4th Grade - Gateway 2

The instructional materials reviewed for Ready Grade 4 meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Students use pictures, manipulatives, and models to demonstrate conceptual understanding.

The Teacher Resource Book contains a section called Concept Extension that provides teachers with additional ways to support building students’ conceptual understanding. Examples of how the Concept Extension supports teachers in building conceptual understanding include:

• Unit 1 Lesson 3 provides guidance to teachers on how to connect regrouping in addition to regrouping in subtraction, using base ten blocks. (4.NBT.4).
• Unit 2 Lesson 7 provides guidance to teachers on how to use patterns to find all the factor pairs for a number (4.OA.4).
• Unit 4 Lesson 15 provides guidance to teachers on how to use number lines to help extend student understanding of decomposing fractions (4.NF.3).

Clusters 4.NF.A, 4.NF.B, and 4.NF.C focus on understanding fractions as numbers and fraction equivalence.

• In Unit 4 Lesson 14 Compare Fractions (4.NF.2) students use note cards and divide them into thirds, fourths, fifths, and tenths to compare fractions. During this lesson, groups of students also use number lines to compare fractions.
• In Unit 4 Lesson 16 Add and Subtract Fractions (4.NF.3) students cut a piece of notebook paper into 12 equal strips and practice adding twelfths. Students use number lines to add fractions. Students divide paper plates into eighths, shade ⅝, and cut out 2 eighths using scissors to determine how much is remaining.
• In Unit 4 Lesson 20 Fractions as Tenths and Hundredths (4.NF.5) students use pennies and dimes to model tenths and hundredths.

The instructional materials reviewed for Ready Grade 4 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials provide opportunities to attend to procedural skill and fluency throughout the course, including fluency of adding and subtracting within 1,000,000 (4.NBT.4).

All lessons provide an opportunity for students to use computation skills. Each lesson contains a Building Fluency section which is designed to be used twice during a lesson. The Student Practice and Problem-Solving Book contain fluency practice pages, and Math Center Activities are included in the Teacher Toolbox which include activities for both procedural skill and fluency.

• Unit 1 Lesson 3 addresses adding and subtracting whole numbers within 1,000,000 (4.NBT.4).
• Unit 1 Math in Action integrates adding and subtracting within 1,000,000 (4.NBT.4).
• In Unit 4 Lessons 16 and 17 students add fractions with like denominators and compare fractions with different numerators and denominators (4.NF.3).

iReady Door 24 Plus is a free iPad app for fact fluency practice and is only available on the Apple platform. The game includes the fluencies for Grade 4.

The instructional materials reviewed for Ready Grade 4 meet the expectation for being 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.

Opportunities to work with engaging applications are provided throughout the instructional materials. Each unit contains a Math in Action where students are exposed to non-routine problems that contain many points of entry, have more than one possible solution, integrate multiple standards, and include a Persevere on Your Own section. During guided practice Try It and independent practice Practice by Myself, students apply what they have learned to solve real-world problems. Most lessons also have an online interactive tutorial for students which features real-world applications. In addition, there are Problem Solving Connection problems where students solve multi-step word problems with whole numbers (4.OA.3), solve real world problems using addition and subtraction of fractions (4.NF.3d), and solve real world problems using multiplication of fractions (4.NF.4c). The following are examples of the applications included in the instructional materials:

• In Unit 2 Lesson 10 students are presented with multiple opportunities to solve problems posed with whole numbers using the four operations in contexts that include miles walked in a 3-week period, planting flowers in pots, and reading books.
• In Unit 4 Lesson 17 Practice & Problem Solving Book students solve real-world problems using addition and subtraction of fractions in contexts that include mixing together a fruit salad, distance traveled during a trip, and determining how much fabric is used in a costume.
• Unit 4 Lesson 20 Practice & Problem Solving Book page 214: “Mila has 100 math problems to finish this week. She solved 2/10 of the problems on Monday and 25/100 of the problems on Tuesday. Which day did she solve more problems? What fraction of the problems did she solve on Monday and Tuesday? Has she completed more than ½ her problems?”

The instructional materials reviewed for Ready Grade 4 meet the expectations for balancing the three aspects of rigor. Overall, the three aspects of rigor are not always treated together and are not always treated separately within the materials.

Each lesson contains opportunities for students to build conceptual understanding, procedural skills and fluency, and apply their learning in real-world problems. Lessons are designed so students engage with all three components of rigor at different points in the lesson. During Guided Instruction and Guided Practice, students explore alternative solution pathways to master procedural fluency. During Independent Practice, students apply the concept in real world applications where they need to use both the procedural skills and their understanding of the concept to solve problems with multiple solutions and explain/compare their solutions.

For example, in Unit 5 Lesson 24 Time and Money students explain procedures for finding elapsed time, total time, conversion between units of time, money spent/saved and conversion between units of money. Through Guided and Independent Practice, students demonstrate multiple strategies to solve time and money problems, building fluency. In Independent practice, students also demonstrate conceptual understanding through error analysis. Teachers receive guidance to set up a classroom store, and students use the store as a place to apply what they have learned about money to a real-world, ongoing application. In the Unit 5 Lesson 24 Quiz students encounter routine procedural problems, evaluate the veracity of numerous expressions to represent a situation, and, using the information from the procedural and conceptual tasks, determine if the goal will be reached.

Math in Action lessons occur at the end of most units. These lessons focus on application problems where students apply procedural fluency and conceptual understanding to solve problems in a non-routine, real-world context.

The instructional materials reviewed for Ready Grade 4 meet the expectations for identifying the Mathematical Practices (MPs) and using them to enrich the mathematics content within and throughout the grade.

The MPs for each lesson are identified in the CCSS Focus section as part of the Lesson Overview. SMP TIPs are found in the Teacher Resource Book throughout the lessons, and these tips highlight the integration of particular MPs within the lessons. The MPs are also identified for each lesson in the Table of Contents for the Teacher Resource Book on pages A4-A7.

Some examples of where the MPs are identified and used to enrich the mathematics content include:

• Unit 2 Lesson 9: MPs 1, 2, 4, 5, 6, and 7 are identified in the Lesson Overview. The SMP TIP in the lesson for MP2 states, “Writing multi-step equations to represent word problems requires students to reason abstractly and quantitatively. To successfully complete this problem, students need to see that 2 + 1 is more than just an addition fact to be computed. They must see 2 + 1 as the quantity of hours worked in one day that will then be multiplied by the number of days worked.”
• Unit 4 Lesson 14: MPs 1, 2, 4, 5, and 7 are identified in the Lesson Overview. The SMP TIP for MP4 states, “Some students may be tempted to use the model alone as justification for saying that Adriana ate more. Encourage them to use the model mathematically by reasoning about the number of parts and sizes of the parts. Discuss how the number of parts in each bar relates to the denominators and how the number of shaded parts relates to the numerators.” The SMP TIP for MP7 states, “Students must apply knowledge of equivalent fractions to solve this problem. Encourage them to look for and make use of structure as they search for a viable common denominator. Ask: Could you rewrite 2/100 as tenths? [No. You would have to divide 2 by 10.] How could you rewrite 8/10 as hundredths? [Multiply both the numerator and denominator by 10 to get 80/100.]” The SMP TIP in the lesson for MP5 states, “Help students use benchmark fractions strategically. When students discuss their comparison, guide them to consider what benchmark fraction will be most helpful in solving a given problem. For example, in problems 10-13, 1 is a useful benchmark because both fractions in the problem are near 1 on a number line.”
• Unit 5 Lesson 23: MPs 2, 5, 6, and 8 are identified in the Lesson Overview. The SMP TIP in the lesson for MP8 states, “Point out to the students that there is repeated reasoning involved when converting from a larger unit to a smaller unit. The number given in the larger unit will always be multiplied by the number of smaller units that make up one larger unit.”

The instructional materials reviewed for Ready Grade 4 meet the expectation for carefully attending to the full meaning of each practice standard. Overall, the materials attend to aspects of the mathematical practices (MPs) during different lessons throughout the grade, so when taken as a whole, the instructional materials attend to the full meaning of each MP.

Examples of where the instructional materials attend to each of the MPs include:

• MP1: In Unit 5 Lesson 26 students make sense of and persevere in solving problems while finding missing side lengths when the perimeter is given.
• MP2: In Unit 2 Lesson 5 students reason abstractly and quantitatively as they translate between words, symbols, and bar models that represent the same multiplicative comparison. In Unit 2 Lesson 10 students reason abstractly and quantitatively as they reason about the remainder and what it means.
• MP4: In Unit 3 Lesson 12 students solve problems that involve division of whole numbers. The students can model the division problems with equations, bar diagrams, or an area model, and the students can use the model they choose to revise their initial solution if there are errors or it doesn’t make sense.
• MP5: In Unit 5 Lesson 25 students have the opportunity to choose from different tools as they engage with problems involving length. The SMP TIP in the lesson for MP5 states, “Encourage students to use and/or draw visual models when confronted with various problems. A visual model can be a useful tool in deciding the necessary steps to finding a solution. Visual models often make the problem seem more ‘real’ and less abstract.”
• MP7: In the Unit 1 Math in Action students discuss where they see place value in their solution to a problem and how their understanding of place value helps them understand the solution. In Unit 4 Lesson 13 students use the structure of fractions to determine what happens when fractions are multiplied or divided.
• MP8: In Unit 6 Lesson 32 students use repeated reasoning to classify a rhombus. The SMP TIP in the lesson for MP8 states, “Students have had many opportunities to identify right, acute, and obtuse angles. Students should be able to correctly place a rhombus in the Venn diagram using the patterns they have seen with other quadrilaterals.”

The instructional materials reviewed for Ready Grade 4 meet the expectation for prompting students to construct viable arguments and analyze the evidence of others. Overall, the materials offer students multiple opportunities to construct viable arguments and/or analyze the arguments of others throughout the materials.

Examples where students are prompted to construct viable arguments and analyze the arguments of others include:

• Unit 1 Math in Action students construct an argument as to why their estimate is reasonable for the expected number of visitors to a blog post about video games.
• Unit 3 Lesson 12 Question 16 students choose the correct answer out of four for a word problem. Then students answer: “Awan chose D as the correct answer. How did he get the answer? How can you tell that Awan’s answer doesn’t make sense?”
• Unit 4 Lesson 14 Practice and Problem Solving Workbook students are asked, “Can two fractions with the same numerator and different denominators be equal? Explain using words and numbers.”

The instructional materials reviewed for Ready Grade 4 meet the expectation for assisting teachers to engage students in constructing viable arguments and analyzing the arguments of others concerning key grade level mathematics detailed in the content standards. The materials provide teachers with SMP TIPs to help facilitate students to construct arguments and/or analyze the arguments of others.

Examples where teachers are supported to help students construct viable arguments and analyze the arguments of others include:

• Unit 2 Math In Action has a note for teachers during Plan It and Solve It that states, “Put students in pairs to discuss solution ideas. Ask them to also discuss the Reflect questions about Mathematical Practices. Remind students that there are always different ways to answer these questions.”
• Unit 3 Math In Action SMP Tip states: “Encourage students to explain and justify their ideas and to ask each other questions about their suggestions. Prompt them to demonstrate how their ideas could lead toward a solution.”

The instructional materials reviewed for Ready Grade 4 meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of Mathematics.

• The Student Practice and Problem Solving Book has notes at the bottom of the pages where mathematical vocabulary is defined.
• The Teacher Resource Book has Lesson Vocabulary for each lesson with mathematical terms and their definitions.
• Teachers are prompted in the Teacher Resource Book to have students use precise mathematical language. For example, Unit 4 Lesson 20 Mathematical Discourse 1 states, “Discuss the meaning of “4/10 of a dollar” [40 cents, or 4 dimes] and “50/100 of a dollar” [50 cents, or 5 dimes].”
• Each lesson has an “English Language Learners” section in the Teacher Resource Book that contains some ways to support vocabulary development for all students.
• Lessons contain language objectives. For example, Unit 2 Lesson 7 states, “Orally define and use in discussion the key mathematical terms factor, factor pair, multiple, composite number, and prime number.”