Kindergarten - Gateway 2

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

The materials include problems and questions that develop conceptual understanding throughout the grade level. For example:

• In Module 1, Lesson 5, students develop conceptual understanding that the last number name mentioned tells the number of objects counted. Students sort pictures into three weather categories and count the number of pictures in each category. The teacher is prompted to ask the following questions, “T: Great Job! I wonder how many sunny pictures we found? Let’s count them. (Number each picture as it is counted.) How many sunny pictures? S: 5. T: What number did I write beside the last picture? S: 5.” (K.CC.4)
• In Module 4, Lesson 8, students engage in grade-level mathematics when using clay to decompose the numeral 7. The Application Problem states, “Ming has 5 raisins. Represent her raisins with the clay. Dan has 2 raisins. Represent his raisins, too. How many raisins are there in all? Put Ming’s raisins into a 5-group. Now, put Dan’s raisins in a row underneath Ming’s raisins like this. Do you still have 7 raisins? Hide the bottom 2 raisins. How many raisins do you see now? Talk about the raisins with your friend. Draw a number bond to represent Ming’s and Dan’s raisins.” (K.OA.3)
• In Module 4, Lesson 22, students develop conceptual understanding of decomposing numbers less than or equal to 10. Students reflect on prior strategies that decompose numbers with drawings. The teacher is prompted to ask the following questions, “T: Put your cubes away. We learned another way to show 6 this year with our 5-groups. Does anyone remember how we could draw 6 the 5-group way? T: Let’s roll the die to see how many we should take away from our 6. How many? S: 3. T: I will cross off 3 to show the ones we are taking away. How many are left? S: 3. T: What would my number sentence be? S: 6 – 3 = 3. T: How could we make a number bond about our picture and then show that we are taking part away?” (K.OA.3)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. For example:

• In Module 1, Lesson 7, students independently demonstrate conceptual understanding of the relationship between numbers and quantities. Students match linear configurations of objects up to 5 with numerals on cards by coloring with the same color. The Problem Set states, “Count and color.”
• In Module 5, Lesson 6, students independently demonstrate conceptual understanding of place value. Students model with number cards to show their knowledge of place value. The teacher is prompted to ask the following questions, “T: Watch this magic. Here is my 10. Here is my 8. I push them together, and I have ten 8. This is how we write ten 8. T: Talk to your partner. What happened to the 0 of the 10 ones? S: It went under the 8, it disappeared. It isn’t there anymore, it is hiding. T: Yes! It is hiding. I’m going to write the number without the cards. (Write 18.) It is like there is a 0 hiding under this 8. T: I want each of you to write this number on your personal white board. When I say to show me your board, show me. T: Show me! T: Here is a bag with a set of these cards for you. Partner A, open the bag, and put all the numbers on your work mat. With your partner, put them in order from 1 to 10. T: Partner B, show me ten 8 with your cards. Be sure to hide the zero!” (K.NBT)

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

The instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In Module 3, Lesson 10, students develop procedural skill and fluency when using cotton balls and white boards to touch and count. The teacher is prompted to ask the following questions, “T: Put your cotton balls in a line across the top of your personal white board like this. T: Now, touch and count them. S: 1, 2, 3, 4, 5. T: You might start at the top of your board and draw tally marks to match your cotton balls like this. (Demonstrate moving from left to right while drawing one tally to match one cotton ball.)
• In Module 5, Lesson 1, students develop procedural skill and fluency when decomposing the number 3. The teacher is prompted to ask the following questions, “The use of wait time encourages students to subitize rather than having them touch and count. T: (Before beginning the activity, place 2 bananas and 1 bear on a desk or table behind the screen.) Peek-a-Boo! (Raise and lower the screen.) Peek-a-Boo! (Again.) T: Wait for the signal. How many things did you see in all? (Signal when ready.) S: 3. T: Wait for the signal. How many bananas? (Signal when ready.) S: 2 bananas. T: Wait for the signal. How many bears? (Signal when ready.) S: 1 bear. T: Very good. Let’s play again! Continue with other decompositions of 3 (e.g., 1 banana and 2 bears, 3 bears and 0 bananas). As students progress, determine if they can remember the number for longer periods of time. Encourage them to show the number on their fingers the Math Way instead of saying it.”

The instructional materials provide opportunities to demonstrate procedural skill and fluency independently throughout the grade level. For example:

• In Module 4, Lesson 6, students independently demonstrate procedural skill and fluency of addition within 5 by creating a story that matches a given number bond. The Problem Set states, “Tell a story that matches the number bond. Draw pictures that match your story. 3 + 1 = 4”
• In Module 4, Lesson 4, students independently demonstrate procedural skill and fluency of subtraction within 5 by creating a story that matches a given picture. The Problem Set states, “Look at the picture. Tell your neighbor a story about the dogs standing and sitting. Draw a number bond, and write numbers that match your story.”

Students build fluency for adding and subtracting to 5 in 5-10 minute fluency practice activities before lessons. These fluency practices are provided in three of the six modules. For example:

• In Module 4, Lesson 1, students use math cards with partners to create addition facts to 5.
• In Module 5, Lesson 17, students create combinations of addends to 5 with partners.
• In Module 6, Lesson 5, students color worksheets coded by sums and differences through 5, as well as practice facts orally with the teacher.

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

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. For example:

• In Module 3, Lesson 3, students listen to directions and apply their knowledge of lengths to draw a picture. “Draw a monkey with a very long tail. Draw a monkey with a very short tail. Now, draw a yummy banana for the monkeys to share. Is the banana longer than or shorter than the tail of the first monkey? Is it longer than or shorter than the tail of the second monkey? Tell your partner what you notice.”
• In Module 6, Lesson 6, students engage in grade-level mathematics when applying an understanding of shapes to the world around them. The Application Problem states, “You are going to be a detective today! First, look around the classroom to see if you can find things made of more than one shape, like we did yesterday. Second, draw one thing on your whiteboard. Third, use your marker to draw the shapes inside. (If necessary, give hints about items such as tiles, bricks, window panes, and so on. Encourage students to look for and highlight the shapes within shapes on their boards. T: Turn and talk to your partner about the hidden shapes that you found!” (K.G.2)

The instructional materials provide opportunities for students to demonstrate independently the use of mathematics flexibly in a variety of contexts. For example:

• In Module 4, Lesson 32, students independently demonstrate the use of mathematics by solving both addends unknown word problems. The Problem Set states, “Listen to the word problem. Fill in the number sentence. Cecilia has 9 bows. Some have polka dots, and some have stripes. How many polka dot and how many striped bows do you think Cecilia has? 9 = ___ + ___” (K.OA.3)
• In Module 4, Lesson 34, students independently demonstrate the use of mathematics by solving subtraction word problems. The Problem Set states, “Fill in the number sentences and number bonds. There are 9 babies playing. 2 crawl away. How many babies are left? 9 - 2 =___.” (K.OA.2)
• In Module 5, Lesson 22, students independently demonstrate the use of mathematics by comparing numbers less than 10. The Application Problem states, “Lisa has 5 pennies in her hand and 2 in her pocket. Matt has 6 pennies in his hand and 2 in his pocket. Who has fewer pennies—Lisa or Matt? How do you know?” (K.CC.C)

The instructional materials for Eureka Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The lessons include components such as Fluency Practice, Concept Development, and Application Problems. Conceptual understanding is addressed in Concept Development. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students engage in practicing procedures and fact fluency while modeling and solving these concepts. Fluency is also addressed as an independent component within most lessons. Lessons may contain an Application Problem which connects previous learning to what students are learning for the day. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 1, Lesson 25, students practice fluency when counting and drawing dots on a ten-frame from a given number. Fluency Practice - Five Shortcut states, “T: I’m going to say a number, and I want you to draw that many dots. Remember to start at the top, filling in the rows from left to right, the same way we see on our 5-group cards!” (K.CC.4)
• In Module 3, Lesson 21, students develop conceptual understanding when comparing sets of shapes using vocabulary such as more, less, and fewer. The Problem Set Question states, “Color the shapes. Count how many of each shape is in the shape robot. Write the number next to the shape. Look at the robot. Color the shape that has more. Are there more rectangles or circles?” (K.MD.3)
• In Module 4, Lesson 16, students engage in the application of mathematics by solving a word problem involving addition. The Application Problem states, “Note: A set of 10 linking cubes for each student deliberately gives students more cubes than necessary to model the story so that they can select those needed from the larger set. 3 airplanes were flying in the air. Use your cubes to show the planes. 3 more airplanes came to join the flying fun. Show the airplanes with your cubes. Now, with your cubes, show how many airplanes were flying in the air. Talk to your partner about what the number sentence would look like. Note: This problem sets the stage for solving add to with result unknown word problems in today’s lesson.” (K.OA.2)

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

• In Module 3, Lesson 2, students develop conceptual understanding of non-standard measurement while applying it to objects within the classroom. Students compare the length or height of items in the classroom to the length of a piece of string. Students find at least five things that are longer than their string and at least five things that are shorter than their string, and then record them on charts. (K.MD.2)
• In Module 5, Lesson 4, students develop conceptual knowledge of tens (The Ten Way) and fluently count to 10. The Application Problem states, “At recess, 17 students were playing. 10 students played handball while 7 students played tetherball. Draw to show the 17 students as 10 students playing handball and 7 students playing tetherball.” (K.CC.4)

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

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. For example:

• In Module 2, Lesson 2, the students create a triangle on their geoboard and prove to another student that what they created is a triangle.
• In Module 5, Lesson 8, the Application problem states, “Peter drew a number bond of 13 as 10 and 3. Bill drew a number bond, too, but he switched around the 10 and 3. Show both Bill’s and Peter’s number bonds. Draw a picture of thirteen things as 10 ones and 3 ones. Explain your thinking to your partner about what you notice about the two number bonds.”
• In Module 5, Lesson 14, in the Debrief students prove that the number of objects is the same regardless of the configuration. They discuss with partners and share out which way they think is the most convincing.

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

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others, frequently throughout the program. The teacher materials consistently provide teachers with question prompts for student discussion and possible student responses to support that discussion. For example:

• In Module 2, Lesson 2, teachers are prompted to engage students in constructing an argument by having students create a triangle on geoboards and stating their reasoning for why it is a triangle. “T: Now, create your own triangle on your geoboard, and then show your partner. Be sure to tell how you know it is a triangle! (Allow time for sharing and discussion.)”
• In Module 3, Lesson 11, teachers are prompted to engage students in constructing and analyzing an argument by having students use a pan balance to compare the weights of equal pieces of clay. The students discuss whether the two sides of their scales are the same size or the same weight and how they know. “Talk to your partner. What do you think will happen when you put the two smaller balls back on the scale? T: Okay. Put the balls back on the scale. T: How are they the same? Are they the same number? The same size? “Let’s try another experiment. Partner A, take your ball and quickly make it into three smaller balls. T: Talk to your partner. What will happen this time when Partner A puts his or her part back on the scale?”
• In Module 5, Lesson 20, teachers are prompted to engage students in constructing an argument by having students discuss the two parts that can make up a number in the teens and the best method for displaying how the two parts make a whole number. “Talk to your partner. When we solved our story problem today we had two parts. What is another way you already know to show a number in two parts?”

The instructional materials reviewed for Eureka Kindergarten meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently-introduced mathematical terms that will be used throughout the module. A compiled list of the terms along with their definitions is found in the Terminology tab at the beginning of each module. Each mathematical term that is introduced has an explanation, and some terms are supported with an example.

The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols. For example:

• In Module 2, Lesson 5, the Notes on Multiple Means of Representation states, “Support students who struggle by partnering key words such as next to, below, above, and below with modeling the actions for them.”
• In Module 2, Lesson 7, the Notes on Multiple Means of Action and Expression states, “As the vocabulary terms cone, face, cube, corners, and edges come up in the lesson, the teacher can use gestures like touching her face and then the face of the solid while saying the word face in order to enrich English language learners’ experience and make it easier for them to access the content of the lesson.”
• In Module 3, Lesson 27, the Notes on Multiple Means of Engagement states, “Ask students to verbalize who has more as they take turns every time they play the game. For example, “I have 8 cubes, and you have 3 cubes; 8 is more than 3.” Or, “I have 4 pennies, and you have 7 pennies; 4 is less than 7.” English language learners benefit from the practice and can be easily observed as to which students might be confused between more and less.”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. For example:

• In Module 4, Lesson 20, the materials use precise terminology of minus and support students in using the term when showing an example. The Concept Development states, “Put 3 linking cubes in your hand, and take them away. How many are left on the table? Yes, 5 take away 3 is 2. There is a special Math Way to write what we just did. We had 5 cubes.
• I will write the number 5 to show all of the cubes together. (Demonstrate.) There is a special sign we can use when we want to show that we are removing some cubes. It looks like this. (Write the minus sign.) How many did we take away?”
• In Module 3, Lesson 8, the mathematical term weight is in bold writing within a question listed in the Student Debrief section. These questions guide teachers in leading a class discussion. “How did you decide which objects on the Problem Set would be heavier? Could you make a prediction even though you couldn’t feel their weight?”
• In Module 5, Lesson 3, the materials use accurate terminology when students learn the concept of teen numbers. The Problem Set states, “Early finishers: Write your own teen number in the box. Draw a picture to match your number.”