4th Grade - Gateway 2

The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 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 Think and Grow section where students develop conceptual understanding of key mathematical concepts through teacher-led activities. Explore and Grow contains one to three problems where students model math and discuss their understanding through guided questions from the teacher. Think and Grow reinforces and extends the learning of the Explore and Grow section. For example:

• Chapter 4, Lesson 1, “Multiply by Tens” addresses standard 4.NBT.6, “Find whole number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and or the relationship between multiplication and division.” In the Explore and Grow section, students are asked to “Model each product. Draw a model. [2 x 3, 2 x 30, 2 x 300, 2 x 3,000]” and further asked, “What pattern do you notice in the products? How can the pattern help you find 20 x 30?” In the Think and Grow section, students explore using place value and properties to multiply two-digit numbers. The teaching notes suggest the use of base-ten blocks for students to build the product.
• Chapter 7, Lesson 3, “Generate Equivalent Fractions by Dividing” addresses 4.NF.1, “Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.” In the Explore and Grow section, students “Shade the second model in each pair to show an equivalent fraction. Then write the fraction.” Students “Describe the relationship between each pair of numerators and each pair of denominators. How can you use division to write equivalent fractions? Explain. Then use your method to find a fraction that is equivalent to 610.” In the Think and Grow section, students extend this understanding to build from the concrete to abstract. The teacher introduces the term “common factor” as a way to find an equivalent fraction through division. In the Teacher’s Guide, “Supporting Learners,” teachers are directed to “Provide grid paper and a multiplication table. Have students draw the fraction first.” This is a way to support students who need a concrete model to demonstrate their conceptual understanding.
• Chapter 10, Lesson 2, “Understand Hundredths” addresses 4.NF.6, “Use decimal notation for fractions with denominators of 10 and 100.” In Explore and Grow, students work on “How many pennies have a total value of one dollar? Draw a model. One penny is what fraction of one dollar? Write your answer in words and as a fraction.” In addition, they answer “How is one tenth related to one hundredth? How do you think you can write 1/100 in a place value chart?” In the Think and Grow section, students shade a model and use a place value chart to demonstrate their understanding of hundredths.

The instructional materials provide limited opportunities for students to demonstrate conceptual understanding independently throughout the grade-level. The Apply and Grow, and Homework and Practice sections do not engage students in conceptual understanding. For example:

• Chapter 3, Lesson 6, Apply and Grow, Problem 14, students independently write a multiplication equation shown by an area model. In the Homework and Practice sections, no opportunities are evident for students to demonstrate their conceptual understanding independently.
• Chapter 5, Lesson 1, “Divide Tens, Hundreds, and Thousands” addresses 4.NBT.6. In the Apply and Grow and Homework and Practice sections, no opportunities are evident for students to demonstrate their conceptual understanding independently. 
• Chapter 11, Lesson 7, Apply and Grow, students are given problems with the directions, “Find the equivalent amount of time.” Problem 7, “3 wk = ____”. In the Homework and Practice section students have similar conversion problems, but no opportunities to demonstrate their conceptual understanding independently.

The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 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, and/or students are given steps to solve the problem. For example:

• In Chapter 3, Lesson 2, Think and Grow: Modeling Real Life, “An aquarium has 7 bottlenose dolphins. Each dolphin eats 60 pounds of fish each day. The aquarium has 510 pounds of fish. Does the aquarium have enough fish to feed the dolphins? Step 1: How many pounds of fish do all of the dolphins eat? 7 x 60 = ____. All the dolphins eat ____ pounds of fish. Step 2: Compare the number of pounds of fish all of the dolphins eat to the number of pounds of fish the aquarium has. The aquarium ____ have enough fish to feed the dolphins.” 

The materials present opportunities for students to independently demonstrate routine application of mathematics. Examples of routine and non-routine applications include:

• Chapter 9, Lesson 1, Think and Grow: Modeling Real Life, “A piece of rope is 8/5 meters long. You cut the rope into 1/5 meter long pieces. How many pieces do you cut?” 
• Chapter 5, Performance Task, “The students in fourth grade go on a field trip to a planetarium.” 
• “1) The teachers have $760 to buy all of the tickets for the teachers and students. They receive less than $6 in change. a. Each ticket costs $6. How many tickets do the teachers buy? b. Exactly how much money is left over? c. There are 6 groups on the field trip. Each group has 1 teacher. There are equal number of students in each group. How many students are in each group? d. Two groups can be in the planetarium for each show. The planetarium has 7 rows of seats with 8 seats in each row. How many seats are empty during each show?
• 2) The groups will be at the planetarium from 11:00 A.M. until 2:30 P.M. During that time, they will rotate through 7 events: the planetarium show, 5 activities, and lunch. The planetarium show lasts 45 minutes. Each activity lasts 22 minutes. Students have 5 minutes between each event. How long does each group have to eat lunch?” This is a non-routine application." 
• Chapter 10 Performance Task, “You have a recipe to make one loaf of homemade whole wheat bread. You want to make 8 loaves of bread.” 
• “1) You need between 6.5 cups and 7 cups of whole wheat flour for one loaf of bread. a. So far, you measure 3 1/4 cups of flour for one loaf. What is the least amount of cups you need to add? b. There are about 4 cups of flour in 1 pound. How many 5-pound bags of whole wheat flour should you buy to make all of the bread? c. You use a $10 bill to buy enough bags of whole wheat flour for 8 loaves. What is your change? [there is a picture of a bag of flour at $2.60] 
• 2) You need to add 2 1/4 cups of warm water for one loaf of bread. The temperature of the water should be about 110°F. a. How many cups of water do you need for all of the bread? b. You find the temperatures of 3 different samples of water. Which sample of water should you use. Explain [Table provided].” Students need to reason and apply the skills of the chapter independently. This is a non-routine application."
• Chapter 6, Lesson 5, Think and Grow: Modeling Real Life, Problem 19, “You start with 128 pictures on your tablet. You take 6 pictures and delete 3 pictures each day. How many pictures do you have on your tablet after 6 days.” (4.OA.3) 
• Chapter 2, Lesson 4, Problem 16, “Students at a school want to recycle a total of 50,000 cans and bottles. So far, the students recycled 40,118 cans and 9,862 bottles. Did the students reach their goal? If not, how many more cans and bottles need to be recycled?” (4.NBT.4)

The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 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 1, Lesson 3, Dig In, in the Explore and Grow and Think and Grow sections, students demonstrate conceptual understanding through the use of a place value chart in order to compare multi-digit numbers. The mathematics becomes procedural as they compare numbers by looking at place value both in isolation and within the context of story problems.
• In Chapter 4, Lesson 5, Divide by 6 or 7, students use multiple strategies to multiply and divide by 6 or 7. In the Apply and Grow and Homework and Practice sections, students have multiple opportunities to develop fluency with questions that ask students to “Find the Quotient”, reinforcing division facts for 6 and 7. 
• In Chapter 6, Lesson 3, Think and Grow: Modeling Real Life, “You need 96 balloons for a school dance. Balloons come in packs of 4, packs of 6, and packs of 9. Which packs could you buy so you have no leftover balloons?” Teachers ask “What do we need to find out?”, “How do we decide if 96 is a multiple of 4?”, and “We don’t know the divisibility rule for 4 so let’s divide 96 by 4, 4 is a multiple of 96. This means if packages of 4 are purchased there will be no leftover balloons.” 

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: 

• Chapter 3, Lesson 7, addresses both conceptual understanding and procedural skill and fluency. During Dig In, students use base-ten blocks to build 4 groups of 17. In the Explore and Grow and Think and Grow sections, students demonstrate conceptual and procedural understanding as they explore multiplication. In the Apply and Grow, Think and Grow: Modeling Real Life, and Homework & Practice sections, students engage in procedural understanding and fluency. 
• In Chapter 9, Lesson 3, the Dig In section explores conceptual understanding as students use the fraction templates to model fractions. In the Explore and Grow section, students use models to multiply whole numbers by fractions. In the Think and Grow section, students have one last conceptual model for multiplying a whole number by a fraction. In the remaining sections, procedural skill and fluency are addressed as they multiply whole numbers by fractions with limited opportunities to demonstrate conceptual understanding independently.
• In Chapter 11, Lesson 7, in the Dig In and Explore and Grow sections of the lesson, students explore time as they review the number of seconds in a minute, minutes in an hour and hours in a day and how to convert between units. In the Think and Grow, Apply and Grow, Think and Grow: Modeling Real Life, and Homework and Practice sections, students are given conversion tables and use procedural understanding and fluency to convert measures of time. 

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 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 4, Lesson 2, Explore and Grow, identifies “MP Constructing Arguments”. Students are asked: “Which estimated product do you think will be closer to the product of 29 and 37? Explain your reasoning.” By comparing the estimates and using mathematical justification, students are constructing arguments. 
• Chapter 10, Lesson 6, Apply and Grow: Practice, You Be the Teacher, Exercise 12 does not identify MP3; however, students have an opportunity to analyze the arguments of others: “Your friend has three $1 bills and 2 pennies. Your friend writes ‘I have $3.2’ Is your friend correct? Explain.” Students use mathematics to analyze the reasoning of others and construct arguments to explain the mathematics correctly.
• Chapter 4, Lesson 3, Homework and Practice, You Be the Teacher, Exercise 7 does not identify MP3, but states: “Your friend finds 12 x 42. Is your friend correct? Explain.” The problem includes an area model and an equation. Students need to analyze their friends argument and use a mathematical argument to support their analysis.

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:

• Chapter 8, Lesson 7, Explore and Grow, MP Construct Arguments is noted in the Student Edition.  Students answer: “How can you use the whole number parts and fractional parts to add mixed numbers with like denominators? Explain why your method makes sense.” 
• Chapter 4, Lesson 8, Explore and Grow, MP Construct Arguments is noted in the Student Edition. Students use information about a ferry that can transport 64 cars each time it leaves the port. Students “Make a plan to find how many cars the ferry can transport in 1 week.” 
• Chapter 11, Lesson 6, Explore and Grow, MP Construct Arguments is noted in the Student Edition. Students create a line plot showing the length of each student’s hand length and are asked: “What conclusions can you make from the line plot?”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 partially meet expectations that the instructional materials assist teachers in engaging students in constructing viable arguments and analyzing 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 3, Lesson 8, Think and Grow: Modeling Real Life, teachers are prompted to ask, “Can you answer Exercise 19 by subtracting $152 - $144 = $8 first, then multiplying by 8? Explain.” In this example, the materials do not provide teachers with questions that elicit viable arguments or analyze the arguments of others.  
• Chapter 6, Lesson 6, Think and Grow: Create Shape Patterns, supports the teacher to use a four shape pattern that is repeated twice. Teachers, “Solicit explanations as to how they can find the 42nd shape. Ask other students to critique each other’s reasoning.” The materials do not provide guidance for the teacher to model the conversation, or include additional probing questions to support students to analyze the reasoning of others.  
• Chapter 13, Lesson 6, Explore and Grow, teachers ask, “‘Who can explain how they can tell by looking whether an angle is greater or less than 90 degrees?' Students can recall their mental image of a 90 degree angle. By comparing the given angle to their mental image, they can explain how the angle compares to a right angle.” The materials do not support the teacher to engage students in constructing an argument.

There are examples where the materials do assist teachers in having students develop an argument. For example:

• Chapter 10, Lesson 7, Think and Grow, students are asked to solve four money problems, each requiring a different operation. The materials provide guidance for teachers: “Focus on one example at a time. Ask a volunteer from a group to (a) explain their strategy for solving and (b) why it make sense. The focus is on the process not the answer.” The materials support teachers to engage students in constructing an argument. “Ask volunteers from other groups to critique the reasoning offered. Did they give a valid reason for the strategy they used? Again, focus on the process not the answer. How they carried out the strategy may vary.” The materials provide support support for teachers to engage students to critique the reasoning of others.
• Chapter 3, Lesson 4, Dig In, students use the distributive property to multiply tiles to create arrays. The teacher prompts, “If each partner is given a certain number of tiles to make a rectangle with no dimension equal to 1, will there always be just one rectangle when you combine the tiles?” Teachers engage students to construct an argument based on multiplication and the distributive property.
• Chapter 8, Lesson 1, Explore and Grow, students add fractions using models. The teacher asks the students questions to explain why you add the numerator and not the denominator. For example, “You did not add the denominators to get 16. Can you explain why?”