3rd Grade - Gateway 2

The instructional materials reviewed for Ready Grade 3 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 2 provides guidance to teachers on how to discuss the Commutative Property of Multiplication with students (3.OA.5).
• Unit 4 Lesson 14 provides guidance to teachers on how to have students fold paper to show equal parts (3.NF.1).
• Unit 5 Lesson 28 provides guidance to teachers on how to explore square numbers with students (3.MD.7).

Cluster 3.NF.A focuses on understanding fractions as numbers and fraction equivalence.

• In Unit 4 Lesson 15 Understand Fractions on a Number Line (3.NF.2) students divide number lines into whole numbers and then whole numbers into fractional parts.
• In Unit 4 Lesson 18 Understand Comparing Fractions (3.NF.3) students use shapes to model fractions and compare them.

Standards 3.OA.1 and 3.OA.2 focus on understanding the meaning of multiplication and division.

• In Unit 1 Lesson 1 Understand the Meaning of Multiplication (3.OA.1) students use counters to build groups and solve for multiplication. Students draw arrays to show equal groups.
• In Unit 1 Lesson 4 Understand the Meaning of Division (3.OA.2) students use counters to divide into equal shares. Students draw pictures to show items being divided into equal shares.

The instructional materials reviewed for Ready Grade 3 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 in solving single-digit products and quotients (3.OA.7) and adding and subtracting within 1,000 (3.NBT.2).

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 6 addresses multiplication and division facts (3.OA.7).
• Unit 2 Lesson 9 specifically addresses adding and subtracting within 1000 (3.NBT.2).
• In Unit 2 Tic-Tac-Times-Ten students multiply a 1-digit number by a multiple of 10 (3.OA.7).
• In Unit 4 Lesson 17 students identify equivalent fractions (3.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 does include the fluencies for Grade 3.

The instructional materials reviewed for Ready Grade 3 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 use multiplication and division to solve word problems (3.OA.3) and solve two-step word problems using the four operations (3.OA.8). The following are examples of the applications included in the instructional materials:

• In Unit 1 Math in Action: Use Multiplication and Division students decide how to use pie plates to build robot props, using multiplication and division as well as working with constraints including the maximum number of plates. (3.OA.3)
• In Unit 3 Lesson 13 Practice & Problem Solving Book students use the four operations to solve two-step word problems in the contexts of determining the number of babysitters needed, the amount of money needed to buy song books, and the amount of change received after purchasing puzzles. (3.OA.8)
• In Unit 3 Math in Action: Use the Four Operations students solve several two-step word problems that arise from situations where students are presented with how much different items cost, the number of items needed, and how much money is available to buy the needed items. (3.OA.3 and 3.OA.8)

The instructional materials reviewed for Ready Grade 3 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 6 Lesson 31 Understand Properties of Shapes students explore how sides and angles inform them about the properties of a shape. In Guided Practice students look at various polygons and group them into categories, for example, rectangles and triangles. They identify characteristics that help define the shapes within each category, i.e. rectangles have square corners; triangles have three sides. Students then choose their own categories and group shapes as those that represent the category and those that do not. They describe the differences between the two and identify and explain if there is one shape that fits into two categories. In the Lesson 31 Quiz students fluently use classifying shapes to determine if shapes meet or don’t meet stated criteria, analyze using a Venn diagram with attributes of shapes to determine if a shape is placed correctly, and write an explanation to a student to support their analysis.

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 3 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 1 Lesson 6: MPs 1, 2, 6, 7, and 8 are identified in the Lesson Overview. The first SMP TIP in the lesson for MP7 states, “Students are asked to look at the structure of multiplication and division equations to make sense of the relationship between the two operations.” The SMP TIP in the lesson for MP6 states, “Discussing strategies and patterns gives students the opportunity to practice using precise mathematical terms to communicate their ideas to others.”
• Unit 3 Lesson 12: MPs 1, 2, 4, and 5 are identified in the Lesson Overview. The SMP TIP for MP1 states, “You may wish to have students present their interpretations of the models and describe the relationships between them. Students make sense of problems in different ways and benefit from hearing how others think.” The SMP TIP for MP5 states, “As students learn to use appropriate tools, it is helpful for them to evaluate tools for their usefulness in a given situation. Help them do this for this two-step problem by asking questions such as: Why is it helpful to use words and numbers to model the problem? What can using a drawing show us that words and numbers may not?”
• Unit 4 Math In Action: MPs 1, 3, and 4 are identified in the Lesson Overview. The SMP TIP in the lesson for MP4 states, “Guides students to focus on the models in this problem. Engage students in discussions about how the models relate to the actual flower gardens. Note that they function somewhat like a map of the gardens.”

The instructional materials reviewed for Ready Grade 3 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 23 students use estimation to help determine if their answer makes sense; they persevere in solving the problem by revisiting their process, if the initial solution does not make sense.
• MP2: In Unit 1 Lesson 2 students explain why an expression matches a situation. In the Unit 2 Math in Action students explain how the context relates to the numbers and operations in a problem.
• MP4: In Unit 4 Lesson 16 students model with mathematics as they use “fraction models and number lines to reason about how the parts of a fraction relate to the whole and to other fractions.” By using the different models, students get to interpret their previous work in different ways and validate that the interpretations agree.
• MP5: In Unit 5 Lesson 26 students choose from inch-rulers or yardsticks as they measure earthworms in some problems and the length of their stride in other problems.
• MP7: In the Unit 3 Math in Action students use the structure of language to understand the structure of the mathematics. The SMP TIP for MP7 states, “Help students recognize that the structure of the information about the parts indicates multiplication. For example, 1 deck costs $8, 2 decks cost twice as much, 3 decks cost three times as much, and so forth.”
• MP8: In Unit 2 Lesson 10 students express regularity in repeated reasoning as they use place value to multiply numbers. The SMP TIP in the lesson for MP8 states, “Multiplying by multiples of 10 is an example of a repeated calculation. Encourage students to look for shortcuts for finding the products. Provide several multiplication equations, for example 4 x 40 = 160, 3 x 60 = 180, 7 x 30 = 210, and 8 x 30 = 240. Have students work with a partner to discuss the patterns they find.”

The instructional materials reviewed for Ready Grade 3 meet the expectation for prompting students to construct viable arguments and analyze the arguments 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 Lesson 2 Question 9 students explain what is the same and what is different about the two multiplication equations they wrote.
• Unit 3 Lesson 11 Question 25 students choose the correct answer out of four to solve a word problem. After the students answer the question they are asked, “Harry chose A as the correct answer. How did he get the answer? How did you figure out how Harry got his answer?”
• Unit 6 Lesson 32 Students construct arguments to answer each of the following questions, “Is every square a rectangle?” and “Is every rectangle a square?”

The instructional materials reviewed for Ready Grade 3 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 3 Math In Action SMP TIP states, “As you discuss Sweet T’s solution, prompt students to explain Sweet T’s reasoning. Ask questions such as: How did Sweet T know that he only had $4 left over? or Why did he add $81, $54, and $50?”
• Unit 5 Lesson 21 Mathematical Discourse prompts the teacher to ask: “The hour on the digital clock changed from 4 to 5. Why isn’t the answer more than 1 hour?”

The instructional materials reviewed for Ready Grade 3 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 1 Lesson 1 At A Glance states, “Students focus on the meaning of each number and symbol in a multiplication equation using the terms ‘factor’ and ‘product’.”
• 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 4 Lesson 16 states, “Orally define and use the key mathematical equivalent fraction when reasoning about equivalent fractions with a partner.”