5th Grade - Gateway 2

The instructional materials for Eureka Grade 5 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 7, students develop conceptual understanding of rounding a decimal number to a place value. A vertical number line is used to model rounding to a place value. Problem Set Question 1 states, “Fill in the table, and then round to the given place. Label the number lines to show your work. Circle the rounded number.” (5.NBT.4)
• In Module 2, Lesson 6, students develop conceptual understanding of the distributive property using area models to understand the distributive property of multiplication and the connection to partial products. Problem Set Question 1 states, “Draw an area model. Then, solve using the standard algorithm. Use arrows to match the partial products from your area model to the partial products in the algorithm.” (5.NBT.7)

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

• In Module 4, Lesson 7, students independently demonstrate conceptual understanding of multiplication of fractions. Tape diagrams are used to represent multiplication involving fractions. Problem Set Question 2 states, “Solve using tape diagrams. 2a. There are 48 students going on a field trip. One-fourth are girls. How many boys are going on the trip?” (5.NF.4)
• In Module 5, Lesson 4, students independently demonstrate conceptual understanding of division of fractions. Tape diagrams are used to represent division involving fractions. Problem Set Question 1 states, “Draw a tape diagram to solve. Express your answer as a fraction. Show the multiplication sentence to check your answer.” (5.NF.7)
• In Module 6, Lesson 5, students independently demonstrate conceptual understanding of a coordinate plane. Students demonstrate understanding by using words and pictures to justify their solution. Problem Set Question 7 states, “Adam and Janice are playing Battleship. Presented in the table is a record of Adam’s guesses so far. He has hit Janice’s battleship using these coordinate pairs. What should he guess next? How do you know? Explain using words and pictures.” (5.G.2)

The instructional materials for Eureka Grade 5 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

In A Story of Units Curriculum Overview, 5.NBT.5 is identified as the fluency standard for Grade 5. The standard 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm, is addressed explicitly in Module 2.

The instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In Module 2, Lesson 8, students develop procedural skill and fluency of multiplying multi-digit whole numbers by using the standard algorithm. Students use the standard algorithm to solve problems like “314 × 236.”
• In Module 4, Lesson 25, Problem 3, students develop procedural skill and fluency of dividing a whole number by a unit fraction. The teacher guides students through solving: “Tien wants to cut 1/4 foot lengths from a board that is 5 feet long. How many boards can he cut?”

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

• In Module 1, Lesson 4, students independently demonstrate procedural skill and fluency of converting units and writing equations with exponents. Problem Set Question 1b states, “Convert and write an equation with an exponent. 105 centimeters to meters, 105 cm = ____ m.”
• In Module 2, Lesson 1, students independently demonstrate procedural skill and fluency of multiplying multi-digit whole numbers using the standard algorithm. Problem Set Question 1b states, “Fill in the blanks using your knowledge of place-value units and basic facts. 230 x 20, Think: 23 tens x 2 tens = ____, 230 x 20 = ____.”
• In Module 5, Lesson 9, students independently demonstrate procedural skill and fluency of multiplying multi-digit numbers when using a visual model to calculate the volume of a figure consisting of two non-overlapping rectangular prisms. The Exit Ticket states, “A student designed this sculpture. Using the dimensions on the sculpture, find the dimensions of each rectangular prism. Then, calculate the volume of each prism.”

The instructional materials for Eureka Grade 5 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 2, Lesson 7, students engage in grade-level mathematics when applying knowledge of multiplication of decimals. The Application Problem states, “The length of a school bus is 12.6 meters. If 9 school buses park end-to-end with 2 meters between each one, what’s the total length from the front of the first bus to the end of the last bus?” (5.NBT.7)
• In Module 4, Lesson 23, students engage in grade-level mathematics when applying knowledge of conversion of measurement units and division to solve word problems. The Application Problem states, “Jasmine took 2/3 as much time to take a math test as Paula. If Paula took 2 hours to take the test, how long did it take Jasmine to take the test? Express your answer in minutes.” (5.NF.7)
• In Module 5, Lesson 11, students engage in grade-level mathematics when applying knowledge of area and multiplication of decimals to solve real-world problems. The Application Problem states, “Mrs. Golden wants to cover her 6.5 foot by 4 foot bulletin board with silver paper that comes in 1 foot squares. How many squares does Mrs. Golden need to cover her bulletin board? Will there be any fractional pieces of silver paper left over? Explain why or why not. Draw a sketch to show your thinking.” (5.NBT.7)

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

• In Module 1, Lesson 16, students independently demonstrate the use of mathematics by applying knowledge of decimal operations to solve word problems. Problem Set Question 3 states, “Mr. Hower can buy a computer with a down payment of $510 and 8 monthly payments of $35.75. If he pays cash for the computer, the cost is $699.99. How much money will he save if he pays cash for the computer instead of paying for it in monthly payments?” (5.NBT.7)
• In Module 4, Lesson 11, students independently demonstrate the use of mathematics by applying understanding of multiplication of fractions to solve word problems. Problem Set Question 5 states, “Create a story problem about a fish tank for the tape diagram below. Your story must include a fraction.” (5.NF.6)
• In Module 4, Lesson 24, students independently demonstrate the use of mathematics by applying understanding of multiplication of fractions and mixed numbers to solve various word problems. Problem Set Question 3 states, “Andres completed a 5-km race in 13.5 minutes. His sister’s time was 1 1/2 times longer than his time. How long, in minutes, did it take his sister to run the race?” (5.NF.6)
• In Module 5, Lesson 15, students independently demonstrate the use of mathematics by applying knowledge of area and fractions to solve word problems. Problem Set Question 3 states, “Janet bought 5 yards of fabric 2 1/4 feet wide to make curtains. She used 1/3 of the fabric to make a long set of curtains and the rest to make 4 short sets. Find the area of the fabric she used for the long set of curtains. Find the area of the fabric she used for each of the short sets.” (5.NF.6)

The instructional materials for Eureka Grade 5 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 5, Lesson 7, students engage in the application of mathematics when finding the volume of a rectangular prism. Problem Set Question 1 states, “Geoffrey builds rectangular planters. Geoffrey’s first planter is 8 feet long and 2 feet wide. The container is filled with soil to a height of 3 feet in the planter. What is the volume of soil in the planter? Explain your work using a diagram.”
• In Module 6, Lesson 14, student practice fluency of multiplying multi-digit whole numbers using the standard algorithm. The Fluency Practice - Multiply Multi-Digit Whole Numbers states, “This drill reviews year-long fluency standards. T: Solve 45 × 25 using the standard algorithm. S: (Solve 45 × 25 = 1,125 using the standard algorithm.) Continue the process for 345 × 25, 59 × 23, 149 × 23 and 756 × 43.”
• In Module 3, Lesson 3, students develop conceptual understanding of adding fractions with unlike denominators by drawing models of equivalent fractions. Problem Set Question 1a states, “Draw a rectangular fraction model to find the sum. Simplify your answer, if possible. 1/2 + 1/3 = ___”

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 4, Lesson 9, students develop conceptual understanding of multiplication of fractions by using tape diagrams to solve real-life problems. Problem Set Question 4 states, “A jewelry maker purchased 20 inches of gold chain. She used 3/8 of the chain for a bracelet. How many inches of gold chain did she have left?”
• In Module 2, Lesson 8, students practice fluency of multi-digit multiplication of whole numbers to solve measurement of real-world measurement problems. Problem Set Question 2 states, “Each container holds 1 L 275 mL of water. How much water is in 609 identical containers? Find the difference between your estimated product and precise product.”
• In Module 3, Lesson 4, students develop conceptual understanding of adding fractions with sums between 1 and 2 and practice fluency of adding fractions with unlike denominators. Problem Set Question 1b states, “For the following problems, draw a picture using the rectangular fraction model and write the answer. When possible, write your answer as a mixed number. 3/4 + 2/3 = ___”

The instructional materials reviewed for Eureka Grade 5 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 2, the materials prompt students to analyze a statement involving place-value equivalence and explain the error that was made. Problem Set Question 4 states, “Janice thinks that 20 hundredths is equivalent to 2 thousandths because 20 hundreds is equal to 2 thousands. Use words and a place-value chart to correct Janice’s error.”
• In Module 3, Lesson 10, the materials prompt students to analyze an addition word problem involving mixed numbers with a given solution and prove whether they agree or disagree with the given solution. Problem Set Question 4 states, “Clayton says that 2 1/2 + 3 3/5 will be more than 5 but less than 6 since 2 + 3 is 5. Is Clayton’s reasoning correct? Prove him right or wrong.”
• In Module 4, Lesson 21, the materials prompt students to construct an argument and critique the reasoning of others when multiplying fractions. Problem Set Question 3 states, “Jack said that if you take a number and multiply it by a fraction, the product will always be smaller than what you started with. Is he correct? Why or why not? Explain your answer, and give at least two examples to support your thinking.”
• In Module 5, Lesson 21, the materials prompt students to construct an argument and critique the reasoning of others when working with attributes of two-dimensional figures. Problem Set Question 2 states, “John says that because rhombuses do not have perpendicular sides, they cannot be rectangles. Explain his error in thinking.”

The instructional materials reviewed for Eureka Grade 5 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.

• In Module 1, Lesson 2, teachers are prompted to engage students in constructing an argument by asking students to explain what patterns they noticed when working with place value. “Work with your partner to solve these problems. Write two complete number sentences on your board. Explain how you got your answers. What are the similarities and differences between the two answers?”
• In Module 3, Lesson 7, teachers are prompted to engage students in constructing an argument and analyze the arguments of others by having students discuss with a partner what is different and the same about each strategy used to solve a problem. “T: Turn and share with your partner, and follow each solution strategy step by step. Share what is the same and different about them. S: (Share.) T: If you have to solve a similar problem again, what kind of drawing and solution strategy would you use? Turn and share. S: (Share.)”
• In Module 3, Lesson 13, teachers are prompted to engage students in analyze an expression involving fractions and explain their thoughts to a partner. “Think about this expression without solving it using paper and pencil. Share your analysis with a partner. What do you know about the total value of this expression without solving?”
• In Module 5, Lesson 20, teachers are prompted to engage students in constructing an argument as students justify whether statements involving two-dimensional figures were true or false. “Justify responses to true or false statements about quadrilaterals based on properties. Trapezoids are always quadrilaterals. Quadrilaterals are always trapezoids. T: Talk to your partner about whether the statement is true or false. Justify your answer using properties of the shapes. T: What about this statement? Trapezoids are always quadrilaterals. Are quadrilaterals always trapezoids? Why or why not? Turn and talk.”

The instructional materials reviewed for Eureka Grade 5 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 1, Lesson 1, the Note on Multiple Means of Action and Expression states, “Throughout A Story of Units, place-value language is key. In earlier grades, teachers use units to refer to a number such as 245, as two hundred forty-five. Likewise, in Grades 4 and 5, decimals should be read emphasizing their unit form. For example, 0.2 would be read 2 tenths rather than zero point two. This emphasis on unit language not only strengthens student place value understanding, but it also builds important parallels between whole number and decimal fraction understanding.”
• In Module 2, Lesson 3, the Notes on Multiple Means of Engagement states, “A review of relevant vocabulary may be in order for some students. Words such as sum, product, difference and quotient might be reviewed, or a scaffold such as a word wall in the classroom might be appropriate.”
• In Module 5, Lesson 4, the Notes on Vocabulary states, “While it is true that any face of a rectangular prism may serve as the base, it is not true for other prisms or cylinders. For example, a right triangular prism has two triangular bases, but the remaining rectangular faces are not bases.”

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

• In Module 1, Lesson 3, the materials use precise terminology when teaching about exponents. The Concept Development Problem 1 states, “T: (Write the term exponent on the board.) We can use an exponent to represent how many times we use 10 as a factor. We can write 10 × 10 as $$10^2$$. (Add to the chart.) We say, “Ten to the second power.” The 2 (point to exponent) is the exponent, and it tells us how many times to use 10 as a factor. T: How do you express 1,000 using exponents? Turn and share with your partner.”
• In Module 5, Lesson 1, the materials use precise terminology of volume and support students in using the term when calculating the volume of several figures. Problem Set Question 1 states, “Use your centimeter cubes to build the figures pictured below on centimeter grid paper. Find the total volume of each figure you built, and explain how you counted the cubic units. Be sure to include units.”
• In Module 5, Lesson 17, the materials use precise terminology of symmetrical and support students in using the term. Exit Ticket Problem 2 states, “Use your set square and ruler to draw symmetrical points about your line that correspond to T and U, and label them V and W.”