4th Grade - Gateway 1

The instructional materials for HMH Into Math Grade 4 meet expectations for assessing grade-level content. An Assessment Guide, included in the materials, contains two parallel versions of each module assessment, and the assessments include a variety of question types. In addition, there is a Performance Task for each unit, and there are Beginning, Middle, and End-of-Year assessments.

Examples of assessment items aligned to grade-level standards include:

• Unit 2, Performance Task, Question 2, “Fei is taking a 5,286-mile road trip with two other friends. If they split the drive equally, how many miles will each person drive?” (4.NBT.6)
• Module 12, Form A, Question 6, students determine if a given fraction with a denominator of 100 is equivalent to a given decimal. They are given “$$\frac{9}{100}$$ and 0.9,  $$\frac{40}{100}$$ and 0.40, and $$\frac{80}{100}$$ and .80”. (4.NF.6)
• Module 14, Form A, Question 5, students solve a story problem requiring addition and subtraction of fractions with like denominators. (4.NF.3d)
• Module 19, Form A, Question 9, “Marcus buys an 8-pound pumpkin. He takes it home and removes 12 ounces of seeds and pulp. How many ounces does his pumpkin weigh now?” (4.MD.2)
• End-of-Year-Test, Question 7, “Jamal's class collects 37 board games. Chen's class collects 3 times as many as Jamal's class. The classes donate all the games to 9 local groups. Each group receives about the same number of games. Which is the BEST estimate of how many games each group receives? Students choose from 4, 16, 28 and 32”. (4.OA.3)
• For Module 20, Form A, Question 10, students convert from centimeters to millimeters instead of from millimeters to centimeters requiring students to convert from larger units to smaller units. (4.MD.1) Question 13, students convert from liters to milliliters. (4.MD.1)

Above grade-level assessment items are present but could be modified or omitted without a significant impact on the underlying structure of the instructional materials. These items include:

• Modules 11 and 13, the following problems use denominators outside of the range of possible denominators for 4.NF: Module 11, Form B, Questions 2 and 3, and Module 13, Form A, Question 7.

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

• The number of modules devoted to major work of the grade is 13 out of 21, which is approximately 62%.
• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 70 out of 108, which is approximately 65%.
• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 107 out of 174 days, which is approximately 61%.

A lesson-level analysis is most representative of the instructional materials because this calculation includes all lessons with connections to major work and is not dependent on pacing suggestions. As a result, approximately 65% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for HMH Into Math Grade 4 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Examples of how the materials connect supporting standards to the major work of the grade include:

• Lessons 10.1- 10.4, 4.OA.4 supports the major work of 4.NBT.2. Lesson 2, Question 7, students decide if 7 is a factor of 91 and justify their answer by using division. Lesson 3, Question 4, students solve a story problem by using 4 as a factor. 
• Lesson 11.4, connections are made between the major work of 4.NF.1 and the supporting work of 4.OA.4. In Question 2, students list the factors of 8 and 12, referring to the fraction 8/12. Students use common factors between 8 and 12 to write equivalent fractions. 
• Lesson 19.5, 4.MD.4 supports the major work of 4.NF.2. Question 1, students plot fractions on a line plot requiring students to first create equivalent fractions, then compare and order them correctly along the line plot.
• Lesson 19.5, On Your Own, Question 7, 4.MD.4 supports the major work of 4.NF.3d when students solve fraction addition and subtraction problems with data given in line plots. “A wildlife center records the weights of six reptiles. The fractions (all with a denominator of 8) represent the weights, in pounds, of the reptiles. Make a line plot to display the data. What is the difference in the weight between the heaviest and lightest reptiles weighed?”

The instructional materials for  HMH Into Math Grade 4 meet expectations that the amount of content designated for one grade level is viable for one year. 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. As designed, the instructional materials can be completed in 174 days, 123 days for lessons and 52 days for assessments.

• The Planning and Pacing Guide and Planning pages at the beginning of each module in the Teacher Edition provide the same pacing information. 
• Grade 4 has 7 units with 21 modules that contain 108 lessons. 
• The Planning and Pacing guide designates 8 lessons as 2-day lessons and 100 as 1-day lessons, leading to a total of 116 days. The materials do not define the number of minutes in a lesson or instructional day.
• Each unit includes a Unit Opener, and there are 7 unit openers for Grade 4 (7 days).
• Each lesson includes a variety of supplemental instruction, such as reteaching lessons, flipbook lessons, etc. However, there is no guidance around building in days for differentiation; therefore, no additional days were added.
• This is a total of 123 lesson days.

Assessments included: 

• The Planning and Pacing Guide indicates a Beginning, Middle, and End-of-Year Interim Growth assessment that would require 1 day each (3 days). 
• Each unit includes a Performance Task which indicates an expected time frame ranging from 25-45 minutes. There are 7 Performance Task for Grade 4 (7 days). 
• Each module has both a review and an assessment. There are 21 modules (42 days). 
• Based on this, 52 assessment days can be added.

The instructional materials for HMH Into Math Grade 4 meet expectations for the materials being consistent with the progressions in the Standards. In general, the materials identify content from prior and future grade-levels as well as relating grade-level concepts explicitly to prior knowledge from earlier grades. In addition, the instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. 

The introduction for every module in the Teacher Edition includes Mathematical Progressions which lists standards under the areas of Prior Learning, Current Development, and Future Connections, as well as clarifying student learning statements in these categories. For example, Module18, Lesson 2, Prior Learning, builds upon work done during Grade 2 where students identified and drew two-dimensional shapes. Future Connections notes the work will continue in Grade 5, Module 20, Lesson 1, where students will identify and classify polygons. Additional features of the materials further support the progressions of the Standards. These include:

• In the beginning of each module there is a diagnostic assessment Are You Ready? that identifies prior knowledge needed for the current module. Module 5 shows the link to prior learning for Multiplication Facts as Grade 3, Module 4 and 5 in the Data-Driven Intervention Chart. A narrative is provided for each skill on the page Multiplication Facts. These items assess whether students are able to find the product of two 1-digit numbers using a variety of strategies. In upcoming lessons, students may use these strategies to multiply larger numbers.”
• In each lesson the standard of focus is explicitly connected to work in future and prior grades.  For instance, Module 20, Lesson 2, identifies the lesson focus as “Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.” (4.MD.1) 

There is one instance of off-grade level work that is not clearly marked:

• Lesson 13.4, Task 2 states  “How could you figure out what angle measure is $$\frac{1}{4}$$ of a circle?”  The answer is “Divide 360 degrees by 4”. (4.NF.4)

The materials give students extensive work with grade-level problems. Each lesson includes a Problem of the Day to activate prior knowledge, a Spark Your Learning portion as an introduction to the day’s learning goals that usually embeds partner or group work to solve a problem. Each lesson includes grade-level work in the Build Your Understanding, Step It Out, and On Your Own sections.  Additionally, Reteach and Challenge pages are available for each lesson which provide more practice with grade-level work. For example: 

• Lesson 8.3, Build Understanding, students relate area models to partial products to multiply two-digit by two-digit numbers. For example, Question 1B,  “Show how you can use an area model to represent the problem?” Question 1C, “How can you write multiplication sentences to find the partial products?” During On Your Own, students solve Question 2: “Complete the area model. Write and solve an equation for the area model.” Additional practice is provided in More Practice/Homework. (4.NBT.5)
• Lesson 12.4, Step It Out, students compare decimals to hundredths using hundredths grids, number lines, and place value charts. During On Your Own, students solve Question 3:  “Chris has two kittens, Oscar and Tiger. Which kitten is heavier? Shade the hundredths model for each weight. Locate and label the weights on the number line." (4.NF.7)
• Lesson 19.5, Step it Out, students represent and interpret measurement data in line plots. Question 1:  “The weights of some cell phones and tablets are shown. How can you display the data using a line plot?” The On Your Own section has 6 questions where students create line plots and interpret the data. Question 7: “A local pizzeria held a pizza-eating contest. The fractions below represent the amount of pizza each contestant ate in 5 minutes.  Make a line plot to display the data. How much more pizza did the winner eat than the person who came in last place?” (4.MD.4)

The materials relate grade level concepts to prior knowledge from earlier grades. 

• In the Activate Prior Knowledge section at the beginning of each lesson, content is explicitly related to prior knowledge to help students scaffold new concepts.
• Lesson 1.4, Compare and Order Numbers students build upon prior learning whereby they “understood that the three digits of a three-digit number represent amounts of hundreds, tens, and ones Grade2, Lessons 6.4 and 6.5 ”. The lesson additionally includes a Make Connections section where it suggests Project the Interactive Reteach, Grade 2, Lesson 6.4. And, Complete the Prerequisite Skills Activity where a problem is presented to the students. Students explain how they can compare numbers and how they can place numbers in order from least to greatest.
• Lesson 6.2, Investigate Remainders students build upon their prior knowledge of division by separating objects into equal groups (3.OA.2) and  by representing division using arrays and bar models. (3.OA.3) Students apply this knowledge to investigate remainders. (4.NBT.6) For example, Module 6, Lesson 2, Question 2, students solve a word problem where 35 pencils are needed to make 8 party favor bags. Students are prompted to create a drawing to show how all the pencils can be divided. Then, describe how many total pencils were drawn, how many pencils went into each bag, and how many pencils were left over (remainder) and to tell why.
• Lesson 11.3, Explain Fraction Equivalence Using Visual Models, students to build upon their prior learning where they “recognized equivalent fractions as the same size or point on a number line Grade 3, Lessons 16.1 and 16.2;  generated simple equivalent fractions, Grade 3, Lesson 16.3; and, expressed whole numbers as fractions (Grade 3, Lesson 13.5).” The lesson additionally includes a Make Connections section where it suggests Project the Interactive Reteach, Grade 3, Lesson 16.1. and Complete the Prerequisite Skills Activity where a problem is presented to the students. They are asked to draw a visual model. Next they are to change the visual model to show a fraction that is equivalent to $$\frac{1}{2}$$. Students are asked “to summarize what is the same about all of the equivalent fractions.” 

The instructional materials reviewed for HMH Into Math Grade 4 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

The materials include learning objectives that are visibly shaped by CCSSM cluster headings. Examples include:

• The Unit 1 learning objective is “Place Value and Whole Number Operations” which is shaped by the cluster heading Generalizing Place Value Understanding for Multi-Digit Whole Numbers. (4.NBT.1)
• In Lesson 11.7, the learning objective is comparisons to order fractions which is shaped by 4.NF.1: “Extend understanding of fraction equivalence and ordering.”
• In Lesson 1.2 the learning objective is place value relationships to read and write multi-digit whole numbers to 1,000,000 in different forms which is shaped by 4.NBT.1: “Generalize place value understanding for multi-digit whole numbers.”
• In Lesson 12.4, the learning objective is to compare decimals using visual models, number lines, or place value which is shaped by 4.NF.3: “Understand decimal notation for fractions, and compare decimal fractions.”

The materials include problems and activities that 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. Examples of this include:

• Lesson 14.6, connects 4.NF.5 with 4.NF.1 when students use equivalent fractions to write fractions with denominators of 10 as denominators of 100 and then add like denominators. For example, On My Own, Question 9: “$$\frac{22}{100} + \frac{7}{10}$$.”
• Lesson 19.4, Question 1, connects 4.MD.1 to 4.OA.1. Students draw a visual representation of how quarts, pints, and cups compare to gallons. They then describe their comparisons in words.
• Lesson 11.3 connects 4.NF.2 to 4.OA.1. For example, Question 2, students create a fraction equivalent to $$\frac{1}{2}$$ with 6 as the denominator. Students are then asked to describe the relationship between the numerator and denominator of each fraction.