test, End of Year Test, and tests for each Unit. Each Unit Test includes multiple choice, multiple-select, short answer, constructed response, and a separate performance task assessment. The materials include a form A and form B assessment for each unit.

Digitally available assessments are PARCC and Smarter Balanced aligned practice tests. Each digital platform includes a variety of practice tests. Digital assessments assess grade-level content.

Examples of on-grade level assessment items include:

  • Unit 6, form A, item 4, “Dakota mixes flour and oats for a recipe. He uses 5/6 cup of flour. This is 4/6 cup more than the amount of oats he uses. How many cups of oats does Dakota use? Write an equation. Then solve.” ( 4.NF.3d)     
  • Unit 7, Performance Assessment, Items 1 and 2, “Taxicabs downtown charge a flat fee of $2.50, plus a state tax of $0.50, plus $0.50 for each 1/5 mile they travel with a passenger. 1.) Explain how you could write 1/5 mile as a fraction with 10 in the denominator, a fraction with 100 in the denominator, and as a decimal. 2.) Travis took a taxicab from his office to a meeting on the other side of town. The cab ride cost $8.50, which includes a $2.00 tip. In decimal form, how far is it from his office to the meeting? Show your work.” (4.NF.5, 4.NF.6, 4.NF.3a)
  • Grade 4, Smarter Balanced Test Prep Practice Test, Item 3, “Frank has two same-sized rectangles divided into the same number of equal parts. One rectangle has 3/4 of the parts shaded, and the other has 1/3 of the parts shaded. Part A: Into how many parts could each rectangle be divided? Show your work by drawing the parts of each rectangle and shading the correct amounts.” (4.NF.1)
  • Grade 4, Middle of Year Test, Item 27, “Ron buys 2 1/4  pounds of chicken and 1 3/4 pounds of beef. How many more pounds of chicken does he buy than beef?” (4.NF.3c.)
  • Grade 4, End of Year Test, Item 15, “There were 12,318 tickets sold at a stadium last weekend. There were 12,584 tickets sold this weekend. How many tickets were sold in all?” (4.NBT.4)

Criterion 1b

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.
4/4
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade.

Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.
4/4
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Indicator Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for spending a majority of instructional time on major work of the grade.

  • The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6 out of 8, which is approximately 75%.
  • The number of Big Ideas, CCSSM clusters, devoted to major work of the grade (including assessments and supporting work connected to the major work) is 18 out of 25, which is approximately 72%.
  • The number of lessons devoted to major work (including assessments and supporting work connected to the major work) is approximately 89 out of 115, which is approximately 77%.

A lesson level analysis is most representative of the instructional materials as the lessons include major work, supporting work, and the assessments embedded within each unit. As a result, approximately 77% of the instructional materials focus on major work of the grade.

Criterion 1c - 1f

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
7/8
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for being coherent and consistent with the standards. The instructional materials have content designated for one grade level that is viable for one school year; are consistent with the progressions in the standards; and foster coherence through connections at a single grade.

Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
1/2
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Indicator Rating Details

The instructional materials reviewed for Math Expressions Grade 4 partially meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials include some lessons where supporting work enhances and supports the major work of the grade. However, there are lessons that include missed opportunities for supporting work to enhance the major work of the grade.

Examples of connections between supporting work and major work include the following:

  • In Unit 7, Lesson 7, Student Activity Book, page 319, students work with a line plot that includes fractions between 0 and 1. Students use the line plot to answer a series of problems (4.MD.B). These problems support students work with solving word problems involving adding and subtracting fractions with like denominators (4.NF.B). Problem 4, “How much less sugar is in a recipe with the least sugar than in a recipe with the most sugar?” Problem 5, “Mateo wants to try all the recipes with exactly 5/8 cup of sugar. How much sugar does he need?”
  • In Unit 7, Lesson 11, students solve problems involving measurement and conversion of measurements (4.MD.A) which supports their work with understanding decimal notation (4.NF.C). In the Student Activity Book, Problem 21, students solve problems by representing measurements using decimals. “There are 100 centimeters in 1 meter. A snake crawls 3 meters and 12 more centimeters. What decimal represents the number of meters the snake crawls?”

There are missed opportunities in Unit 5 to make connections between supporting work and major work. For example:

  • In Unit 5, Review/Test, Problem 13, students analyze a line plot (4.MD.B) and answer the question, “How many classmates did Jill ask about the time spent sleeping?” The opportunity to add and subtract fractions is a missed connection (4.NF.B.3).
  • In Unit 5, Lesson 3, students use a table to complete a line plot in fractions of a unit (4.MD.4). However, they are not required to add or subtract fractions after making the line plot (4.NF.B.3).

Supporting standard 4.OA.4 is taught in isolation. Students work with this standard miss the opportunity to connect to 4.NBT.B and 4.NF.A. This connection would support students' work with multi-digit arithmetic and fraction equivalence. In Unit 7, Lesson 4, Problem 2, students respond to the problem, "Maria said, 'You are just fracturing each third into 4 twelfths. You can show what you did using numbers.' Here's what Maria wrote: $$\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}$$. Discuss what Maria did. How does multiplying the numerator and denominator by 4 affect the fraction?" This problem could have connected to 4.OA.B.4 by having the student identify the factor pairs of 8 and 12 in order to determine if this is a viable solution as an intermediary step to determine if multiplying the numerator and denominator by 4 leads to a viable solution.

Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that the amount of content designated for one grade-level is viable for one year.

As designed, the instructional materials can be completed in 150 days. The Pacing Guide can be found on page I18 in the Teacher Edition. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications.

  • The program is designed with eight units and 99 lessons. Most lessons require one day.
  • The Pacing Guide notes 9 lessons that could take two days, but this is not noted in the Day at a Glance for each lesson.
  • All Units designate two days for Unit Assessments.  
  • The instructional materials consist of 25 days of Quick Quizzes and Strategy/Fluency Checks which are listed in the Pacing Guide.
  • Unit 1 designates one day for the Prerequisite Skills Inventory Test.

Teachers start each lesson with a 5-minute Quick Practice and each lesson is comprised of several activities with estimated time ranging from a total of 55-65 minutes per lesson. Math Activity Centers are tailored for all levels of achievement across readiness and learning styles. They can be completed within the lesson or after, however, the time required for the activity is unstated.

Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations for the materials being consistent with the progressions in the Standards. Content from prior and future grades is identified and connected to grade-level work, and students are given extensive work with grade-level problems.

The materials clearly identify content from prior and future grades and connect concepts to grade level work. Each unit includes a Unit Overview providing a Learning Progression. The Learning Progression states connections between the standards of the prior grade, current grade, and future grade. Additionally, each unit contains a Math Background Section. This section contains in depth information for the teacher articulating the learning progressions and the progression of the content between lessons. For example:

  • Unit 1, the Learning Progression chart makes connections between Grade 3, Grade 4, and Grade 5 within the Numbers and Operations in Base Ten Domain as it relates to place value and multi digit addition and subtraction.
  • Unit 5, Math Background quotes from the math Progressions Documents for Measurement/Data. For example, the Math Background for Lesson 3 connects information about elapsed time for the current grade to prior grade. “In the previous grade, students found elapsed time in hours and minutes and used these skills to solve real world problems.”
  • Unit 6, Math Background quotes from the Progressions Documents for Number and Operations Adding Fractions. “This simple understanding of addition as putting together allows students to see in a new light the way fractions are built up from unit fractions. The same representation that students used in Grade 3 to see a fraction as a point on the number line now allows them to see a fraction as a sum of unit fractions: just as 5 = 1 + 1 + 1 + 1 +1, so 5/3 = 1/3+1/3+1/3+1/3+1/3 because 5/3 is the total length of 5 copies of 1/3. Armed with this insight, students decompose and compose fractions with the same denominator. They add fractions with the same denominator.”

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. Within each lesson, students practice grade level problems within Quick Practice, Student Activity Book pages, Homework, and Remembering activities. During modeled and guided instruction, students are given opportunities to engage in the grade level work by doing various examples with teacher and peer support. The independent practice in the Student Activity Book corresponds with the lesson and provides students the opportunity to work with grade level problems to extend concepts and skills. For example:

  • Unit 3, Lesson 1, students discuss and complete activities illustrating the relationship between multiplication and division and are introduced to division with remainders. They use patterns in multiplication with zeros to divide numbers with zeros. In the Student Activity Book, problem 13, “What pattern do you notice when you multiply with zeros?”. Students practice solving division problems including those with patterns of zeros. Students relate the quotient to a multiplication problem to check the division work. (4.NBT.6)
  • Unit 7, Lesson 1, students use prior knowledge of unit fractions to discuss comparing fractions with different denominators. “Encourage students to generalize what they have learned about comparing two fractions with different denominators and the same numerator. Just as with unit fractions, students should be able to reason that for fractions that have the same numerator, the fraction with the lesser denominator is greater.” Students discuss, compare, and order unit fractions using visual models and fraction reasoning. In Activity 2, students determine which fraction is greater, use inequality symbols to compare fractions, and construct a viable argument in the “What’s the Error?”portion of the lesson. (4.NF.2)

Each lesson contains Math Center Activities, as well as Homework and Remembering (spiral reviews) pages which provide additional practice with grade-level problems. For example:

  • Unit 6, Lesson 6, Homework, students write mixed numbers as fractions, add and subtract mixed numbers and fractions, and answer two real world problems involving addition and subtraction of mixed numbers.
  • Unit 7, Lesson 6, Remembering, students write number sentences to convert metric measurements between units, complete addition equations with a missing fraction addend, simplify fractions, and compare three fractions in a real world context.

Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Each unit is structured by specific domains and Big Ideas. Learning objectives within the lessons are clearly shaped by CCSSM cluster headings. For example:

  • Unit 1, Big Idea 1, “Place Value to One Million.” This Big Idea is shaped by 4.NBT.A, “Generalize place value understanding for multi-digit whole numbers.” Examples of lesson objectives in this section include, “Students will learn to identify the place value of numbers through thousands, students will learn how to read, write, and model numbers to a thousand, and students will learn to round and compare multi-digit whole numbers.”
  • Unit 4, “Equations and Word Problems” is shaped by 4.OA.A, “Use the four operations with whole numbers to solve problems.” Students use the four operations with whole numbers to solve equations and multi-step problems.
  • Unit 6, Lesson 8 learning objective states, “Students will solve problems involving multiplying a fraction by a whole number.” This is shaped by 4.NF.B, “Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.”

Materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. For example: 

  • Unit 1, Lessons 6 and 13 connect two domains, 4.NBT.B and 4.MD.A, when students use denominations of money to build addition and subtraction fluency.
  • Unit 3, Lesson 9 connects two domains, 4.OA.A and 4.NBT.B, when students interpret remainders of multi-digit division problems in a variety of ways. In the Student Activity Book, students solve, “Henry’s coin bank holds only nickels. Henry takes $4.42 to the bank to exchange for nickels only. How many nickels will he get from the bank?”
  • Unit 7, Lesson 6 connects two clusters within a domain, 4.NF.A and 4.NF.C, when students compare fractions with unlike denominators using <, >, and = including some of the following: 4/5 __ 75/100, 3/4 ___ 8/10.

Gateway Two

Rigor & Mathematical Practices

Meets Expectations

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Gateway Two Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for Gateway 2, rigor and balance and practice-content connections. The instructional materials meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and they meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

Criterion 2a - 2d

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
8/8
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations, by giving appropriate attention to developing students’ conceptual understanding and procedural skill and fluency. The instructional materials also do not always treat the aspects of rigor separately or together.

Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
2/2
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Indicator Rating Details

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.

Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
2/2
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Indicator Rating Details

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.

Indicator 2c

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
2/2
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Indicator Rating Details

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.” 

Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
2/2
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Indicator Rating Details

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?”

Criterion 2e - 2g.iii

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
8/10
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified and use accurate mathematical terminology. The instructional materials also partially support teachers and students in students constructing viable arguments and analyzing the arguments of others.

Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
2/2
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Indicator Rating Details

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.

Indicator 2f

Materials carefully attend to the full meaning of each practice standard
1/2
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Indicator Rating Details

The instructional materials reviewed for Math Expressions Grade 4 partially meet expectations for carefully attending to the full meaning of each practice standard. 

The materials do not attend to the full meaning of Mathematical Practice 5. 

Mathematical Practice 5: The instructional materials do not meet the full meaning of MP5 as tools are chosen for students, and there are few opportunities for students to choose tools strategically. For example:

  • Unit 4, Lesson 10, MP5 Use Appropriate Tools | Concrete Model. “Discuss the definition of prime number and composite number on Student Activity Book page 192. Start Exercises 8-9 by giving pairs of students 24 counters and have them model arrays for all the factor pairs of 24. Ask students how they know if  a number is prime or composite. Discuss with students that 1 is neither prime or composite.” 
  • Unit 7, Lesson 1, Compare Fractions, students use a chart, Understand Fractions of 1, to find equivalent fractions between 0 and 1, with fraction bars starting at 1/1 and showing equivalent fraction bars up to 20/20. 
  • Unit 8, Lesson 4, Activity 1, MP 5 Use Appropriate Tools | Use a Straightedge. “Students can draw specific types of triangles by starting with an angle. Then students draw the third side to complete the triangle.” Students do not choose a tool, and there is no guidance for teachers on the use of the straightedge in relationship to this Activity.

Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
0/0

Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
2/2
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Indicator Rating Details

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.”

Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
1/2
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Indicator Rating Details

The instructional materials reviewed for Math Expressions Grade 4 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Overall, the teacher materials provide students multiple opportunities to construct viable arguments, however there are missed opportunities to support teachers in engaging students in analyzing the arguments of others throughout the materials.

Throughout the Teacher Edition, the MP3 is identified with explanations and guidance for teachers, either in reference to specific parts of the lesson, or in specific activities such as Math Talks. However, this guidance often supports teachers to engage students in explaining their methods, instead of constructing arguments or critiquing reasoning. For example:

  • Unit 2 Lesson 1, MP3 Construct a Viable Argument |  Compare Models. “Refer students to the models at the top of Student Activity Book page 53. Ask students to describe the characteristics of the first model. Students should see the arrangement of square tiles as an array. Invite students to describe the similarities and differences between the first model and the second model. Students should see that pushing the square tiles together does not change the number of squares in the array, but it does turn the array into a rectangle. Students should see that the third model represents the same rectangle as the second model, but without the square tiles. Tell students that this model is called an area model.” There is no guidance to engage students with MP3.
  • Unit 4, Lesson 8, Problem 1 has students compare the methods used by two characters to solve a multi-step problem. MP3 Construct Arguments | Compare Methods. “Invite students to complete Nicole’s and David’s methods on Students Activity Book page 183. Encourage students to notice that although the methods are different, the solution is the same. Invite students to compare and contrast the two methods. Encourage students to see how the steps in Nicole’s method appear in David’s equation. For Nicole’s method, make sure students notice that first and second steps answer the helping questions identified earlier in the discussion. The last step directly addresses the question in the word problem. For David’s method, help students see that the equation is solved by following the Order of Operations. You would complete the division first in this equation, so there is no need for parentheses.” Students do critique the reasoning; however, there are not explicit prompts to help students see the differences and similarities.

Examples of materials assisting teachers in engaging students in constructing viable arguments:

  • Unit 4, Lesson 5, Puzzled Penguin, What’s the Error, the penguin uses a multiplication equation to solve a division situation. Teachers are provided prompts to engage students in constructing an argument and critiquing the Puzzled Penguin’s work. “Is this and addition comparison problem or a multiplication comparison problem? Explain how you know. What type of equation can you write to solve a multiplication comparison problem? Puzzled Penguin wrote the equation 81 x 9 = s. What was Puzzled Penguin’s mistake? What equation can you write to solve this problem?”
  • Unit 6, Lesson 10, Analyze a Statement, Establish a Position, MP3 Construct a Viable Argument, “Students should establish an arguable position by writing or stating sentence that support a specific point of view. They should give an equation as an example. Students analyze the statement: “The sum of two mixed numbers can be a mixed number or a whole number, but not a fraction less than 1.”

There are instances where MP3 is identified in A Day at a Glance for a lesson, but there is no guidance for teachers on how to engage students to construct arguments or analyze the arguments of others.

Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.
2/2
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Indicator Rating Details

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.”  

Gateway Three

Usability

Meets Expectations

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
8/8
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for being well-designed and taking into account effective lesson structure and pacing. The instructional materials include an underlying design that distinguishes between problems and exercises, assignments that are not haphazard with exercises given in intentional sequences, variety in what students are asked to produce, and manipulatives that are faithful representations of the mathematical objects they represent.

Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials distinguish between problems and exercises.

Materials provide the opportunity for students to learn new mathematics through problem solving activities. In a typical lesson, Activity 1 and Activity 2 develop the new math content of the lesson. Lessons are outlined according to an Inquiry Lesson Path based on four phases: Phase 1 Guided Introduction, Phase 2 Learning Unfolds, Phase 3 Knead Knowledge (practice stage), and Phase 4 Maintaining and Integrating Fluency. Students build mastery through practice problems/exercises. In a typical lesson, during Activity 2 and Activity 3, students complete problems in the Student Activity Book which provide practice with the math content. The purpose of each Activity within a unit is explained in the “Teaching the Lesson Section” found on the first page of each lesson.

Examples include but are not limited to:

  • In Unit 5, Lesson 2, Teaching the Lesson Section, Activity 1, Metric Units of Liquid Volume, is stated as important because “Students develop a sense of the relative sizes of metric units of liquid volume.” Activity 2, Metric Units of Mass, is stated as important because “Students develop a sense of the relative sizes of metric units of mass.”  Activity 3, Real World Problems with Metric Measurements, is stated as imortant because “Students apply their understanding of metric units of liquid volume and mass to solve real world problems.”
  • In the Student Activity Book, Unit 3, Lesson 5, Activity 1, students discuss word problems from Lesson 4 homework: “A fruit stand sells packages containing 1 peach, 1 pear, 1 apple, 1 banana, and 1 mango each. One week they sold a total of 395 pieces of fruit. How many packages did they sell?” In Activity 2, students explore division with 4-digit dividends: “Use any method to solve. 3,248 ÷ 5 =.”

Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide tasks in an intentional sequence.

The design of the assignments follows a natural progression, leading to full understanding and mastery of new mathematics. Lessons follow a consistent pattern of two or three activities per lesson. Activity 1 usually focuses on the new learning. This learning is reinforced in Activity 2, and then students practice the new learning by completing Student Activity Book pages during Activity 3. Activity 3 either reinforces the new skill, or it reviews previously learned content.

Examples include but are not limited to:

  • In Unit 1, Lesson 6, students share addition methods for adding 3-digit whole numbers. Then students practice using various methods of addition in their Student Activity Book: New Group Above, New Groups Below, Subtotal Left to Right, and Subtotal Right to Left.
  • In Unit 4, Lesson 1, students use expressions to simplify and solve equations. Then in the Student Activity Book, students use the identity property to simplify expressions and solve equations using the associative property.
  • In Unit 6, Lesson 4, students practice adding and ordering unit fractions. They discuss uses for mixed numbers and then convert between mixed fractions and fractions greater than one.

Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide varied opportunities for students to present their mathematical knowledge.

Examples of how students produce answers and solutions include but are not limited to:

  • Using Math Mountains to put together and take apart numbers
  • Using Arrays and Area Models to solve multiplication problems
  • Using drawings to make sense of mathematics
  • Using Place Value Charts to compare numbers
  • Providing thinking explanations as they answer Check for Understanding questions in the Student Activity Book
  • Completing fluency checks and practice in the Student Activity Book 
  • Critiquing the Reasoning of others by asking “good thinker questions” and using “good justifications” 
  • Practicing “good explanations” 
  • Identifying the error and correcting it (Puzzled Penguin)
  • Solving problems and exercises in the Student Activity Book

Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide virtual and physical manipulatives that are faithful representations of the mathematical objects they represent and are connected to the written material.

Students use a variety of manipulatives including MathBoards, Secret Code Cards, base-ten blocks, Square Inch Tiles, fraction bars, number lines, Math Mountains, and Math Mountain Cards. Most of the manipulatives are available virtually in the itools found in ThinkCentral. Manipulatives are often connected to written methods when appropriate.

Examples include but are not limited to:

  • Unit 1, Lesson 4, students use Secret Code Cards to build numbers to the millions place then write in standard, expanded, and word form. 
  • Unit 2, Lesson 1, students use MathBoards to make area models to write and solve multiplication problems. 
  • Unit 5, Lesson 1 students make and use meter strips to help them understand and convert metric units of length.

Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the mathematics.

Student Activity Book pages include many exercises per page, but they follow a consistent layout and do not feel cluttered because there are no extra and unnecessary pictures on the pages. Additionally, students are provided ample space to show their work. When needed, models, which are consistent with the materials used in the lesson, are included on the pages. For example, on Student Activity Book page 211, a meter strip is shown, and students use it to convert metric units of length.

In the Teacher Guide, lessons follow a consistent layout, moving from one activity to another. Each Activity includes a large blue box that highlights the mathematical content and practice standards, the focus of the lesson, and materials needed. Parts of the lesson, such as MathTalk, are clearly labeled. For example, in Unit 5, Lesson 3, a MathTalk in Action box shows examples of how students might share their methods for determining elapsed time.

The digital interactive game, Poggles, includes simple, appealing characters that do not distract students as they practice addition and subtraction. Poggles are small squarish characters with animated faces whose appearance can be changed by adding hair and hats to the Poggle squares.

Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
8/8
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for supporting teacher learning and understanding of the CCSSM. The instructional materials include: quality questions to support teachers in planning and providing effective learning experiences, a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials, a teacher edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons, and explanations of the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Examples of teacher support include but are not limited to:

  • Questions for teachers to pose are consistently included in the lesson narrative. They are italicized, making them easily visible. 
  • MathTalk in Action boxes include questions for the teacher to ask and potential student responses. For example, in Unit 4, Lesson 3, the teacher is guided to ask the questions: “What do we know about the problem? What do we need to find?”
  • Teacher Notes are also provided at the bottom of the lesson pages and include questions to deepen students understanding of the mathematics. For example, in Unit 6, Lesson 8, Inquiry Notes state, “As students discuss their work, ask questions like ‘How did you decide what model you would use to represent the problem? Why is an equation an efficient way to represent a problem? Do you see another way to solve the problem? How can you check the answer to the problem?’”
  • Teacher Notes at the bottom of lesson pages also include information about best practices to strengthen teachers questioning techniques. For example, in Unit 1, Lesson 3, the Inquiry note provides the following directions concerning using questions: “The Math Talk sample discussions are intended to provide ideas that enhance an inquiry approach to learning in your classroom. Use questions like those in the sample discourse to guide discussions of mathematical concepts and skills.”

Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials contain a teacher’s edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Materials also, when necessary provide teacher guidance for the use of embedded technology to support and enhance student learning.

Ample guidance is provided in the Teacher Guide for planning. The Pacing Guide provides guidance for each unit. Charts show the Learning Progression for the Content Standards Across Grades for the standards addressed in the Unit. A Planning Chart for each Unit that includes Math Activity Center Resources, Big Idea Resources, and Lesson Resources is provided. The Planning Chart also includes the standards addressed in each lesson, the digital and print resources for each lesson, and the assessments for the Unit. A table of the Standards for Mathematical Practice and the lessons where each is embedded is included. Also, a Table of the Math Content Standards and the lessons where they are taught is provided. Finally, a list of Assessment, Review, and Intervention Resources for the Unit is provided.

Examples include but are not limited to:

  • Each lesson includes guidance on the focus of each Activity and why it is important. For example, in Unit 1, Lesson 4, Activity 2, Understand Greater Numbers, is stated as important because “Students need to read and write numbers to one million in standard form, word form, and expanded form.” 
  • Each Activity includes an explanation of what the teacher should do or say and includes possible correct responses to questions posed by the teacher. 
  • Formative Assessment and Check for Understanding questions are highlighted in the Teacher Guide.
  • Math Practices are highlighted in the lesson narratives.
  • A list of questions that can be used to build a Math Talk community is included at the beginning of each Unit. 
  • Notes at the bottom of each page of the lesson narrative give useful suggestions for implementing the lesson, asking questions, acquiring vocabulary, and building concepts. For example, in Unit 5, Lesson 5, the Teaching Notes for Watch For! states, “Students may confuse an ounce, a unit of weight, with a fluid ounce, a unit of liquid volume. A fluid ounce is often shortened to ounce when the context is known.”
  • Digital Resources for each lesson are highlighted on the first page of the lesson, and itools, which include virtual manipulatives, are shown in the lesson narrative when it may be beneficial to use them. For example, in Unit 4, Lesson 10, a picture of itools Counters are shown because they may be used in the lesson.

Indicator 3h

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials contain a teacher’s edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

Notes are provided at the bottom of each lesson narrative in the Teacher Edition to deepen teacher understanding of the mathematics and to improve instruction. Math Background Notes provide information about the math topic to deepen teacher’s understanding. Watch For! Notes provide information about potential misconceptions and things to watch for as students complete the lesson. What to Expect from Students notes provide information about how students might engage with the math and why the math is important. Building Concepts notes provide explanations of the math and how students learn.

Examples include but are not limited to:

  • Path to Fluency Charts are provided.
  • Chart of the Addition/Subtraction and Multiplication/Division problem types is provided.
  • Table of the Major Work and Major Clusters of the Grade is provided.
  • Table of the Common Core State Standards for Mathematical Content is provided.
  • Table of the Common Core State Standards for Mathematical Practice with an explanation for each Mathematical Practice is provided.
  • The Putting Research into Practice section at the beginning of each unit provides research about best practices in teaching children mathematics.
  • The Math Background section, prior to each unit, includes sections that deepen teacher knowledge of the math in the unit. Examples include Learning Path in the Common Core Standards, Help Students Avoid Common Errors, Effective Practice Routines, Relate Mathematics to the Real World, and Focus on Mathematical Practices. 
  • The Math Background section, prior to each unit, provides excerpts from the Progressions for the Common Core State Standards.
  • The Mathematical Practices section, prior to each unit, provides information on how students will engage with the Practice Standards throughout the unit.
  • A Teacher Glossary is provided.

Indicator 3i

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials contain a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series.

A Path to Fluency: Kindergarten through Grade 6 Chart is provided and highlights the fluency requirements of each grade level, activities that target fluency, and interventions for Grades 3, 4, 5, and 6. Also, a Major Work and Major Clusters of the Grade Chart for Grades K-6 is provided. Finally, for each unit, a Learning Progressions for the Common Core State Standards Chart for the domains addressed in the unit, which includes the current, prior, and next grade level standards is provided.

Indicator 3j

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
0/0
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 provide a list of lessons in the teacher's edition, cross-referencing the standards addressed and providing an estimated instructional time for each lesson, chapter, and unit.

Math Expressions does not include chapters, but rather units which are divided by Big Ideas, which are further divided into lessons. The Pacing Guide provides estimated instructional time for lessons and units. This Pacing Guide provides an estimated number of days for each unit, including lessons that may take two days and the number of days for assessments and quizzes. It should be noted that Lessons identified as taking two days in the Pacing Guide are not identified in the lesson narratives, nor is a breaking point indicated.

Examples include but are not limited to:

  • The Table of Contents provided in the introduction to the materials includes standards for all units’ Big Ideas.
  • The Chart of the Common Core State Standards for Mathematical Content provided identifies the lessons in which each standard will be addressed. 
  • The Chart of the Common Core State Standards for Mathematical Practice provided identifies the lessons in which each Mathematical Practice will be addressed. 
  • A Planning Chart is provided in the Overview for each unit that includes the standards that are addressed in each lesson. 
  • Charts of the Math Content Standards and Math Practice Standards are provided in the Overview for each unit. These charts include a list of each standard and the lessons where they are addressed. 
  • The Content and Practice Standards are identified on the first page of each lesson. The standards are also listed for each Activity within a lesson.

Indicator 3k

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
0/0
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Family Letters for each unit are found in the Student Activity Book. These letters explain content, manipulatives students may use, and an explanation of terminology that may be unfamiliar to parents. Most units include between 1-3 Family Letters. Spanish versions of the letters are also included in the Student Activity Book.

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.
0/0
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The Teacher Edition contains explanations of the program’s instructional approaches and research-based strategies. An Inquiry Learning Path describes the four phases of the Math Expressions classroom: Guided Introduction, Learning Unfolds, Knead Knowledge, and Maintaining and Integrating Fluency. The Putting Research into Practice pages at the beginning of each Unit explain best practices related to the content of the Unit. Excerpts from the Progressions for the Common Core State Standards are included in the Math Background section of each Unit. Research Notes are sometimes included in the Teaching Notes at the bottom of the lesson narrative in the Teacher Edition. For example, the Teaching Notes in Unit 3, Lesson 1, Activity 1, Math Symbols state, “There are many ways to represent division. Students have seen these symbols in previous grades: 

  • 10 ÷ 2 (obelus)
  • $$\frac{10}{2}$$, (fraction bar) Some mathematicians refer to this as a vinculum because it groups digits that are to be interpreted as one unit. 
  • / (slash used on computers) 
  • 2⟌10 (long division bracket or long division symbol) Some mathematicians refer to this as a vinculum because of the horizontal bar placed above the dividend.”


Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
10/10
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for offering teachers resources and tools to collect ongoing data about student progress on the CCSSM. The instructional materials provide strategies for gathering information about students’ prior knowledge, strategies for teachers to identify and address common student errors and misconceptions, and assessments that clearly denote which standards are being emphasized.

Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
2/2
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Examples include but are not limited to:

  • The Assessment Guide contains a Prerequisite Skills Inventory Test, organized by Domains, and a corresponding Prerequisite Skills Inventory Test Correlation document. The correlation aligns each question with a description of the prerequisite skill addressed, as well as the DoK level of the question. This correlation document is formatted as a table so each student’s performance by question/skill can be recorded. The Prerequisite Skills Inventory Test is designed to be administered at the beginning of the school year.
  • When a student completes practice opportunities and tests in the Personal Math Trainer, all of the performance data and adaptive learning information follows each student to the next grade.
  • The Math Background section for each unit alerts teachers to prior knowledge opportunities. For example, in Unit 5, the Math Background for Time states, “Working with time provides opportunities to review what students have learned about fractions. It is common to relate fractions of an hour to minutes in order to determine that 1/4 hour is 15 minutes and 3/4 hour is 45 minutes. Students apply this understanding as they interpret line plots that include fractions of an hour.” 
  • Quick Practice activities at the beginning of each lesson are designed to “provide opportunities for students to call to mind their prior understanding of a topic that has already been discussed in class or to begin to build a prerequisite skill for a topic that is to come later” (Teacher Edition page I4). 
  • Quick Quizzes and Fluency Checks are embedded within the units to check understanding of Big Ideas prior to moving on to the next Big Idea instruction, and to monitor progress toward computational fluency. For example, Fluency Check 1 assesses student addition of multi-digit numbers (4.NBT.4).
  • Students take three progress monitoring assessments to assess grade level skills and concepts students have learned. The Beginning of Year test assesses concepts they will learn throughout the year, the Middle of Year Test shows progress made in the first half of the year, and the End of Year Test measures growth throughout the school year.

Indicator 3n

Materials provide strategies for teachers to identify and address common student errors and misconceptions.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide support for teachers to identify and address common student errors and misconceptions.

Examples include but are not limited to:

  • Common student errors are identified for each Unit Review/Test question along with a direction on how to help students. For example, on the Unit 3 Review/Test, if a student misses Questions 1, 11, or 18, the common error identified states, “Student does not understand how to use division methods.” Teachers are directed to “Review with students the three methods suggested in this Unit. Have students work in pairs to solve the division problems using each of the three methods.”
  • The Math Background section of each Unit provides a narrative called “Help Students Avoid Common Errors”. 
  • Puzzled Penguin activities highlight typical student mistakes and misconceptions by challenging students to find the Puzzled Penguin’s mistake and correct it. Teachers are provided questions in order to lead classroom conversations through a MathTalk format that revolve around the mistake and its correction, helping students understand the mathematics.
  • Watch For! are teaching notes periodically found in each unit. These notes alert teachers to common misconceptions they should be on the lookout for. For example, in Unit 6, Lesson 2, the Watch For! note states, “Students often misunderstand fraction drawings that have parts colored or shaded. Instead of seeing the relationship of the shaded parts to the whole, they see shaded versus unshaded parts. Classroom research has shown that having students write chains of unit fractions and identify fraction partners that make 1 whole can help them to overcome this misunderstanding.”

Indicator 3o

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Examples include but are not limited to:

  • Homework and Remembering pages provide a review of recently taught topics as well as a spiral review throughout the year. The Personal Math Trainer online platform allows students to complete homework tasks for each lesson, receive instant feedback, and  step-by-step guidance if needed.
  • Unit Review/Test and Performance Tasks for each unit are found in the Student Activity Book. The author states, You can use this Unit Review/Test as an end-of-unit review to determine if children have mastered the content of the unit. You can assess children’s knowledge with one of the forms of the Unit 1 Test in the Assessment Guide.” Teachers are provided with a Data-Driven Decision Making Table which suggests specific reteaching activities for students who incorrectly answer the correlated questions, as well as suggestions for which Standards Quiz to assign in the Personal Math Trainer which provides a personalized intervention for the student. The Performance Task includes a detailed scoring rubric which can be used to provide feedback to students.
  • The Personal Math Trainer can be used for homework practice, fluency practice, standards practice, unit pre-tests with instant feedback, and step-by-step guidance when needed. Everything a student completes in the platform helps to improve the adaptive workflow (powered by Knewton Adaptivity) for the student throughout the year.  
  • The Knewton Adaptivity, Homework with Daily Intervention and Enrichment can be used in multiple ways in the classroom. A 5-minute Warm-Up provides students with personalized review prior to the assignment. On-level and advanced students may receive less or no warm-up, as determined by Knewton. After the warm-up, the HMH pre-built assignment is given to students. A 10-minute personalized enrichment is provided for students who demonstrate mastery (95% or higher) on the assignment.  Enrichment shows students proximate, forward-looking concepts based on the assignment content.
  • Other Formative Assessment opportunities include: daily Check Understanding tasks on select Student Activity Book pages, daily observation with anecdotal notes, observations during Math Talk conversations, and analyzing student work samples and student responses in the Student Activity Book. Portfolio suggestions are also provided at the end of each unit.

Indicator 3p

Materials offer ongoing formative and summative assessments:
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Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
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Indicator Rating Details

The materials reviewed for Math Expressions Grade 4 meet the expectation for offering ongoing assessments that clearly denote which standards are being emphasized.

Examples include but are not limited to:

  • Every unit includes two versions of a Unit Assessment, Form A and Form B, found in the Assessment Guide. Both assessments provide PARCC and Smarter Balance question formats and a Standards Correlation Document which can be used to collect student performance data. This document also aligns each question to a DoK Level and Standard(s). 
  • Each unit contains a Performance Assessment which can be found in the Assessment Guide. The standards are clearly noted for the assessment as a whole, and not by specific question. 
  • There are three Benchmark Assessments (Beginning of the Year Inventory, Middle of the Year Inventory and End of Year Assessment) found in the Assessment Guide. Standards for these assessments are clearly noted on the Correlation Document and DoK Levels are noted.

Indicator 3p.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Examples include but are not limited to:

  • Scoring Guides are provided for each Unit Performance Assessment found in the Assessment Guide. Each question is assigned a point value and a rubric is provided to determine Performance Levels 0-3 based on the number of points earned. Additionally, each Performance Level is further defined on a task-specific basis and indicates specifics about student understanding to assist teachers in interpreting student work. Sample student work for each Performance Level is also provided in the Assessment Guide. 
  • Answer keys for the Unit Assessments, Form A and Form B, are located in the back of the Assessment Guide. However, no guidance or suggestions for follow-up instruction are included in the Assessment Guide. 
  • The online Personal Math Trainer can be utilized to administer Beginning, Middle and End of Year Tests, Unit Assessments, and Fluency Checks. The data from these assessments is collected and analyzed, and a Personal Study Plan is prescribed through Adaptive Workflow settings (through Knewton Adaptivity) based on the data and the mastery threshold percentage established for the assessment. The primary use is for end of the unit assessments, or to provide targeted students with occasional review, intervention, and re-assessment opportunities. Students must complete an initial assignment (test). Students who do not demonstrate mastery receive a Personal Study Plan, consisting of a personalized review and intervention assignment lasting 15 minutes. After completing the Personal Study Plan, the initial assignment is given again, but numbers in the assessment are changed.

Indicator 3q

Materials encourage students to monitor their own progress.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 do not encourage students to monitor their own progress and do not provide direction for teachers to encourage students to monitor their progress.

Criterion 3r - 3y

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4 meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners and strategies for meeting the needs of a range of learners. The materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations, and they provide opportunities for advanced students to investigate mathematics content at greater depth. The instructional materials also suggest support, accommodations, and modifications for English Language Learners and other special populations and provide a balanced portrayal of various demographic and personal characteristics.

Indicator 3r

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Teachers guide students through an inquiry path to become mathematically proficient. The four stages of the path to learning are guided introduction, learning unfolds, knead knowledge through practice, and maintain fluency. As stated by the publisher, “Within the curriculum, a series of learning progressions reflect research on students’ natural learning stages when mastering concepts such as computation and problem-solving strategies. These learning stages informed the order of concepts, the sequence of units, and the positioning of topics in Math Expressions.” 

Examples include but are not limited to:

  • Unit 2, Lesson 1, students use arrays and area models. Prompts are given for EL students at three different levels: emerging, expanding, bridging. A teaching note is included to help teachers diagnose if students are at an Emerging, Expanding, or Bridging level. Teachers are instructed, “Draw a 4 x 2 rectangle showing 4 unit squares down and 2 unit squares across. Write array and area on the board.” Emerging: “The array of unit squares is 4 down by 2 across. The area of the rectangle is 4 x 2. What is 4 x 2?” Expanding: “What is the array of unit squares? How do you find the area of the rectangle? What is the area?” Bridging: “Students can work in pairs. One partner draws an array of unit squares and the other names the area.”
  • In Unit 5, Lesson 1, the Universal Access/Extra Help teaching note instructs teachers, “If students have difficulty overlapping the ends to make their meter strips, have them cut off the ends and butt each part to the next.”
  • In Unit 6, Lesson 2, students are adding and subtracting fractions with like denominators. The Universal Access/Special Needs teacher note states, “For students who have difficulty reading a problem line by line, give them an index card they can use to uncover one line of a problem at a time.”

Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Examples include but are not limited to:

  • An explanation of differentiated instruction is provided in the Teacher Edition.
  • A list of intervention resources is provided for each unit in the Unit Overview Assessment.
  • Math Activity Centers resources for on-level, challenge, and intervention are provided for each unit’s lessons. 
  • Teaching notes for English Learners are provided for emerging, expanding, and bridging students and are provided for each unit’s lessons.
  • Some lessons have Differentiated Instruction notes provided for universal access/extra help.

Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations.

MathTalks provide “an inquiry environment that encourages constructive discussion of problem-solving methods through well-defined classroom activity structures. . . comprises four components: questioning, explaining math thinking, contributing math ideas, and taking responsibility for learning” (Teacher Edition page I3). Initially, teachers model MathTalks and then students run the MathTalk. For example, Unit 2, Lesson 4, MathTalk, Best Practices states, “Review with the class what makes a good explanation. 1. Write your work so everyone can see it. 2. Talk loud enough for other students to hear. 3. Use a pointer to point to your work. 4. Say how you arrived at the answer, not just your answer. 5. Stand to the side of your work when you talk.”

Indicator 3u

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Examples include but are not limited to:

  • Scaffolding of vocabulary is provided. For example, in Unit 4, Lesson 1, the word simplify is explained for EL students. Teachers are instructed to “Write ‘5n + 9 - 2 = 5n + 7’. Write ‘simplify’. Simplify means ‘to make easier’. Point to the 9 and then the 2. These are called like terms. To simplify combine like terms. Have students repeat.”
  • Extra support is provided for EL students. For example, in Unit 2, Lesson 5, students use rounding to find estimates. The teacher prompt states, “If students have difficulty recognizing the rounding frames, suggest they use color pencils or markers to highlight the 4 x 70 and 4 x 60 rounding frames,”
  • Each unit lesson contains a Math Activity Center with activities and resources for students who are on-level and those needing challenge and intervention.
  • Teaching notes included in some lessons provide specific guidance for teachers to support students who are emerging, expanding, and bridging language acquisition.

Indicator 3v

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Examples include but are not limited to:

  • Math Lessons contain Differentiated Instruction Math Activity Centers. Challenge Resources specify which Activity Card will challenge advanced students.
  • The online Personal Math Trainer provides personalized enrichment with learning supports.
  • Challenge worksheets for each lesson are available in print and digitally and are noted on the Differentiated Instruction page for each lesson.
  • Math Readers, books in the Math Activity Center, place math content in the context of stories and support higher levels of critical thinking.

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics. 

Examples include but are not limited to:

  • Puzzled Penguin appears throughout the unit to provide opportunities to help students avoid common errors. These errors are presented as letters to students. Students teach Puzzled Penguin the correct way and explain why the penguin is wrong. 
  • Math Readers contain a variety of animals, children, and adults.

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 provide opportunities for teachers to use a variety of grouping strategies.

Examples include but are not limited to:

  • Math Activity Centers are provided at the conclusion of each lesson and students can be grouped individually, in pairs, or in groups to complete the Activity Cards. For example, in Unit 6, Lesson 1, Intervention Activity card 6-1, students work with a partner to fold the paper and make different unit fractions. 
  • Math Writing Prompts are part of the Math Activity Centers and provide opportunities for students to work individually, in pairs, or in groups. For example, in Unit 2, Lesson 2, the Challenge Math Writing Prompt states, “Explain how you can use the relationship between place value and multiplication to know how many hundreds 10 x 23 gives you.” 
  • MathTalks provide various grouping structures. During Solve and Discuss, 4-5 students go to the board and solve the problem while the rest of the class is solving independently or as part of a small group consisting of 2-3 students. During Scenarios, a group of students act out a particular mathematical situation for other students to see.

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 sometimes encourage teachers to draw upon home language and culture to facilitate learning.

Family Letters for each unit are found in the Student Activity Book. Spanish versions of these letters are also included in the Student Activity Book. However, instructional materials do not encourage teachers to draw upon home language and culture to facilitate learning. English Learner notes in the Teacher Edition do not reference Spanish vocabulary to facilitate learning.

Criterion 3z - 3ad

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
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Criterion Rating Details

The instructional materials reviewed for Math Expressions Grade 4: integrate technology in ways that engage students in the Mathematical Practices; are web-­based and compatible with multiple internet browsers; include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology; are intended to be easily customized for individual learners; and do not include technology that provides opportunities for teachers and/or students to collaborate with each other.

Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Examples include but are not limited to:

  • The Student Activity eBook provides audio, ability to submit answers online, a drawing tool for math drawings, guided practice to help students solve problems, and virtual manipulatives.
  • The Personal Math Trainer is an online adaptive assessment and personalized learning system for students. It analyzes student activity to determine strengths, weaknesses, learning style preferences, and pace. It provides a personalized learning path for students and generates reports for teachers to inform instruction.
  • The online Math Activity Center provides online differentiated instruction opportunities for practice, reteach, and challenge. Teachers can assign RTI assignments to students who struggle on Big Idea Quick Quizzes. Fluency Builders develop students’ basic facts and automaticity.
  • OSMO is an interactive gaming system for iPads to build students' fluency and problem-solving skills. It offers physical manipulatives and provides immediate feedback.

Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 are web-based and compatible with multiple internet browsers. In addition, materials are platform neutral and allow the use of tablets and mobile devices.

Web-based instructional materials for both teachers and students can be accessed using multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, both students and teachers can use multiple devices to access instructional materials (desktop computer, tablet, iPad, Smartboard, laptop, or cellphone). Students with disabilities can use mobile devices, assistive technology, or PCs to access materials. For example, non-readers have the option to have the entire text in an audio format. Additionally, the materials are platform-neutral for a variety of operating systems.

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 provide opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

Online assessments are available. Teachers have the ability to create their own assessments or customize those provided by the program. A variety of assessment types are provided: multi-step, fill in the blank, multiple-choice, or teacher-created questions. For example, teachers giving the computer adaptive test may edit the format and/or values of the text causing the corresponding complexity of the lesson to change accordingly.

The Personal Math Trainer is an online adaptive assessment and learning system of mathematical understanding and procedural skill/fluency. Teachers can identify question types, assignment type, or standard tested. Once students have completed the task or assessment, various charts and graphs can be generated based on standards to inform instruction. Reports are available for individual students and the entire class.

Indicator 3ac

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.

Teachers can manipulate the Personal Math Trainer to create learning experiences for students targeting their needs. Additionally, teachers can create lesson materials that are specific to the learning targets for specific unit lessons. For example, in Unit 6, Lesson 2, students can use the digital Fraction Bars to find pairs of fractions that equal one whole.

The instructional materials for Math Expressions Grade 4 can be easily customized for local use. 

Digital materials include adaptive technological innovations for teachers to personalize learning for students. Digital materials can be differentiated based on individual student’s needs. For example, when using the Personal Math Trainer, teachers can add or modify existing tasks to a student’s personalized learning path. Additionally, adaptive technology allows teachers to provide two flexible differentiated styles (Daily Intervention and Enrichment or Personal Study Plan) for students.

Indicator 3ad

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
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Indicator Rating Details

The instructional materials for Math Expressions Grade 4 do not include reference technology that provides opportunities for teachers and/or students to collaborate with each other.

Materials do not provide opportunities for students and teachers to participate in discussion groups using technology.

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Additional Publication Details

Report Published Date: 09/04/2019

Report Edition: 2018

Title ISBN Edition Publisher Year
CCSS Homework and Remembering BLM Grade 4 9781328703590 Houghton Mifflin Harcourt 2018
CCSS Assessment Guide BLM Grade 4 9781328703668 Houghton Mifflin Harcourt 2018
Teacher Resource Book Grade 4 9781328703736 Houghton Mifflin Harcourt 2018
CCSS Teacher Edition Collection Grade 4 9781328741448 Houghton Mifflin Harcourt 2018
CCSS Softcover Consumable Student Activity Book Collection w/Mathboards Grade 4 9781328764270 Houghton Mifflin Harcourt 2018

About Publishers Responses

All publishers are invited to provide an orientation to the educator-led team that will be reviewing their materials. The review teams also can ask publishers clarifying questions about their programs throughout the review process.

Once a review is complete, publishers have the opportunity to post a 1,500-word response to the educator report and a 1,500-word document that includes any background information or research on the instructional materials.

The publisher has not submitted a response.

Educator-Led Review Teams

Each report found on EdReports.org represents hundreds of hours of work by educator reviewers. Working in teams of 4-5, reviewers use educator-developed review tools, evidence guides, and key documents to thoroughly examine their sets of materials.

After receiving over 25 hours of training on the EdReports.org review tool and process, teams meet weekly over the course of several months to share evidence, come to consensus on scoring, and write the evidence that ultimately is shared on the website.

All team members look at every grade and indicator, ensuring that the entire team considers the program in full. The team lead and calibrator also meet in cross-team PLCs to ensure that the tool is being applied consistently among review teams. Final reports are the result of multiple educators analyzing every page, calibrating all findings, and reaching a unified conclusion.

Rubric Design

The EdReports.org’s rubric supports a sequential review process through three gateways. These gateways reflect the importance of standards alignment to the fundamental design elements of the materials and considers other attributes of high-quality curriculum as recommended by educators.

Advancing Through Gateways

  • Materials must meet or partially meet expectations for the first set of indicators to move along the process. Gateways 1 and 2 focus on questions of alignment. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?
  • Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom. In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Key Terms Used throughout Review Rubric and Reports

  • Indicator Specific item that reviewers look for in materials.
  • Criterion Combination of all of the individual indicators for a single focus area.
  • Gateway Organizing feature of the evaluation rubric that combines criteria and prioritizes order for sequential review.
  • Alignment Rating Degree to which materials meet expectations for alignment, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.
  • Usability Degree to which materials are consistent with effective practices for use and design, teacher planning and learning, assessment, and differentiated instruction.

Math K-8 Rubric and Evidence Guides

The K-8 review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The K-8 Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

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