Kindergarten - Gateway 2

The instructional materials for Math Expressions Kindergarten meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade where called for in the standards. The Teacher Edition states, “Through using objects, drawings, conceptual language, and real-world situations, students build mathematical ideas that make sense to them.”

Materials utilize MathBoards (laminated boards) for students to make their own drawings to communicate their conceptual understanding. Students have access to iTools to model conceptual understanding. Math Talk activities provide “frequent opportunities for students to explain their mathematical thinking and to ask questions of other explainers deepens their understanding of concepts.” Additionally, “What’s The Error” activities with the Puzzled Penguin provide students the opportunity to identify errors, discuss why it is incorrect, and how to correct it. Finally, Learning Paths, found in each unit, explain how students will build understanding of concepts throughout the unit.

Examples include but are not limited to:

• In Unit 2, Lesson 2, in the Student Activity Book, students identify a number (6-10) and look at a finger representation showing a group of 5 fingers on one hand and then identifying the additional number of fingers shown on the second hand to complete a “5 + ___” addition equation (K.OA.1).  
• Unit 2, Lesson 19, “What’s The Error,” Puzzled Penguin uses a 10 strip to order numbers 1-10 incorrectly. He asks, “Did I make a mistake?” and students write the numbers in the correct order (K.CC.2).
• In Unit 3, Lessons 2, 5, 6, 8, 13, 15, and 17 through 20, students count a teen number of objects, then group 10 objects as a 10-group and see how many more ones are in the number. Students use a 10-Counter Strip to represent 10 ones or one 10, and use Number Tiles to make a number with tens and ones. Students use a 10-Counter Strip to represent 10 ones or one ten, and use Number Tiles to make a number with tens and ones. For example, to make 12, students place a 2 tile over the 0 on the 10 tile to show 12 consists of 10 and 2. A classroom Teen Display is created with an equation “14 = 10 + 4” and students use stickers or simple drawings to represent the equation as a group of 10 and a group of 4 (K.NBT.1).
• In Unit 3, Lesson 3, students use Break-Apart Sticks and square tiles to find all the partners of the numbers 2 through 6, then write the equation for the total and the partners. In the Student Activity Book, students find various decompositions of 4, 5 and 6 using illustrated tiles pictured (K.OA.3).
• Unit 3, Lesson 16, Math Talk, students play Unknown Partners game for 6 and 7. Student pairs sit beside each other and face their workspace which contains a Counting Mat, Number Tiles, and Operation Tiles. A number is called out. One partner uses centimeter cubes to represent the number, then removes break apart sticks leaving the remaining cubes in place. The partner opens their eyes and must identify the break apart partner that was removed (K.OA.1).
• The Unit 4, Learning Path, explains how students continue to build understanding of addition and subtraction. “Children also continue to build their understanding of adding and subtracting by telling and solving such problems with larger numbers in the class grocery store and by solving addition equations without a given scenario” (K.OA.2).

The instructional materials for Math Expressions Kindergarten meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop and provide independent opportunities for procedural skill and fluency throughout the grade-level. Math Expressions includes a Path to Fluency for each grade level. “This plan provides targeted practice in the Student Activity Books, Teacher Editions, Teacher Resource Books, Math Activity Centers, as well as Fluency Checks in the Student Activity Books” (TE I12). In Kindergarten the fluency plan contains practice problems in the Student Activity Book identified by a Path to Fluency icon, Fluency Checks beginning in Unit 4, Reteach and Practice sheets, Daily Quick Practice Routines, and online resources. Examples include but are not limited to:

• Unit 2, Lesson 6, Student Activity Book, Connect the Dot activities help students recognize and order numbers (K.CC.2).
• Unit 3, Lessons 16-21, Daily Quick Practice Routines, students complete +1 in Addition Stories. Students use the strategy of adding a one to quickly find sums (K.OA.A).
• Unit 3, Lessons 4, 5, 6, 7, 12, 14, 18, and 19, Student Activity Book, students begin regular practice to fluently add and subtract within 5. Students add or subtract numbers as shown in equation form (K.OA.5).
• Unit 4, Lesson 17, Homework 4-17, Practice Sheets, students add within ten. These Homework workbook pages found throughout the series spiral back to prior work and help build fluency (K.OA.5).
• Fluency Checks, Volume 2, Fluency Check 5, page 326, students add and subtract within 5 (K.OA.5).
• Online Resource, Poggles MX, interactive game providing addition and subtraction fluency practice.

The instructional materials for Math Expressions Kindergarten meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level and to independently demonstrate the use of mathematics flexibly in a variety of contexts. Opportunities for contextual problem solving and non-routine problems are found in Math Talks. Students are provided real-world problem scenarios throughout each lesson. Performance Tasks at the end of each unit, provide students the opportunity to solve real world situations. Also, Math Readers embed math learning in a context appropriate story. Finally, online games provide problem solving practice. For example:

• Math Talks provide opportunities for students to engage with more detailed problems. For example, in Unit 2, Lesson 3, the teacher introduces verbal math story problems about real things that happen with families, such as setting the table for dinner and foods eaten for dinner. Students then tell stories about a meal and create and answer addition and subtraction questions from their story.
• In Unit 3, Lesson 1, students solve “what if” story problems set in a park. “There is 1 child playing with some balls. Imagine that more children come to play. What if 5 more children come? How many children are playing with the balls now?” Students are expected to make math drawings and to begin using symbols to write expressions and equations to match (K.OA.2).
• Performance Tasks at the end of each unit, provide students with the opportunity to solve a real world task. For example, on the Unit 4 Performance Task, students “show a set of partners for 10, use the partners to make up an addition story problem about the grocery store, make a drawing to show the story problem, and write the numbers in the boxes to show the addition equation.” Next, they show a different set of partners for 10 and create subtraction story problem.
• Online activities provide additional opportunities for students to apply mathematical knowledge and skills to real-world contexts. For example, for the In-Depth Inquiry Based Task in Unit 4, Stock and Shop, students create a grocery store, sort and count the items, and create subtraction word problems related to the items in the store.
• Students create their own story problems. For example, in Unit 5, Lesson 2, students create their own addition and subtraction story problems. Students start by using numbers for adding and subtracting within five and towards the end of the unit create for within 10.
• Math Readers provide opportunities for students to solve word problems in a variety of contexts. Unit 5’s Reader, “Under the Umbrellas” includes addition and subtraction story problems based on the pictures in the story.

The instructional materials for Math Expressions Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

Instructional materials attend to conceptual understanding, procedural skill and fluency, and application independently to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level materials. The three aspects of rigor can be found separately and together in the Daily Routines, Quick Practice, Math Talks, Fluency Checks, Puzzled Penguin, Lessons, and Homework. For example:

• Fluency can be found in any of the 6 Fluency Checks. For example, Fluency Check 5 has students adding and subtracting within 5 (K.OA.5).
• Conceptual Understanding can be found in any of the Puzzled Penguins, “What’s the Error?” Questions. For example, in Unit 4, Lesson 19, Student Activity Book, page 254, students must look at what Puzzled Penguin wrote, “6 = 4 - 2” help him, and discuss with classmates how he wrote the equation incorrectly.
• Application can be found in Math Talk in Action activities. For example, in Unit 4, Lesson 4, teachers present the following Math Talk problem, “The fruit bowl has 7 bananas. Jeff takes out 3 bananas to peel. How many bananas are in the fruit bowl now? Ask children how they might solve this problem and what tools, if any they would use to help them.” (TE 380)

Examples where the aspects of rigor are treated together include, but are not limited to:

• In Unit 2, Lesson 5, students act out addition and subtraction stories from family experiences. Students build a conceptual understanding by discussing addition and subtraction using familiar scenarios. The remaining lesson involves procedural skill as students practice writing the number six and drawing groups of six.
• In Unit 3, Lesson 1, students observe a park scene and use procedural skill to identify groups of 1-10 objects and color them with one color. Students use conceptual understanding to see how objects are arranged in the picture and identify partners for each number (nine birds total are pictured as a group of four and a group of five). Students use pictures and apply their learning to create story problems, such as, “There is 1 child playing with some balls. What if 5 more children come? How many children are playing with balls now?”
• In Unit 4, Lesson 2, students’ fluency prior knowledge is activated through Partner Practice with Finger Wiggles and review on how to find partners and record them with an equation. Conceptual understanding is built using centimeter cubes, number tiles, and the plus sign in the Unknown Partner game for six or seven. Students use a break-apart stick to find the partners of six and then record the expression using the plus sign. Next, within the context of sleeping/awake bugs students use inch tiles to apply their learning and find partners of 10. Lastly, the activities progress to more of a skill-oriented activity where students find partners of 10 and record them as equations in their Student Activity Book.

The instructional materials reviewed for Math Expressions Kindergarten meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level; however there are instances when the Mathematical Practices are over-identified.

Materials clearly identify Mathematical Practices being used in each lesson and are embedded in the content to enrich the mathematics. Instructions are provided for teachers on how to implement Mathematical Practices within the lesson. While Mathematical Practices are not identified in the student materials, the Teacher Edition does provide highlighted narratives for Mathematical Practice activities found in the Student Activity Book.

Examples include, but are not limited to:

• The Teacher Edition provides guidance on how to implement the Mathematical Practices in the Student Activity Book. For example, in the Student Activity Book, Unit 3, Lesson 13, students identify the error the Puzzled Penguin made as he circled groups of five (MP3).
• All Mathematical Practices identified in the materials have notes for the teacher. For example, Unit 2, Lesson 1, Activity 2, teacher notes include, "MP6 Attend to Precision. In this unit, children will focus on numbers 6 through 10. Children will draw scenes with groups of a specified number (6, 7, 8, 9, 10) for homework. Children can add to their scenes by drawing groups of small things.”  
• Focus on the Mathematical Practices lessons are the last lesson in each unit. These lessons engage students in all eight practices, however, the practices are often over-identified. For example, in Unit 4, Lesson 22, Focus on the Mathematical Practices, students engage with MP8 as they describe the attributes of objects they find, and MP3 as they justify the attributes of the objects they circled.
• The Overview of every unit contains “Common Core State Standards for Mathematical Practices in this Unit”.  A table is provided listing every Mathematical Practice along with corresponding lessons where the practice is embedded.
• The “Using the Common Core Standards for Mathematical Practice” section contains a description of the Mathematical Practice along with examples of where to find it within the unit. For example in Unit 2, Lesson 2-5, MP1 Make Sense of Problems: Act It Out. “Act out another story about a meal in someone’s family. Ask, “Who can tell a story about a family dinner?  Then we’ll find some math in the story.” “Focus children’s attention on the addition found in the story. Act out the situation in the front of the room. Show the adding with fingers. Lead children to see partner sets of numbers, saying 4 is 2 and 2 or 3 and 1.” Additional examples are provided for each Mathematical Practice.
• Every lesson identifies and provides explanations for the Mathematical Practice addressed in the lesson.  For example, Unit 4, Lesson 4-7, “MP7 Look for Structure | Identify Relationships After children have placed 10-sticks and centimeter cubes under all of the numbers, ask them to discuss different things they notice about the board. MP8 Use Repeated Reasoning | Generalize, There are many observations that the class can make. For example, children may notice that each of the teen numbers has one 10-stick. They may see that each number has one more cube than the number before it. Children might also discern that the number in the right column matches the number of centimeter cubes and the number in the left column matches the number of 10-sticks. Elicit as many observations from them as you can.”

The instructional materials reviewed for Math Expressions Kindergarten meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Math Talk is a critical component of the instructional materials and presents opportunities for students to use a consistent structure: Solve, Explain, Question, and Justify. Math Talk activities are identified in the Teacher Edition, and the structure is a familiar routine for students. In addition, students are presented with opportunities to use pictures to create math stories. Students need to explain how their story represents the picture. Examples include but are not limited to: 

• In Unit 2, Lesson 11, Tell A Math Story, The Neighborhood Garden presents a picture of a garden in which there are birds, pears on a tree and on the ground, rabbits, tulips, carrots, tomatoes, and flowers. Students use a variety of math strategies to add, choose to count on, combine quantities, etc. to tell their story.
• In Unit 5, Lesson 23, students compare heights of animals. This is extended to compare other attributes. Students use the attributes of shapes and length to construct arguments as they compare. 
• In Unit 3, Lesson 3, students work with partners of numbers that make a larger number; for example 2 and 2 are partners of 4. MP3 is identified and students discuss their partners. “Which group of children was able to show the most partners? Which group showed the fewest partners? Why do you think this is true?” Students need to construct an argument to explain why. 

Puzzled Penguin problems are found throughout the materials and provide students an opportunity to correct errors in the penguin’s work. These tasks focus on error analysis, and many of the errors presented are procedural. Examples of Puzzled Penguin problems include:

• In Unit 1, Lesson 6, Puzzled Penguin “Am I correct?” students look at a picture of three dogs and answer “How many cats are there?” The Puzzled Penguin writes 1 cat. Students “Count the cats. Ring the number of cats.” (either 0 or 2.) “Count the dogs. Ring the number of dogs.” (either 3 or 5).
• Unit 3, Lesson 11, Puzzled Penguin. “Help Puzzled Penguin. Look at Puzzled Penguin’s answers. Am I correct?” There are five diagrams with circles inside of a rectangular box. Students count the number in each box and identify the error. 
• In Unit 4, Lesson 17, MP3 Construct Viable Arguments | Critique the Reasoning of Others, Puzzled Penguin, “Look at what the Puzzled Penguin wrote. Help Puzzled Penguin.” The Puzzled Penguin wrote the number sentence 4 + 1 = 3. Students use a picture of four apples and one apple to find 4 + 1 = 5. They explain that the Puzzled Penguin subtracted instead of adding.

The instructional materials reviewed for Math Expressions Kindergarten meet expectations that materials use accurate mathematical terminology.

There are instances where materials use “A classroom research-based term developed for Math Expressions.” These terms are used in the Student Activity Book. Examples include but are not limited to: 

• “Break-Apart Stick” is a simple stick, such as a coffee stirrer, that children can use to help break apart numbers. Children lay out a certain number of counters and then use the Break-Apart Stick to separate the counters into two groups.
• “Math Mountains” are used to represent a total number and its decomposition into number partners. The total appears at the top, and the two partners that are added to produce the total are below to the right and left.
• “Number Parade” is a large connected sequence of numbers from 1 to 10 that is displayed permanently in the classroom. Above each number is a 5-structured grouping of dots.
• “Switch the Partners” refers to changing the order of the partners in an addition equation (commutative property).

In addition to a Glossary in the Student Activity Book, there are Teaching Notes on vocabulary and language. These are found in Unit 1 as students are introduced to mathematical language. For example:

• In Unit 1, Lesson 7, Vocabulary includes: “partners, how many, add, subtract.”
• In Unit 1, Lesson 9, Teaching Note, Language and Vocabulary states “To demonstrate the word horizontal, children can each hold out an arm horizontally. Have them pretend the sun (their other hand made in a fist) is coming over the horizon. Connect the two words. To demonstrate the word vertical, children can each move an arm so it is straight up. Help children visualize things they do horizontally (sleep, lie down on the beach). Also visualize things they do vertically (walk, stand).”