Kindergarten - Gateway 2

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. The Focus portion of the lesson provides opportunities for students to explore, engage in, and discuss conceptual understanding of mathematical content. Examples include:

• In Lesson 1-10, Focus: Introducing Quick Looks, “Present the dot images in order from Cards 1 to 10. Flash each image and ask: ‘What did you see/How did you see it?’ To move children beyond counting, highlight strategies that involve decomposing the number by asking questions such as: ‘Did everyone understand Tamika’s strategy of seeing groups? Can someone say it for us again? Can you try her way on the next card?’” This activity supports conceptual understanding of K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way” and K.CC.4, “Understand the relationship between numbers and quantities.”
• In Lesson 2-5, Focus: Solving Pocket Problems, each student has ten counters to help them solve pocket problems. The teacher demonstrates by putting three counters in the pocket. Then the teacher shows one more counter and adds it to the pocket. Students use their counters to show how many are in the pocket now. The teacher then takes all the objects out of the pocket and leads the class in counting the total. After practicing as a class adding to or taking away from the pocket, students work in pairs giving each other pocket problems. This activity supports conceptual understanding of K.OA.1, “Represent addition and subtraction with objects, fingers, mental images, drawings, or sounds.”
• In Lesson 4-8, Focus: Decomposing Numbers, “Children use connecting cubes to compose and decompose numbers in multiple ways.” At the conclusion of the lesson, students share their results, and the teacher asks, “What did you notice? Did you see any patterns?” leading to the concepts of turnaround pairs and doubles. This activity supports conceptual understanding of K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way.”
• In Lesson 5-5, Focus: Representing Teen Numbers, “Hold up the 10 card from the Class Number Card set and have all children hold up 10 fingers. Ask: What number comes next? Hold up the 11 card and ask if anyone can think of a way to show 11 fingers. If no one suggests it, call on two children to work together. Choose one child to hold up all 10 fingers. Ask the other child how many fingers he or she must hold up so that together they show 11 fingers. Repeat with the number 15, having one child show 10 fingers and another child show 5 fingers.” This activity supports conceptual understanding of K.NBT.1, “Compose and decompose numbers from 11 to 19 into ten ones and some further ones.”
• In Lesson 7-9, Focus: Exploring Number Combinations, “Model how to make a counting loop by placing beads on a chenille stem and twisting (or tying) the ends together to close and fasten the loop. Have each child take one chenille stem, put 7 to 9 same-color beads on it, and make a loop. (Children will make bead combinations that add to 10 in Lesson 8-9, Practice.) Direct them to group their beads and write number sentences for four different groupings on the ‘My First Math Book’ page. Challenge children to divide their beads into three groups for the last box on the page.” This activity supports students’ conceptual understanding of K.OA.1, “Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.”

Games, Daily Routines, and Math Masters provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• In Routine 1: Number of the Day, “Use think aloud to briefly review the total days in terms of tens and ones: first count the bundles of tens and then count the ones. Confirm that the total number of straws or sticks matches the number of days in school so far.” The teacher asks, “How many days have we been in school? How is this shown on the Class Number Line? How is it shown by the straws (or sticks or whatever material your class uses) in our Concrete Number Count?” This activity provides continuous conceptual understanding practice of K.OA.1, “Represent addition and subtraction with objects.”
• In Lesson 1-3, Game: Gotcha, students use one-to-one correspondence and the cardinal principle as they engage in a counting game. In this game, students “catch” the teacher making counting mistakes such as “saying the number words in the wrong order, not saying one number word for each object the teacher points to, and saying the wrong number for the total of the set (for example 1, 2, 3, 4; that’s 3 objects!).” Students signal with a “thumbs up” if the teacher is counting correctly and switch to “thumbs down” when the teacher makes a mistake. Each time the teacher makes an error, the students explain the mistake and model the correct counting. This activity provides practice of conceptual understanding of K.CC.4a, “When counting objects, say the number names in standard order, pairing each object with one and only one number name and each number name with one and only one object” and K.CC.4b, “Understand that the last number name tells the number of objects counted.”
• In Lesson 2-4, Math Masters, students independently use counters and a blank number board to cover the spaces on their board with the appropriate number of objects. This supports conceptual understanding of K.CC.4, “Understand the relationship between numbers and quantities; connect counting to cardinality.”
• In Lesson 3-2, Math Masters, students toss 10 pennies and sort into groups of “heads” and “tails” and put them on a ten frame. Then they count the number of heads and tails and record the numbers. This activity is repeated three more times and supports the conceptual understanding of K.CC.4, “Understand the relationship between numbers and quantities; connect counting to cardinality.”
• In Lesson 5-10, Math Masters, students take turns with a family member telling and solving number stories that use addition. They are encouraged to use pennies or other small objects, and the addition symbol to act out or model their stories. This activity supports the conceptual understanding of K.OA.1, “Represent addition with objects, fingers, acting out, or equations.”

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Daily Routines, Focus, and Practice portions of the lesson. Examples include:

• In Lesson 6-11, Practice: Counting to the Number of the Day, “First have children choral count by 1s to the Number of the Day. To practice counting on from different numbers, stop children during the sequence; then skip to a new number and have them restart. Next have children count by 10s and then 1s to the Number of the Day (10, 20, 30, 40, 50, 60, 61, 62…). Encourage children to use the number line to help them count if needed.” This activity provides an opportunity for students to develop fluency of K.CC.1, “Count to 100 by ones and by tens,” and K.CC.2, “Count forward beginning from a given number within the known sequence (instead of having to begin at 1).”
• In Lesson 7-12, Focus: Playing Dice Addition, students play with a partner and each roll a set of the Addition Dice. Once they roll their dice they state the resulting addition equations such as, “2 + 3 = 5.” The student with the higher total colors a square on their ten frame, and the first student to color all ten spaces wins. This activity provides an opportunity for students to develop fluency of K.OA.5, “Fluently add and subtract within 5.”
• In Lesson 8-11, Focus: Playing Addition Top-It, students play with a partner and each take two cards from the top of the deck and place them faceup. Then they add the two numbers and state their total. The player with the higher total takes all 4 cards and the player with the most cards at the end wins. This activity provides an opportunity for students to develop fluency of K.OA.5, ”Fluently add and subtract within 5.”
• In Routine 5: Survey, each week the teacher poses a survey question and students record their responses. Teachers choose how students record their answers. They can use a magnet and place it in the appropriate column, designate colored cubes for responses and students choose the cube that corresponds to their response, or students write their initials in the column on chart paper corresponding to their response. Once all responses are collected, the teacher appoints a Survey Helper to lead the class in counting and recording the results. This weekly activity provides an opportunity for students to develop fluency of K.CC.1, “Count to 100 by ones and by tens,” and K.CC.3, “Write numbers from 0 to 20.”
• In Routine 3: Daily Schedule and Monthly Calendar Routine, at the end of the month students take down the calendar in preparation for the next month. Dismantling the Calendar states, “Dismantling the calendar is a rich whole-class activity that provides an opportunity to enhance number skills and awareness of calendar patterns.” Teacher prompts change in complexity as the year progresses. For example, students are prompted to, “Remove or erase the number that equals 2 + 5. Remove or erase all pairs of date numbers that add to 7, or remove or erase two date numbers that add to 10.” This monthly activity provides an opportunity for students to develop fluency of K.OA.5, “Fluently add and subtract within 5,” and K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way.”

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Practice portions of the lesson and the Math Masters. The materials provide only partner or group activities for fluency of K.CC.2, “Count forward beginning from a given number,” and independent opportunities are only found in assessments. Examples include:

• In Lesson 5-11, Math Masters, students cut out an addition symbol, then they put some snacks, like goldfish, on the table and count them. Then students put another group of snacks on the other side of the addition symbol. Finally, students remove the addition symbol, put all of the snacks together and count to find out how many. This activity is repeated several times and provides an opportunity for students to independently demonstrate the procedural skill of K.CC.5, “Count to answer ‘how many’ questions about as many as 20.”
• In Lesson 7-7, Practice: Playing Roll and Record with Dot Dice, students roll a set of dice, determine the sum and write number sentences on slates such as, “3 + 2 = 5”. This activity provides an opportunity for students to develop fluency of K.OA.5, “Fluently add and subtract within 5.”
• In Lesson 8-12, Math Masters, students roll two die, determine their total, and then record the total on the Roll and Record Grid. This activity provides an opportunity for students to develop fluency of K.OA.5, “Fluently add and subtract within 5.”
• In Lesson 9-3, Practice: Counting the Class Collection, students count the items in the class collection using groups or counting-on strategies and record in their My First Math Book. The teacher asks, “What do you notice about the number of ___ in our collection since we started it? Where was the largest jump in our total? Can you see it on the table? On the thermometer display? What else do you notice about the data and the displays? This activity provides an opportunity for students to independently demonstrate procedural skill of K.CC.3, “Write numbers from 0 to 20,” K.CC.5, “Count to answer how many questions,” and K.CC.7, “Compare 2 numbers between 1 and 10 presented as written numerals.”

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. 

The materials attend to conceptual understanding. Examples include:

• In Lesson 5-8, Focus: Playing Teens on Double Ten Frames, “Distribute a blank double ten frame to each child. Discuss what children notice about it, making connections to the ten frames they have used before. Hold up a number card between 10 and 19 and ask children to use counters to show that number on their double ten frame. Direct children to fill one ten frame first (ten ones, or the ten) and then add counters to the second ten frame to complete the number.” This activity helps develop conceptual understanding of K.NBT.1, “Compose and decompose numbers 11 through 19 into ten ones and some further ones.”
• In Lesson 8-5, Focus: Playing Dice Subtraction, each student is given a blank ten frame and a pair of Dice Subtraction dice. Students take turns rolling the dice and subtracting the smaller number from the larger number, then state the subtraction equation and the difference to their partner. The student with the smallest difference colors one space on their ten frame. The winner is the student who fills their ten frame first. This activity develops conceptual understanding of K.OA.1, “Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.”

The materials attend to procedural skill and fluency. Examples include:

• In Lesson 4-2, Practice: Playing Roll and Record, students play a dice game using a Roll and Record Grid. Students roll the dice and fill in the grid to see what number gets filled first, or students can play with a partner to see who gets their number filled on the grid first. This develops the procedural skill of K.CC.5, “Count to answer ‘how many’ questions about as many as 20 things.”
• In Lesson 8-11, Focus: Playing Addition Top-It, students play with a partner using a deck of cards. Each player takes 2 cards, lays them faceup, and adds the 2 numbers stating their total. The students with the greater total takes all 4 cards. The player with the most cards wins. This develops the procedural skill and fluency of K.OA.5, “Fluently add and subtract within 5.”

The materials attend to application. Examples include: 

• In Lesson 4-13, Practice: Comparing Capacities, students are shown various containers of different sizes filled with beans or other pourable materials, and a reference container. The materials state, “Have a child choose one. Ask: Do you think this container holds more or less than the mug? How can we find out? As needed, model pouring the beans from the mug into the other container to compare capacities. Have children work in a small group to compare various containers to the reference container.” This activity provides the opportunity to apply the understanding of K.MD.2, “Directly compare 2 objects with a measurable attribute in common to see which object has more or less of the attribute.”
• In Lesson 6-8, Practice: Using the Subtraction Symbol, students use a craft stick, a sheet of paper, and about 10 counters to model subtraction story problems, “Levi saw 5 birds. Three birds were red. The rest were orange. How many birds were orange.” Students write a “-” symbol on their craft sticks and model the number stories with the craft sticks and counters. As they model, they discuss how to write the number models for the stories. This activity provides the opportunity to apply the understanding of K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10.”

Multiple aspects of rigor are engaged in simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Lesson 4-3, Focus: Favorite Colors Graph, students group themselves according to their favorite colors and create a graph to represent and analyze the results. After students create a Favorite Colors Graph, the teacher leads a class discussion analyzing and making comparisons of the graphs, “How many more children chose red than green? How did you figure that out?” Students practice fluency of K.CC.5, “Count to answer ‘how many’ questions about as many as 20 things,” and application of K.MD.3, “Classify objects into given categories, count the number of objects in each category and sort the categories by count.”
• In Lesson 4-6, Focus: Counting and Moving with Teens, students use a number line to extend the counting sequence beyond 10 to include teen numbers. Students are asked, “What is the same about these numbers? How are they different from the numbers 1 through 9? Why do you think they are called teen numbers?” Students then read numbers from 10-19 Class Number cards. Students develop conceptual understanding of K.CC.4, “Understand the relationship between numbers and quantities; connect counting to cardinality,” and fluency of K.CC.A, “Know number names and the count sequence.”
• In Lesson 8-13, Focus: Making Name-Collection Posters, the teacher writes the number 10 at the top of chart paper and draws a filled in ten frame and writes 5 = 5 on the paper. Students are asked to share other ways to show or name 10 and the teacher adds their responses. Students develop conceptual understanding of K.OA.A, “Understand addition as putting together and adding to, and subtraction as taking apart and taking from,” while applying understanding of K.NBT.1, “Compose and decompose numbers from 11 to 19.”

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

All MPs are clearly identified throughout the materials, with few or no exceptions. Examples include:

• The mathematical practices are listed on pages xi-xiii in the Kindergarten Volume I Teacher’s Lesson Guide as “Correlation to the Mathematical Process and Practices” which states, “Everyday Mathematics is a standards-based curriculum engineered to focus on specific mathematical content, processes, and practices in every lesson and activity. The chart below shows complete coverage of each mathematical process and practice in the core program throughout the grade level.” 
• Each Section Organizer contains a Mathematical Background: Processes and Practices component identifying the MPs addressed in the section and in individual lessons. Additionally, “The authors created Goals for Mathematical Practice (GMP) that unpack the practice standards, operationalizing them in ways that are appropriate for elementary students.” 
• Within each lesson description, GMPs appear in bold print and teacher side notes identify the MPs that are addressed in the lesson.

The majority of the time the MPs are used to enrich the mathematical content. Examples include:

• In Lesson 1-10, Focus: Introducing Quick Looks, “Present the dot images in order from Cards 1 to 10. Flash each image and ask: What did you see? How did you see it? To move children beyond counting, highlight strategies that involve decomposing the number by asking questions such as: Did everyone understand Tamika’s strategy of seeing groups? Can someone say it for us again? Can you try her way on the next card?” The mathematical content in this activity is enriched by MP1.
• In Lesson 2-11, Focus: Getting to Know Rectangles, students are shown shape cards of several rectangles to look for similarities and differences. Teachers ask, “How are all these rectangles alike? How are all these rectangles different from one another? How can all these shapes be rectangles when they look different from one another? and What other shapes have we learned about that have lots of different types?” The mathematical content in this activity is enriched by MP7.
• In Lesson 6-6, Focus: Playing “What’s My Rule?” Fishing, students try to figure out what rule the teacher is applying as they fish and catch students (wearing red). Teachers ask, “What is the same about the fish I caught? What is the same about the fish who are not in my net? If I continue to follow my rule, who else can I catch? Why? and Can you state my sorting rule?” The mathematical content in this activity is enriched by MP8.
• In Lesson 6-13, Focus: Relating Symbols to Number Stories, students make sense of “join” or “take away” word problems. For example, “I have 5 stickers, 3 are red and the rest are yellow. How many stickers do I have?” In their My First Math Book on page 10, students draw or write one join and one take-away number story and use a number model to represent the story. The mathematical content in this activity is enriched by MP4.
• In Lesson 8-6, Focus: Bundling Craft Sticks, students work in pairs to estimate the number of sticks in a bag (10 - 19). Students then bundle the sticks into 10 sticks and some more sticks. For example, “Have children bundle the sticks into groups of 10 and re-count the sticks by 10s and 1s. Model how to record their findings by showing the number as 10 and some 1s, on a double ten frame, and as an equation.” The mathematical content in this activity is enriched by MP2.

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. Lessons offer Differentiated Options where students work in small groups or with partners as teachers facilitate discussions so students have opportunities to construct viable arguments and critique the reasoning of others. Open Response and Reengagement Lessons provide opportunities for students to critique the open response answers of other students. 

Students construct arguments. Examples include:

• In Lesson 4-2, Focus: Identifying Attributes of Shapes, “Introduce the Feely Box and note that there are different shapes inside the box that children will touch but not see. Explain that you will give clues to help them choose shapes from the box, and they will then explain why they chose the shape. Select from the activities below, choosing prompts that are best suited to children’s skill levels.” Two of the possible activities state, “Have a child pick out two shapes that feel the same. Have the child show the shapes to the class and name them. Ask: How do you know that the shapes were the same? Name a shape and have a child find it by touch. Ask: How do you know you found a _____?”
• In Lesson 5-2, students play a game of Roll & Record. After the game, a class chart is made recording each child’s winning number. Students analyze the results of the game and respond to questions, “Why did the middle numbers win most often? How many ways are there to get 2? Are there more ways to get 8? Why?”
• In Lesson 5-7, Focus: Solving the Open Response Problem, teachers pose the following question, “There were 6 coats hanging on hooks. Two children put on their coats. I think there are 4 coats still hanging on hooks. Am I right? How do you know?” Teachers explain that two students tried to solve the problem and display Child 1’s solution and ask, “Does this child think there are 4 coats left on hooks? How can you tell? Sample answer: Yes. The child drew 4 coats hanging on hooks and said ‘Yes’ on his or her paper.”

Students critique the reasoning of others. Examples include:

• In Lesson 3-12, Practice: Solving Number Stories, “After they have a chance to solve each problem, invite children to share and discuss how they solved them. Elicit a variety of different approaches such as counters, drawing pictures, counting, and using derived facts. Have children show and compare their various methods, rather than just describe them.”
• In Lesson 8-7, Focus: Reengaging in the Problem, students answer an open response problem about Birds on Wires. The materials state, “Review the Birds on Wires open response problem. Tell children that today they will look at different ways some of them solved the problem. Begin by showing some correct and incorrect solutions you found in their work. Prompt children to describe and compare the two solutions by asking: Can both of these solutions be correct? Why or why not? How can we figure out the number of birds on the second wire if we have 4 birds on the first wire? Does the number of birds drawn on each wire match the numbers below it?”
• In Lesson 9-1, Focus: Playing Make My Design, students play a partner game where one student creates a design with pattern blocks, then uses shape and positional language to describe the design to the other partner. The materials state, “Encourage the other partner to try to re-create the design from the instructions, asking for further description and clarification as needed.”

The instructional materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Examples of assisting teachers in engaging students to construct viable arguments include:

• In Section 5 Organizer, Mathematical Background: Process and Practice, Standard for Mathematical Process and Practice 3, “Children have been informally “justifying their conclusions and communicating them to others” all year. In Section 5, children have opportunities to ‘construct arguments using concrete referents such as objects, drawings, diagrams, and actions.’ For example, when graphing sums of dice rolls (Lessons 5-2 and 5-7), children explain why they think there are more combinations for some totals than for others and use examples from their recording sheets to justify their explanations. Children also make a mathematical argument in the Lesson 5-7 Open Response task as they explain and justify their solution to a number story. As they work with Shape Cards (Lesson 5-13), children ‘make conjectures’ about how to combine shapes to make new shapes.”
• In Lesson 4-9, Differentiation Options, Extra Practice, Predicting and Testing Weights, “To provide additional practice comparing and describing the weights of objects, encourage children to predict which of the two objects is heavier and which one is lighter. Have them explain their reasoning and then use a pan balance to test their predictions.”
• In Lesson 7-4, Focus: Playing Solid-Shapes Match Up, “Show children an object from the Solid-Shapes Museum, and invite them to name the object (sphere or cone, for example) and describe it in detail. Ask whether the object is 2- or 3-dimensional, and prompt children to explain their thinking. Review and discuss the definitions of these terms as needed. You might review the comparisons they made in lesson 6-5 (between a circle and a sphere, and a square and a cube) to illustrate that 2-dimensional (or 2-D) shapes are flat and 3-dimensional (or 3-D) shapes are not.”

Examples of assisting teachers in engaging students to analyze the arguments of others include:

• In Lesson 3-12, Practice: Solving Number Stories, students solve a variety of number stories such as, “There were 6 apples in the bowl. Carlos took 3 of the apples. How many apples were left in the bowl?” Teachers are prompted, “Provide children with counters and writing materials. After they have a chance to solve each problem, invite children to share and discuss how they solved them. Elicit a variety of different approaches, such as using counters, drawing pictures, counting, and using derived facts. Have children show and compare their various methods, rather than just describe them.”
• In Lesson 5-7, Focus: Solving the Open Response Problem, “Explain that you are going to share how two children solved this problem and justified, or tried to ‘prove’, their answers. Display Child 1’s solution from the top of Math Masters, p.81, and read the dictation under the drawing: Yes, I knew it in my brain. Ask: Does this child think there are 4 coats left on hooks? How can you tell? Does this child show the whole story?”
• In Lesson 8-13, Focus: Making Name-Collection Posters, students make posters to represent equivalent names for a number between 5 and 20. At the conclusion of the activity, teachers are directed, “Have groups share their posters. Help other children make sense of each poster by asking questions such as: Which names for 10 did you recognize quickly? Which ones are harder to figure out? Why? Did this group represent their number in a way that you didn’t use for your name collection? If time permits, allow groups to revisit their posters and add new representations they learned as a result of sharing.”