1st Grade - Gateway 2

The materials reviewed in Grade 1 for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.

The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open-ended questions that require evidence to show their thinking and reasoning.

The following are examples of attention to conceptual understanding of 1.OA.4:

• In Unit 3, Module 1, Session 5, students use the number rack as a tool to model and solve subtraction word problems. Problem types include Take From-Change Unknown, Take From-Start Unknown, and Compare-Difference Unknown. These varying problem structures provide opportunities for students to develop a deep understanding of the relationship between addition and subtraction.

The following are examples of attention to conceptual understanding of 1.OA.7:

• In Unit 6, Module 1, Session 2, students are provided with equations with a box for the missing addend. They solve various equations and also determine if equations are true.

The following are examples of attention to conceptual understanding of 1.NBT.B:

• In Unit 4, Module 4, Session 2, students build conceptual understanding of bundles of ten within 100 using concrete materials and the structure of a ten-frame.
• In Unit 7, Module 1, Session 2, students count popsicle sticks, grouping 10 ones into groups of 10. Then, two index cards are labeled "10's" and "1's," and students reorganize their sticks so that they can be counted more easily. Students collaborate to model groupings in our base ten number system helping them develop a deep understanding of place value.
• In the February Number Corner Calendar Collector, a ten-strip model is used to build conceptual understanding of place value with tens and ones.

The following are examples of attention to conceptual understanding of 1.NBT.C:

• In Unit 2, Module 2, Session 4, the 100s Number Grid is observed, and students share some things they notice (you can count by 10s in the last column, there are all 0's in that column, there are 1's under 1's and 2's under 2's, etc.). Students use the Number Grid as a scaffold/tool to help solve the "Change Cards" game problems. As students play the game, "Change Cards," they are adding and subtracting multiples of 10 (Cards: 25 and 35 = rule of +10). They then discuss the "rule" and pair-share to make predictions for the next group of cards.
• In Unit 3, Module 3, Sessions 1-4, students are using cube trains of ten and single cubes to represent addition two-digit equations equations.
• In Unit 8, Module 3, Sessions 3-6, students use unifix cube trains

The Grade 1 materials meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.

• Students spend a significant amount of time and have a variety of opportunities to fluently add and subtract throughout Number Corner activities.
  • In the Number Corner September Computational Fluency, in Activity 1, students are using the double 10-frame to solve addition problems by matching the equation with the cards.
  • In the Number Corner October Computational Fluency, in Activity 2, students play Ten-Frame Flash and show, with their fingers, how many dots are on the 10-frame cards.
  • In the December Number Corner Computational Fluency Routine, the routine for this month involves investigating double facts greater than 10 using a double ten-frame.
  • In the March Number Corner Days in School Routine, the hundreds grid is used as a visual model to assist students in explaining their mental reasoning for ten more or ten less.

• Fluency is developed throughout the sessions of the Grade 1 instructional materials.
  • In Unit 1, Module 2, Session 3, students respond to representations of the 10-frame as the teacher flashes the 10-frame cards with totals within 10.
  • In Unit 4, Module 1, Session 5, students solve addition and subtraction combinations such as 3+2 and 5-2. Students are using their fluency to help deepen their understanding about the relationship between addition and subtraction.
  • In Unit 2, Module 2, Session 3, students identify strategies they used in adding numbers represented on a domino. Teacher led discourse elicits student thinking using counting on, doubling, and decomposing strategies.
  • In Unit 1, Module 3, Session 1, students solve missing-addend problems on their number racks. They use 5 as a landmark and find doubles and then count on.
  • In Unit 3, Module 4, Sessions 1 and 2, students use unifix cubes to represent equations up to 10 with various combinations of two or three addends.
  • In Unit 6, Module 2, Session 2, students use ten and double ten frames to represent addition up to twenty in the game "I have, who has" to create the addition equations.

Materials meet the expectations for having engaging applications of mathematics as they 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.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide single step contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. The majority of the application problems are done with guiding questions elicited from the teacher through whole group discussions that build conceptual understanding and show multiple representations of strategies. Materials could be supplemented to allow students more independent practice for application and real-world contextual problems that are not teacher guided within discussions.

The instructional materials include problems and activities aligned to 1.OA.1 and 1.OA.2 that provide multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. Examples of these applications include the following:

• In Unit 2, Module 2, Session 2, the materials provide story problems to investigate the relationship between addition and subtraction equations. Students write their own story problems and equations as an extension of the learning.
• In Unit 3, Module 1, Session 5, students use the number rack as a tool to model and solve subtraction word problems (problem types - Take From-Change Unknown, Take From-Start Unknown, and Compare-Difference Unknown). These varying problem structures provide opportunities for students to develop a deep understanding of the relationship between addition and subtraction and apply mathematical knowledge and skills in a real-world context.
• In Unit 4, Module 1, Session 3, the teacher provides frog word problem to students and they solve using the number line.
• In Unit 4, Module 2, Session 3, the lesson utilizes a floor number line to investigate the use of addition and subtraction on the number line through contextual story problems.
• In Unit 7, Module 3, Session 2, students solve addition story problems with sums to 20 involving adding to, put together with unknowns in all positions.
• In the Number Corner October Calendar Grid, the teacher provides a word problem (add to, result unknown problem type), and students solve.
• In the Number Corner February Computational Fluency, students began to add to ten in the context of themed story problems and application within the Number Corner computational fluency routine. Thinking within these contextual situations is extended toward building conceptual understanding of subtraction as a missing addend problem.

The materials reviewed in Grade 1 meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Grade 1 materials all three aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the unit sessions and in Number Corner activities. For example, in Unit 3 Module 3 Session 1 and Unit 6 Module 1 Session 4 all aspects of rigor are present. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve conceptual understanding.

The instructional materials reviewed for Grade 1 meet the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level Standards for Mathematical Content, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Grade 1 Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The chart is broken down into four categories: habits of mind, reasoning and explaining, modeling and using tools, and seeing structure and generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Grade 1?" guide for teachers (AG, page17).

Each session clearly identifies the MPs used in the Skills & Concept section. Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation of the connection between the indicated MP and the Standards for Mathematical Content for the teacher. Examples of the MPs in the instructional materials include the following:

• Unit 1, Module 4, Session 1 references MP2 and MP5. Unit 1, Module 4, Session 2 references MP5 and MP6. Unit 1, Module 4, Session 3 addresses MP4 and MP6. In Unit 1, Module 4, Sessions 4 and 5 both address MP4 and MP7. There is a "Math Practices In Action" reference in Session 1 and Session 3.
• In Unit 2, Module 1 in the Skills & Concepts section, two sessions (3 and 4) list MP2, Session 4 lists MP3, Session 5 lists MP6, two sessions (1 and 2) list MP7, and Session 2 lists MP8.
• In Unit 6, Module 1, Sessions 1 and 3 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In Unit 7, Module 2, four of the five sessions address MP7.
• In Unit 7, Module 3, Session 1 references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In the September Number Corner, MP4 is addressed in Calendar Grid, Days in School, and Computational Fluency; MP6 is addressed in Calendar Collector, MP7 is addressed in Calendar Grid, Days in School, Computational Fluency, and Number Line, and MP8 is addressed in Number Line.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These sessions that focus entirely on MPs include the following:

• Unit 1, Module 2, Session 1
• Unit 6, Module 4, Session 4
• Unit 8, Module 4, Session 5

The materials meet the expectations for attending to the full meaning of each practice standard. Each Session clearly identifies the MPs used in the Skills & Concept section of the session. Typically there are two MPs listed for each session, so there is not an overabundance of identification.

Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, the MPs are not discussed or listed in unit overviews or introductions (major skills/concepts addressed); however, they are listed in the section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.

In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Days in School, Computational Fluency, Number Line). The MP are also listed in the assessment section of the introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of the MPs within Number Corner.

The following are examples of times that the instructional materials attend to the full meaning of the MPs:

• MP7 is addressed in Unit 1 Module 3 Session 2. This session focuses on the pattern and structure of the unit of ten using number lines, 10-frames and the place value relationship between the ones unit and tens unit.
• MP1 is addressed in Unit 3, Module 1, Session 5. Students are presented with very challenging number rack story problems in the Problems & Investigations section of this lesson. The wide range of problem types makes this session cognitively challenging for students in Grade 1. They are supported in their efforts to solve the problems and grow accustomed to devoting significant time and effort to persevere in solving them.
• MP3 is addressed in Unit 6, Module 2, Session 5. Students are working together to play the game "Pick Two to Make Twenty." Students have to pick two cards that total a number closest to 20. Students share their ideas and learn to construct viable arguments and critique the reasoning of others. Playing together as a team against the teacher motivates students to listen carefully to one another so they have the best chance at winning (Math Practices In Action, page 31). Students are invited to make a case for the combination they thinks works best by explaining their thinking to the class, and they can demonstrate on the number rack to help justify their thinking.
• MP5 is addressed in Unit 8, Module 4, Session 2. This session is titled, "How We Have Grown." Students work to compare the lengths of two strings that represent the length of an average baby with the length of an average student in Grade 1. Students are asked what strategies they might used to figure out the difference between the two lengths, and possible strategies are discussed, such as using Unifix cubes to represent lengths and then laying them next to each and count the extras on the big one, a number line, and models. Students are using appropriate tools strategically when they compare the lengths using strings, Unifix cubes, and the number line. They think carefully about how to use Unifix cubes and how to use efficient jumps on the number line. They are making choices about which tools to use and how to use them based on the problem at hand. (Math Practices in Action).

However, at times the materials only partially attend to the meaning of MP4. The intent of this practice standard is to apply mathematics to contextual situations in which the math arises in everyday life. Often when MP4 is labeled students are simply selecting a model to represent a situation. For example, in Unit 3, Module 1, Session 1, MP4 is indicated, but students are simply representing a number on a 10-frame. The Math Practices in Action note states that "Students will use drawings, numbers, expressions, and equations to model with mathematics."

The materials reviewed for Grade 1 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.

• In Unit 2, Module 1, Session 4, students work with the teacher to create a class "Addition Strategies Chart." As they review the Domino Add & Compare game, students discuss some of their strategies they can use to find the total number of dots on a domino. After the chart is created, students are asked to share the advantages and disadvantage of each strategy. Since all of the strategies are written on the chart, students are able to critique and compare the strategies of others in a safe manner.
• In Unit 3, Module 4, Session 2, students are asked to show their thinking about how to decompose the number seven into two addends in multiple ways using addition and subtraction.
• In Unit 4 Module 3 Session 5, students are given a contextual word problem to compare two number sets involving the unit of one with the tool of pennies. They have to show their thinking with a visual representation.
• In Unit 6, Module 2, Session 5, students are introduced to a new game, "Pick Two to Make Twenty." When students share their ideas about how to get as close to 20 as possible, they are learning to construct viable arguments and critique the reasoning of others.
• In Unit 6, Module 4, Session 2, students are asked to show how they solved a contextual problem using pictures, numbers and words to explain their thinking.
• In Unit 7, Module 2, Session 1, students are making trains of Unifix cubes that total 120. Before beginning, students are asked to discuss with their partner about how many groups of 20 cubes will be needed to make the train. Volunteers share and explain their thinking. The next part of the lesson students are asked again to explain their thinking regarding locating a specific cube within the train.

The instructional materials reviewed for Grade 1 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 that is present throughout the materials.

The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon and trapezoid and using operations and algebraic thinking terminology such as equation, difference, and 10-frame.

• Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
• The online teacher materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved them, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
• At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
• Unit 3 Module 4, Session 1 investigates the relationship between numbers with the commutative property of operations. The language within the lesson reinforces and contextually uses the terms equations, equal and strategies to explain the reasoning between visual representations.
• In Unit 5, Module 2, Session 4, during a Problems & Investigations activity with 3-dimensional shapes, the teacher asks students how they identified a cube just by touch. The teacher guide gives specific direction to the teacher: "Model the vocabulary of geometry as you field their responses." The student's response includes the word "corner," and the teacher restates the response by saying, "What do you mean the corners - in math we call them vertices..."
• The Number Corner, October Calendar Collection investigates patterns of geometrical shapes using the terms hexagon, rhombus and trapezoid.