4th Grade - Gateway 2

The instructional materials for Eureka Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. For example:

• In Module 1, Lesson 1, students develop conceptual understanding of a multiplication equation as a comparison. The teacher is prompted to facilitate student discussion by saying, “T: Discuss the patterns you have noticed with your partner. S: 10 ones make 1 ten. 10 tens make 1 hundred. 10 hundreds make 1 thousand. Every time we get 10, we bundle and make a bigger unit. We copy a unit 10 times to make the next larger unit. If we take any of the place value units, the next unit on the left is ten times as many.” (4.NBT.A)
• In Module 1, Lesson 8, students develop conceptual understanding of rounding to a place value. A vertical number line is used to model rounding to a place value. Problem Set Question 1 states, “Complete each statement by rounding the number to the given place value. Use the number line to show your work. 1a. 53,000 rounded to the nearest ten thousand is _______.” (4.NBT.3)
• In Module 3, Lesson 16, students develop conceptual understanding of division with a remainder. Students use place-value disks to solve two-digit dividend division problems with a remainder in the ones place. The teacher is prompted to facilitate student discussion by saying, “T: (Point to the place value chart.) We divided 6 ones and have no ones remaining. 6 ones minus 6 ones equals 0 ones. (Write the subtraction line.) What does this zero mean? S: There is no remainder. All the ones were divided with none left over. We subtracted the total number distributed from the total number of ones. T: We can see the 3 groups of 2 both in our model and in our numbers and know our answer is correct since 3 times 2 equals 6.” (4.NBT.B)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. For example:

• In Module 1, Lesson 15, students independently demonstrate conceptual understanding of place value. Students use tape diagrams to demonstrate conceptual understanding of place value to solve word problems. Problem Set Question 2 states, “Use tape diagrams and the standard algorithm to solve the problems below. Check your answers. 2. David is flying from Hong Kong to Buenos Aires. The total flight distance is 11,472 miles. If the plane has 7,793 miles left to travel, how far has it already traveled?” (4.NBT.4)
• In Module 4, Lesson 6, students independently demonstrate conceptual understanding of fractions. Students relate fractions as division to fraction of a set. The teacher is prompted to facilitate student discussion by saying, “T: Make an array with 6 counters turned to the red side, and use your straws to divide your array into 3 equal parts. T: Write a division sentence for what you just did. S: 6 ÷ 3 = 2. T: Rewrite your division sentence as a fraction, and say it aloud as you write it. S: (Write 6/3=2.) 6 divided by 3 equals 2. T: If I want to show 1 third of this set, how many counters should I turn over to yellow? Turn and talk.” (4.NF.B)
• In Module 5, Lesson 20, students independently demonstrate conceptual understanding of fractions. Students use tape diagrams when adding fractions. Problem Set Question 1 states, “Use a tape diagram to represent each addend. Decompose one of the tape diagrams to make like units. Then, write the complete number sentence.” (4.NF.3)

The instructional materials for Eureka Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

In A Story of Units Curriculum Overview, 4.NBT.4 (fluently add and subtract multi-digit whole numbers using the standard algorithm), is addressed explicitly in Module 1. For example:

• The Lesson 11 Problem Set students gain procedural skills with multi-digit addition problems using the standard algorithm.
• In Lesson 13, students develop procedural skill and fluency of subtraction of multi-digit whole numbers.
• In Lesson 16, students solve two-step word problems using the standard algorithm. Problem Set problem #2, “A gas station has two pumps. Pump A dispensed 241,752 gallons. Pump B dispensed 113,916 more gallons than Pump A. a. About how many gallons did both pumps dispense? Estimate by rounding each value to the nearest hundred thousand and then compute. b. Exactly how many gallons did both pumps dispense? C. Assess the reasonableness of you answer in (b). Use you estimate from (a) to explain.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. For example:

• In Module 1, Lesson 5 students build fluency of multiplication by 4 during the Sprint.
• In Module 5, Lesson 6 Sprint students build fluency with addition of whole numbers and unit fractions, and multiplication of unit fractions by whole numbers.
• In Module 6, Lesson 13, students engage in Fluency Practice ordering decimal numbers, and writing in decimal and fraction notation.

The instructional materials for Eureka Grade 4 meet expectations that the materials are 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.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. For example:

• In Module 1, Lesson 18, students engage in grade-level mathematics when using subtraction to solve a word problem. The Application Problem states, “In all, 30,436 people went skiing in February and January. 16,009 went skiing in February. How many fewer people went skiing in January than in February?” (4.NBT.4)
• In Module 2, Lesson 3, students engage in grade-level mathematics when solving a word problem involving conversion of measurements and subtraction. The Application Problem states, “A liter of water weighs 1 kilogram. The Lee family took 3 liters of water with them on a hike. At the end of the hike, they had 290 grams of water left. How much water did they drink? Draw a tape diagram, and solve using an algorithm or a simplifying strategy.” (4.MD.2)
• In Module 5, Lesson 13, students engage in grade-level mathematics when using equivalent fraction knowledge to solve word problems. The Application Problem states, “Mr. and Mrs. Reynolds went for a run. Mr. Reynolds ran for 6/10 mile. Mrs. Reynolds ran for 2/5 mile. Who ran farther? Explain how you know. Use the benchmarks 0, 1/2 , and 1 to explain your answer.” (4.NF.2)

The instructional materials provide opportunities for students to demonstrate independently the use of mathematics flexibly in a variety of contexts. For example:

• In Module 3, Lesson 2, students independently demonstrate the use of mathematics by applying knowledge of area and perimeter to solve multiplicative comparison word problems. Problem Set Question 4 states, “The area of Betsey’s rectangular sandbox is 20 square feet. The longer side measures 5 feet. The sandbox at the park is twice as long and twice as wide as Betsy’s. a. Draw and label a diagram of Betsy’s sandbox. What is its perimeter? b. Draw and label a diagram of the sandbox at the park. What is its perimeter?” (4.OA.3)
• In Module 5, Lesson 19, students independently demonstrate the use of mathematics by applying understanding of addition and subtraction of fractions to solve word problems. Problem Set Question 1 states, “Sue ran 9/10 mile on Monday and 7/10 mile on Tuesday. How many miles did Sue run in the 2 days?” (4.NF.3d)
• In Module 5, Lesson 40, students independently demonstrate the use of mathematics by applying their knowledge of multiplication and fractions to solve real-world problems. Problem Set Question 3 states, “Six of the players on the team weigh over 300 pounds. Doctors recommend that players of this weight drink at least 3 3/4 quarts of water each day. At least how much water should be consumed per day by all 6 players?” (4.NF.4c)

The instructional materials for Eureka Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The lessons include components such as: Fluency Practice, Concept Development, and Application Problems. Conceptual understanding is addressed in Concept Development. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students engage in practicing procedures and fact fluency while modeling and solving these concepts. Fluency is also addressed as an independent component within most lessons. Lessons may contain an Application Problem which serves as an anticipatory set for the concept or standard that is the focus of the lesson. This Application Problem connects previous learning to what students are learning for the day. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 3, Lesson 7, students practice fluency of multi-digit multiplication. Sprint Question 5 states, “2 x 3,000 = ____”
• In Module 5, Lesson 5, students develop conceptual understanding of equivalent fractions by decomposing rectangles. Problem Set Question 1 states, “Draw horizontal lines to decompose each rectangle into the number of rows indicated. Use the model to give the shaded area as both a sum of unit fractions and as a multiplication sentence. 2 rows. 1/4 = 2/__, 1/4 = 1/8 + ___ = ___, 1/4 = 2 x ___ = ___.”
• In Module 7, Lesson 4, students engage in the application of mathematics by solving word problems involving multiplicative comparison of measurement units. Problem Set Question 1 states, “Beth is allowed 2 hours of TV time each week. Her sister is allowed 2 times as much. How many minutes of TV can Beth’s sister watch?”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• In Module 1, Lesson 10, students develop conceptual understanding of place value by solving real-world problems. Problem Set Question 3 states, “Empire Elementary School needs to purchase water bottles for field day. There are 2,142 students. Principal Vadar rounded to the nearest hundred to estimate how many water bottles to order. Will there be enough water bottles for everyone? Explain.”
• In Module 3, Lesson 11, students develop conceptual understanding and practice fluency of multi-digit multiplication by using partial products and the area model to solve problems. Problem Set Question 1a states, “Solve the following expressions using the standard algorithm, the partial products method, and the area model. 425 x 4, 4 (400 + 20 + 5), (4 x ___) + (4 x ___) + (4 x ___).”
• In Module 3, Lesson 32, students practice fluency of division by solving real-world problems. Problem Set Question 1 states, “A concert hall contains 8 sections of seats with the same number of seats in each section. If there are 248 seats, how many seats are in each section?”

The instructional materials reviewed for Eureka Grade 4 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. For example:

• In Module 1, Lesson 4, the materials prompt students to determine which multi-digit number is written in word form correctly and explain their analysis for their choice. Problem Set Question 4 states, “Black rhinos are endangered, with only 4,400 left in the world. Timothy read that number as “four thousand, four hundred.” His father read the number as “44 hundred.” Who read the number correctly? Use pictures, numbers, or words to explain your answer.”
• In Module 1, Lesson 10, the materials prompt students to construct an argument stating why an answer can be the same when rounding a number to different place values. Exit Ticket Question 1 states, “There are 598,500 Apple employees in the United States. Round the number of employees to the given place value. Thousand: ___, ten thousand: ___, hundred thousand: ___. Explain why two of your answer are the same.”
• In Module 4, Lesson 4, the materials prompt students to determine whether a statement about the attributes of a triangle is correct and construct a viable argument for their thinking. Problem Set Question 5 states, “True or false? A triangle cannot have sides that are parallel. Explain your thinking.”
• In Module 5, Lesson 18, the materials prompt students to analyze two different strategies to add fractions and explain which strategy they like best. Problem Set Question 2 states, “Monica and Stuart used different strategies to solve 5/8 + 2/8 + 5/8. Whose strategy do you like best? Why?”

The instructional materials reviewed for Eureka Grade 4 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others, frequently throughout the program. The teacher materials consistently provide teachers with question prompts for student discussion and possible student responses to support that discussion. For example:

• In Module 4, Lesson 4, teachers are prompted to engage students in constructing an argument by asking students about the properties of a rectangle. “What do you notice about sides of a rectangle and parallel lines? Is this true for all rectangles? Does the length of the opposite sides of a rectangle change the fact that they are parallel?”
• In Module 7, Lesson 5, teachers are prompted to encourage students to critique a partner's work. “T: Work with a partner to complete the Problem Set. When you are finished solving and creating a word problem to go along with each diagram, turn to your partner and share. Use the peer share and critique form to take notes about your work and your partner’s work.”
• In Module 7, Lesson 12, teachers are prompted to engage students in constructing an argument by asking students to explain their alternative solution of finding an equivalent fraction other than using a tape diagram. “Talk to your partner. Instead of just using the tape diagram, how can we use what we know about finding equivalent fractions to find the number of twelfths equal to 1/2 foot? Again, how many inches are equal to 1/2 or 6/12 foot? Work with your partner to find how many inches are equal to 1/4 foot. (Allow students time to work.) How did you figure it out?”

The instructional materials reviewed for Eureka Grade 4 meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently-introduced mathematical terms that will be used throughout the module. A compiled list of the terms along with their definitions is found in the Terminology tab at the beginning of each module. Each mathematical term that is introduced has an explanation, and some terms are supported with an example.

The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols. For example:

• In Module 5, Lesson 9, the Notes on Multiple Means of Expression states, “As the conceptual foundation for simplification is being set, the word simplify is initially avoided with students as they compose higher-value units. The process is rather referred to as composition, the opposite of decomposition, which relates directly to their drawing, work throughout the last two lessons, and work with whole numbers. When working numerically, the process is referred to at times as renaming, again in an effort to relate to whole number work.”
• In Module 6, Lesson 9, the Notes on Terminology state, “Mass is a fundamental measure of the amount of matter in an object. While weight is a measurement that depends upon the force of gravity (one would weigh less on the moon than one does on Earth), mass does not depend upon the force of gravity. Both words are used here, but it is not important for students to recognize the distinction in mathematics at this time.”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. For example:

• In Module 2, Lesson 1, the mathematical term kilometer is in bold writing within a question listed in the Student Debrief section. These questions guide teachers in leading a class discussion. “What pattern did you notice in the equivalences for Problems 1 and 2 of the Problem Set? How did converting 1 kilometer to 1,000 meters in Problem 1(a) help you to solve Problem 2(a)?”
• In Module 3, Lesson 22, the materials use accurate terminology and support students in using the term factor pairs. Problem Set Question 4 states, “Sheila has 28 stickers to divide evenly among 3 friends. She thinks there will be no leftovers. Use what you know about factor pairs to explain if Sheila is correct.”