2nd Grade - Gateway 2

The instructional materials reviewed for Ready Grade 2 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 3 Lesson 17 guides teachers to help students use string to measure around boxes (2.MD.1).
• Unit 3 Lesson 22 guides teachers to help students compare lengths in a line plot (2.MD.6).

Cluster 2.MD.A focuses on measuring and estimating lengths in standard units.

• In Unit 3 Lesson 16 Understand Length and Measurement Tools students consider the strengths and limitations of measuring with paperclips to develop an authentic need for a standard unit. Students are then given concrete manipulatives that are the same length as standard units and are asked to consider through mathematical discourse the reason that standard units are needed. Students answer questions such as “Why do we need standard units such as inches and centimeters?” and “What if you use inches to measure your shoe, and your friend uses paper clips to measure her shoe? Can you compare the lengths of the two shoes?”

Clusters 2.NBT.A and 2.NBT.B address conceptual understanding of place value and using the properties of operations to add and subtract.

• In Unit 2 Lesson 10 Understand Three-Digit Numbers the hundreds chart and connecting cubes are used to build one hundred and other three-digit numbers. Students are taught to represent connecting cubes with simple visual models that can be used in future learning. Students are engaged in discussion to consider the representation using questions such as “How do the models help you think about the number 345?”

The instructional materials reviewed for Ready Grade 2 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 adding and subtracting within 20 (2.OA.2) and adding and subtracting within 100 using strategies (2.NBT.5).

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.

• The Student Practice and Problem-Solving Book contains fluency practice pages for adding and subtracting within 20 (2.OA.2) and adding and subtracting 2-digit and 1-digit numbers within 100 (2.NBT.5).
• In Unit 1 Lesson 3 students develop fluency in adding and subtracting within 20 by utilizing the strategy of making 10 (2.OA.2).
• In Unit 3 Lesson 23 Math Center Activity 2.46 Draw and Use a Bar Graph students create a bar graph and answer addition and subtraction questions about the graph (2.OA.2).

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 2.

The instructional materials reviewed for Ready Grade 2 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. Beginning in Grade 2, each unit contains a lesson called 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 are exposed to situations to apply problems involving addition and subtraction. The following problems are examples of the application problems included in the instructional materials:

• Unit 1 Lesson 6 Try It: “Gus had 7 shells. Then he found 4 more. Then some shells broke. Now Gus has 9 shells. How many shells broke?” (2.OA.1)
• Unit 2 Lesson 15 Independent Practice: “Pablo loses the 29 card. Taj picks three of the cards that are left. His cards have a sum less than 100. What are the three cards?” The cards at the top of the page are 27, 48, 43, 29, 34, and 35.
• Unit 3 Lesson 25 Independent Practice: “A bookmark costs 68 cents. Haley uses 3 quarters to pay for it. Which coins should she get back in change?”

The instructional materials reviewed for Ready Grade 2 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 1 Lesson 5 Day 1 Using Arrays students exhibit conceptual understanding by building arrays using the terms rows and columns. Students exhibit procedural skill by building arrays with a specific number of rows and columns. Students model how they add using an array, and students also apply their knowledge of arrays to demonstrate different addition facts and skip counting.

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 2 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 3: MPs 1, 3, 4, 5, 7, and 8 are identified in the Lesson Overview. The SMP TIP in the lesson for MP7 states, “Formal exploration of the associative property validates the structure informally used by students and provides a foundation for future applications of the property.” The SMP TIP in the lesson for MP8 states, “Encourage students to look for consistencies in patterns by following the structure of the Concept Extension, but beginning with a different set of numbers like 8 + 8 = 10 + _____.”
• Unit 2 Math in Action: MPs 1, 2, and 3 are identified in the Math In Action Overview. The SMP TIP for MP4 states, “Invite students to use a variety of representations (quick drawings, place-value charts, base-ten blocks) as they try different numbers of cookies. Prompt them to compare how each representation shows how many there are of each kind and the total.” The SMP TIP for MP2 states, “Engage students in a discussion about why they might want to use more or fewer boxes. Encourage them to use mathematical concepts to bolster their arguments.”
• Unit 3 Lesson 17: MPs 3, 4, 5, 6, and 7 are identified in the Lesson Overview. The SMP TIP in the lesson for MP5 states, “Throughout this lesson, students use tiles, rulers, yardsticks, and meter sticks to measure. This gives them the opportunity to find out how the different tools look and feel and to practice using appropriate tools strategically.” The SMP TIP in the lesson for MP6 states, “Lining up the 0 mark of a measuring tool correctly is a good opportunity for students to attend to precision. You may want to have students move objects to the left and right of the 0 mark to see how this affects the measurement.” The SMP TIP in the lesson for MP4 states, “In Problem 10, when students add 30 cm + 30 cm + 30 cm, they are modeling a situation with an equation.”

The instructional materials reviewed for Ready Grade 2 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 3 Lesson 21 the SMP TIP for MP1 states, “As students complete problem 7, emphasize the importance of taking the time to understand what the problem is asking. Also, encourage students to look back at the problem after they have solved it to make sure the model they used and the answer they found make sense.”
• MP2: In Unit 1 Lesson 6 students decontextualize a problem about pears to reason abstractly and recognize the relationship among equations and then reason quantitatively about the quantities in the equations. In Unit 3 Lesson 23 students use a bar graph to represent information and to interpret information.
• MP4: In Unit 4 Math in Action students determine how they will cut three cakes into pieces that represent different fractions of the whole. Students are then presented with the solution of a fictitious student and are supposed to discuss their solution and the fictitious one with a partner. Students are given the opportunity to analyze their original solution, the solution of the fictitious student, and determine if both solutions are appropriate and why or if either solution needs to be revised.
• MP5: In Unit 3 Math in Action has students measuring 8 scraps of wood and answering various questions about the measurements they obtain. Students are given the opportunity to choose what tool they will use to measure the scraps of wood, what units they will measure the scraps in, and whether exact measurement is needed or if estimation is sufficient.
• MP7: In Unit 2 Lesson 7 students look for and use place value structure to add numbers within 100. Students examine pictures of objects and place them into groups of ten in order to help with addition. Students also use place value structure as they work with base ten blocks and open number lines to add within 100.
• MP8: In Unit 2 Lesson 15 students look for and express regularity in repeated reasoning as they use different number facts when adding up to four two-digit numbers. The SMP TIP for MP8 states, “As students analyze strategies for adding numbers, they recognize the structure and regularity of those strategies. By generalizing and applying strategies to new situations, they gain fluency in mental computation.”

The instructional materials reviewed for Ready Grade 2 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 2 Lesson 7 students choose the correct answer out of four for a word problem. They are then asked: “Brady chose A as the answer. This answer is wrong. How did Brady get his answer?”
• Unit 2 Lesson 10 students are given the following prompt: “Lana did this homework problem. What did she do wrong?” In the problem the student wrote 2 hundreds (206) as 2 tens (26).
• Unit 4 Lesson 26 students are shown a shape that is a rectangle. They are told that Bruce says the shape is a square, and they construct an argument for whether or not they agree with Bruce’s claim.

The instructional materials reviewed for Ready Grade 2 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:

• In Unit 1 Lesson 1 teachers have students share strategies about how they created fact families and then discuss the similarities and differences with each other.
• The SMP Tip in Unit 2 Lesson 14 says to “(h)ave students compare the strategies they used to calculate. Discuss the similar elements and differences of each one and ask students to justify their computational approaches.”

The instructional materials reviewed for Ready Grade 2 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 2 Lesson 14 Mathematical Discourse 2 states, “To regroup when subtracting, we can break apart, or decompose, 1 ten into 10 ones.”
• 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 11 states, “Read aloud three-digit numbers. Write three-digit numbers in expanded form. Write a three-digit number shown with base ten blocks.”