3rd Grade - Gateway 2

The instructional materials for Big Ideas Math: Modeling Real Life Grade 3 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. 

Each lesson begins with an Explore and Grow and a Think and Grow section, where students develop conceptual understanding of key mathematical concepts through teacher-led activities. The Explore and Grow section contains one to three problems where students model math and then discuss their understanding through guided questions from the teacher. The Think and Grow section reinforces and extends the learning of the Explore and Grow section. For example:

• In Chapter 1, Lesson 3, Explore and Grow, teachers direct students to “Put 24 counters into equal rows. Draw your model.” “Put 24 counters into a different number of rows. Draw your model.” “Compare your models. How are the models the same? How are they different?” This example builds conceptual understanding under the cluster heading, “Represent and solve problems involving multiplication and division” and specifically the standards 3.OA.1 and 3.OA.3. In the Think and Grow section, students model an array with 2 rows and 7 columns and build both a repeated addition equation (7 + 7 = 14) and a multiplication equation (2 x 7 = 14). 
• In Chapter 8, Lesson 7, Explore and Grow, students use a hundred chart and a number line for students to subtract. Students “Color to find 79 - 47. Then model your jumps on the number line.”...“How can finding 79-47 help you find 379-47?” Throughout the lesson, students use the number line to model addition and subtraction. This example builds conceptual understanding for 3.NBT.A “Use place value understanding and properties of operations to perform multi-digit arithmetic.” In the Think and Grow section, students learn and model “count back” and “count on” strategies prior to moving to independent work for the lesson. 

While there are some opportunities for students to demonstrate conceptual understanding independently, the instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. Within the Apply and Grow and Homework and Practice sections, students have limited opportunities to independently demonstrate conceptual understanding. For example:

• Chapter 2, Lesson 1, Explore and Grow, students “Model 3 x 2 using equal groups.” They answer: “How can you use the model to find 4 x 2?” Throughout the Think and Grow and Apply and Grow sections, students are given representations and tools to show both repeated addition and the math facts. In the Homework and Practice section, students have limited opportunities to use arrays, models, and repeated addition to demonstrate conceptual understanding of multiplication independently. In Problems 3 through 13 students “Find the Product.” 
• Chapter 4, Lesson 3, “Divide by 2, 5, or 10", Explore and Grow, students “Use the number line to model 10 ÷ 2” and answer ‘In your number line model, does 2 represent the number of equal groups or the size of the groups? Explain.’” Students have no additional opportunities to show their conceptual understanding in this lesson.
• Chapter 11, Lesson 7, “Compare Fractions", Explore and Grow, students “Use a strategy to find the greater fraction.” Students are given the fractions “2/3” and “2/8”. Students are directed to “Use a different strategy to check your answer. Tell your partner which strategy you prefer. Explain.” In the Think and Grow section, students are provided fraction strips to compare two fractions during the teacher directed activity. As students transition to the independent work in the Apply and Grow section, they are asked to complete many problems with the directions to “Compare,” with examples such as “1/4 ____ 2/4." Scaffolding is available in the Laurie’s Notes section, however, students are not required to demonstrate conceptual understanding during their independent work. The Homework and Practice problems provide limited opportunity for students to demonstrate conceptual understanding using fraction strips (3.NF.3d).

The instructional materials for Big Ideas Math: Modeling Real Life Grade 3 partially 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 series includes limited opportunities for students to independently engage in the application of routine and non-routine problems due to the heavily scaffolded tasks.

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher led questions. During the Dig In, Explore and Grow, and Think and Grow sections of lessons, teachers are provided with explicit guidance to support students to engage with applications of mathematical content. For example:

• In Chapter 3, Lesson 8, Think and Grow, students solve the problem, “There are 26 students in your class. Your teacher brings in 4 boxes of muffins. Each box has 4 packages with 2 muffins in each package. Are there enough muffins for the class?” In the Teacher Notes, teachers are given guiding questions: “How could we draw a picture of this situation? Which container is the largest, the box or the package? How many boxes should we draw? How many packages do you see? What is inside one of the packages? How can we adjust our equation to show we have 4 full boxes? Where would you place grouping symbols? Why?...Complete the multiplication using your choice of grouping. Decide if there are enough muffins for each of the students to have one. Compare your answer with your partner and discuss how you solved it.” 
• In Chapter 9, Lesson 4, Explore and Grow, students solve, “A box of 8 burritos costs $9. How much does it cost a group of friends to buy 40 burritos?” The Teacher Notes include guiding questions and directions: “Point to What do you know? and read the first statement. Demonstrate how to return to the problem, reading it again to find the information and complete the statements. Do the same for What do you need to find? ‘Now we are ready to make a plan.’ Show how to use the information in the Understand the Problem to complete the How will you solve? ‘Why does the plan write a division equation to find b? What other equations might a student use?’ The 40 burritos are boxed in equal-sized groups of 8. Division can be used to find the number of groups...As you work through Steps 1 and 2 with students, ask ‘What does b represent? What does c represent? How is the value for b used to find c? What if we get an answer of $10, does this seem reasonable?...What about 1 million dollars?”

The materials present opportunities for student to independently demonstrate routine application of mathematics; however, there are few opportunities for students to independently demonstrate application of grade-level mathematics in non-routine settings. Examples of routine applications include:

• In Chapter 6, Lesson 3, Show and Grow, Problem 11: “A sign has an area of 18 square feet. The sign is 6 feet long. How wide is the sign?” This is a routine application problem requiring students to find a missing factor. 
• In Chapter 4 Lesson 5, Think and Grow: Modeling Real Life,  “There are 42 students in gym class. The teacher divides the students into 7 teams. How many more students would be on each team if the teacher divides the students into 6 teams?” This is an example of a routine application as students need to revise the problem to represent a different divisor.

The instructional materials for Big Ideas Math: Modeling Real Life Grade 3 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Students engage with each aspect of rigor independently. For example:

• In Chapter 11, Lesson 4, Explore and Grow, students engage in conceptual understanding as they solve “Color each fraction. Circle the greater fraction. Explain to your partner how you can compare fractions with the same denominator.” Two examples with 2 equally partitioned parts are provided.
• Chapter 3, Lesson 4, Apply and Grow: Practice, students engage with procedural skill and fluency as they solve 11 multiplication problems.  
• Chapter 2, Lesson 5, Question 9, Show and Grow, students engage with application as they solve, “A joke book has 20 pages. There are 5 jokes on each page. You read 16 jokes. How many jokes do you have left to read?”

The instructional materials present opportunities for students to engage in multiple aspects of rigor within a lesson, however, these are often treated separately. For example:  

• In Chapter 3, Lesson 4, Dig In, Explore and Grow and Think and Grow sections, students demonstrate conceptual understanding through the use of number lines and models for multiplying by 7s. In Apply and Grow Practice, students engage with conceptual knowledge (using arrays), as well as procedural skills and fluency. During the Think and Grow: Modeling Real Life, and Show and Grow, students engage with application. In Homework and Practice, students engage in conceptual understanding, procedure skill, and fluency. 
• In Chapter 7, Lesson 3, during the Dig In and Explore and Grow sections, students engage in conceptual understanding using a hundreds chart and number line. The Think and Grow, Apply and Grow, Think and Grow: Modeling Real Life, and Homework and Practice sections focus on procedural understanding. 
• In Chapter 10, Lesson 3, students focus on conceptual understanding as they determine how many parts in a whole (denominator) and how many of the parts are shaded (numerator). However, within the context of story problems, students are introduced to the key words used to determine how many parts in a whole, thus proceduralizing the concepts. During Review & Refresh, students complete two problems that involve fact families involving multiplication and division. 
• In Chapter 11, Lesson 7, conceptual understanding of equivalent fractions is addressed in the Dig In and Explore and Grow sections with students using fraction strips to explore comparing fractions. In the Think and Grow section, the focus shifts to procedural understanding by providing students with a process to compare fractions. In the Apply and Grow, Think and Grow: Modeling Real Life, and Homework and Practice sections, the mathematics is predominantly procedural (3.NF.3.d).

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 3 partially meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples of where students engage in the full intent of MP3 include the following:

• Chapter 15, Lesson 2, You be the Teacher section is not identified with a MP; however, students need to construct an argument: “Descartes says that a square will always have a greater perimeter than a triangle because it has more sides. Is he correct? Explain.” The explanation requires the use of the properties of quadrilaterals.
• Chapter 7, Lesson 3 Homework and Practice, You be the Teacher does not identify a MP; however, students construct a viable argument: “Descartes says that a number rounded to the nearest ten can be greater than the same number rounded to the nearest hundred. Is Descartes correct? Explain?” 
• Chapter 7, Lesson 5, students estimate the differences “173-63” and “263-197”. The bottom of the page is marked “MP Construct Arguments”, as students “Compare your answers to your partner’s answers. Explain why they are the same or why they are different.” 
• Student Edition, Chapter 5, Lesson 3, Explore and Grow, students complete a multiplication table. MP Critique Reasoning is noted: “Describe how you completed the table. Compare your method to your partner’s method. How are they the same? How are they different?” 

The Student Edition labels Standards of Mathematical Practices with “MP Construct Arguments”, however, these noted activities do not always indicate that the students are constructing arguments or analyzing arguments of others. For example:

• In Chapter 11, Lesson 4, Explore and Grow, students “Explain to your partner how you can compare fractions with the same denominator.” Students do not need to construct an argument. 
• Chapter 4, Lesson 5, Explore and Grow, students “Complete the statements and the models: 42 6 and 42 7." Students answer, “Without solving, which quotient is greater? Explain how you know.” The students had already solved it, and the model that is created for them shows the answer. There is no thinking, analyzing or arguing that occur.
• In Chapter 4, Lesson 8, Explore and Grow, students solve 36 ÷ 4, “What other strategies can you use to solve? Explain the strategy to your partner.” In this example, students explain a strategy, they do not need to construct an argument.

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 3 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• Chapter 8, Lesson 2, Think and Grow: Adding on a Number Line, students find “245 + 38”. The teacher is guided with the following question, “Could we start at 38 and count on 245? Would you? Explain.” 
• Chapter 13, Lesson 4, Think and Grow: Drawing Quadrilaterals, students “draw a quadrilateral that has four right angles and name the quadrilateral.” The teacher is directed to poll the class for the results “rectangle or square.” “Explain to your partner why square is a more specific name than the rectangle.” This is a missed opportunity for the teacher to ask the students to provide a different name for their quadrilateral and argue which name best describes the quadrilateral, rather than telling the student that a “square is more specific."
• Chapter 14, Lesson 1, Explore and Grow, students read and interpret a picture graph. MP3 is identified in the teaching guide, “Explain to your partner how you found the total.” The materials do not assist teachers in having students construct viable arguments and analyze the arguments of others.

There are occasions where the materials do assist teachers to engage students to construct and/or analyze an argument. For example:

• Chapter 3, Lesson 9, Think and Grow: Using the Problem-Solving Plan, teacher guidance includes: “How do we know we should multiply the 8 and 3 to find the number of feathers used?” The materials encourage the teacher to “prompt justifications with questions: ‘Why not 30 x 3? Why subtract the product from 30?” 
• Chapter 10, Lesson 2, Think and Grow: Modeling Real Life, teachers are guided to have students look at flags from different countries, and based on their understanding of fractions, teachers ask: “How can you argue that there is only one flag in Exercise 13 that represents a unit fraction for the amount of red? Present your argument to your partner. Answer any questions your partner asks about your choice.”
• Chapter 13, Lesson 3, Dig In, Circle Time, teachers create a rectangle on a geoboard and ask: “How do you know it is a rectangle?” The teacher is prompted to “ask students to write how they know and show their partner. Do their reasons agree?” Based on the teacher’s observations they are then asked to “solicit reasons from students”.