Kindergarten - Gateway 2

The instructional materials reviewed for Ready Kindergarten 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 shows teachers how to deepen students’ understanding of the relationship between counting and the amount left over (K.CC.B).
• Unit 4 Lesson 16 shows teachers how to connect subtraction and addition (K.OA.A).
• Unit 5 Lesson 22 shows teachers how to find number pairs with teen numbers (K.CC.5).

Standard K.OA.1 focuses on representing addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.

• In Unit 4 Lesson 14 Understand Addition students draw pictures of things they might add together (2+1), and then they add two of the items to another item using the words “add” and the “addition sign.” Students are given two different colors of connecting cubes to add. Students also use their cubes to write addition number sentences in order to connect what they made and what they write. Finally, students work with a partner to find the pairs of addends to make 4.
• In Unit 4 Lesson 16 Understand Subtraction students use counters to physically take a counter away when subtracting 5-1. Students also model number sentences with counters.

Cluster K.NBT.A addresses conceptual understanding of working with numbers 11-19 to gain foundations for place value.

• In Unit 5 Lesson 21 Understand Teen Numbers students first show how to count to 11 using their fingers and then use a partner to show an additional finger. Next they use ten blue counters and then add on 1 more red counter to show 11 and come to the understanding that 11 is 10 and 1 more. Students then use counters and 10-frames to add 10 and 1 more to conclude that they can count on from 10 instead of starting at 1. Finally, students fill a 10-frame with counters and add 1.

The instructional materials reviewed for Ready Kindergarten 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 to add and subtract within 5 (K.OA.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.

• In Unit 4 Lesson 15 students are provided opportunities to practice finding the sum of related facts (K.OA.5).
• In Unit 2 Lesson 8 students show combinations within 5 (K.OA.5) and practice rote counting to 30 on day 1 and day 3.
• In Unit 7 Lesson 29 Problem-Solving Connections includes three problems:
  • 1. “A dog has 3 spots on his ear and 5 spots on his body. How many spots does he have in all?
  • 2. There are 6 people. Some wear orange coats and the rest wear blue coats. How many people can wear blue coats and how many can wear orange coats?
  • 3. There are 8 leaves on the ground. 3 leaves are under a cone and the rest are not. How many leaves are not under a cone?”

iReady Door 24 Plus is a free iPad app for fact fluency practice and is only available in the Apple platform. The game does include the fluencies for Kindergarten.

The instructional materials reviewed for Ready Kindergarten 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.

Each lesson includes guided practice Practice Together and independent practice Practice by Myself where 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 addition and subtraction (K.OA.2). The following problems are examples of the application problems included in the instructional materials:

• Unit 2 Lesson 8: “How can you tell how many counters you need to draw to finish each picture so that it shows 6?”
• Unit 2 Lesson 10: “There are 5 jars of jam on a shelf. 3 more are put on the shelf. How many jars of jam are on the shelf now?”
• Unit 4 Lesson 15: Students think of an addition story problem they could tell using given numbers.
• Unit 4 Lesson 17: “There are 5 balloons. Some pop. Now there are 4 balloons left. How many balloons popped?”
• Unit 5 Lesson 21: “Bob has 8 blocks. He gets 2 more blocks. How many blocks does he have now?”

The instructional materials reviewed for Ready Kindergarten 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 students use what they know (conceptual understanding) about whether a friend has more, the same, or fewer number of snacks. Students then use counters to identify the values. During Guided Practice students exhibit procedural skill and fluency as they count, identify sets, and compare sets to written numbers. Students then apply their knowledge during Independent Practice by counting a set, writing the number, and modeling a set that is either less than or more than.

In Unit 4 Lesson 14 students develop an understanding of the addition sign (conceptual understanding), see ‘plus’ in place of ‘and,” and use the plus sign under the word plus (procedural skill).

The instructional materials reviewed for Ready Kindergarten 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 3 Lesson 11: MPs 1, 2, 3, 4, 6, 7, and 8 are identified in the Lesson Overview. The SMP TIP in the lesson for MP1 states, “Allowing children sufficient time to think through the problem on their own encourages them to try different approaches if their first or second attempt does not work. This builds confidence in finding ways to use what they learned from those attempts to revise their thinking on subsequent attempts.” The SMP TIP in the lesson for MP6 states, “In counting the groups on this page to identify those showing 10, children must attend to precision, being sure to count each object only once to maintain the relationship between sequential number names and counted objects.” The SMP TIP in the lesson for MP7 states, “The Hands-On Activity requires children to use structure to determine where the count sequence breaks down as a means to identify the missing number. Alternate regular counting activities with ones like this to help reinforce structure.”
• Unit 4 Lesson 20: MPs 1, 2, 4, 6, 7, and 8 are identified in the Lesson Overview. The SMP TIP for MP2 states, “Using number sentences to describe story problems shows that children can reason quantitatively. They see that 2 and 3 is 5 when adding windows and that 4 and 1 is 5 when adding children. Encourage children to tell addition and subtraction story problems throughout the day to show that they can make sense of quantities and their relationships.” The SMP TIP for MP7 states, “Seeing the relationship between addition and subtraction facts helps children to look for and make use of structure. Encouraging children to use this relationship between addition and subtraction will help them to add and subtract more fluently.”
• Unit 5 Lesson 25: MPs 1, 6, 7, and 8 are identified in the Lesson Overview. The SMP TIP in the lesson for MP8 states, “Asking children to explain how they decided which numbers were wrong encourages them to express regularity in repeated reasoning. Continuing to have children count aloud and then identify mistakes in counting will help them look for and recognize repetitive actions in counting.”

The instructional materials reviewed for Ready Kindergarten 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 1 Lesson 1 students complete several word problems. The SMP TIP for MP1 in the lesson states, “Allowing children sufficient time to think through the problem on their own encourages them to try different approaches if their first or second attempt does not work. This builds confidence in finding ways to use what they learned from those attempts to revise their thinking on subsequent attempts.”
• MP2: In Unit 1 Lesson 3 students match a group of 4 counters to 4 apples helping them make sense of the abstract idea of 4 as a quantity. In Unit 3 Lesson 12 students develop an understanding that 10 is greater than 8 no matter the objects being counted or how they are arranged by comparing a quantity of 10 items to a quantity of 8 items.
• MP4: In Unit 4 Lesson 16 students model a subtraction problem by drawing a set of items and showing that some items are being taken away by circling those items or putting an X through them. Teachers are prompted to have students share their models and consider which models might be more useful than others.
• MP5: In Unit 4 Lesson 15 students complete addition problems and are able to use two-sided counters or 5-frames as tools for helping them complete the problems.
• MP7: In Unit 1 Lesson 6 students are asked to make observations about train cars colored yellow and blue. The SMP TIP in the lesson for MP7 states, “Asking children to make observations about how the cars are the same helps them to see single objects as being composed of smaller units in different ways. Encourage children to make similar observations about the different number pairs throughout the lesson.”
• MP8: In Unit 5 Lesson 24 students are developing skill in counting numbers up to 100. The SMP TIP in the lesson for MP8 states, “The Talk About It question encourages children to look for patterns in numbers. If time allows, ask children to describe other patterns they see in the hundreds chart.”

The instructional materials reviewed for Ready Kindergarten 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:

• In Unit 1 Lesson 1 students look at the counting depicted on the student practice page and think about whether the objects on the page were counted correctly. The students discuss, “What is the mistake in counting the soccer balls? What is the mistake in counting the children?”
• In Unit 1 Lesson 5 students listen to the work of their peers’ and analyze their peers’ work by determining how their own explanations are similar and different.
• In Unit 3 Lesson 12 students listen to their peers and then ask questions or make comments about the explanations given for drawing an appropriate picture based on the number given.

The instructional materials reviewed for Ready Kindergarten 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 4 Lesson 19 the teacher is provided with the following prompt to help children develop a deeper understanding of problems completed involving subtraction within 10, “How are all the problems the same? How are they different? Why are the answers all the same?”
• In Unit 7 Lesson 32 Talk About It notes for the teacher include, “When children have completed the page, say: 'Work with a partner. How are your pictures alike? How are your pictures different?' Pictures will be alike in that they both show triangles and squares, but how those triangles and squares are combined can produce a variety of differences.”

The instructional materials reviewed for Ready Kindergarten meet the expectation for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers provide 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 14 Step By Step states, “Introduce the term 'add' as a word for joining or putting together numbers.”
• 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 15 states: “Tell put-together and add-to addition problems to match a given picture. Count pictures to find the total for an addition sentence.”