4th Grade - Gateway 2
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Rigor & Mathematical Practices
| Score | |
|---|---|
Gateway 2 - Partially Meets Expectations | 61% |
Criterion 2.1: Rigor | 5 / 8 |
Criterion 2.2: Math Practices | 6 / 10 |
The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 partially meet the expectations for rigor and mathematical practices. The materials partially meet the expectations for rigor by reflecting the balances in the Standards and giving appropriate attention to procedural skill and fluency. The materials partially meet the expectations for practice-content connections, they identify the Standards for Mathematical Practices, and attend to the specialized language of mathematics, but do not attend to the full intent of each practice standard.
Criterion 2.1: Rigor
Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 partially meet the expectations for rigor and balance. The instructional materials give appropriate attention to procedural skill and fluency, but lack opportunities for students to independently demonstrate conceptual understanding and application. The materials also partially address the three aspects of rigor with balance, treating them separately but never together.
Indicator 2a
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
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.
Indicator 2b
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.
The instructional materials attend to the CCSSM fluency standards for Grade 4 including 4.NBT.4, Fluently add and subtract multi-digit whole numbers using the standard algorithm". For example:
- Chapter 2, Lesson 2, Add Multi-Digit Numbers, students use a step-by-step breakdown for adding multi-digit numbers within 1,000,000. For example, “307,478 + 95,061.” Students use place value to line up the addends, and “add the ones, then the tens, and then the hundreds. Regroup if necessary.” Then they add the thousands, one place value at time and regroup. Students use this strategy during the Apply and Grow and Homework and Practice sections, where they have multiple opportunities to practice adding multi-digit numbers (4.NBT.4).
- Chapter 2, Lesson 3, Subtract Multi-Digit Numbers, students use place value to line up the numbers and subtract each place value beginning with the ones, regrouping as necessary. Students are provided multiple opportunities to practice the procedural skill in Think and Grow, Apply and Grow, and Homework and Practice sections. For example, Homework and Practice problems include procedure practice such as “9,127 - 6,753” (4.NBT.4).
- Chapter 5, Lesson 1, Divide Tens, Hundreds, and Thousands, students divide a multiple of ten, one hundred, or one thousand by a one-digit number, applying their understanding of place value and division facts. In Think and Grow, procedural understanding is developed. For example, Problem 1, “Find 2,400 ÷ 6”. Students are directed to think about the fact “24 ÷ 6”. Then students are encouraged to think of 5,600 as “56 hundreds ÷ 8” which is equal to “7 hundreds” or “700”. Students answer additional questions using this procedure. For example, Problem 13, “2400 ÷ 6 = ____”; Apply and Grow, Problem 24, “There are 240 students visiting a fair. They are divided equally among 8 barns. How many students are in each barn?”; Homework and Practice, Problem 3, “Find each quotient. 12 ÷ 2=___, 120 ÷ 2=___, 1200 ÷ 2=___,” and Problem 14, “50 ÷ ___=10” (4.NBT.4).
In addition to the Student Print Edition, Big Ideas Math: Modeling Real Life Grade 4 has a technology package called Dynamic Classroom. The Dynamic Student Edition includes a game library where students can practice fluency and procedures. For example, the game “Race to the Moon” allows students to practice subtracting multi-digit numbers. “Multiplication Quest” allows students to practice multiplying one-digit numbers. Additionally, the Dynamic Student Edition includes videos that explain procedures and can be accessed through the QR Code in the Student Edition.
Indicator 2c
Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
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)
Indicator 2d
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
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.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials for Big Ideas Math: Modeling Real Life Grade 4 partially meet the expectations for practice-content connections. The materials identify the practice standards and explicitly attend to the specialized language of mathematics. However, the materials do not attend to the full meaning of each practice standard.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 meet expectations for identifying the Mathematical Practices (MPs) and using them to enrich the Mathematical Practices. The MPs are identified in both the Teaching Edition and Student Edition and the practices are connected to the mathematical content.
The Standards for Mathematical Practice (MP) are identified in the digital Teacher's Edition on page vi. The guidance for teachers includes the title of the MP, how each MP helps students, where in the materials the MP can be found, and how it correlated to the student materials using capitalized terms. For example, MP2 states, "Reason abstractly and quantitatively.
- "Visual problem-solving models help students create a coherent representation of the problem.
- Explore and Grows allow students to investigate concepts to understand the REASONING behind the rules.
- Exercises encourage students to apply NUMBER SENSE and explain and justify their REASONING."
The MPs are explicitly identified in Laurie’s Notes in each lesson, and are connected to grade-level problems within the lesson. For example:
- Chapter 14, Lesson 1, Line Symmetry - (MP4) Dig In, “Line Symmetry” is marked with MP4 - Model with Mathematics. Teacher asks students, “How many diagonals does each quadrilateral have? Fold across one diagonal in each of these shapes. Keep the quadrilateral folded, don’t unfold it.” The teacher is then instructed to have students discuss what they notice about the folded quadrilaterals and provide them with some possible conversations that may occur.
- Chapter 8, Lesson 4, Use Models to Subtract Fractions (MP4), students are asked to create and use a model to find 9/12 - 5/12. The teacher is directed, “Ask several volunteers to share their models. You will likely see many number line diagrams. Students may recall how to show subtraction with an area model by crossing out the subtrahend.”
The MPs are identified in the digital Student Dashboard under Student Resources, Standards for Mathematical Practice. This link takes you to the same information found in the Teacher Edition. In the Student Edition, the MPs are identified in the Explore and Grow, Apply and Grow: Practice, and Homework and Practice Sections. For example:
- Chapter 11, Lesson 7, Units of Time - Explore and Grow, MP - Structure is identified in the Student Edition with the following question, “You know the amount of time in minutes. Without using a clock or a stopwatch, how can you find the amount of time in seconds?” The teaching notes identify the MP and the teacher is directed to ask, “When you know an amount of time in minutes, how can you find the amount of time in seconds?” (MP7 - assumed)
- Chapter 2, Lesson 2, Add Multi-Digit Numbers - Explore and Grow: MP - Reasoning is identified in the Student Edition. Students are given several addition problems and asked to determine which problems show a correct way to find the answer, focusing on the vertical alignment of the digits with in the number. Students are asked the question, “Why do you need to use place value when adding? Explain.” There are no teaching notes for this question to support teacher facilitation of the MP. (MP2 - assumed)
- Chapter 3, Lesson 1, Homework and Practice, MP - Precision is identified in a question where students are asked to “Compare the door’s height to the desk’s height using multiplication and addition.” This item does not meet the full intent of MP6. While students do use symbols to create a comparison between the height of the door (8 ft) and the height of the desk (2 ft), they are not communicating this to others. The teaching notes do not identify this specific MP. (MP6 - assumed)
MP5 and MP8 are under-identified.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 do not meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. The materials do not attend to the full meaning of three or more Mathematical Practices.
The instructional materials do not present opportunities for students to engage in MP1: Make Sense of Problems and Persevere in Solving Them, MP4: Model with mathematics, MP5: Use appropriate tools strategically, and MP8: Look for and express regularity in repeated reasoning.
MP1: The instructional materials present few opportunities for students to make sense of problems and persevere in solving them. For example:
- Chapter 3, Lesson 1, Laurie’s Notes, Think and Grow: Modeling Real Life, “Circulate as students discuss the model and then solve the problem. Students should recognize that there are 5 parts total for the amount of liquid. Knowing that the 5 parts represent 10 tablespoons of liquid helps them reason that each part must be 2 teaspoons.” Students are not given the opportunity to make sense and persevere independently as the problem is scaffolded.
- Chapter 10, Lesson 6, Laurie’s Notes, Think and Grow: Modeling Real Life, “‘How can the two money amounts be compared?’ Write the fraction of a dollar (Newton’s money) as a decimal dollar amount or write the price tag as a fraction of a dollar.” Students compare 85/100 and $0.99.
MP4: The instructional materials present few opportunities for students to model with mathematics. For example:
- Chapter 3, Lesson 10, Laurie’s Notes, Think and Grow, students solve, “A coach buys 6 cases of sports drinks and spends $60. Each case has 28 bottles. A team drinks 85 bottles at a tournament. How many bottles are left?” Laurie’s Notes state, “A tape diagram helps us to visualize the 6 cases of sports drinks. The letter k represents the unknown product, the number of bottles in 6 cases. If we find out what k equals, will we be able to find out n, the number of bottles left?” Students do not model to solve this problem and the tools are provided for them.
- Chapter 5, Lesson 4, Laurie’s Notes, Think and Grow: Use Partial Quotients to Divide, students use an area model and partial quotients to find 235 ÷ 5. Laurie’s Notes state, “The area model shows how we reasoned through the problem. We don’t know the missing factor. Why? It is not a division fact we know, so we estimate. We use a fact we do know, 5 x 20, we subtract and the difference is 135. We can subtract another 5 x 20. We continue to subtract multiples until the remainder is less than the divisor.” Students do not need to model with mathematics.
MP5: While the Dynamic Student Edition includes tools for students, the instructional materials present few opportunities for students to choose their own tool, therefore, the full meaning of MP5 is not being attended to. MP5 is only identified a total of four times throughout the instructional materials and only in two chapters. Big Ideas Math: Modeling Real Life Grade 4 presents limited opportunities for students to choose tools strategically, as the materials indicate what tools should be used. For example:
- Chapter 8, Lesson 6, Laurie’s Notes, Dig In, students hold a rope and stand at points 0, 1, 2, and 3, with each whole segment divided into two equal parts. Teachers are directed to “Ask a volunteer to locate the fraction 1/2 and explain how they know they are correct. Label the location 1/2 with a sticky note. ‘Now I want to divide the length between 1 and 2, and between 2 and 3 into two equal parts.’ Clip two sticky notes at 1 1/2 and 2 1/2 but label them A and B. ‘Tell your partner what names we should write for these two locations.’ Elicit responses from students.” Students do not select tools strategically. The activity is fully guided by the teacher.
- Chapter 5, Lesson 6, Laurie’s Notes, Dig In, students use “base 10 blocks to find a quotient where regrouping of 1 ten and 10 ones is needed. The figures show the stages of thinking that connect to the written record. The first figure shows three groups of ten with 1 ten and 2 ones left to be divided. Then 1 ten is regrouped as 10 ones plus 2 more ones for a total of 12 ones. Then 12 ones can be divided evenly into the three groups. The quotient is 4. It is important for students to manipulate the base ten blocks and explain aloud how they are using the blocks to find 42÷3. Have students write on their whiteboards to show how partial quotients can be used to solve 42 ÷ 3.” Students do not choose tools strategically as they are given a tool.
Indicator 2g
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Indicator 2g.i
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
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?”
Indicator 2g.ii
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
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?”
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 4 meet expectations that materials use precise and accurate terminology and definitions when describing mathematics and the materials support students to use precise mathematical language. For example:
- In the beginning of each chapter is “Laurie’s Overview.” In this section, the mathematics of the chapter is described. For example, Chapter 7, Laurie’s Overview states, “Vocabulary terms equivalent and equivalent fractions are introduced and explored in the first three lessons. Models used to develop understanding of equivalent fractions are fraction strips, area models, and number lines. In all three models, students need to be able to see the whole.”
- Each chapter contains Vocabulary Cards for students and a vocabulary activity to introduce and reinforce the terms. For example, the Chapter 5 vocabulary cards include the terms partial quotients and remainder. The reverse side of each card gives a definition and an example.
- Teachers are provided explicit instructions in utilizing accurate mathematical terminology. For example, in Chapter 13, Lesson 3, Homework and Practice, teachers are provided the following note: “Proficient students are generally able to understand how a given quadrilateral can have two different names. They need to attend to this possibility at all times. Sometimes a proficient student will forget and only select the most obvious category.” In Chapter 10, Lesson 1, Think and Grow, the following is noted, “The decimal point is read as and, not point. Read 2.8 as ‘two and eight tenths.’”
- “MP Precision” is labeled in the student book and highlights the precise use of numbers, symbols, and terminology. For example, in Chapter 8, Lesson 2, Apply and Grow, students are asked to “Match each fraction with an equivalent expression.” In Chapter 3, Lesson 1, Homework and Practice, student are given the height of a door and desk and asked to “Compare the door’s height to the desk’s height using multiplication and addition.” Students are asked to be precise in their description, such as “the door is 4 times as tall as the desk” and “The door is 6 feet taller than the desk.”
Overall, the materials accurately use numbers, symbols, graphs, and tables. Students are encouraged throughout the materials to use accurate mathematical terminology. The teaching guide reinforces the use of precise and accurate terminology.