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

All units begin with a Unit Organizer, Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. The Focus portion of each lesson introduces new content, designed to help teachers build their students’ conceptual understanding through exploration, engagement, and discussion. The materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards. Examples include:

• Lesson 1-10, Quick Looks, 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 the 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.”

• Lesson 2-5, Pocket Problems, 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. Then the teacher 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. “Divide children into pairs and give each pair counters and an envelope to use as a paper pocket. Have one partner begin by posing a pocket problem, and the other partner use counters to represent and solve the problem. Then have partners reverse roles. Encourage them to show, rather than tell, what they are doing, just as you did when you modeled the problems.” This activity supports conceptual understanding of K.OA.1, “Represent addition and subtraction with objects, fingers, mental images, drawings, or sounds.”

• Lesson 4-8, Building Numbers, Focus: Decomposing Numbers, students compose and decompose numbers using connecting cubes and share their results. To assist students in making the connection between turnaround pairs and doubles the teacher asks, “What did you notice? Did you see any patterns?” This activity supports conceptual understanding of K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way.”

• Lesson 5-6, Teen Partners, Focus: Representing Teen Numbers, students use their fingers to compose and decompose 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.”

• Lesson 7-9, Bead Combinations, Focus: Exploring Number Combinations, students solve addition and subtraction problems, “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.”

Home Links and Games provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Lesson 2-4, Number Board, Focus: Building a Number Board, students use counters and a blank number board to cover the spaces on their board with the appropriate number of objects. “Direct children to cover the spaces on their board with the appropriate number of objects. Circulate and confer with children about their counting and the ‘one more’ pattern.” This supports conceptual understanding of K.CC.4, “Understand the relationship between numbers and quantities; connect counting to cardinality.”

• Lesson 3-2, Ten-Bean Spill, Home Link, students toss 10 pennies and sort them 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. “Gently toss 10 pennies. Sort the pennies into groups of ‘heads’ and ‘tails’ and put them on the ten frame. Count the number of heads and the number of tails. You may want to record the numbers you find on the back of this page. Repeat at least three more times.” This activity supports the conceptual understanding of K.CC.4, “Understand the relationship between numbers and quantities; connect counting to cardinality.”

• Lesson 5-10, The Addition Symbol (+), Home Link, students take turns with a family member telling and solving number stories that use addition. “Cut out the addition symbol (+). Take turns telling and solving number stories that use addition. For example: Two children were on the playground, and three more come to play. How many children were there altogether? Use pennies or other small objects and the addition symbol (+) to act out, or model, the stories. Draw or write one of your number stories below.” This activity supports the conceptual understanding of K.OA.1, “Represent addition with objects, fingers, acting out, or equations.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

All units begin with a Unit Organizer, Planning for Rich Math Instruction. This component indicates where procedural skill and fluency exercises are identified within each lesson of the Unit. The Mental Math Fluency exercises found at the beginning of each lesson develop fluency with basic facts and other skills that need to be automatic while engaging learners. The Practice portion of the lesson provides ongoing practice of skills from past lessons and units through activities and games. Examples include:

• Lesson 6-11, Hiding Bears, Practice: Counting to the Number of the Day, students count by 1s and 10s. “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 in 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).”

• Lesson 7-12, Dice Addition, 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, “.” “The player with the highest total colors one space on a blank ten frame. If players have the same totals, they both color a space on their ten frames.” “The game ends when one player fills a ten frame.” This activity provides an opportunity for students to develop fluency in K.OA.5, “Fluently add and subtract within 5.”

• Lesson 8-11, Addition Top-It, Focus: Playing Addition Top-It, students play with a partner and each takes two cards from the top of the deck and places them face up. “Players add the two numbers and take turns stating their total. (4 plus 2 equals 6!)” 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 in K.OA.5, ”Fluently add and subtract within 5.”

• 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, “Appoint a Survey Helper who will make sure all children respond to the survey and can take the lead on counting and recording the results.” This weekly activity provides an opportunity for students to develop fluency in K.CC.1, “Count to 100 by ones and by tens,” and K.CC.3, “Write numbers from 0 to 20.”

Math Boxes, Home Links, Games, and Daily Routines provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade. Examples include:

• Lesson 5-11, Growing Train, Home Link, students add snacks, “Put a small number of snacks, such as cereal or raisins, on a table and count them. Cut out the addition symbol (+) and put it next to the snacks. Put another group of snacks next to the addition symbol and count them. Remove the addition symbol and put all the snacks together in one pile. Count the snacks and say how many there are all together.” 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, Representing Survey Data, Practice: Playing Roll and Record with Dot Dice, students roll a set of dice, determine the sum and write number sentences on slates, “.” This activity provides an opportunity for students to develop fluency in K.OA.5, “Fluently add and subtract within 5.”

• Lesson 8-12, Function Machines, Practice: Playing Roll and Record with Dot Dice, students roll two dice, determine their total, and then record the total on the Roll and Record Grid. “As they play, help children develop addition fluency by encouraging them to solve as many combinations as they can from memory or using another efficient strategy.” This activity provides an opportunity for students to develop fluency in K.OA.5, “Fluently add and subtract within 5.”

• Lesson 9-3, “What’s My Rule?” with Numbers, Practice: Counting the Class Collection, students count the items in the class collection using groups or counting-on strategies and record them in their My First Math Book. “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 the 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 materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Focus activities introduce new content, provide routine exercises, review recent learning, and provide challenging problem-solving tasks that help build conceptual understanding, procedural skill and fluency, and application of mathematics. Open Response lessons provide challenging problems that involve more than one strategy or solution. Home-Links relate to the Focus activity and provide informal mathematics activities for students to do at home. Examples of routine and non-routine applications of the mathematics include:

• Lesson 2-12, Number Stories, Focus: Telling and Acting out Number Stories, students solve change-to-more, change-to-less, and parts-and-total problems using counters, their fingers, or use drawings to model the story as the teacher tells it. “Davon was the snack helper today. He carried 3 apples to the table. Then he got 2 more apples. How many apples does Davon have now?” Students apply their understanding of K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10.”

• Lesson 3-6, Obstacle Course Positions, Home-Link: Simon Says, students play “Simon Says” using position words. “Play Simon Says with your family. Use positional words such as above, below, next to, in front of, and behind. Take turns being “Simon” (the leader). Use clues such as these: Simon says, put your finger below your chin. Simon says, put your foot next to your knee. Simon says, shake your hands behind your back. Wiggle your fingers above your head. (Don’t follow this command. Simon didn’t say!)” Students apply their understanding of K.G.1, “Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.”

• Lesson 5-3, Ten Bears on a Bus, Focus: Playing Ten Bears on a Bus, students play the game Ten Bears on a Bus to generate number combinations that add to 10. “4 yellow bears are on the bus, how many red bears must get on the bus to fill all 10 seats?” Students apply their understanding of K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way.”

• Lesson 5-7, Seats at the Party, Focus: Solving the Open Response Problem, students solve a comparison number story. “I was having a party. I put 4 chairs at the table. The doorbell rang, and I saw 7 friends at the door. Do I have enough chairs for all my friends? How do you know?” This activity provides the opportunity for students to apply their understanding of K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10.”

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Independent Problem Solving provides “additional opportunities for children to apply the content they have learned during the section to solve non-routine problems independently. These problems often feature: applying math in the real world, multiple representations, drawing information or data from pictures, tables, or graphs, and opportunities for children to choose tools to support their problem-solving.” Examples of independent demonstration of routine and non-routine applications of the mathematics include:

• Independent Problem Solving 3a, “to be used after Lesson 3-4”, Problem 1, students compare numbers. “Jonah and Carla went on a walk to collect rocks. They dumped their rocks into 2 piles.” Students are shown a picture of 2 piles of rocks. 1 pile has 6 rocks, and 1 pile has 7 rocks. “Who found more rocks? Use words or pictures to explain how you know.” This activity provides the opportunity for students to independently demonstrate K.CC.6, “Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.”

• Independent Problem Solving 5b, “to be used after 5-11”, Problem 1, students write and represent addition number stories from a picture with birds, flowers, and butterflies. “Draw and write an “adding” number story that goes with the picture above.” Problem 2, “Solve your number story. Draw and use words to explain how you got your answer.” This activity provides the opportunity for students to independently demonstrate understanding of K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10.”

• Independent Problem Solving 6a, “to be used after Lesson 6-8”, Problem 1, students write a number story to represent subtraction using a given image. “Draw and write a “take-away” (subtracting) number story that goes with the picture above.” Students independently demonstrate their understanding of K.OA.1, “Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations and K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10.”

• Independent Problem Solving 9b, “to be used after Lesson 9-9”, Problem 2, students create their own number story using a number sentence. “Draw and write a number story about animals or toys that could go with this number sentence: $4+$ ___ $=5$.” This activity provides the opportunity for students to independently demonstrate K.OA.2, “Solve addition and subtraction word problems, and add and subtract within 10,” K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way” and K.OA.4, “For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

All three aspects of rigor are present independently throughout the grade. Examples where materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Lesson 4-2, Shapes By Feel, Practice: Playing Roll and Record, students play a dice game using a Roll and Record Grid. Students roll a die, and fill in a box in a column with the matching number, “Have children fill a Roll and Record grid independently (at least until one column is full) or play Dice Race with a small group or partner. In Dice Race, children choose a target number and see who fills that column on their grid first.” This activity develops the procedural skill of K.CC.5, “Count to answer ‘how many’ questions about as many as 20 things.”

• Lesson 4-13, Number-Grid Exploration, Practice: Comparing Capacities, students are shown various containers of different sizes filled with beans or other pourable materials and a reference container to compare quantities, “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.”

• Lesson 8-5, Dice Subtraction, 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, “As you model several rounds of the game, ask children to share strategies for subtracting.” This activity develops a 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.”

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

• Lesson 4-6, Moving With Teens, Focus: Counting and Moving with Teens, students use a number line to extend the counting sequence beyond 10 to include teen numbers, “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 a 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.”

• Lesson 6-10, Attribute Spinner, Focus: Playing Attribute Spinner, students play a game with three attribute spinners: size, color, and shape, “Players take turns spinning and choosing the block that has the attributes shown on all three spinners (the large, blue triangle, for example). To end their turn, players describe the block to confirm that it matches all the attributes on the spinners.” Students develop a conceptual understanding of K.G.2, “Correctly name shapes regardless of their orientations or overall size,” and application of K.MD.1, “Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.”

• Lesson 8-13, Name-Collection Posters, Focus: Making Name-Collection Posters, the teacher writes the number 10 at the top of the 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, “Ask the class to think of other ways they can name, or show 10. As they share, record or have children record their ideas on the chart paper.” 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 an understanding of K.NBT.1, “Compose and decompose numbers from 11 to 19.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 3-12, Monster Squeeze, Focus: Playing Monster Squeeze, students use a variety of strategies to identify a number between two other numbers. Using the 0-10 section of the Growing Number Line and a monster to the left of 0 and on top of 11, the teacher facilitates students guessing his/her number. “If they guess too high, reply: Your number is too high. My number is less than that number. Move the right-hand monster along the number line until it covers the number guessed.” “If they guess too low, say: Your number is too low. My number is greater than that number. Move the left-hand monster to cover the number that was guessed. As children play, help them make sense of the game and strategic guessing by discussing questions such as: Why did you guess that number? What would be a good next guess? Why? Could the ‘mystery’ number be___? Why or why not?”

• Lesson 4-5, Ten-Frame Quick looks, Focus: Taking Ten-Frame Quick Looks, students practice mentally composing and decomposing numbers on ten frames to develop fact strategies by understanding the information presented. “Flash each image for about three seconds before removing (or covering) it and ask children to remember what they saw. To elicit flexible ways of thinking about each image, ask: What did you see? How did you see it?”

• Lesson 9-12, Subtraction Top-It, Focus: Preparing for a Math Celebration, students make sense of answers when determining how many items will be needed for the class party using a guest list. “Figure out how many chairs, tables, napkins, plates, utensils, and food items are needed for the party using the list of expected guests.”

• Independent Problem Solving 7b, “to be used after Lesson 7-10”, Problem 1, students use a variety of strategies to solve a word problem. “Marco had 7 balloons on a windy day. Two of Marco’s balloons popped when the wind blew them into a branch. a) How many are left? ___ balloons b) Write a number sentence to model what happened to the balloons.”

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 1-7, Class Birthdays, Practice: Getting to Know Numbers (3), students attend to the meaning of quantities when collecting items of 3 for their number-collection bags. “Revisit Lesson 1-5, pages 56 - 59. Focus on 3 as the featured number. Add a 3-cube tower and the label 3 to your growing connecting-cube display. Ask: What will tomorrow’s tower look like? How do you know? Have pairs of children create and save number-collection bags with 3 items. (Children who need support creating 3 collections may use the number strip on Math Masters, page TA3.) Use as many of the other suggested “featured number” activities from the original lesson as time permits. Consider reading, telling, or acting out a story that features 3 characters, such as “The Three Little Pigs,” “Goldilocks and The Three Bears,” or “The Three Billy Goats Gruff.” Ask children why they think so many stories feature the number 3!” 

• Lesson 2-2, Top-It With Dot Cards, Focus: Playing Top-It with Dot Cards, students demonstrate understanding of mathematical representations as they turn over dot cards and make comparisons to determine who has more dots. The player with more dots takes both cards, “If the cards have an equal number of dots (the same number), both players turn over another card and the player with the greatest number of dots takes all four cards.” As the children play the game, the teacher asks, “How do you know how many dots each card has? How can you figure out which card has more, or a greater number of, dots? How can you be sure? Sample answers: I count the dots. I know that 5 comes after 3, so 5 is more than 3. I see that this card is more full than that card.”

• Lesson 8-6, Craft-Stick Bundles, Focus: Bundling Craft Sticks, students discuss number representations as they work in pairs to estimate the number of sticks in a bag, then bundle the sticks into 10 sticks and some more sticks, “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.” 

• Independent Problem Solving 3b, “to be used after Lesson 3-10”, Problem 1, students represent the number 4 in four different ways. “Show four different 4 cards. Use dots for the blank cards and ten frames for the others.” Students are given a picture with four playing cards, 2 blank and 2 with ten frames on them.

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:

• Lesson 4-2, Shapes By Feel, Focus: Identifying Attributes of Shapes, students construct viable arguments as they choose specific shapes based on attributes. “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 _____?”

• Lesson 5-2, Roll and Record with Dot Dice, Focus: Playing Roll & Record with Dot Dice, students construct viable arguments during 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?”

• Independent Problem Solving 2a, “to be used after Lesson 2-3”, Problem 2, students construct viable arguments when shown different shapes and instructed to put an “X” on the shapes that are not triangles. “Draw one of the shapes you put an X on. Explain why it is not a triangle.”

Materials provide support for the intentional development of MP3 by providing opportunities for students to critique the reasoning of others in connection to grade-level content. Examples include:

• Lesson 8-7, Open Response Birds on Wires, Focus: Reengaging in the Problem, students critique the reasoning of others when they answer an open response problem about Birds on Wires. “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?”

• Lesson 9-1, Make My Design, Focus: Playing Make My Design, students critique the reasoning of others as they play a partner game where one student creates a design with pattern blocks, then uses that shape and positional language to describe the design to their partner. “Encourage the other partner to try to recreate the design from the instructions, asking for further description and clarification as needed.”

• Independent Problem Solving 1b, “to be used after Lesson 1-11”, Problem 1, students critique the reasoning of others when comparing two groups of orange slices. “Akesha and Micah are supposed to get the same amount of snacks. Akesha says this is fair. Micah says he doesn’t have as much. Who do you agree with and why?”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP4 to meet its full intent in connection to grade-level content. Students model with mathematics to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson  7-10, Class Number-Story Book, Focus: Creating Number Stories, students model a situation with pictures and mathematical symbols as they solve number stories. “Review how number stories can be recorded with pictures, numbers, and symbols. Tell a number story, such as: There were 4 squirrels on the ground. 1 squirrel ran up a tree. How many squirrels were still on the ground? Draw a picture to illustrate the story and invite the class to help you write a number sentence to model it, such as $4-1=3$ or $4=1+3$.”

• Independent Problem Solving 5a, “to be used after Lesson 5-4”, Problem 2, students model a situation with an appropriate representation as they put two shapes together. “Draw an object you see that is made from 2 or more shapes put together. Label the shapes that make up the object.”

• Independent Problem Solving 9b, “to be used after Lesson 9-9”, Problem 1, students use the math they know to solve problems and everyday situations when they use a number sentence to model a number story. “Draw and write a number story about snacks or people that could go with this number sentence: $5-2=$ ___.”

Materials provide intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 8-4, Interrupted Counting, Practice: Playing Hiding Bears, students choose appropriate strategies as they find the number of hiding bears. “Encourage them to use strategies such as counting on, modeling with fingers, or using known combinations of 10 to find the number of hiding bears.”

• Independent Problem Solving 2b, “to be used after Lesson 2-13”, students choose appropriate tools and strategies as they solve word problems involving addition and subtraction. “Problem 1, Kayla sorted the buttons in a box by shape. There were 2 square buttons, 3 round buttons, and 0 triangle buttons. Draw a picture of Kayla’s sorted buttons. How many buttons were there altogether? Problem 2, Kayla’s dad trades a round button for a square one. How many square and round buttons are there now? You may use counters, fingers, a drawing, or another tool to figure it out. Show or explain what you did.”

• Independent Problem Solving 4a, “to be used after Lesson 4-8”, Problem 1, students recognize both the insight to be gained from different tools/strategies and their limitations. “You may use cubes, counters, your fingers, or another tool to help you solve the problem below. 1a) Marisol and Nico are having a picnic. Draw one way Marisol and Nico can split up all 6 crackers. They don’t need the same amount. 1b) Draw a different way they can split up the crackers.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

Each lesson targets one to three MPs. Math Practices are identified for teachers in several places: Pathway to Mastery Correlation to the Mathematical Processes and Practices, Focus, Student Math Journals, Student Reference Book, Independent Problem Solving Masters, and Practice. Materials refer to the Mathematical Practices as GMPs (Goals for Mathematical Practice). 

Materials provide intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure throughout the sections as they describe, and make use of patterns within problem-solving as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 4-8, Building Numbers, Focus: Decomposing Numbers, students look for patterns or structures as they compose and decompose numbers using connecting cubes. “Direct them to build the number in as many ways as they can using the two colors and to record each combination on their sheets. Afterward ask: What do you notice? Do you see any patterns?”

• Lesson 7-13, Mystery Block, Focus: Playing Mystery Block, students look for and explain the structure within mathematical representation using shape attributes to correctly guess the mystery block. “After children ask a question, have them remove blocks according to the response. For example, if the answer to ‘Is it thick?’ is yes, then the child who asked the question removes all the blocks except the thick blocks. After children remove blocks, follow up with questions such as: Why did you take away all the thin blocks? Why didn’t you need to ask whether it was thin? What rule were you following? Have children think strategically about the questions that will be most helpful for figuring out the mystery block given the blocks that remain.” 

• Independent Problem Solving 9a, “to be used after Lesson 9-6”, Problem 1, students make use of structure by solving a problem in 3 different ways. “Bria loves sports. She plays softball, basketball, and soccer. She has 8 trophies that she wants to arrange on 2 shelves. Show 3 different ways she could arrange them. Write a number sentence that fits each way.”

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning throughout the sections to make generalizations and build a deeper understanding of grade level math concepts as they work with the support of the teacher and independently throughout the units. Examples include:

• Lesson 2-7, Introduction to Sorting, Focus: Solving the Open Response, students sort and classify objects in different ways. “Explain that pairs of children will sort a collection of objects in a way that makes sense to them and that others can understand. Remind them that all objects in a group must share one or more attributes, or be the same in one or more ways. Give each pair a sorting mat and a collection of objects (or send groups of children to their first sorting station). As children work, circulate and provide support and guidance for sharing objects and working together, as well as for sorting. What is your rule for sorting your objects?”

• Lesson 4-1, Attribute Blocks, Assessment Check-In, students evaluate the reasonableness of their answers and thinking to sort and classify objects based on their attributes. “Use this lesson to assess whether children can identify and describe multiple attributes of objects and sort and classify objects by those attributes. Expect most children to be able to describe a single block using at least two attributes, such as color and shape, and to sort attribute blocks by color. Many children will be able to sort by other attributes. For children who struggle, provide more chances to work with attribute blocks; reduce the number of variables initially (for example, by removing all the thick blocks).”

• Lesson 5-12, Number Scrolls, Focus: Making Number Scrolls, students notice repeated calculations as they use number patterns to fill in a blank number scroll. “Distribute blank number scrolls to children and support them as they begin to write numbers. As you circulate, note how children fill in their scrolls. Do they write the numbers in order? Do they use patterns to complete the scroll (for example, filling in all the numbers in one column)?”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Teacher’s Lesson Guide, Welcome to Everyday Mathematics, explains how the program is presented. “Throughout the Kindergarten program, emphasis is placed on building from and connecting with children's informal, everyday experiences with mathematics; problem solving in everyday situations and mathematical contexts; an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning; distributed practice through routines, games, and other activities; teaching that supports ‘productive struggle’ and maintains high cognitive demand; and playful lessons and activities that engage all children and make mathematics fun!”

• Implementation Guide, Guiding Principles for the Design and Development of Everyday Mathematics, explains the foundational principles. “The foundational principles that guide Everyday Mathematics development address what children know when they come to school, how they learn best, what they should learn, and the role of problem-solving and assessment in the curriculum.”

• Unit 5, Exploring Teen Numbers, Organizer, Coherence, K.OA.4, provides an overview of content and expectations for the unit. “Earlier in Kindergarten, children explored number pairs that add to 10 through ten-frame activities and games like Ten-Bean Spill. Children’s fingers provided a natural context for these explorations in Pre-K and early Kindergarten as well. In Section 5, children compose and decompose numbers 11 through 19 on double ten frames and with partners using two pairs of hands. In Sections 7 and 8, they record these types of decompositions with drawings and equations. In Grade 1, children will apply this concept as they extend their understanding of place value to include all 2-digit numbers.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Examples include:

• Implementation Guide, Everyday Mathematics Instructional Design, “Lesson Structure and Features include; Lesson Opener, Mental Math and Fluency, Daily Routines, Math Message, Math Message Follow-Up, Assessment Check-In, Summarize, Practice, Math Boxes, and Home-Links.”

• Lesson 2-2, Top-It with Dot Cards, Focus: Assessment Check-In, teacher guidance supports students in identifying and comparing quantities. “As children play Top-It with Dot Cards, observe whether they can correctly identify how many dots are on the cards and which card has more dots. Expect most children to be able to count and compare quantities for representations (both arranged and scattered) of at least 1 through 5 dots; many children will be able to compare larger sets. Children will continue to practice comparing sets in the sorting and graphing activities in Lessons 2-7, 2-10, and 3-1, as well as in later lessons.”

• Lesson 4-11, Counting by 10s, Focus: Counting by 10s, Professional Development, teacher guidance enhances instruction by explaining the importance of how to introduce groups of 10. “Counting concrete groups of 10 develops foundational place-value concepts. Delay using manipulatives (such as base-10 rods) that are already connected into groups of ten and cannot be broken apart into ones. Instead, have children count and group sets of ten discrete objects, such as fingers or straws to promote understanding that 1 ten is the same as 10 ones.”

• Lesson 8-2, Marshmallow and Toothpick Shapes, Focus: Assessment Check-in, teacher guidance supports students with modeling 2- and 3-dimensional shapes. “Expect most children to model basic 2-dimensional shapes with toothpicks and marshmallows, not all will be able to model 3-dimensional shapes without help. Also expect children to identify, name, and describe basic 2- and 3-dimensional shapes in their own or other children’s work.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Each Unit Organizer Coherence table provides adult-level explanations and examples of complex grade/course-level concepts so teachers can improve their content knowledge. Professional Development side notes within Lessons support teachers in building knowledge of key mathematical concepts. Examples include:

• Lesson 1-12, Describing Shapes, Professional Development, supports teachers with concepts for work beyond the grade. “This lesson focuses on describing shapes. The emphasis is not yet on formal shape terminology (such as vertex or parallel), but hearing these terms helps children build them into their vocabulary. Gesturing can be a powerful teaching tool. When two shapes have straight sides, gesture along that side on each shape to highlight the common feature. Later, gestures such as these can help children learn new words such as vertex and angle.”

• Lesson 2-13, More Number Stories, Focus: Exploring More Number Stories, Professional Development, teacher guidance explains the different problem-solving situations. “Exposing children to a variety of problem types encourages them to think flexibly about the meanings and solutions of problems, instead of carrying out rote operations with the numbers. Generally, children find end-unknown problems the easiest to solve and start-unknown problems the most challenging.”

• Lesson 3-7, Comparing Representations, Focus: Solving the Open Response Problem, Professional Development, teacher guidance explains the different representations for the same number. “During this task, children create a variety of representations for the same number, which develops their quantitative reasoning in a variety of ways. First, they must think about what can change from one representation to the next (the objects, arrangement, grouping, and format) and what cannot change (the quantity). Then they must apply this knowledge to their own representations.”

• Lesson 6-7, Tall Enough to Ride?, Professional Development, supports teachers with concepts for work beyond the grade. “This lesson gives children an opportunity to explore using same-size units as a tool to measure and compare heights. Children engage with the idea that effective measurement requires iterating, or repeating, units without gaps or overlaps. They also share and make sense of their results in order to solve a problem. This early problem-solving experience lays the foundation for measurement lessons in later grades, when children learn about standard measurement units (such as inches, centimeters, milliliters, minutes, and grams) and tools (such as rulers, clocks, and scales).”

• Lesson 7-12, Dice Addition, Focus: Playing Dice Addition, Professional Development, teacher guidance explains how students develop fluency. “Dice Addition helps children develop fluency for sums that total 5 or less. To avoid having children view addition as simply a rote process, emphasize understanding before focusing on speed. With practice and encouragement, children will develop efficiency and fluency.”

• Lesson 9-10, Doubles on Double Ten Frames, Focus: Working with Doubles, Professional Development, teacher guidance explains how students will use doubles in future grades. “In this lesson, children begin learning small doubles facts using Quick Looks. In Grades 1 and 2, children will use their knowledge of doubles, which are among the easiest additional facts to learn, to derive more challenging addition and subtraction facts.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the Correlations to the Standards for Mathematics, Unit Organizers, Pathway to Mastery, and within each lesson. Examples include:

• Kindergarten Math, Correlation to the Standards for Mathematics Chart includes a table with each lesson and aligned grade-level standards. Teachers can easily identify a lesson when each grade-level standard will be addressed. 

• Mastery Expectations, K.CC.5, “First Quarter: Count arranged and scattered sets of up to 10 objects. Second Quarter: Count arranged sets of up to 20 objects. Count scattered sets of up to 10 objects. Count out up to 10 objects. Third Quarter: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects. Fourth Quarter: Ongoing practice and application.”

• Lesson 4-1, Attribute Blocks, core standards are identified for the Focus: K.CC.5, K.CC.6, K.MD.1, K.MD.3, K.G.2, and the Practice: K.CC.3, K.CC.5. Lessons contain a structure that includes Before You Begin, Terms to Use, Materials, Daily Routines, Focus, Practice, Assessment Check-in, and Home Link. This provides an additional place to reference the standards within each lesson.

Each Unit Organizer Coherence table includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Examples include:

• Unit 1, Foundational Counting Principles and Skills, Unit 1 Organizer, Coherence, K.G.2, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In PreK, Children explored mostly 2-dimensional shapes in various sizes and orientations through tactile, kinesthetic, and visual activities. In Grades 1 and 2 they will further these understandings by thinking about defining and non-defining attributes of particular shape categories and by identifying additional shape categories, such as quadrilaterals.”

• Unit 3, Reading, Writing, and Using Numbers; Making Comparisons, Unit 3 Organizer, Coherence, K.CC.6, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “In Kindergarten Sections 1 and 2, children used visual, counting and matching strategies to compare small sets during sorting and graphing activities and games. Children had similar experiences in PreK, starting with even smaller sets. This will lead to comparisons of numerals later in Kindergarten and into Grade 1, when children will order and compare numbers using the number line and will be formally introduced to inequality symbols.”

• Unit 7, Addition and Subtraction Strategies; Expanding Number Sense, Unit 7 Organizer, Coherence, K.OA.2, includes an overview of how the content in Kindergarten builds from previous grades and extends to future grades. “Since the beginning of Kindergarten, children have solved addition and subtraction word problems. They have also practiced addition and subtraction in games and activities using dominoes and dice. They began by adding and subtracting within 5, and increased the range to within 10, beginning in Section 5. In Grade 1, children will model and solve problems involving addition or subtraction of two numbers within 20.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Instructional approaches to the program are described within the Teacher’s Lesson Guide. Examples include:

• Teacher’s Lesson Guide, Welcome to Everyday Mathematics, The University of Chicago School Mathematics Project (UCSMP) describes areas of the Everyday Mathematics 4 classroom. “Building from and connecting with children’s informal, everyday experiences with mathematics; problem-solving in everyday situations and mathematical contexts; an instructional design that revisits topics regularly to ensure depth of knowledge and long-term learning; distributed practice through routines, games, and other activities; teaching that supports ‘productive struggle’ and maintains high cognitive demand; and playful lessons and activities that engage all children and make mathematics fun!” 

• Teacher’s Lesson Guide, About Everyday Mathematics, An Investment in How Your Children Learn, The Everyday Mathematics Difference, includes the mission of the program as well as a description of the core beliefs. “Decades of research show that children who use Everyday Mathematics develop deeper conceptual understanding and greater depth of knowledge than children using other programs. They develop powerful, life-long habits of mind such as perseverance, creative thinking, and the ability to express and defend their reasoning.”

• Teacher’s Lesson Guide, About Everyday Mathematics, A Commitment to Educational Equality, outlines the student learning experience. “Everyday Mathematics was founded on the principle that every child can and should learn challenging, interesting, and useful mathematics. The program is designed to ensure that each of your children develops positive attitudes about math and powerful habits of mind that will carry them through college, career, and beyond. Provide Multiple Pathways to Learning, Create a System for Differentiation in Your Classroom, Access Quality Materials, Use Data to Drive Your Instruction, and Build and Maintain Strong Home-School Connections.”

• Teacher’s Lesson Guide, About Everyday Mathematics, Problem-based Instruction, approach to teaching skills helps to outline how to teach a lesson. “Everyday Mathematics builds problem solving into every lesson. Problem solving is in everything they do. Warm-up Activity: Lessons begin with a quick, scaffolded Mental Math and Fluency exercise. Daily Routines: Reinforce and apply concepts and skills with daily activities. Math Message: Engage in high cognitive demand problem-solving activities that encourage productive struggle. Focus Activities: Introduce new content with group problem solving activities and classroom discussion. Summarize: Discuss and make connections to the themes of the focus activity. Practice Activities: Lessons end with a spiraled review of content from past lessons.” 

• Teacher’s Lesson Guide, Everyday Mathematics in Your Classroom, The Everyday Mathematics Lesson, outlines the design of lessons. “Lessons are designed to help teachers facilitate instruction and engineered to accommodate flexible group models. Embedded Rigor and Spiraled Instruction: Each lesson weaves new content with the practice of content introduced in earlier lessons. The structure of the lessons ensures that your instruction includes all elements of rigor in equal measure with problem solving at the heart of everything you do.”

Preparing for the Module provides a Research into Practice section citing and describing research-based strategies in each unit. Examples include:

• Implementation Guide, Everyday Mathematics & the Common Core State Standards, 1.1.1 Rigor, “The Publishers’ Criteria, a companion document to the Common Core State Standards, defines rigor as the pursuit, with equal intensity, of conceptual understanding, procedural skill and fluency, and applications (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CCSSO], 2013, p. 3).

• Implementation Guide, Differentiating Instruction with Everyday Mathematics, Differentiation Strategies in Everyday Mathematics, 10.3.3, Effective Differentiation Maintains the Cognitive Demand of the Mathematics, “Researchers broadly categorize mathematical tasks into two categories;  low cognitive demand tasks, and high cognitive demand tasks. While the discussion of cognitive demand in mathematics lessons is discussed widely, see Sten, M.K., Grover, B.W. & Henningsen, M. (1996) for an introduction to the concept of high and low cognitive demand tasks.”

• Implementation Guide, Open Response and Re-Engagement, 6.1 Overview, “Research conducted by the Mathematics Assessment Collaborative has demonstrated that the use of complex open response problems “significantly enhances student achievement both on standardized multiple-choice achievement tests and on more complex performance-based assessments” (Paek & Foster, 2012, p. 11).”

• The University of Chicago School Mathematics Project provides Efficient Research on third-party studies. For example:

  • Use of Student Constructed Number Stories in a Reform-Based Curriculum.

  • An Action-Based Research Study on How Using Manipulatives Will Increase Student’s Achievement in Mathematics.

  • Differentiating Instruction to Close the Achievement Gap for Special Education Students Using Everyday Math.

  • Implementing a Curriculum Innovation with Sustainability: A Case Study from Upstate New York.

  • Mental Computation of Students in a Reform-Based Mathematics Curriculum.

  • ARC Center Tri-State Achievement Study.

  • Teacher-Initiated Differentiation.

  • The Impact of Two Standards-Based Mathematics Curricula on Student Achievement in Massachusetts.

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

A year-long list of materials needed is provided in the Teacher’s Lesson Guide, Getting to Know Your Classroom Resource Package, Manipulative Kits, and eToolkit. “The table below lists the materials that are used on a regular basis throughout Kindergarten Everyday Mathematics.” Each section includes a Materials Overview section outlining supplies needed for each lesson within the unit. Additionally, specific lessons include notes about supplies needed to support instructional activities, found in the overview of the lesson under Materials. Examples include:

• Lesson 3-8, Spin a Number, Materials, “Focus: walk-on gameboard and Spin a Number Gameboards and spinners (see Before You Begin); game markers Practice: Math Masters pp. 43-44; crayons or markers Home Link: Math Masters p. 51.”

• Section 4, Advanced Counting; Composing/Decomposing Numbers and Shapes; Measurable Attributes, Section 4 Organizer, Section 4 Materials, each lesson has materials listed under the following categories: Math Masters My First Math Book, Activity Cards, Manipulative Kit, Other Materials, and Literacy Suggestions. For example, Lesson 4-1, listed materials, My First Math Book: “MM, p.55-56”, Activity Card: “30”, Manipulative Kit; “attribute blocks”, Other Materials: “prepared Number Cards -10, Literacy Suggestions: “3 Little Firefighters, “The Button Story” (Frog and Toad Are Friends).” 

• Lesson 7-5, Count and Skip Count with Calculators, Materials, “Focus: calculators Practice: Math Masters p. TA12 (optional); bear counters (10 per pair); plastic cups (1 per pair); slates Home Link: Math Masters, p.96.”

• Section 9, Measurement and Spatial Thinking, Section 9 Organizer, Section 9 Materials, each lesson has materials listed under the following categories: Math Masters My First Math Book, Activity Cards, Manipulative Kit, Other Materials, and Literacy Suggestions. For example, Lesson 9-7, listed materials, My First Math Book: “MM, p.TA11”, Activity Card: “84”, Manipulative Kit; “counters, dice”, Other Materials: “materials for a model of the classroom; large paper; camera (optional); children’s work from Day 1; prepared Car Race gameboards (Lesson 8-8)”, Literacy Suggestions: “books about maps.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. Implementation Guide, Differentiating Instruction with Everyday Mathematics, 10.1 Differentiating Instruction in Everyday Mathematics: For Whom?, “Everyday Mathematics lessons offer specific differentiation advice for four groups of learners. Students Who Need More Scaffolding, Advance Learners, Beginning English Language Learners, and Intermediate and Advanced English Language Learners.” Differentiation Lesson Activities notes in each lesson provide extended suggestions for working with diverse learners. Supplementary Activities in each lesson include Readiness, Enrichment, Extra Practice, and English Language Learner. 

For example, the supplementary activities of Section 4, Advanced Counting; Composing/ Decomposing Numbers and Shapes; Measurable Attributes, Lesson 6, include:

• Readiness, “To prepare children to count into the teens, identify a teen number on the Growing Number LIne as the counting target. Have children count to the “target” number, starting at either 1 or 10. You may wish to point, or have a child point, to each number on the Growing Number Line as children say it. Repeat for several different teen numbers.”

• Enrichment, “To extend skills from the lesson, invite children to create a movement sequence that totals the number on a ten number card. For example, a ‘12 dance’ might include 8 side steps, 2 forward steps, and 2 jumps. Have children record their sequence with symbols, pictures, or words, then repeat it multiple times to create a dance. Let them teach their teen dances to others!”

• Extra Practice, “To provide additional practice with reading and counting teen numbers, have partners or small groups take turns picking a teen number card and choosing an action to perform that many times while they count aloud, one movement per count. Alternatively, children may count out a set of objects to match the teen number on their card.”

• English Language Learner, Beginning ELL, “Use gestures to help children understand the word after. For example: Make hopping gestures, moving your hand to the right along a number line as you ask: What number comes after ___? What number comes next? Point to specific numbers, and ask the same questions. Model making corresponding statements after each example with statements such as: The number seven comes after six. Twelve comes after eleven. Encourage early-production children to respond to questions by gesturing from one number to the next on a number line and using sentence frames such as: ‘____ comes after ___.’”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity rather than doing more assignments. The Implementation Guide, Differentiation Instructions with Everyday Mathematics, 10.4 Working with Advanced Learners “Nearly all Everyday Mathematics lessons include a set of high cognitive demand tasks with mathematical challenges that can be extended. Every regular lesson includes recommended enrichment activities related to the lesson content on the Differentiation Options page at the end of the lesson. Everyday Mathematics lessons incorporate varied grouping configurations which enable the kind of flexibility that is helpful when advanced learners in heterogeneous classrooms. The 2-day Open Response and Re-Engagement lesson rubrics provide guidance for students in Exceeding Expectations. Examples include:

• Lesson 3-10, Number-Card Activities, Enrichment, “To extend the Focus activity, have children lay the cards faceup in any order or configuration. While one partner closes his or her eyes, the other partner removes a card from the set. With eyes open, the first partner tries to figure out which card is missing. Use higher number cards if children are ready.”

• Lesson 7-5, Count and Skip Count with Calculators, Enrichment, “To extend the activity, have children use a calculator to skip count by 5s and 2s. Have them substitute 5 or 2 for 10 in the key sequences described on page 450.”

• Lesson 8-7, Birds on Wires (Day 1), Focus: Solving the Open Response Problem, Adjusting the Activity, “If children quickly find many combinations for the 10 birds, challenge them to find every solution to the problem. You may wish to have them discover how many possible combinations there are (11) or provide them with the total and ask them to find them all. Remind them that duplicate solutions do not count toward the total. Discuss how they might organize their work to know that they have found every combination without any duplicates.”

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Teacher’s Lesson Guide and ConnectED Teacher Center include guidance for the teacher in meeting the needs of English Language Learners. There are specific suggestions for making anchor charts or explaining new vocabulary. The ConnectED Teacher Center offers extended suggestions for working with diverse learners including English Language Learners. Most lessons contain “ELL Support.” Examples include:

• Lesson 1-6, Count and Sit, Focus: Playing Count and Sit, ELL Support, “Have children recite numbers in unison, in small groups, and in pairs before asking them to recite the numbers individually.”

• Lesson 4-12, Top-It with Number Cards, Focus: Playing Top-It with Number Cards, ELL Support, “Involve children in practice rounds of Top-It using modeling and think-aloud statements and making sure children understand how to determine who takes the cards at the end of each round. Use gestures, pictures, and the number line to reinforce the terms more, less, and greater.”

• Lesson 7-11, Class Collection, Focus: Collecting Objects, ELL Support, “Help children learn that item is a general term and not the name of a specific object. Also, review the term collection (see Lesson 5-1). Show several collections of objects and pose prompts such as: How many items are in this collection? What are the names of the items in that collection? What is your favorite item in the collection?”

• The online Student Center and Student Reference Book use sound to reduce language barriers to support English language learners. Students click on the audio icon, and the sound is provided. Questions are read aloud, visual models are provided, and examples and sound definitions of mathematical terms are provided. 

• The Differentiation Support ebook available online contains Meeting Language Demands providing suggestions addressing student language demands for each lesson. Vocabulary for the lesson and suggested strategies for assessing English language learners’ understanding of particularly important words needed for accessing the lesson are provided.

The materials reviewed for Everyday Mathematics 4 Kindergarten meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include: 

• Lesson 2-9, Ten Frames, Focus: Exploring Ten Frames, materials reference use of ten-frames and counters. “Remind children about representing numbers on five frames (Lesson 1-11). Explain that today they will show, or represent, numbers using a ten frame. Give each child a ten frame and 10 counters. Ask: What do you notice about this tool? Why do you think it is called a ten frame? How is a ten frame similar to a five frame? How is it different from a five frame?”

• Lesson 4-1, Attribute Blocks, Focus: Exploring Attribute Blocks, materials reference the use of attribute blocks. “Divide the class into small groups and give each group a handful of same-color blocks. (If needed, remove the thick blocks to reduce the number of variables.) Have each group identify an attribute (besides color) by which to sort their blocks.”

• Lesson 7-9, Bead Combinations, Focus: Exploring Number Combinations, materials reference use of bead counters. “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.”