3rd-5th Grade - Gateway 1

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Assessment Guide, summative assessments, including Course-Level, End-of-Module, End-of-Topic, and Performance Tasks are designed to measure mastery of concepts and standards proficiency. These are present across Grades 3-5 and consistently align with grade-level content, providing multiple opportunities to assess student understanding throughout the year.

The materials for Grade 3 contain five End-of-Module assessments, fourteen End-of-Topic assessments, fourteen Performance Tasks, and one End-of-Year assessment. An example of an End-of-Topic Assessment includes:

• Grade 3, Module 2, Topic 5, End-of-Topic Assessment, Question 8, a hexagon shape is divided into 3 equal parts. “This hexagon represents 1 whole and is partitioned into equal pieces. Which unit fraction does each piece represent? a. 1 third, b. 1 fourth, c. 1 sixth, d. 1 eighth” (3.G.2)

The materials for Grade 4 contain five End-of-Module assessments, thirteen End-of-Topic assessments, thirteen Performance Tasks, and one end-of-year assessment. An example of an End-of-Module Assessment includes:

• Grade 4, Module 5, End-of-Module Assessment, Question 6, a mosaic is shown, and measurements are noted on each side. “Emerson makes a rectangular piece of mosaic tile art that is 7 feet wide and 12 feet long. What is the area of Emerson’s mosaic? Explain your reasoning.” (4.MD.2)

The materials for Grade 5 contain five End-of-Module assessments, fourteen End-of-Topic assessments, fourteen Performance Tasks, and one End-of-Year assessment. An example of an End-of-Topic Assessment includes:

1. Grade 5, Module 4, Topic 9, End-of-Topic Assessment, Question 4, “Morgan has 15 total math problems to complete. She has completed \frac{3}{5} of the math problems so far. How many math problems has Morgan completed so far? ___ math problems” (5.NF.4a)

The materials reviewed for ClearMath Elementary Grades 3 through 5 meets expectations for including assessment information that indicates which standards are assessed. The materials identify mathematical practices for the assessment items within each grade level Assessment Guide, titled Habits of Minds.

Formal assessments, including End-of-Module, End-of-Topic, and End-of-Year Assessments, consistently align with grade-level content standards. Assessment Blueprints in the Assessment Guide include a chart with columns describing the item number, CCSS, item type, DOK, recommendations for EOY Activities, and connections to learning in a future grade level. For example:

• Grade 3, Module 1, Topic 2, End-of-Topic Assessment, Question 3, “Josiah has 7 pages of stickers. Each page has 5 stickers. How many stickers does Josiah have in all? _____stickers” (3.OA.3)

• Grade 4, Module 3, Topic 6, End-of-Topic Assessment, Question 5, “Oliver and Imani go apple picking. Oliver picks 9 apples. Imani picks 3 times as many apples as Oliver. How many apples does Imani pick? Imani picks ____ apples.” (4.OA.2)

• Grade 5, Module 2, End-of-Module Assessment, Question 1, “Round 5.467 to the nearest hundredth.” (5.NBT.4)

The materials refer to the Standards for Mathematical Practice (SMPs) as “Habits of Mind”. The materials map each “Habit of Mind” to a corresponding SMP, and an alignment chart shows where students are expected to reflect on these habits. The SMPs appear primarily in the Re-Engagement Lessons within the Mindset Reflections. For example: 

• Grade 3, Module 2, Topic 4, Lesson 14 identifies MP5 and MP6.

• Grade 4, Module 1, Topic 1, Lesson 8 identifies MP1, and MP8. 

• Grade 5, Module 4, Topic 11, Lesson 3 identifies MP2 and MP4.

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative assessments occur at the lesson level through structures such as ReadyCheck Assessments, Ongoing Assessments, Purposeful Questions, and Snapshot Assessments. Summative assessments include End-of-Topic Assessments, End-of-Module Assessments, and Performance Tasks that capture student thinking through writing, illustrating, demonstrating, and modeling. These assessments provide opportunities for students to demonstrate understanding of grade-level content standards through a variety of item types, including choice tables, data displays, extended response, fill-in-the-blank, graphing, highlighting, hotspot, label images, matching, multiple choice, multiple select, number line, shading, and short answer. However, while the assessments regularly address the full intent of grade-level content standards, they do not consistently provide opportunities for students to demonstrate the full intent of the grade-level/course-level practices across the series.

An example of an assessment question from Grade 3 includes:

• Module 3, Topic 7, End-of-Module Assessment, Question 5a, and Module 3, Topic 8, End-of-Topic Assessment Question 9, develop the full intent of 3.OA.5, (Apply properties of operations as strategies to multiply and divide.) “5. Joon buys 2 pairs of pants that cost $9 each and 2 shirts that cost $5 each. a. Which expression shows how much money Joon spends on the 4 clothing items? a.  2\times(9+2) b. 9\times(2+5) c. (2\times2)+(9\times5) d. (2\times9)+(2\times5).” Students continue working toward the full intent of this standard in Topic 8, End-of-Topic Assessment, Question 9, “Complete each equation to show a way to use the Distributive Property to multiply 8\times7. I can multiply 8\times7 by using 8\times ___ =40 and 8\times ___ = ___. So, the product of 8\times7= ___.”

An example of an assessment question from Grade 4 includes:

• Module 1, Topic 1, Snapshot Assessment, Question 5, develops the full intent of 4.NBT.1, (Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.) “Choose the number that completes the sentence. In the number 1,220, the value of the digit in the hundreds place is ____ times the value of the digit in the tens place. a. \frac{1}{10},  b. 1,  c. 10  d. 100.”

While the assessments regularly address the full intent of grade-level content standards, they do not consistently provide opportunities for students to demonstrate the full intent of one course-level practice across the series. Materials partially include opportunities for students to demonstrate full intent of MP5 across the assessments. Examples include:

• Grade 3, The Performance Tasks, Reflect activities, and End-of-Topic Assessments do not require students to choose appropriate tools or strategies, recognize the insight or limitations of different tools, use technological tools to deepen their mathematical understanding, or decide whether to use tools at all. In the Module 5, Topic 12 Performance Task students shade pre-drawn containers and write equations, and the only material provided is a “straightedge” for Question 1, which does not support tool choice or analysis. In Module 2, Topic 6, Lesson 7 Reflect, Reflect and Summarize students are provided a ruler on paper and use it to tell the length of an object, but they do not select a tool or decide whether to use tools at all. In the Module 1, Topic 6 Performance Task students are directed to “Use the broken ruler” to partition and draw earthworms, limiting them to a single predetermined tool with no opportunity to select or compare tools. In the Module 2, Topic 6 End-of-Topic Assessment students use the printed ruler and clock embedded in the task to measure or partition, but they do not choose tools or engage in tool-based decision-making.

• Grade 4, The Performance Tasks, Reflect activities, and End-of-Topic Assessments do not require students to choose appropriate tools or strategies, recognize the insight or limitations of different tools, use technological tools to deepen their mathematical understanding, or decide whether to use tools at all. In the Module 5, Topic 11 Performance Task students are told to use a protractor and number line, and the task states to “Use a protractor to determine the measure,” which limits them to a single predetermined tool with no opportunity to select or compare tools. In the Module 3, Topic 7, Performance Task students complete an area model and solve problems using the representations already provided, but they do not choose tools or engage in tool-based decision-making. In the Module 1, Topic 2 and Topic 3 End-of-Topic Assessments students work with pre-drawn number lines, models, and grids to compare decimals and fractions, yet no variety of tools is available and no task asks students to select or justify a tool. In Module 1, Topic 2, Lesson 4, Reflect, students are provided rods to solve problems. In Module 1, Topic 3, Lesson 2, Reflect, students are given a hundreds grid and asked decimal questions based on it. 

• Grade 5, The Performance Tasks, Reflect activities, and End-of-Topic Assessments do not require students to choose appropriate tools or strategies, recognize the insight or limitations of different tools, use technological tools to deepen their mathematical understanding, or decide whether to use tools at all. In Module 3, Topic 6, Lesson 4, Reflect, students are provided an open number line to subtract a decimal. In Module 3, Topic 7, Lesson 1, Reflect, students are provided grids to help them solve problems involving multiplying whole numbers by decimals. In the Module 5, Topic 14 Performance Task students classify and draw quadrilaterals using “grid paper,” but the task does not offer a variety of tools or require tool selection. In the Module 1, Topic 1 Performance Task students build a prism using “a maximum of 75 connecting cubes,” limiting them to a single fixed tool with no opportunity to select or compare tools. In the Module 4, Topic 9 and Module 3 End-of-Module Assessments students work with area models, tables, and diagrams that are already embedded in the tasks, and they do not choose tools or decide whether tools are needed. In the Module 2, Topic 3 End-of-Topic Assessment students complete algorithms and area models provided on the page, but they do not engage in selecting tools or recognizing the insight or limitations of different tools.

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide students with consistent opportunities to engage in the full intent of all Grade 3-5 standards. Each lesson includes an Activate, Explore, and Reflect. Modules also include Ready Checks, which are short assessments that address the prerequisite skills students need to access the module and topic content. The data can be used to determine when and how the teacher should re-engage students as they move through the modules and topics. Student Practice Books give students additional opportunities to reinforce the knowledge and skills developed throughout the topics with extra practice problems. The Review Center in Grades 1-5 includes Spaced Review activities, which provide students with opportunities to revisit previously learned content to reinforce or prepare for upcoming content.

The Mathematical Progressions and Connections Handbook provides extensive work with module-level Mathematical Progression maps showing how standards and concepts connect across the modules in the course. Arrows illustrate how mathematical ideas in one module relate to those in others. The topic-level Mathematical Progression maps highlight the standards addressed within each topic, and arrows show how concepts build from earlier work and extend into later topics. Examples of extensive work with grade-level problems to meet the full intent include:

• Grade 3, Module 5, Topic 13, Lessons 6, 7, and 9, students engage with the full intent of 3.MD.8 (Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters). In Lesson 6, Explore 1, students apply strategies to calculate the perimeter of each polygon by adding its side lengths. Students calculate the perimeters of a variety of polygons without relying on grids, for example, “Determine the perimeter of each shape.” In Explore 2, students use measurement tools to determine and compare the perimeters of real-world objects. Students measure classroom items and discuss results, reinforcing the idea that perimeter is additive. During Reflect, students measure and label the side lengths of each shape and then calculate the perimeter, for example, “Use a ruler to measure the side lengths of each shape using centimeters. Then, determine the perimeter of each shape.” Students solidify their understanding of the relationship between side lengths and perimeter. In Lesson 7, Student Practice Book, Question 6, a complex figure is given with five of the six sides marked with unit side lengths. Students must determine the sixth unit side length to then calculate the area and perimeter of the figure. In Lesson 9, Explore, Center 1, students play a teacher-facilitated game called Perimeter and Area Builder. The purpose of this game is to relate perimeter and area: “1. For each round, both players roll two number cubes. 2. Each player adds the number cubes together and multiplies the sum by 2. This value determines the perimeter of a figure they will draw. 3. Each player draws a figure with the given perimeter on their gameboard. 4. Players score 1 point for each square unit of area inside the figure. 5. The player with the most points after 6 rounds wins the game.”

• Grade 4, Module 3, Topic 6, Lesson 3, students engage with the full intent of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). In the Student Resource Book, the lesson begins with Activate, students distinguish between additive comparison and multiplicative comparison by identifying which situations show one quantity as being a certain number of times larger than another. Students discuss examples to clarify the difference between additive and multiplicative situations. In Explore 1, students match each multiplicative comparison description to its corresponding equation, and students connect verbal descriptions with symbolic equations. In Explore 2, students use models to multiply or divide to solve each word problem. Students solve problems such as Question 3, “A bag of trail mix costs $6, which is 3 times as much as a bottle of juice. How much does a bottle of juice cost?” During Reflection, students multiply or divide to solve each multiplicative comparison word problem.

• Grade 5, Module 4, Topic 9, Lesson 3, students engage with the full intent of 5.NF.3 (Interpret a fraction as division of the numerator by the denominator [\frac{a}{b}=a\div b]. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem). In the Student Resource Book, the lesson begins with Activate as students model a scenario by dividing the distance in a relay race equally. Students consider how dividing a whole into equal parts leads to fractional results. In Explore 1, students write division expressions that are equivalent to fractions. Students investigate how the quotient changes when the same dividend is divided by different divisors. In Explore 2, students create models and write division equations leading to a fractional answer. They solve contextual problems such as Question 7, “Henry uses 6 ounces of cheese to make 8 sandwiches. How many ounces does each sandwich get?” During Reflection, students solve word problems involving division of whole numbers with fractional results..

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade:

In Grade 3:

• The approximate number of modules devoted to the major work of the grade (including assessments) is 5 out of 5, approximately 100%.

• The number of topics devoted to the major work of the grade (including assessments) is 11 out of 14, approximately 79%.

• The number of lessons devoted to the major work of the grade (including assessments) is 106 out of 136, approximately 78%.

• The number of instructional days devoted to major work of the grade (including assessments) is 106 out of 136, approximately 78%.

In Grade 4:

• The approximate number of modules devoted to the major work of the grade (including assessments) is 4 out of 5, approximately 80%.

• The number of topics devoted to the major work of the grade (including assessments) is 10 out of 13, approximately 77%.

• The number of lessons devoted to the major work of the grade (including assessments) is 98 out of 135, approximately 73%.

• The number of instructional days devoted to the major work of the grade (including assessments) is 98 out of 135, approximately 73%.

In Grade 5:

• The approximate number of modules devoted to the major work of the grade (including assessments) is 4 out of 5, approximately 80%.

• The number of topics devoted to the major work of the grade (including assessments) is 10 out of 14, approximately 71%.

• The number of lessons devoted to the major work of the grade (including assessments) is 100 out of 136, approximately 74%.

• The number of instructional days devoted to the major work of the grade (including assessments) is 100 out of 136, approximately 74%. 

An instructional day analysis across Grades 3 through 5 is most representative of the instructional materials, as the lessons include major work, supporting work connected to major work, and embedded assessments. As a result, approximately 78% of the materials in Grade 3, 73% of the materials in Grade 4, and 74% of the materials in Grade 5 focus on the major work of the grade.

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations that supporting content simultaneously enhances focus and coherence by engaging students in the major work of the grade.

Materials are designed so that supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers in their Mathematical Progressions and Connections handbook, on the Course-Level Coherence Map. Examples of a connection include:

• Grade 3, Module 1, Topic 3, Lesson 6, Explore 1, connects the supporting work of 3.MD.3 (Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs) to the major work of 3.OA.8 (Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). In this lesson, students apply their understanding of scaled picture graphs and bar graphs to display and analyze data sets as part of a treasure hunt. Directions state, “Task, Tell students they have $500 to spend on supplies for their trip. Explain they have to select their supplies from the lists provided. Have students work in pairs to create the list of items they want to take on the treasure hunt. Tell them they can purchase as many as they want of each item as long as they stay within their $500 budget. Tell students they can use a data display to visualize how much money they plan to spend on each supply type: essentials, food, entertainment, and other. Have students work in pairs to create a bar graph representing the total amount of money they plan to spend in each category. Monitor pairs as they work.”

• Grade 4, Module 4, Topic 10, Lesson 7, Explore 2, Question 1, connects the supporting work of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale) to the major work of 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies, including rounding). In this lesson, students write stories representing division expressions. They analyze strategies for solving multi-step word problems, making sense of the context. Students then solve multi-step word problems involving unit conversions and division. Directions state, “Mr. Brown has 3 hamsters. He has 9 grams of food to feed all the hamsters. Mr. Brown gives the same amount of food to each hamster. How many milligrams of food will he give each hamster?”

• Grade 5, Module 4, Topic 11, Lesson 12, Reflect, Experimenting with Water, Question 2, connects the supporting work of 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}). Use operations on fractions for this grade to solve problems involving information presented in line plots.) to the major work of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). The materials give a line plot partitioned into four equal parts, beginning with the number 1 and ending with the number 4, with fractional representations given. Directions state, “Determine the total amounts of water in all of the beakers.”

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Connections among the major work of the grade are present throughout the materials where appropriate. These connections are listed for teachers in the Mathematical Progressions and Connections section of the Course-Level Coherence Map, and they may also appear in one or more parts of a typical lesson (Activate, Explore, Additional Centers, or Reflect). Examples of a connection include:

• Grade 3, Module 3, Topic 9, Lesson 11, Activate, Question 1, connects the major work of 3.OA.A (Represent and solve problems involving multiplication and division) to the major work of 3.OA.B (Understand properties of multiplication and the relationship between multiplication and division). Students decompose dividends as they develop an understanding of the Distributive Property of Division. First, students determine the number of groups hidden underneath a cloud. Then, they use models and formalize the Distributive Property by decomposing equations. Directions state, “What is behind the cloud? There are a total of 16 oranges.” The picture shows two groups of four oranges with a cloud covering additional oranges. Students draw the missing groups.

• Grade 4, Module 1, Topic 3, Lesson 10, Activate, Question 1, connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NF.C (Understand decimal notation for fractions, and compare decimal fractions). Students write a comparison statement for each pair of numbers and explain their reasoning. Directions state, “Write a comparison statement for each pair of numbers. Be prepared to explain your reasoning. 0.5 and \frac{3}{4} ” Students compare a decimal with a fraction, choosing to convert decimals to fractions, fractions to decimals, or reason about their size using appropriate symbols and explanations.

• Grade 5, Module 4, Topic 9, Lesson 9, About the Math, connects the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.B (Apply and extend previous understandings of multiplication and division to multiply and divide fractions). In this lesson, students determine the product of a fraction with a fraction and calculate the product of a fraction and a whole number. Students use area models to solve fraction multiplication problems in mathematical and real-world contexts. Directions state, “Students use their understanding of whole number multiplication and previous experiences with paper folding and sketched models for fraction multiplication to interpret fraction multiplication as calculating part of a part. Students build from their understanding of area and experiences with tiling to represent fraction products as rectangular areas. They analyze these representations of fraction multiplication to generalize the sequence of operations when multiplying fractions.”

The materials reviewed for ClearMath Elementary Grades 3 through 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

The Module Overviews and Topic Overviews identify connections to future work and prior knowledge, including how these connections relate to module- and topic-level work and, where appropriate, to prior or future grades. The Mathematical Progressions and Connections document further supports teachers in developing their understanding of the Common Core State Standards by highlighting connections within and across grade levels. “About the Math” sections within the Module Overview, Topic Overview, and Lesson Overview extend teachers’ understanding beyond individual standards to make sense of these connections. 

An example of a connection to future grades includes:

• Grade 3, Module 4, Topic 11, Topic Overview, highlights the connections to future grades. In this topic, students represent time and measurement data in multiple ways. They use number lines to create line plots (3.MD.4) and determine start time, end time, and elapsed time (3.MD.1). They tell time to the nearest minute and solve one- and two-step word problems involving elapsed time (3.MD.1). Students also create line plots with fractional intervals to display and interpret measurement data (3.MD.4). Connection to Future Learning, “In Grade 4, students create line plots to display measurement data including whole numbers, halves, fourths, and eighths. They also solve addition and subtraction problems by analyzing the data.”

• Grade 4, Module 3, Topic 6, Topic Overview, highlights the connections to future grades. In this topic, students examine the differences between additive and multiplicative comparisons and build on their understanding of multiplicative relationships (4.OA.1, 4.OA.2). Module Overview, Connection to Future Learning, “In Grade 5, students will learn about using strategies and models to multiply multi-digit numbers, multiplying multi-digit numbers using the standard algorithm, and applying various multiplication strategies to multiply decimal numbers or fractions.”

• Grade 5, Module 1, Topic 1, Topic Overview, highlights the connections to future grades. In this topic, students explore volume as an attribute by packing rectangular prisms with unit cubes (5.MD.3). They identify the volume of a rectangular prism as the product of the prism’s dimensions (5.MD.5). Students also use the idea of layers to understand that they can multiply the area of the base by the height to calculate volume (5.MD.5). Connection to Future Learning, “In Grade 6, students pack rectangular prisms using unit cubes with fractional side lengths. They further explore solids and define the surface of a solid figure.”

An example of a connection to prior grades includes:

• Grade 3, Module 4, Topic 10, Topic Overview, highlights the connections to prior grades. In this topic, students generate equivalent fractions using area models and fraction strips (3.NF.3). They locate fractions on number lines to compare fractions (3.NF.2). Students identify benchmark fractions and reason about numerators and denominators to compare fractions (3.NF.1, 3.NF.3). In the Module Overview, Connection to Prior Learning, “In Grade 2, students learned about telling time to the nearest 5 minutes and creating line plots with whole numbers based on measurements.”

• Grade 4, Module 3, Topic 7, Topic Overview, highlights the connections to prior grades. In this topic, students use place value understanding and multiplicative reasoning to multiply multi-digit numbers (4.NBT.5). They find the product of a one-digit number and multiples of 10, 100, and 1,000. Students also develop and apply strategies to multiply a one-digit number by numbers up to four digits, as well as to multiply two-digit numbers by two-digit numbers (4.NBT.5). Connection to Prior Learning: “In Grade 3, students multiplied 1-digit whole numbers by multiples of 10 using the Associative and Communicative Property.”

• Grade 5, Module 1, Topic 2, Topic Overview, highlights the connections to prior grades. In this topic, students investigate the rules for carrying out operations when an expression involves more than one operation (5.OA.1). They express whole numbers from 2 to 50 as a product of prime factors. Students look for patterns when using operations and grouping symbols (5.OA.1) and learn to define, write, and interpret numerical expressions (5.OA.2). Connection to Prior Learning: “In Grade 3, students encountered grouping symbols as they multiplied numbers using the Associative Property of Multiplication. In Grade 4, they interpreted multiplication equations.”