Kindergarten - Gateway 2

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

STEMscopes materials develop conceptual understanding throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Elementary, Conceptual Understanding and Number Sense, STEMscopes Math Elements, this is demonstrated. “In order to reason mathematically, students must understand why different representations and processes work.” Examples include:

• Scope 3: Represent Numbers to 10, Explore, Explore 3–Represent 8 with Objects and Pictures, Procedure and Facilitation Points, with teacher guidance, students develop conceptual understanding that the last number named tells the number of objects by counting cubes. “Read the following scenario: Marcy loves to visit the train station by her house. She notices how strong the train engines are. They can push some train cars while pulling others at the same time. One day while visiting the trains, Marcy noticed the same engine was pushing and pulling a total of 8 train cars at once. What are all of the possible combinations of the 8 train cars being pushed and pulled? Direct students’ attention to the bag of 8 same-color linking cubes, 1 different-color linking cube with a train engine picture attached, and 1 chenille stem. Allow the students and their partners a few moments to discover the manipulatives and experience how they work. Ask the following question: DOK-1 What does a train look like? How does it move? … Invite the students to create trains with the linking cubes. Discuss how the cubes are connected together like a train and how a conductor in the engine is in charge of the train. Explain to students that the linking cubes represent the train cars and the cubes with the pictures on them represent the train engines. Have students work with partners to do the following: They explore the actions of pushing and pulling a train by creating train engines with train cars using the linking cubes and chenille stems. Support students as they place their linking cubes on their chenille stems in different ways to represent a total of 8 train cars. They can put as many cars in front of or behind as they want, but they must use all 8 linking cubes and no extra linking cubes. They will place their chenille stems on the tracks on their Train Story Mats. Encourage students to practice writing the numbers on their Story Mats by using a dry-erase marker to describe the train car placement. During this time, students should also practice stating equations to their partners that represent the train setups they have constructed. For example: If a student creates a concrete model with 3 train cars being pushed and 5 train cars being pulled, they would verbally state "3+5=8" to a partner. Students may then count the trains to check their calculations. They should always end on the number 8, indicating there are 8 train cars.” (KCC.4b)

• Scope 4: Compare Numbers to 10, Explore, Skill Basics–Matching and Counting Strategies, Procedure and Facilitation Points, students develop conceptual understanding as they count and compare numbers using a matching strategy. (Sample answers follow some questions.) “1. Divide the class into pairs. Give each pair of students a set of 10 blue and 10 yellow linking cubes. Instruct students to break apart their stacks of cubes into individual cubes. 2. Review key vocabulary words: compare, greater than, less than, and equal to. Allow students to share the meaning of each term and address any misconceptions. a. Say, ‘Today I am going to show you a matching and counting strategy that you can use to compare two numbers. First, I would like you to create a group of 6 blue linking cubes and 4 yellow linking cubes.’ 3. Walk around the room, and make sure each pair of students has counted out the correct number of cubes. 4. Using the document camera, allow one student volunteer to count out 6 blue linking cubes. Allow a different volunteer to count out 4 yellow cubes. a. Say, ‘Now that we have represented each number using objects, we are going to learn the new strategy for comparing.’ 5. Model lining up the blue cubes horizontally. As you do so, count each cube aloud.6. Instruct students to do the same thing with their blue linking cubes. a. Say, ‘We have counted our blue cubes, and now we need to compare this amount to the number of yellow cubes. To do this, I’m going to match my yellow cubes to the blue cubes by lining them up underneath until I do not have any more yellow cubes.’ (As you do so, reiterate how you are matching a yellow cube to a blue cube.) 7. Instruct students to do the same thing with their yellow linking cubes. Once the cubes have been matched up, discuss the following questions: a. What do you notice about your model? b. Are these two sets of objects equal? c. How do you know? d. Which color linking cubes has more? e. How can you tell? f. Which color linking cubes has less? g. How can you tell? h. Compare the number of blue cubes to the number of yellow cubes using the terms greater than, less than, or equal to. i. Compare the number of yellow cubes to the number of blue cubes using the terms greater than, less than, or equal to. 4 is less than 6. Repeat the steps above using various numbers up to 10. Students should see the one-to-one correspondence between cubes that are matched and understand any cubes that do not have a match show that the set is larger than the other set.” (K.CC.6)

• Scope 5: Join and Separate, Explore, Skill Basics–Acting Out Word Problems and Drawing Models, Procedure and Facilitation Points, with teacher guidance, students develop conceptual understanding of representing addition and subtraction. “7. Read the first story problem again aloud to the class, and use questioning to guide students: a. Say, ‘Aden had 3 books. Let’s draw a quick picture of what that might look like.’ Model for students how to draw this first piece of information. Draw 3 squares on your chart paper while students draw the same image by using dry-erase markers on their work mats. b. Continue to read, ‘Suzy gave him 2 more books. How can I show the 2 more books?’ Draw 2 more squares on the chart paper while students continue to add to their pictures on their work mats. c. Say, ‘Now that we have our picture, we have to show our answer.’ Model how to count all the squares, and have the students follow along by pointing to each square they drew. ‘How many books does Aden have now? We have 5 squares, so that means he had 5 books.’ 8. Discuss the solution to the story problem as a whole group, and have students determine whether the problem was joining (adding to) or separating (taking from).” (K.OA.1)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

• Scope 5: Join and Separate, Show What You Know, Part 3: Decompose with Objects and Drawings, students represent a number of objects with a written numeral. “Use the fish at the bottom to help you solve the problem. Write two possible ways Patrice could have shared all her fish with Richard and Tamika. Patrice had 7 fish in her fish tank to give away. She gave Richard some fish and Tamika some fish. How many fish could Richard and Tamika have? Richard could have ___ fish. Tamika could have ___ fish. Tamika could have ___ fish. Richard could have ___ fish.”  (K.OA.3)

• Scope 6: Represent Numbers to at Least 20, Part 4: Composing Tens and Ones, Show What You Know, students decompose numbers 11-19. Students are given two ten frames, a picture of a basketball, and the number 14. “Draw a (circle) in the ten frames to represent each (basketball). The team made ___ points. 10 and ___ equals ___. ___ is the same as 10 and ___.” (K.NBT.1)

• Scope 11: Data Analysis, Show What You Know-Part 2: Labeling a Sort and Drawing Conclusions, Student Handout, students sort food items and answer questions using the data. On the handout, students see three tables, one each for doughnuts, waffles and pancakes.  Across the bottom, students see small icons that can be cut out and glued to one of the labeled tables. “Part 2: Labeling a Sort and Drawing Conclusions Data Analysis Part 2 1 Cut out the breakfast foods and sort them. Organize and glue them onto the correct tabletops. Count how much food is on each table. ___Doughnuts, ___Waffles, ___Pancakes, How did you sort the breakfast food? (Circle one.) Size, Color, Type, Which table has the most breakfast food?___, Which table has the least breakfast food?___” (K.MD.3)

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

STEMscopes materials develop procedural skills and fluency throughout the grade level. In the Teacher Toolbox, STEMscopes Math Philosophy, Elementary, Computational Fluency, STEMscopes Math Elements, these are demonstrated.  “In each practice opportunity, students have the flexibility to use different processes and strategies to reach a solution. Students will develop fluency as they become more efficient and accurate in solving problems.” Examples include:

• Fact Fluency: Addition and Subtraction, Fact Fluency–Mini-Lesson, Procedure and Facilitation Points, this is demonstrated in the following example:  “5. Ask the following questions: a. How many counters did I grab the first time? Answers vary from 0-5. b. How many did I grab now? Answers vary from 0-4. c. What strategy did you use to know how many I grabbed? Answers will vary. I counted them one by one. I just looked at them and knew there were 3. d. How can we join the counters to find the sum? Answers will vary. We can line them up. We can count them one by one. We can count on from ___. e. How can you describe what you have just done? Answers will vary. Three and two is five, or two plus three equals five.” (K.OA.2)

• Scope 2: Count Objects, Engage, Foundation Builder, Procedure and Facilitation Points, students develop counting fluency. “1. Project the slide with the button counters. Ask students to count how many they see. 2. Discuss the following questions with the class: a. How many buttons did you count? Answers may vary. 8 buttons, b. Was it hard to count the buttons on the screen? Answers may vary. Yes, it was hard to keep track of how many I already counted. No, I got it right. c. What would make it easier to count the buttons? If we had the actual buttons, they would be easier to count. If we had our own copy to touch as we count, they would be easier to count. 3. Give each pair of students a bag of buttons. Instruct students to count the buttons aloud to themselves and a partner. Discuss the following questions with the class: a. Was it easier to count the buttons? Yes, b. How did you count the buttons? Answers will vary. I counted them and slid them as I counted them. I used a number path. 4. Give each pair of students a bag of bear counters. Instruct students to count the bear counters aloud to themselves and a partner. Discuss the following questions: a. How many bear counters did you count? 10 bear counters, b. How did you count the bear counters? I counted them and slid them as I counted them. I used a number path. 5. Give each pair of students a bag of pattern blocks. Instruct students to count the pattern blocks aloud to themselves and a partner. Encourage students to use a different strategy to count. Discuss the following questions with the class: a. How many pattern blocks did you count? 6 pattern blocks, b. How did you count the pattern blocks? I counted them and slid them as I counted them. I used a number path.” (K.CC.4)

• Scope 4: Compare Numbers to 10, Engage, Foundation Builder, Procedure and Facilitation Points, students develop their understanding of greater than and less than. “1. Project the slide with the purple circles in the boxes. Ask students to observe the pictures and think about how they would decide which box has the greatest number of circles. 2. Discuss the following questions: What strategy did you use to figure out which had the most?I looked to see how many circles were in each box and found out it was 4 and 1.Which box has the greatest amount, and how do you know? The box with 4 had the most because 4 is a greater number than 1. Display 5 linking cubes. Place 2 linking cubes in one pile and 3 linking cubes in another pile. Show students how to match up the two piles one to one to see which pile has more. Discuss the following questions: Does each linking cube have a partner? No, there is one that does not have a partner.How many linking cubes are in each pile? There are 2 linking cubes in one pile and 3 linking cubes in the other pile.Which pile has more? Which pile has less? 3 linking cubes is more than 2 linking cubes. 2 linking cubes is less than 3 linking cubes.Bring the linking cubes back together into one pile. Place 2 linking cubes in one pile and 2 linking cubes in another pile. (Push the linking cube not being used away so as not to confuse the students.) Again, show students how to match up the two piles one to one to see which pile has more. Discuss the following questions:Does each linking cube have a partner? Yes, each linking cube has a partner.How many linking cubes are in each pile? There are 2 linking cubes in one pile and 2 linking cubes in the other pile.Which pile has more? Which pile has less? They are equal or the same. 2 linking cubes is equal to 2 linking cubes.” (K.CC.6)

The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Scope 5: Join and Separate, Elaborate, Fluency Builder–Four in a Row, students demonstrate addition and subtraction fluency. “Description, Students play this game in pairs. They take turns solving problems involving addition and subtraction. For each correct answer, students mark a game space. The first student to mark four game spaces in a row wins.” (K.OA.5)

• Scope 7: Two-Dimensional Shapes, Explore, Explore 1–Sorting 2-D Shapes, Exit Ticket, students independently classify shapes regardless of their orientation. “Circle the shape that belongs in each group.” Given 4 quadrilaterals in the box for Group 1 and the choice of a heart and a rectangle. “Circle the shape that belongs in Group 1.” (K.G.2)

• Scope 11: Data Analysis, Elaborate, Fluency Builder, Four in a Row, Preparation and Procedure and Facilitation Points, students develop fluency sorting objects into different categories including size, color, etc. Preparation, “Print the Game Board. Print the Student Recording Sheet back-to-back. Print and cut out sets of double-sided Playing Cards. It is suggested that you laminate the cards and place them in an envelope or resealable bag for long-term use. Place a sticky note over each answer on the Playing Cards so students solving the problem can view the card without seeing the answer.” The game cards have different objects sorted using different criteria. Students need to determine how the objects were sorted. Procedure and Facilitation Points, “1. Demonstrate playing the game with a student partner. a. The dealer shuffles the deck of Playing Cards, and deals them equally between the players. Each player may view all dealt cards at once. b. Each player chooses which side of the two-color counter he or she wants to use to mark his or her spot. c. Players alternate turns. During each turn, a player chooses one card and shows it to the other player. The opponent solves the problem on the Student Recording Sheet, and the first player checks the answer by looking under the sticky note. d. Each time a player solves a problem correctly, the player places a counter in one game space. The player who successfully covers four connected spaces in a row (horizontally, vertically, or diagonally) wins. e. If all Playing Cards are used and no player has covered four spaces in a row, then the game results in a draw.  2. Distribute materials. 3. Have the students play the game.” (K.MD.3)

The materials reviewed for STEMscopes Math Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics.

STEMscopes materials include multiple routine and non-routine applications of mathematics throughout the grade level, both with teacher support and independently. Within the Teacher Toolbox, under STEMscopes Math Philosophy, Elementary, Computational Fluency, Research Summaries and Excerpt, it states, “One of the major issues within mathematics classrooms is the disconnect between performing procedural skills and knowing when to use them in everyday situations. Students should develop a deeper understanding of mathematics in order to reason through a situation, collect the necessary information, and use the mechanics of math to develop a reasonable answer. Providing multiple experiences within real-world contexts can help students see when certain skills are useful.” 

This Math Story activity includes both routine and non-routine examples of engaging applications of mathematics. For example:

• Scope 5: Join and Separate, Elaborate, Math Story–Field Trip to the Science Museum, students solve both routine and non-routine problems with teacher support. “First, we see 3 owl butterflies way up at the top of the sky dome. They are joined by 4 Atlas moths. Now, how many are there in all? Draw and solve after giving it some thought. Next, my friend Alex points out six hissing cockroaches. He says, “They are so ugly, I bet nobody approaches!” But just like that, three ran off as a praying mantis got close. Now how many cockroaches do you note? Draw and solve.” (K.OA.1)

Engaging routine applications of mathematics include:

• Scope 3: Represent Numbers to 10, Evaluate, Skills Quiz, students independently demonstrate application as they solve addition and subtraction problems. “Finish the drawings by filling in the box with the correct number of objects. If nothing needs to be added, leave the box blank. 3. given (6 triangles) + (blank box) = 10” (K.OA.2)

• Scope 6: Represent Numbers to At Least 20, Explore, Explore 1, Procedure and Facilitation Points, “1. Read the following scenario: Maria is getting ready for a celebration. She is making gift bags for her guests. Her brother decided to help her, but he never pays attention to detail. Maria wanted exactly 15 objects in each bag, but her brother just grabbed a handful of items and stuffed them in each bag. Can you help Maria sort out this mess by counting the objects in each bag? 2. Give each student a copy of the Student Journal. Allow each group to select a bag to count first. When they are finished counting that bag, have them bring it back and select another one. 3. Instruct students to carefully pour out the objects from the bag onto the table. Have students lay out the objects and take turns counting the objects one by one. Once they have counted the collection, have students write the bag number in the space provided on the Student Journal. Then, have them draw a picture and write the numeral that represents how many objects were counted by completing the sentence stem.” (K.CC.2)

Engaging non-routine applications of mathematics include:

• Scope 4: Compare Numbers to 10, Engage, Hook–Counting Watermelon Seeds, Procedure and Facilitation Points, students solve non-routine problems with teacher support as they compare numbers up to 10. “Part I: Pre-Explore, 1. Introduce this activity toward the beginning of the scope. The class will revisit the activity and solve the original problem after students have completed the corresponding Explore activities. 2. Show the Phenomena video. Ask questions such as, ‘What do you notice?’ and ‘Where can you see math in this situation?’ Allow students to share all ideas. 3. Explain the situation: Hayden, Anika, and Devon are each enjoying a juicy, cold slice of watermelon on a hot summer day. They notice the black seeds in their watermelon slices and want to see who has the most. Hayden counts 6 seeds in his slice of watermelon. Devon counts 1 more seed in his slice than Hayden has in his slice. Anika counts 1 fewer seed in her slice than Hayden has in his slice. How many seeds did Devon and Anika count in their slices? 4. Discuss the following: a. DOK-1 What information do we know? b. DOK-1 What information do we need to find out? 5. Give each student a bag of black beans, and tell them that the beans represent watermelon seeds. Let them explore and try to determine the answer. There will only be 10 beans in each bag…” (K.CC.4c)

• Scope 9: Create and Compose 2-D Shapes and 3-D Solids, Elaborate, Problem-Based Task– Shape Composer, Student Handout, students independently solve non-routine problems as they use smaller shapes to make larger shapes. Students have a large area on the paper to create their new shape as well as shapes to cut out to use to make the larger shapes. “Shape Composer, You and your partner are shape composers! Use the two-dimensional shapes at the bottom of the next page to compose some awesome new shapes. Cut out the shapes. Glue 2, 3, or 4 of them together to create a brand new shape. Count the sides around the new shape, and write the shape’s name below it. Think About, How can you arrange your shapes? How will you know what new shape you have composed? How many new shapes can you make? My New Shape, Cut out the shapes. Glue 2-4 together to compose a new shape. Write the name of the new shape below. Shape Name: Number of Sides:” (K.G.6)

The materials reviewed for STEMscopes Math 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 instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Fact Fluency: Addition and Subtraction, Making Tens, Fact Fluency–Mini Lesson, Procedure and Facilitation Points, students develop fluency making 10 with counters and addition equations. “1. Distribute a bag of counters, a Ten Frame, and a Recording Sheet to each student. 2. Ask the following discussion question: a. How many squares are on your mat? 10, 3. Instruct students to put their counters on the ten frame. Ask students to turn all red sides up. 4. Ask the following discussion questions: a. How could you use the two-color counters to show a way to make 10? Allow students time to create 1 way to make 10. Answers will vary. 5 red and 5 yellow counters, b. Are there any other ways to make 10? Allow students to find an additional way to make 10. Answers will vary. 4 red and 6 yellow; 3 red and 7 yellow; 2 red and 8 yellow; 1 red and 9 yellow (also the opposite of each of these). 5. Ask students to work with their counters to make ten in several different ways. Students will record each way to make 10 using a number sentence on the Recording Sheet. 6. Allow time for students to share the number sentences they recorded with a partner. 7. After completing this mini-lesson, have students move on to station activities and fact fluency games.” (K.OA.4)

• Scope 3: Represent Numbers to 10, Elaborate, Math Story–Saturday at the Panadería with Papá, students apply their understanding of decomposing numbers to solve addition and subtraction problems. “This Saturday, we don’t have any of the big bags to fill the orders. Can you help me figure out the different ways to put the tasty treats into two small boxes instead? Señora Ramírez orders seven orejas, and I get to work. I run off quickly with a skip and a smirk. Draw and write how I could put the orejas in the two boxes to equal seven.” (K.OA.3)  Students solve routine problems as they solve addition and subtraction word problems. “Right behind her comes Don José. Papá asks, ‘How many empanadas for you today?’ Don José holds up 9 fingers, and I quickly get to it. If I put 5 in the first box, how many empanadas go in the other?” (K.OA.2)

• Scope 6: Represent Numbers to at Least 20, Explore, Explore 2–Counting Objects and Organizing Counts, Math Chat, students develop conceptual understanding with teacher support and guidance to gain foundations for place value. Given 14 linking cubes, “Look at Collection 2. DOK-1 How many groups of 10 are in the number for Collection 2? One group of 10, DOK-1 How many other objects did you count after that group of 10 was made? DOK-1 When the one group of 10 and the 4 others are combined, what does the total number look like when it is written? DOK-1 How do you say this number? DOK-3 Does finding a group of ten when you are counting collections make counting easier? Why? DOK-4 How can using Counting Mats and knowing how to organize and count collections of items help you with tasks at home?” (K.NBT.1)

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

• Scope 5: Join and Separate, Evaluate, Skills Quiz, students demonstrate application alongside conceptual understanding as they decompose numbers to solve a problem. “6. How can you decompose 7 in two different ways? Draw a picture and write an equation for each way.” (K.OA.3)

• Scope 7: Two-Dimensional Shapes, Evaluate, Show and Tell, students demonstrate application alongside conceptual understanding as they describe objects and their relative positions. “Teacher Prompt–Card 4, 1. Place Student Card 2 on the table in front of the students. 2. Ask, 'What shapes are around the Sun and in the center of the Sun?' 3. Ask, 'What shapes are between the triangles on the bird?' 4. Ask, 'What shape are the wheels on the bicycle?' 5. Ask, 'What shape are the windows on the car?' 6. Ask, 'What shapes are at the bottom of the picture?' 7. Ask, 'Where are the triangles on the bicycle in relation to the circle on the Sun?' 8. Ask, 'Where is the hexagon in relation to the rectangles on the car?' 9. Ask, 'Where are the  squares in relation to the triangles on the car?’” (K.G.1)

• Scope 9: Create and Compose 2-D Shapes and 3-D Solids, Elaborate, Problem-Based Task-Shape Composer, Student Handout, students apply their knowledge of shapes and solids alongside conceptual understanding. Students have a large area on the paper to create their new shape as well as shapes to cut out to use to make the larger shapes. “Shape Composer, You and your partner are shape composers! Use the two-dimensional shapes at the bottom of the next page to compose some awesome new shapes. Cut out the shapes. Glue 2, 3, or 4 of them together to create a brand new shape. Count the sides around the new shape, and write the shape’s name below it. Think About, How can you arrange your shapes? How will you know what new shape you have composed? How many new shapes can you make? My New Shape, Cut out the shapes. Glue 2-4 together to compose a new shape. Write the name of the new shape below. Shape Name: Number of Sides:” (K.G.6)

The materials reviewed for STEMscopes Math 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.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the scopes. MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 1, Count Objects, Explore, Explore 4–Counting Forward and Backward Within 20, students build experience with MP1 as they work to understand counting forward and backward. Procedure and Facilitation Points, “1. Read the following scenario to the class: The Watsons flew to New York City for winter vacation. To get around the city without a car, they will need to ride the subway. These underground trains do not turn around; they just go forward and backward. Can you help the family get to their destinations and back safely? 2. Divide the class into groups of 2 or 3 students, and give each group a set of Task Cards and a Number Path. 3. Instruct students to pull a Task Card and count forward to the station and then backward to return. Encourage students to use the Number Path to assist in counting forward and backward. 4. Give each student a Student Journal. Ask students to fill in the missing numbers for each Task Card. 5. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 What is the starting station number? b. DOK-1 What is the number you are counting to? c. DOK-2 How do you count to this number? d. DOK-2 How would you count backward from this number? e. DOK-3 What strategies or tools did you use? 6. After students have completed all of the Task Cards and their Student Journals, bring the class together as a whole group…” In Scope 1, Count Objects, Content Scope, Applying Mathematical Practices, the program identifies work with MP1. “MP.1 Make sense of problems and persevere in solving them: Students demonstrate counting using one-to-one correspondence when counting a group of objects. They persist with counting in a correct sequence.”

• Scope 5: Join and Separate, Explore, Explore 4–Writing Equations and Explaining Strategies, Standards of Mathematical Practice, students engage with MP.1 as they, “...understand that mathematics involves solving problems and discussing how to solve them. They use concrete objects or pictures to help them conceptualize and solve problems.” Procedure and Facilitation Points, “3. Project the Task 1 slide from the Task Cards document. Read the word problem aloud as students follow along. After reading the task card, go through each step together as a whole group. a. Step 1: On the left column of the table, where it says Build it!, there is a template of how to arrange the pattern blocks for each living thing. Have students place their pattern blocks on top of the template to see the turtle. Once they have assembled their turtle, they can remove the blocks and use crayons to color the shapes to match where the pattern blocks were. b. Step 2: Once students have built their models using the pattern blocks, have them represent what is happening in the problem by drawing circles. One circle of the correct color should be drawn for each pattern block that was used to build the turtle. This is done in the box labeled Draw it! on the Student Journal. c. Step 3: Ask students to write equations that describe their pictorial models. They write them in the boxes labeled “Write it!” First, students must decide whether joining or separating is occurring. Have students discuss this with their partners and explain why the scenario shows joining or separating. Then have them put plus or minus signs in the small squares to describe their actions. Make sure that students know that a + (plus) is for joining and a – (minus) is for separating. Then have them fill in the lines that are provided for them to write the numbers to complete the equation. d. Step 4: Have students continue solving the problem by explaining their strategies to their partners. Partners should compare their strategies. Have them complete the sentence in the Explain it! section by filling in the blanks with words and numbers. The students should fill in the first blank with the word added or subtracted to describe their models and equations. Then they fill in the numbers used and the answer to the problem.”

• Scope 9: Create and Compose 2-D Shapes and 3-D Solids, Explore, Explore 2–Composing 2-D Shapes, students make sense of problems and persevere in solving them while they compose shapes. Procedure and Facilitation Points, “4. Instruct students to look at the directions for their station on the Student Journal. Students need to compose the shape by using their manipulatives. Students might not use all of the pattern blocks provided. Have them draw their designs on the Student Journals and record how many triangles, squares, and hexagons they used to compose the new shape. 5. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-2 What is one way you were able to compose a new shape using the pattern blocks? b. DOK-2 Is there another way you can compose this shape using different pattern blocks? c. DOK-2 Is there another way you can use the same shapes to compose a different shape? d. DOK-2 Can you compose this shape in a different way? e. DOK-2 If so, how are the compositions alike/different? f. DOK-2 How is your model similar to your neighbor’s model? g. DOK-2 How is it different?...”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the scopes. Examples include:

• Scope 3: Represent Numbers to 10, Explore, Explore 3–Represent 8 with Objects and Pictures, Standards of Mathematical Practice, “Students recognize that a number represents a specific quantity. Then they represent that quantity using the written numeral.” Procedure and Facilitation Points, “4. Have students work with partners to do the following: a. They explore the actions of pushing and pulling a train by creating train engines with train cars using the linking cubes and chenille stems. b. Support students as they place their linking cubes on their chenille stems in different ways to represent a total of 8 train cars. They can put as many cars in front of or behind as they want, but they must use all 8 linking cubes and no extra linking cubes. They will place their chenille stems on the tracks on their Train Story Mats. c. Encourage students to practice writing the numbers on their Story Mats by using a dry-erase marker to describe the train car placement. d. During this time, students should also practice stating equations to their partners that represent the train setups they have constructed. For example: If a student creates a concrete model with 3 train cars being pushed and 5 train cars being pulled, they would verbally state "3+5=8" to a partner. Students may then count the trains to check their calculations. They should always end on the number 8, indicating there are 8 train cars. 5. Once students have had time to use the manipulatives to find the different ways to represent 8, give each student a copy of the Student Journal.”

• Scope 4: Join and Separate, Content Support, and Explore 3–Decompose with Objects and Drawings, Exit Ticket, In Content Support, the program identifies the work within Scope 4 related to MP2. “MP.2 Reason abstractly and quantitatively: Students recognize that a number represents a specific quantity and connect the quantity to written symbols.” Exit Ticket, students determine the number of objects needed to be drawn to represent a number of two types of books. “Draw a picture to solve. Fill in the blank to explain your answer. Noel had 6 library books. Some of the books were fiction. Some of the books were nonfiction. What is one way to show her two types of books? A total of ___ (symbol for books)is equal to ___ and ___.” 

• Scope 11: Data Analysis, Explore, Explore 2–Labeling a Sort and Drawing Conclusions, Standards of Mathematical Practice, “Students recognize that the number of objects in a set represents a specific quantity.” Procedure and Facilitation Points, “5. Instruct students to look inside of each basket and discuss how the objects were sorted. Have students work together as a group to decide what label should be given to each basket. Have them write each label on the baskets found on their Student Journals. Assist students with spelling as needed. 6. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 What are some different ways objects can be sorted? b. DOK-2 How were these objects sorted? How do you know? c. DOK-1 Into how many groups were the objects sorted? d. DOK-1 How many objects are in each basket?”

The materials reviewed for STEMscopes Math 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.

The materials provide opportunities for student engagement with MP3 that are both connected to the mathematical content of the grade level and fully developed across the grade level. Mathematical practices are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. Students construct viable arguments and critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the Scopes. Examples include:

• Scope 5: Join and Separate, Explain, My Math Thoughts, students construct viable arguments as they justify their addition and subtraction strategies. “Sofia saw 6 hummingbirds in her garden. Four of them flew away. Use the drawing above to show how many hummingbirds stayed in the garden. Then use your cubes to model the same problem. Sofia saw that ___ hummingbirds stayed in the garden. Draw a picture and write a number sentence below to show how you solved the problem.”

• Scope 6: Two-Dimensional Shapes, Standards for Mathematical Practice and Explore, Explore 1–Sorting 2-D Shapes, Math Chat, Standards for Mathematical Practice, “MP.3 Construct viable arguments and critique the reasoning of others: Students reason about and justify whether or not various shapes fit into categories. If a shape fits into a particular category, then it must have certain attributes and vice versa.” Math Chat, “DOK-2 If we put all of these shapes together in a group, how could you describe something they all have in common? DOK-3 How did you and your partner sort the objects in your bag? DOK-3 Is there another way to sort the shapes in the bag? Explain. DOK-3 Were there some objects in your bag that did not seem to fit in any categories? Why? DOK-2 What do shapes A and B have in common? DOK-3 Can you draw a new shape that would fit into each of your boxes in your Student Journal? How can you plan this? DOK-2 How are shapes C and F different? DOK-2 What do shapes C and M have in common? DOK-1 What are two-dimensional shapes?”

• Scope 8: Three-Dimensional Solids, Explore, Explore 4–Identifying 3-D Solids and Their Positions in the Real World, Standards for Mathematical Practice “MP.3 Construct viable arguments and critique the reasoning of others: Students reason about and justify whether or not various solids fit into categories. If a solid fits into a particular category, then it must have certain attributes and vice versa.” Procedure and Facilitation Points, “4. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-2 What do you notice about this real-world solid? b. DOK-2 In what column should we write this object? c. DOK-2 Why do you think this is a cylinder (cube, cone, sphere)?”

• Scope 10: Measurement, Explore, Explore 1 – What Can Be Measured? Identifying Measurable Attributes, students have the opportunity to critique the reasoning of others during group discussion about measurable attributes. Procedure and Facilitation Points, “3. Direct students’ attention to the items in the containers at their stations. Allow students a few moments to discover the manipulatives and experience how they work with their groups. 4. Instruct students to explore each item in the container and discuss attributes that they notice about each item. 5. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 What are some attributes you observe about this item? b. DOK-2 Are all of those attributes measurable? c. DOK-2 What do you think makes an attribute measurable? d. DOK-2 What do you think makes an attribute non measurable? e. DOK-2 When you are looking for measurable attributes, what might you look for? f. DOK-3 Would length be a measurable attribute for a sale item? Why? g. DOK-3 Would cuteness be a measurable attribute for a sale item?...”

The materials reviewed for STEMscopes Math 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.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 3: Represent Numbers to 10, Home, Content Support, “MP.4 Model with mathematics: Students represent problem situations using numbers, words, drawings, and concrete objects” while working with addition. Elaborate, Problem-Based Task–House Designers, students are given the outline of a house. “This house needs windows and doors. Draw windows and doors to equal any number up to 10. Write your equation below.” 

• Scope 4: Compare Numbers to 10, Explore, Explore 2–Compare Sets, Exit Ticket, students build experience with MP4 when counting and comparing groups of objects by matching and counting objects. Students see a set of 5 and 6 unifix cubes in the first problem and a set of 7 and 3 cubes in the second problem. “Count the (cubes) in each set. Write the number. Compare the numbers. Circle the number that is greater. ___ is greater than ___. Circle the number that is less. ___ is less than ___.”

• Scope 6: Represent Numbers to at Least 20, Skills Quiz, students model the situation with an appropriate representation while working with place value. Student Journal, “3. Draw 14 squares and write an equation to represent the number of tens and ones.” 

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 5: Join and Separate, Explore, Explore 4–Writing Equations and Explaining Strategies, students use manipulatives and paper/pencil to solve word problems and represent a number of objects with a written numeral. Procedure and Facilitation Points, “1. Read the following scenario: Rika and Javier went fishing last week. At the lake, they saw a turtle, a duck, and a crab. They also noticed beautiful wildflowers on the bank. But they forgot to take pictures. Can you help Rika and Javier remember what they saw on their fishing trip by solving the following word problems? 2. Give each student a copy of the Student Journal and a bag of pattern blocks. Allow the students a few moments to examine the Student Journal, to discover the manipulatives, and to experience how to use them. 3. Project the Task 1 slide from the Task Cards document. Read the word problem aloud as students follow along. After reading the task card, go through each step together as a whole group. a. Step 1: On the left column of the table, where it says Build it!, there is a template of how to arrange the pattern blocks for each living thing. Have students place their pattern blocks on top of the template to see the turtle. Once they have assembled their turtle, they can remove the blocks and use crayons to color the shapes to match where the pattern blocks were. b. Step 2: Once students have built their models using the pattern blocks, have them represent what is happening in the problem by drawing circles. One circle of the correct color should be drawn for each pattern block that was used to build the turtle. This is done in the box labeled Draw it! in the Student Journal. c. Step 3: Ask students to write equations that describe their pictorial models. They write them in the boxes labeled “Write it!” First, students must decide whether joining or separating is occurring. Have students discuss this with their partners and explain why the scenario shows joining or separating. Then have them put plus or minus signs in the small squares to describe their actions. Make sure that students know that a + (plus) is for joining and a – (minus) is for separating. Then have them fill in the lines that are provided for them to write the numbers to complete the equation. d. Step 4: Have students continue solving the problem by explaining their strategies to their partners. Partners should compare their strategies. Have them complete the sentence in the Explain it! section by filling in the blanks with words and numbers. The students should fill in the first blank with the word added or subtracted to describe their models and equations. Then they fill in the numbers used and the answer to the problem.” 

• Scope 9: Create and Compose 2-D Shapes and 3-D Solids, Explore, Skill Basics–How to Use a Geoboard, students learn to use a geoboard while they identify and describe shapes. Procedure and Facilitation Points, “1. Divide the class into pairs. 2. Give a set of Geoboard Shape Cards, 10 rubber bands, and two geoboards to each pair of students. 3. Instruct groups to find the Geoboard Shape Cards with the blue squares. 4. Model on a geoboard using the rubber band to create the square shape. Be sure to count the dots to make sure that you are accurately creating the shape. All sides must be equal. Challenge students to make the same shapes on their geoboards. 5. Facilitate a discussion with the students by asking the following questions: a. How did I use the rubber bands and geoboard to make a shape? We counted the dots and made sure that we put the rubber band on the correct dots. b. What shape did we make? We made a square. c. Describe the attributes of this shape. A square has 4 sides and all the sides are the same length. 6. Challenge students to work with their partners to create the remainder of the shapes using their cards, rubber bands, and geoboards. 7. Continue to guide students in discussion by asking the questions from step 5. a. Teacher note: Students do not need to know the shape names for parallelogram, pentagon, and hexagon. For these shapes, only ask questions related to their number of sides and vertices. 8. When this activity is complete, move on to Explore 1 for students to apply their knowledge of the skills they just learned.” (K.G.2) 

• Scope 10: Measurement, Explore, Explore 3–Comparing Height, Procedure and Facilitation Points, students use given Handprint Cutouts to measure and compare. “7. Instruct students to measure the height of the child’s outline on their butcher paper using their set of Handprint Cutouts. They will record the number of handprints tall the child is on their Student Journals. 8. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-2 Explain how you are measuring ___ (child’s name from scenario) height. b. DOK-1 How many handprints tall is the child you are measuring? c. DOK-3 What do you notice about the height of ___ compared to the height of ___? d. DOK-2 How can you determine which child is the shortest/tallest? e. DOK-3 What strategy can you use to determine how many handprints shorter/taller one child is compared to another child? 9. Rotate the pieces of butcher paper until each group has measured and recorded the height for all 5 traced outlines. Once all heights have been recorded, students will complete the comparison statements at the bottom of their Student Journals. They will determine how many handprints shorter or taller one child is compared to another. They will use the word bank to complete the comparison statements. Assist with reading if needed.”

The materials reviewed for STEMscopes Math Kindergarten meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP6 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students attend to precision as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 4: Compare Numbers to 10, Engage, Foundation Builder, Procedure and Facilitation Points, students attend to precision and the specialized language of mathematics as they count objects and compare numbers. “3. Display 5 linking cubes. Place 2 linking cubes in one pile and 3 linking cubes in another pile. Show students how to match up the two piles one to one to see which pile has more. Discuss the following questions: a. Does each linking cube have a partner? No, there is one that does not have a partner. b. How many linking cubes are in each pile? There are 2 linking cubes in one pile and 3 linking cubes in the other pile. c. Which pile has more? Which pile has less? 3 linking cubes is more than 2 linking cubes. 2 linking cubes is less than 3 linking cubes. 4. Bring the linking cubes back together into one pile. Place 2 linking cubes in one pile and 2 linking cubes in another pile. (Push the linking cube not being used away so as not to confuse the students.) Again, show students how to match up the two piles one to one to see which pile has more. Discuss the following questions: a. Does each linking cube have a partner? Yes, each linking cube has a partner. b. How many linking cubes are in each pile? There are 2 linking cubes in one pile and 2 linking cubes in the other pile. c. Which pile has more? Which pile has less? They are equal or the same. 2 linking cubes is equal to 2 linking cubes” Possible Preconceptions, “Students may confuse the terms more/greater than with less/fewer than. Suggested Solution: Using hand motions, say the words more/greater than with arms wide open and less/fewer than with hands close together. Work with students to make vocabulary posters with examples of more/greater than and less/fewer than.”

• Scope 8:Three-Dimensional Solids, Explore, Explore 1–Sorting 3-D Solids, Math Chat, students attend to precision and the specialized language of mathematics as they use geometric language to identify and describe the attributes of three-dimensional solids. “How are 3-D solids different from 2-D shapes you have learned about before? What do you notice about how 3-D solids are similar to 2-D shapes? How did your group sort the objects in your bag? How do you know the Shape Cutouts are 3-D solids? What do you notice about the solids A, F, and N? How are they similar/different? Did the color, size, or orientation (direction the solid faces) of the solids matter when you were sorting? Why?”

• Scope 10: Measurement, Explore, Explore 1–What Can Be Measured? Identifying Measurable Attributes, students attend to precision and the specialized language of mathematics as they communicate observations about the measurable attributes of objects in their everyday lives. Student Journal, Box 1, given a table with labeled pictures of an item and 3 choices, “Circle the item’s measurable attributes. Cross out the item’s nonmeasurable attributes.” Row 1 shows, “Book, Softness, Weight, Length.” Row 2 shows, “Water Bottle, Capacity, Height, Color.” Row 3 shows, “Stuffed Animal, Height, Color, Softness”

The materials reviewed for STEMscopes Math 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 the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with the Math Practices across the year and are identified for teachers within the Standards of Mathematical Practice within the Explore sections of the Scopes. MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 5: Represent Numbers to at Least 20, Explore, Explore 3–Generate a Set of Objects, Exit Ticket, students build experience with MP7 as they look for structure knowing that any teen number starts with a one in the tens place, and they can decompose a number in different ways. Students see two bowls with a number on each bowl. They draw counters to match the number and fill in the blanks to make true mathematical sentences. The numbers are 12 and 15. “Draw (a circle) to match the number of crackers in the bowl. Circle groups of ten. Decompose the total into ten and ones. Write the equation. 12, 10 and ___ equals ___. ___ is the same as 10 and ___. ___ =10+ ___, 15, 10 and ___ equals ___. ___ is the same as 10 and ___. ___ =10+ ___” 

• Scope 7: Two-Dimensional Shapes, Explore, Explore 2–Classifying and Identifying 2-D shapes, Exit Ticket, students build experience with MP7 as they identify different shapes using rules. Students circle the shapes that match the name in each set. Students see five shapes in each set. “Circle all the triangles. Circle all the rectangles. Circle the shapes with 0 straight sides. Circle the shapes with 3 or more vertices.”

• Scope 8: Three-Dimensional Solids, Explore, Explore 3–Classifying 3-D Solids, Exit Ticket, students build experience with MP7 as they classify and match three-dimensional solids with the correct name. Students see four pictures of a solid, cylinder, sphere, cube and cone and words in the second column to match with the correct picture. “Match the three-dimensional solid to the name of the solid. Cube, Cone, Cylinder, Sphere”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with the support of the teacher and independently throughout the Scopes. Examples include:

• Scope 2: Count Objects, Explore, Explore 3–Counting Forward and Backward within 10, Procedure and Facilitation Points, students build experience with MP8 as they count forward and backward to 10 from any given number to find the correct number. “Read the following scenario to the class: The Watsons are on summer vacation. The children decide to play the elevator game in the hotel. They enter the elevator at the lobby on the first floor and push a button. They count all the way up to that floor. Then, they ride the elevator back down to the lobby and count all the way down. Would you like to play the elevator game with them? 2. Instruct students to place the counter on 1 and then roll the die to decide what floor they will visit first. If students roll a 1, then they need to roll again. 3. Depending on what number is rolled, ask students to count forward from 1 to the number while moving their counter on the Elevator Buttons. 4. After students reach that floor, instruct them to count backward to 1 while moving their counter on the Elevator Buttons. 5. Give each student a Student Journal, and instruct them to fill in the buttons to record how they counted forward and backward. Students will color the starting number green and ending number red. 6. Repeat these steps several times to practice counting forward and backward from 1. Students do not record while practicing. 7. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-1 What number did you roll? b. DOK-2 How do you count to this number? c. DOK-2 How would you count from this number back to 1? c. DOK-3 Explain how you used the Elevator Buttons to help you. 7. Instruct students to place the counter on 3 and then roll the die to decide what floor they will visit next. If students roll a 1, 2, or 3, then they need to roll again. 8. Depending on what number is rolled, ask students to count forward from 3 to the number while moving their counter on the Elevator Buttons. 9. After students reach that floor, instruct them to count backward to 3 while moving their counter on the Elevator Buttons. 10. Instruct students to fill in the buttons to record how they counted forward and backward on their Student Journals. Students will color the starting number green and ending number red. 11. Repeat these steps several times to practice counting forward and backward from 3. Students do not record while practicing. 12. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-3 How was starting at 3 different from starting at 1? b. DOK-1 What number did you roll?  c. DOK-2 How do you count to this number? d. DOK-2 How would you count from this number back to 3?” 

• Scope 3: Represent Numbers to 10, Explore, Explore 5–Represent 10 with Objects and Pictures, Procedure and Facilitation Points, students build experience with MP8 as students use repeated calculations with addition problems to 10 and recognize that the next number in a counting sequence is one more. “1. Read the following scenario: Rhea and Charles are neighbors who love to collect bugs during the summer. They love to take old jars and their bug net out on their hunts. They like to search for only two types of bugs at a time for a total of 10 bugs in each jar. So far, they have collected enough bugs to fill 11 jars. Your job is to figure out the different combinations of bugs they found on their hunt. 2. Divide the class into groups of 2 or 3 and hand each student a copy of the Student Journal. 3. Have each small group begin at one of the 11 jars that have been set up at a station. Tell students to open the jar and dump out the bug images. Ask the following questions: b. DOK-1 How many bugs are in your jar? c. DOK-1 How many types of bugs are in your jar? d. DOK-1 How many bugs of each type are in your jar? 4. Project a Ten Frame Mat and model how to place bug cutouts on the ten frame. They will count how many of each type of bug and complete the equation at the bottom of the ten frame using a dry-erase marker. 5. Once they have created their concrete models, students can complete pictorial models on their Student Journals. Explain how to find the sections on the Student Journal that correspond to the jar numbers at their stations. Be sure that students understand that they will be coloring the circles either red or yellow based on which bug they are counting. For example, in Jar 1 there are 0 butterflies and 10 dragonflies. Therefore, they will color 0 circles red and 10 circles yellow. 6. Give students about 3–5 minutes at each station before rotating. Have them repeat steps until their Student Journals are complete. 7. Monitor and talk with students as needed to check for understanding by using the following guiding questions: a. DOK-3 How did you decide how many of each bug was in your jar? b. DOK-3 What are you noticing about the equations you are creating?”

• Scope 4: Compare Numbers to 10, Explore, Explore 1–Generate a Number that is One More or One Less, Exit Ticket, students build experience with MP8 as students look for a number one larger and one smaller than the number given. Students are given a number, four for the first question and six for the second question and must find the number one before and one after and complete the given table. “Draw and write the numbers to show one less and one more.  One Less, Number, One More, ___ is one more than ___. ___ is one less than ___. One Less, Number, One More, ___ is one more than ___. ___ is one less than ___.”