Kindergarten - Gateway 2

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

At the beginning of each chapter, Laurie’s Notes identify a whole group lesson with a manipulative or specific strategy to develop conceptual understanding. The Explore and Grow “give students a hands-on approach to develop conceptual understanding.”

Cluster K.CC addresses representing addition and subtraction with objects, fingers, mental images, etc. Many lessons in Chapters 1-7 explore ways to develop and independently demonstrate conceptual understanding of adding and subtracting using different representations. Examples from these chapters include:

• Chapter 1, Lesson 6, Explore and Grow, the teacher reads the book “At The Pond.” After reading, students look at the page in their math book with a log and a pond. Students use counters to show how many turtles are in the story, then write the number of how many turtles are in the story (5). This develops understanding the relationship between numbers and quantities; connect counting to cardinality (K.CC.4) and writing numbers from 0 to 20 (K.CC.3).
• Chapter 2, Lesson 1, Think and Grow, students “Draw paint spots so that the numbers of paint tubes and paint spots are not equal. Draw lines between the objects in each group to show that you are correct.” This develops understanding of 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 (K.CC.6).
• Chapter 3, Lesson 11, Apply and Grow, students “Place counters in the ten frame as you count forward to 10. Trace and write the missing numbers.” This develops the understanding when counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object (K.CC.4a).
• Chapter 4, Lesson 3, Think and Grow, students “Compare the numbers. Draw a line through the number that is less than the other number. Draw to show how you know.” For example, students compare 8 and 4 by drawing circles to represent each number, and draw lines to match them together to see which has more. This develops understanding of identifying 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 (K.CC.6).

Chapters 5, 6, and 7 address K.OA.A, understand addition and subtraction as putting together and adding to, and understand subtraction as taking apart and taking from. The materials provide opportunities for students to develop conceptual understanding and independently demonstrate their understanding. For example:

• Chapter 5, Lesson 3, Think and Grow, students solve, “There are 6 tomatoes at a farm stand. Newton and Descartes buy all of them. Newton buys fewer tomatoes than Descartes.” Students have the opportunity to show a variety of answers such as “5 + 1 = 6; 2 + 4 = 6”, etc. K.OA.3. (Decompose numbers less than or equal to 10 into pairs in more than one way.)
• Chapter 5, Lesson 5, Think and Grow, students solve, “Put the marbles into 2 groups. Circle the groups. Then complete the number bond to match your picture.” Students circle “4 + 4 = 8” or “5 + 3 = 8”, etc. K.OA.3. (Decompose numbers less than or equal to 10 into pairs in more than one way.)
• Chapter 6, Lesson 1, Explore and Grow, directions, “Use counters to act out the story. There are 3 students on the bus. Write the number. 2 more students get on the bus. Write the number. Tell how many students are on the bus now.” This supports conceptual understanding as it aligns with K.OA.1, as students show the mathematics using counters.
• Chapter 6, Laurie’s Notes (2nd bullet under Additional Support), “Ask students to use their fingers on both hands to represent the whole. The fingers on each hand are the partner numbers for the whole.” This supports conceptual understanding of K.OA.1, because students represent addition with fingers.
• Chapter 7, Laurie’s Notes (Teaching Notes, Supporting Learners), “Each problem can be modeled using a five frame and counters, using fingers, or having students act it out.” This supports conceptual understanding of K.OA.1, because students subtract using objects, fingers, and acting out.
• Chapter 7 Lesson 2, Think and Grow, students solve “Some bats in a cave fly away. Cross out the bats that fly away….” Students cross out 8 of 9 bats. This supports conceptual understanding of K.OA.1, because students subtract using objects.

Within the Apply and Grow and Homework and Practice sections, students have opportunities to independently demonstrate conceptual understanding. For example:

• Chapter 3, Lesson 6, Apply and Grow, Practice, students “Count the objects. Say the number. Write the number” for objects with 8 connecting cubes, 8 worms, 6 flowers, and 8 ants.
• Chapter 6, Lesson 3, Apply and Grow, Practice, students “Circle the groups to put them together. Then write an addition sentence to tell how many objects there are in all.” Problem 2 shows 3 red balls and 5 yellow cubes.
• Chapter 9, Lesson 6, Apply and Grow, Practice, students “Count the objects in each group. Write each number.” Problem 2, “Circle the number that is greater than the other number.” Students have a group of 12 yellow balls and 11 red balls.
• Chapter 12, Lesson 2, Practice, Problem 1, students “Look at the solid shapes on the left that stack. Circle the other solid shapes that stack.” Students are given a cylinder, cube, and ball.

The instructional materials for Big Ideas Math: Modeling Real Life 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, especially where called for by the Standards, and present opportunities for students to independently demonstrate procedural skill and fluency. 

In Explore and Grow, there is “teacher-guided instruction with opportunities for student involvement to ensure all levels of learners attain procedural fluency.” Apply and Grow Practice presents some opportunities for independent practice to allow students to further their procedural skills.

The instructional materials present opportunities for students to develop procedural skill and fluency, especially for K.OA.5 (Fluently add and subtract within 5). For example:

• Chapter 6, Lesson 6, Apply and Grow, “Complete the addition sentence. Tell how you found your answer. Complete the addition sentences. Tell what you notice.” In Laurie’s Notes, Preparing to Teach, “This is the first section in Kindergarten where the focus is on fluency.” 
• Chapter 7, Lesson 5, Apply and Grow Practice, subtraction within 5. “Complete the subtraction sentence. Tell how you found your answer. Circle all of the subtraction problems that equal 3.”
• Chapter 7, Lesson 6, Think and Grow, students write an addition or subtraction equation based on a picture. “Tell whether the pictures show addition or subtraction. Then write the addition or subtraction sentences to show the related facts.” 
• Chapter 7, Lesson 7, Explore and Grow, “You have 3 yellow linking cubes and 2 green linking cubes. How many linking cubes do you have in all? Write an addition sentence to match the picture. You have 5 linking cubes. You give your friend 2 linking cubes. How many linking cubes do you have left? Write a subtraction sentence to match the picture.” 
• Online Skills Trainer provided one section that addresses K.OA.5.
• One Center game, “Numberland”, has addition and subtraction facts to 5 (K.OA.5).

Students independently demonstrate procedural skill and fluency in some lessons, while most online Center Games present opportunities for students to practice procedural skills and fluency. These include:

• Chapter 6, Lesson 5, Apply and Grow, student solve “9 + 1 = 10.” During Practice, students solve problems such as Problem 3, “___ + ___ = 4, and Problem 4, ” ___ + ___ = 5” where students practice adding with 0 and 1. 
• Chapter 7, Lesson 5, Apply and Grow, students solve subtraction problems within 10 (K.OA. 2) without using visual aids.

The instructional materials for Big Ideas Math: Modeling Real Life Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The instructional materials present opportunities for students to engage in each aspect of rigor, as well as multiple aspects of rigor, in each lesson. For example:

• Chapter 1, Lesson 7, students develop conceptual understanding of zero. They guess the numbers on dot card models displaying 1-5. After a few rounds, the 0 card is displayed. Students are asked, “What can we call it when we have nothing?” Students then discuss examples of the quantity zero developing their understanding of 0. During the Explore and Grow, students hear a “Story Time” about an empty box. In the Think and Grow, students compare real-world objects that consist of “two tree limbs”. One tree limb has one leaf, the other has zero leaves. Students practice writing 0. In Apply and Grow Practice, students count fireflies in a jar, then write the numeral next to the jar. 
• Chapter 6, Lesson 6, Apply and Grow, “Practice Addition”, focuses on procedural skills and fluency of adding within 5. 
• Chapter 9, Lesson 6, Think and Grow: Modeling Real Life, students solve routine application problems to find “who has more.” For example, “You have 14 balls, Your friend has a number of balls that is 1 more than 12. Draw the balls. Write the numbers. Circle the number that is greater than the other number.”

Students have opportunities to engage in multiple aspects of rigor. For example:

• Chapter 4, Lesson 1, Think and Grow, “Is the number of toothbrushes equal to the number of toothpaste tubes? Circle the thumbs up or thumbs down.”
• Chapter 6, Lesson 4, Practice, Problem 4, “There are 6 paint spots on a paint tray. Some are purple and some are green. Color the paint spots to show partner numbers that make 6. Then color the paint spots to show another way to make 6. Write the addition sentence to match your pictures.”
• Chapter 9, Lesson 3, Think and Grow: Modeling Real Life, “You have 20 coins in your piggy bank. You drop and break your piggy bank. Did you find all your coins? Circle the thumbs up for yes or the thumbs down for no.”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Kindergarten meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level. 

The Standards for Mathematical Practice (MP) are identified in the digital Teacher's Edition on page vi. The guidance for teachers includes the title of the MP, how each MP helps students, where in the materials the MP can be found, and how it correlated to the student materials using capitalized terms. For example, MP2 states, "Reason abstractly and quantitatively.

• "Visual problem-solving models help students create a coherent representation of the problem.
• Explore and Grows allow students to investigate concepts to understand the REASONING behind the rules.
• Exercises encourage students to apply NUMBER SENSE and explain and justify their REASONING."

The MPs are explicitly identified in Laurie’s Notes in each lesson, and are connected to grade-level problems within the lesson. For example:

• In Chapter 1, Lesson 8, Think and Grow: Modeling Real Life, Laurie’s Notes, identifies MP3, Construct Viable Arguments and Critique the Reasoning of Others. For example, “Encourage students to count and then check with partners to see whether they agree about the number of each color star. Students should feel comfortable talking with partners and then self-correcting when they recognize a mistake.” 
• In Chapter 4, Lesson 3, Think and Grow, Laurie’s Notes, identifies MP8, Look for and Express Regularity in Repeated Reasoning as students compare numbers by drawing counters and pairing them for each number. Guidance for the teacher states “How did you compare 10 and 7?’ I know 10 comes after 7, so 10 is greater. ‘How did you compare 1 and 6?’ I know 6 comes after 1, so 6 is greater. If 6 is greater, then 1 is less.” 
• Chapter 5, Lesson 1, Laurie’s Notes, students show different ways to make a number. “Is there more than one way to make the number 5? How do you know?” (MP7).

The MPs are identified in the digital Student Dashboard under Student Resources, Standards for Mathematical Practice. This link takes you to the same information found in the Teacher Edition.

There are some instances where the MPs are over or under-identified. For example:

• MP2 is identified in most lessons.
• MP5 is under-identified in the materials. 
• MP8 is under-identified as students begin to generalize addition and subtraction as ‘add to’ and ‘take from’.

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Kindergarten meet expectations that materials use accurate mathematical terminology. In the Instructional Resources Grade K, there are vocabulary cards for each chapter.

At the beginning of each chapter, there are vocabulary cards in the student materials that have the word on one side, and a model of the word on the other. In the teaching materials, there are instructions for use. For example, on page 2, the materials state, “Offer envelopes or small bags for students to store their vocabulary cards. Students can reference these cards as needed throughout the chapter.” Also on this page, the materials provide an activity to do with the students, “Have the students repeat as you say and show a vocabulary word and model the action provided below. To demonstrate the count card, hold up one hand and act like you are counting on your fingers to five. The number card can be acted out by pointing to a sign in the room that has a number on it. Stomp or clap to act out the numbers 1 through 5. Use a closed hand for 0. For the five frame card, draw and then point to a five frame on the board. Have students get in line to act out the order card.”

The following are examples where the materials use precise vocabulary with the students:

• Chapter 2, Lesson 4, Laurie’s Notes, Teaching Notes, “How do you know the groups are not equal?” This problem gives students an opportunity to use language and definitions accurately, but does not give students the opportunity to use mathematical symbols as students are not yet using symbols with the mathematics they are learning at this point in Kindergarten. 
• Chapter 2, the beginning of chapter materials state, “Students lay their vocabulary cards in front of them with the picture side facing up. Show students a vocabulary card, say the word, and describe the picture definition. Students find the corresponding card. Students tell their partners how they know the picture is a match to the vocabulary word.”  
• Chapter 3, Lesson 5, Attend to Precision, states, “Have students use precise language to explain how they count. Ask, ‘How can you find out whether there are eight objects in a group?’ ‘How can you use counters to help you count the objects?’”
• Chapter 4, Lesson 1, Laurie’s Notes, Think and Grow, “How do you know the groups are equal?” Listen for an understanding that the quantities are the same. 
• Chapter 7, the beginning of chapter materials start with a “Vocabulary Review”,  reviewing the words “parts, whole, and number bond”. Included is an activity for the new vocabulary words, “Say a vocabulary word and then pass the echo ball to a student. The student echoes, or repeats the word, and then passes the echo ball back to you. Students need to listen carefully to repeat the word correctly. Consider challenging the students by having them define the words after they echo them.” 
• Chapter 7, Lesson 9, Laurie’s Notes, Think and Grow, MP6 is identified and the materials state, “When students say ‘4 take away 4 is none,’ acknowledge their correct thinking and say, ‘The number for none is 0. So 4 take away 4 is 0.’”
• Chapter 8, Lesson 3, Laurie’s Notes, Think and Grow, Attend to Precision, “Students may write 5 = 10 + 15. They did not pay attention to the symbols already written, or are confused about the equal sign and addition symbol. Ask the student(s) to explain what their addition sentence means.”
• Chapter 9,“Vocabulary Review”. Students review the words “equal” and “greater than”. Then, “Have students lay out their vocabulary cards in front of them with the picture side facing up. Say the word on the vocabulary card, show the word, and describe the picture definition to students. Have students find the corresponding card. Have students take turns showing the card and telling a partner about the word and its picture definition.”
• Chapter 9, end of chapter materials, has a game called “Number Boss”. MP6 is identified and the materials state, “As you played the game, how did you know if the numbers were equal or not equal? How did you know who had the greater number?”