3rd Grade - Gateway 2

The instructional materials for Eureka Grade 3 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 10, students develop conceptual understanding of the distributive property of multiplication. Students practice decomposing arrays to model the distributive property of multiplication. (3.OA.B)
• In Module 3, Lesson 19, students develop conceptual understanding of multiplying by multiples of 10. Students use place value charts when multiplying by multiples of 10. Problem Set Question 2 states, “Use the chart to complete the blanks in the equations. 2 x 4 tens = ____ tens.” (3.NBT.A)
• In Module 5, Lesson 27, students develop conceptual understanding of fractions. During guided practice the teacher is prompted to ask the following questions, “T: Label the fractions in each model. S: (Label.) T: What is different about these models? S: They all started as thirds, but then we cut them into different parts. The parts are different sizes. T: Yes, they’re different units. T: What is the same about these models? S: The whole. T: Talk to your partner about the relationship between the number of parts and the size of parts in each model. S: 3 is the smallest number, but thirds have the biggest size. As I drew more lines to partition, the size of the parts got smaller. That’s because the whole is cut into more pieces when there are ninths than when there are thirds.” (3.NF.3a)

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

• In Module 1, Lesson 15, students independently demonstrate conceptual understanding of the commutative property of multiplication. Students use tape diagrams and arrays to show the commutative property of multiplication. Problem Set Question 1 states, “Label the tape diagrams and complete the equations. Then, draw an array to represent the problems.” (3.OA.B)
• In Module 5, Lesson 8, students independently demonstrate conceptual understanding of fractions. Problem Set Question 1 states, “Show a number bond representing what is shaded and unshaded in each of the figures. Draw a different visual model that would be represented by the same number bond.” (3.NF.3)
• In Module 5, Lesson 23, students independently demonstrate conceptual understanding of fractions. Students use number lines to show equivalent fractions. Problem Set Question 3 states, “List the fractions that name the same place on the number line.”(3.NF.2)

The instructional materials for Eureka Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

In A Story of Units Curriculum Overview, 3.OA.7 and 3.NBT.2 are identified as the fluency standards for Grade 3. The standard 3.OA.7, fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations is addressed explicitly in Modules 1 and 3. The standard 3.NBT.2, fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction, is explicitly addressed in Module 2.

3.OA.C - Multiply and Divide within 100

• In Module 1 students work with multiplication and division within 100.
• In Module 2 contains fluency practice on multiplication and division with 100.
• Module 3 continues to build on multiplication and division within 100 by having students work with factors of 2,3,4,5, and 10.
• Lessons within each module build on fluency facts for a particular unit before moving to other units. According to the overview of Module 3, “The factors are sequences to facilitate systematic instruction with increasing sophisticated strategies and patterns.” Students revisit the commutative property, then arithmetic patterning, then skip count,, then the distributive property, then the associative property.

3.NBT.2 - Fluently add and subtract within 1,000 using strategies and algorithms based on pace value, properties of operations, and/or the relationship between addition and subtraction.

• In Module 2, Topic D, students add two- and three- digit metric measurements, apply place value concepts as they round, and compose units multiple times. For example, in Lesson 15 Homework students “find the sums below. Choose mental math or the algorithm. A. 75cm + 7cm b. 39kg + 56kg c. 362mL + 229mL…”
• In Module 2, Topic E, students subtract two- and three-digit measurement using the standard algorithm. In Lesson 18 students work with unlabeled place value chart templates in their personal white boards and solve problems like “825mL - 132 mL.”

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

• In Module 1, Lesson 17, Sprint, students solve as many problems that they can involving multiplication and division by 4. These problems are not presented in sequence so that students develop fluency with number facts. (3.OA.7)
• In Module 2, Lesson 18, students subtract within a 1,000 when subtracting measurements. Problem Set Question 1h states, “Solve the subtraction problems below. 307g + 234g” (3.NBT.2)
• In Module 3, Lesson 4, students multiply and divide when counting by six. Problem Set Question 2 states, “Count by six to fill in the blanks below. Complete the multiplication equation that represents the final number in your count-by. Complete the division equation that represents your count-by.” (3.OA.7)

The instructional materials for Eureka Grade 3 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 4, students engage in grade-level mathematics when multiplying equal groups. The Application Problem states, “The student council holds a meeting in Mr. Chang’s classroom. They arrange the chairs in 3 rows of 5. How many chairs are used in all? Use the RDW (Read-Draw-Write) process.” (3.OA.1)
• In Module 3, Lesson 8, students engage in grade-level mathematics when solving two-step word problems. The Application Problem states, “Richard has 2 cartons with 6 eggs in each. As he opens the cartons, he drops 2 eggs. How many unbroken eggs does Richard have left?” (3.OA.8)
• In Module 4, Lesson 15, students apply knowledge of area to determine areas of rooms in a given floor plan. Students are engaged in grade-level mathematics when they discuss with a partner their strategy and solution to the Problem Set Questions. The Problem Set Question states, “1. Make a prediction: Which room looks like it has the biggest area? 2. Record the areas and show the strategy you used to find each area. 3.Which room has the biggest area? Was your prediction right? Why or why not? 4. Find the side lengths of the house without using your ruler to measure them, and explain the process you used.” (3.MD.7)
• In Module 5, Lesson 28, students engage in grade-level mathematics when using equivalent fractions to solve word problems. The Application Problem states, “LaTonya has a 2 equal-sized hot dogs. She cut the first one into thirds at lunch. Later, she cut the second hotdog to make double the number of pieces. Draw a model of LaTonya’s hotdogs. How many pieces is the second hotdog cut into? If she wants to eat ⅔ of the second hotdog, how many pieces should she eat?” (3.NF.3)

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 21, students independently demonstrate the use of mathematics by applying knowledge of multiplication and division to solve two-step word problems. Problem Set Question 2 states, “Miss Lianto orders 4 packs of 7 markers. After passing out 1 marker to each student in her class, she has 6 left. Label the tape diagram to find how many students are in Miss Lianto’s class.” (3.OA.8)
• In Module 3, Lesson 18, students independently demonstrate the use of mathematics by applying their understanding of all operations to solve two step word problems. Problem Set Question 3 states, “Pearl buys 125 stickers. She gives 53 stickers to her little sister. Pearl then puts 9 stickers on each page of her album. If she uses all of her remaining stickers, on how many pages does Pearl put stickers?” (3.OA.8)
• In Module 7, Lesson 1, students independently demonstrate the use of mathematics by solving various word problems using a letter to represent the unknown. The problems apply their knowledge of multiplication to real-life situations. For example, students are given the price of adult and child hayrides and solve “a. Lena’s family buys 2 adult tickets and 2 child tickets for the hayride. How much does it cost Lena’s family to go on the hayride?“ (3.OA.3)

The instructional materials for Eureka Grade 3 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 6, students develop conceptual understanding of the distributive property by using tape diagrams and number bonds to illustrate their thinking. Problem Set Question 1c states, “Label the tape diagrams. Then, fill in the blanks below to make the statements true: 8 x 6 = ___, (5 x 6) = ___ (___ x 6) = ___, 8 x 6 = (5 + ___) x 6, = (5 x 6) + (___ x 6), = 30 + ___ = ___”
• In Module 4, Lesson 2, students practice fluency of multiplication facts within 100. The Fluency-Pattern Sheet includes 60 problems with an unknown product when multiplying the numbers 1 through 10 by 4.
• In Module 6, Lesson 9, students engage in the application of mathematics by creating and analyzing data in a line plot to solve word problems. Problem Set Question 3 states, “Ms. Pacho’s science class measured the lengths of blades of grass from their school field to the nearest ¼ inch. The lengths are shown below. Make a line plot of the grass data. Explain your choice of scale. How many blades of grass were measured? Explain how you know. What was the length measured most frequently on the line plot? How many blades of grass had this length? How many more blades of grass measured 2 ¾ inches than both 3 ¾ inches and 2 inches combined?”

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 3, students develop conceptual understanding of multiplication using equal groups while applying that knowledge to solve real-world problems. Problem Set Question 1 states, “Solve Problems 1–4 using the pictures provided for each problem. There are 5 flowers in each bunch. How many flowers are in 4 bunches?”
• In Module 2, Lesson 20, students engage in the application of mathematics and practice fluency of addition and subtraction within 1,000 to solve real-world measurement problems. Problem Set Question 3 states, “The weight of a pear, apple, and peach are shown to the right. (500 g) The pear and apple together weigh 372 grams. How much does the peach weigh? Estimate the weight of the peach by rounding each number as you think best. Explain your choice. How much does the peach actually weigh? Model the problem with a tape diagram.”
• In Module 3, Lesson 2, students practice fluency in multiplication while developing conceptual understanding of the commutative property of multiplication. Problem Set Question 1 states, “Each cube has a value of 7. Unit form: 5 ___, Facts: 5 x ___ = ___ x 5, Total = ___”

The instructional materials reviewed for Eureka Grade 3 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 1, the materials prompt students to analyze a multiplication model and explain their reasoning on whether the model correctly represents the given equation. Problem Set Question 2 states, “The picture below shows 2 groups of apples. Does the picture show 2 x 3? Explain why or why not.”
• In Module 3, Lesson 4, the materials prompt students to analyze the argument of a fictional student's solution and explain their thinking. Problem Set Question 5 states, “Julie counts by six to solve 6 x 7. She says the answer is 36. Is she right? Explain your answer.”
• In Module 5, Lesson 12, the materials prompt students to choose a drawing that best represents the unit fraction ¼ to create 1 whole and to explain their thinking. Exit Ticket Question 3 states, “Aileen and Jack used the same triangle representing the unit fraction ¼ to create 1 whole. Who did it correctly? Explain your answer.”
• In Module 7, Lesson 14, the materials prompt students to analyze two solutions to a perimeter problem and explain which solution is correct. Problem Set Question 5 states, “Mr. Spooner draws a regular hexagon on the board. One of the sides measures 4 centimeters. Giles and Xander find the perimeter. Their work is shown below. Whose work is correct? Explain your answer.”

The instructional materials reviewed for Eureka Grade 3 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 3, Lesson 3, teachers are prompted to engage students in constructing an argument by asking students why a strategy worked for a multiplication problem. “One way we’ve learned to solve 9 x 8 is by breaking 9 eights up into 5 eights plus 4 eights. Why did it work well?”
• In Module 4, Lesson 11, teachers are prompted to ask students the following questions to lead a discussion about area and side lengths of rectangles. “Discuss your answer to Problem 4 with a partner. What would the rectangle look like if the difference between side lengths was 0? How do you know? Compare your answer to Problem 4(c) with a partner’s. Did you both come up with the same side lengths? Why or why not? Explain to a partner how to use the strategy we learned today to find all possible whole number side lengths for a rectangle with an area of 60 square units.”
• In Module 5, Lesson 22, teachers are prompted to engage students in constructing an argument by allowing time for students to talk with their partner regarding equivalent fractions. “Now, I want you to work with a partner to look at your fraction strips again. See if you can find other equivalent fractions, shaded or unshaded. Draw and label them on your personal white board. For example, using my fraction strips, I can see that 2/2 and 4/4 are equivalent. Fourths are just halves cut in half again. Be ready to explain how you know, just like I did.”
• In Module 7, Lesson 2, teachers are prompted to guide students with questions that can be used when critiquing the work of others. “Have students share their work in groups of three or four. Encourage group members to practice asking questions of the presenter. They might ask some of the questions listed below. I’m not sure what you mean. Can you say more about that? Why did you decide? What do you think about instead? Which other way did you try to draw the problem?”

The instructional materials reviewed for Eureka Grade 3 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 7, the Notes on Multiple Means of Representation states, “Students need not master the term commutative property (3.OA.5). However, they will need to be familiar with the vocabulary moving forward in this module.”
• In Module 6, Lesson 1, the Notes on Vocabulary state, “Students are familiar with tally marks and tally charts from their work in Grades 1 and 2. In Grades 1 and 2 they also used the word table to refer to these charts.”
• In Module 7, Lesson 10, the materials guide the teacher through the teaching of the concept of perimeter. Problem 1 states, “T: Use your finger to trace around the edge of the piece you cut out. We call the boundary of the shape its perimeter. Say the word to yourself as you trace. S: Perimeter. (Trace with finger.)”

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

• In Module 3, Lesson 1, the materials use precise terminology of commutative property and support students in using the term when showing an example. The Concept Development states, “T: This is an example of the commutative property that we studied in Module 1. What does this property tell us about the product and its factors? S: Even if the order of the factors changes, the product stays the same!”
• In Module 4, Lesson 10, the mathematical term tiled is in bold writing within a question listed in the Student Debrief section. These questions guide teachers in leading a class discussion. “How is the rectangle in Problem 1(a) similar to the rectangle you tiled in today’s lesson? How is it different?”
• In Module 5, Lesson 3, the materials use accurate terminology when students learn the concept of equal parts. Problem Set Questions 1-3 state, “1. Each shape is a whole divided into equal parts. Name the fractional unit, and then count and tell how many of those units are shaded. The first one is done for you. 2. Circle the shapes that are divided into equal parts. Write a sentence telling what equal parts means. 3. Each shape is 1 whole. Estimate to divide each into 4 equal parts. Name the fractional unit below.”