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

Materials develop conceptual understanding throughout the grade level. According to the Snappet Teacher Manual, 1. Deeper Learning with Snappet Math, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Each lesson embeds Conceptual Learning as the foundation and is designed to progress students along the learning path that begins with Student Discovery, transitions to Applying in Concrete pictorial representations, and then provides opportunities for Processing in Abstract representations.” According to the Kindergarten Teacher's Edition Volume 1, “Snappet’s Student Discovery Phase of the lesson design helps teachers present important math concepts using hands-on manipulatives, games, and classroom activities. Virtual manipulatives are also provided for guided practice and adaptive practice. The lesson design includes Concrete Pictorial Representations that utilize models and visuals during the lesson instruction. This approach helps teachers deliver high-quality instruction and builds a deeper understanding of math concepts for students.” Examples include:

• Unit 3: Numbers to 20, Lesson 3.3, Instruction & Guided Practice, Example 1f, students develop conceptual understanding as they compose numbers from 11-19 with a ten and some further ones. “How many in this group? (10) How many in this group? (8) How many strawberries? (18)” Teacher tip, “Ask: How many pieces are in the top two rows? [10] How can you decide how many? [Sample answer: Count the rest starting with 11.]” K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones; understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.)

• Unit 5: Understand Addition Within 10, Lesson 5.3, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding as they solve word problems by acting out situations. “Game: In the movies! “Have students act out addition word problems. Explain they will pretend to be in a movie. Say, ‘Scene one, two children are at the door. Three more children come to the door.’ Say, ‘Let's write the numbers in the scene on the board.’ Write $2+3$. Repeat with other students for several scenes using different addition sentences.” Teacher tip, “Ask: What did you just model? [We modeled an addition sentence.] Repeat the exercise using different numbers and different addition sentences. Next, write an equation, such as $2+4$, on the board and have the students create their own scene. Ask the students to describe the scene as the teacher did in Step 1. Ask: What did you create? [We created an addition word problem.]” K.OA.1 (Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g. claps), acting out situations, verbal explanations, expressions, or equations.)

• Unit 7: Addition and Subtraction Strategies, Lesson 7.5, Instruction and Guided Practice, Exercise 1i, students develop conceptual understanding as they demonstrate addition as putting together and adding to, and understand subtraction as taking apart and taking from. “Break apart 10.” A number bond is shown with 10 in the top box, 7 in one of the bottom boxes, and the second bottom box blank. Images of one net with 7 soccer balls and a second net with 3 soccer balls are shown. Students choose from 1 through 10 to drag into the blank number bond space. Teacher tip, “Inform students that they can drag the number of soccer balls from the left box into the left net. Ask: What do you do with the remaining soccer balls? [Sample answer: drag them to the right box and count].” K.OA.4 (For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.)

According to Snappet, “Student Discovery, Lessons begin with hands-on learning. Research supports that new concepts are best learned using manipulatives in real, informal situations. This learning serves as the basis for conceptual understanding. Apply in Concrete Actual situations are presented as a concrete representation using models and visuals. Students learn to establish the relationship between the actual situation and the concrete representation.” The teacher is given guidance to use with struggling students to complete the Independent Practice Items.  In the Snappet Teacher Manual, Section 3.2, states, “When the teacher has completed the instruction for the day, by demonstrating the opportunity to practice independently on their new skills. Each lesson includes approximately ten practice problems scaffolded for difficulty and are common for the whole class. Students are then presented with ten adaptive exercises that are customized to their skill levels. While students are working on their practice problems, the teacher can monitor the progress of their class in real-time. If the teacher notices a student or groups of students struggling with their exercises, they can intervene and provide support targeted to the needs of the students. At the same time, students that are “getting it” can move directly into adaptive practice and receive more challenging practice problems customized to their skill levels.” Examples include:

• Unit 2: Numbers to 10, Lesson 2.4, Independent Practice, Exercise 2h, students count and drag soccer balls into a box to match a given number. The teacher can support struggling students with teacher direction: “Guide the students to drag the correct number of objects to the box to represent 10. Encourage them to notice how the structure of the objects changes as they add additional objects.” K.CC.4 (Understand the relationship between numbers and quantities; connect counting to cardinality.)

• Unit 6: Understand Subtraction Within 10, Lesson 6.2, Independent Practice, Exercise 2j, students represent situations involving subtraction with expressions. Students are given a subtraction problem with four pictures. “Choose the picture that matches the math problem. $4-3$.” The teacher can support struggling students with teacher direction: “Ask: What problem with birds can be represented by $4-3$? [Sample answer: Four birds are on a branch. Three fly away.]” K.OA.1 (Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.)

• Unit 7: Addition and Subtraction Strategies, Lesson 7.7, Independent Practice, Exercise 2c, students add within 5 using strategies. “Drag the correct number. 1 + yellow circle = ___, 2 + yellow circle = ___, 3 + yellow circle = ___. Students choose from 2, 3, 4, 5. The teacher can support struggling students with teacher direction: “Remind students that they are adding 1 to each number.” K.CC.4c (Understand that each successive number name refers to a quantity that is one larger.)

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

According to the Snappet Teacher Manual, “In Snappet, students will build understanding by problem-solving using Models, Number Sentences, and Word Problems to develop mathematical fluency.” Process in Abstract: “Concrete situations are replaced with abstract mathematical symbols such as dashes, squares, or circles. Different schemas, models and step-by-step plans are often used for this. Learning takes place at a higher, more abstract level, preparing students for practicing procedural skills, developing fluency, and applying concepts flexibly to different situations.” The Instruction & Guided Practice problems provide ongoing practice of procedural skills within lessons. Examples include: 

• Unit 2: Numbers to 10, Lesson 2.9, Instruction & Guided Practice, Exercise 1h, students develop procedural skill and fluency as they count to answer how many. “How many?” 10 flags are shown. Teacher tip, “Ask: What are some ways you can count objects in a group? [Sample answer: I can use markers to match the objects. I can make smaller groups from the bigger group of objects.]” K.CC.5 (Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.)

• Unit 4: Numbers to 100, Lesson 4.2, Instruction & Guided Practice, Exercise 1p, students develop procedural skill and fluency as they count by 10s. “Count by 10s. Drag the numbers to the correct box. Think about a hundreds chart.” Teacher tip, “Inform students that there are four boxes and five numbers. One number will not be included. Ask: What is the pattern when counting by 10s? [Each number will have one more 10 than the number before.]” K.CC.1 (Count to 100 by ones and by tens.)

• Unit 7: Addition and Subtraction Strategies, Performance Task, Exercise 1a, Question 1, students develop procedural skills and fluency as they practice counting to add and find how many. “‘4 black cats. 3 white cats. How many total cats? ___ cats.” Teacher tip, “Students will likely count to find the answer. Encourage students to start at 4 black cats and count on the white cats. Ask: How can you find how many?” K.OA.5 (Fluently add and subtract within 5.)

In the Snappet Teacher Manual, Lesson Structure, “Automating and memorizing, Automating and memorizing is embedded in the learning goals of the Snappet program where this skill is important. The moment that Snappet recognizes the student has mastered the arithmetic knowledge and skill of the learning goal, the system automatically switches to tasks aimed at automation and memorization. This is accomplished by using exercises that students must completed in a given amount of time. Using this method, identifies whether a student knows the answer by automation or memorization or if they are still working out the calculations. If the student does not provide the correct answer in the given amount of time, then the program will allot more time for that exercise on the next attempt. The Snappet program will recognize when a student has sufficiently automated and memorized a goal and will adapt accordingly.” Students have opportunities to independently demonstrate procedural skills and fluency throughout the grade. Examples include:

• Unit 2: Numbers to 10, Lesson 2.8, Independent Practice, Exercise 2g, students demonstrate procedural skill and fluency as they count the number of cubes and write the number. “How many? Write the number.” K.CC.3 (Write numbers from 0 to 20. Represent a number of objects with a written number 0-20 (with 0 representing a count of no objects).)

• Unit 3: Numbers to 20, Lesson 3.8, Independent Practice, Exercise 2c, students demonstrate procedural skill and fluency as they compare two numbers. “18 ___ 20” Students selected from answer choices: “19, 11, 15, 13, 14, 10, 12, 16, 17.” K.CC.7 (Compare two numbers between 1 and 10 presented as written numerals.)

• Unit 7: Addition and Subtraction Strategies, Lesson 7.8, Independent Practice, Exercise 2e, students demonstrate procedural skill and fluency as they subtract. “Drag the correct answer. $3-2=$___. Answer choice: 1, 2, 3.” K.OA.5 (Fluently add and subtract within 5.)

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

Students have opportunities to engage with multiple routine and non-routine application problems with teacher support and independently. Snappet Teacher Manual, Performance Tasks, “Each grade-level course includes Performance Task Lessons that are designed to be a cumulative lesson encompassing multiple mathematical concepts. These lessons are designed as group projects or whole class discussion opportunities.” 

Examples of teacher-supported routine and non-routine applications of mathematics include:

• Unit 5: Understand Addition Within 10, Lesson 5.3, Instruction & Guided Practice, Exercise 1h, students add to solve a word problem in a routine application. “John has 3 grapes. He gets 2 more.” Teacher tip, “Remind students to drag the numbers and signs to the boxes. Ask: In what ways is a picture helpful? [Student answers should show understanding of using models to help solve problems]” K.OA.2 (Solve addition and subtraction word problems and add and subtract within 10, e.g., by using objects or drawings to represent the problem.)

• Unit 6: Understand Subtraction Within 10, Lesson 6.6, Instruction & Guided Practice, Exercise 1c, students solve take apart subtraction word problems in a non-routine application. “Mystery bag! Rules: Students play in pairs. Each pair has an opaque bag filled with 5 erases and 5 pencils. Students are not allowed to look in the bag. The teacher calls out a number from 1-10. Student pairs take turns pulling items out of the bag and counting the number of school supplies to equal the teacher’s number. The teacher asks, ‘How many school supplies all together?’ ‘How many erasers?’ ‘How many other school supplies left?’” Teacher tip, “Have students form pairs. Give each pair an opaque bag with five erasers and five pencils. Tell students not to look inside the bag. Call out a number from 1 to 10. Partners take turns pulling items out of the bag and counting until they reach the number you said. Ask: How many school supplies all together? How many erasers? How many other school supplies are left?” K.OA.2 (Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawing to represent the problem.)

• Unit 7: Addition and Subtraction Strategies, Performance Task, Exercise 1b, Problem 4, students compare two numbers in a routine application problem. “8 dogs in all. Some dogs are white. Some are brown. How many brown dogs? How many white dogs? Show two ways to represent your answers.” K.OA.2 (Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawing to represent the problem.)

• Unit 9: Geometry, Performance Task, Exercise 1c, Problem 9, students make a new shape from shapes given in a non-routine application. “Use the shapes. Put them together. Draw the new shape. The shapes provided are square, rectangle, triangle.” Teacher tip, “Students’ shapes will vary. Students may use some or all of the shapes shown, and may use more than one of each shape. Make sure that students draw a composite shape with no gaps between shapes or overlapping shapes.” K.G.6 (Compose simple shapes to form larger shapes.)

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples of independent demonstration of routine and non-routine applications of mathematics include:

• Unit 5: Understand Addition Within 10, Lesson 5.6, Independent Practice, Example 2b, students solve addition problems in a routine application. “$$1+2=3$$, $3+2=5$, $2+2=4$, $2+4=6$; 2 birds fly in from the west and 3 birds fly in from the east. How many birds are there all together?” K.OA.2 (Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.)

• Unit 6: Understand Addition Within 10, Lesson 6.5, Independent Practice, Exercise 2j, students use the “take apart” strategy to solve subtraction word problems in a non-routine application. “I have 4 towels. 2 are blue. How many other color towels?” K.OA.2 (Solve addition and subtraction word problems and add and subtract within 10, e.g., by using objects or drawings to represent the problem.)

• Unit 8: Measurement and Data, Lesson 8.3, Independent Practice, Exercise 2k, students use words to compare two objects in a non-routine application. Students see a cell phone and a candle. “The candle is… the phone. thinner than, as wide as, wider than.” K.MD.2 (Directly compare two objects with a measurable attribute in common, to see which object has “more of”/”less than” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.)

• Unit 9: Geometry, Lesson 9.1, Independent Practice, Exercise 2k, students use direction terms to solve problems in a routine application. “Drag the book to the shelf below the glue.” K.G.1 (Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.)

The materials reviewed for Snappet Math Kindergarten meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The materials address the aspects of rigor, however, not all are addressed equally. Heavy emphasis is placed on conceptual understanding, procedural skills, and fluency. All three aspects of rigor are present independently throughout the materials. Examples include:

• Unit 2: Numbers to 10, Lesson 2.2, Instruction & Guided Practice, Exercise 1n, students extend their conceptual understanding as they represent the numbers using their fingers and concrete objects. Students are given the number 3 and asked, “How many fingers?” Students select the correct answer from images that show fingers making the numbers three, four, five, and six. K.OA.4 (For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.)

• Unit 3: Numbers to 20, Lesson 3.5, Independent Practice, Exercise 2g, students develop procedural skill and fluency as they count the number of bees and write the number three times. “How many? Write the number 3 times.” 14 bees are shown. K.CC.3 (Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).)

• Unit 5: Understand Addition Within 10, Lesson 5.7, Instruction & Guided Practice, Exercise 1k, students apply their understanding of addition as they solve routine word problems. “Five birds and two birds fly onto a branch. How many birds altogether? $5+$___$$=$$___.” K.OA.2 (Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study. Heavy emphasis is placed on procedural skills, fluency, and teacher-guided conceptual understanding. Examples include:

• Unit 1: Numbers to 5, Lesson 1.1, Instruction & Guided Practice, Exercise 1g, students develop conceptual understanding alongside procedural skill and fluency as they keep track of objects counted in order to count groups up to 20 accurately. “Let’s count together!” Teacher tip, “Point from left to right, top to bottom. As you point, count aloud, and have students count with you.” K.CC.4a (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.)

• Unit 3: Numbers to 20, Lesson 3.7, Independent Practice, Exercise 2a, students develop conceptual understanding alongside procedural skill and fluency as they count a number of objects and select the corresponding number to match. “How many?” Seven buttons are arranged for students to count. Students select the total number of buttons with choices from 0 through 10. Teacher tip, “Ask: How can you keep track of which buttons you already counted so you only count each button once? [Sample answers: I can start at the button that is at the top right and work my way around the circle. I can put  my finger on the first button I count, and count the rest until I reach the first one again.]” K.CC.5 (Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.)

• Unit 5: Understand Addition Within 10, Lesson 5.1, Instruction & Guided Practice, Exercise 1c, students develop conceptual understanding alongside procedural skill and fluency as they add using objects or drawings. “Game: One More! Play together. One student shows a number of fingers. The other looks closest to how many fingers there are and puts one more than that number of makers on the table. Both students check to see if it’s right. After that, roles are switched. The teacher can continue the game by changing the rules to two more or three more.” Teacher tip, “Split students up into partners. Have them use their fingers or their counters. Ask: What did you and your partner just show? [We showed one more than (original number).] Repeat the exercise, having the students show 2 more, then 3 more. Ask the same question. Ask: What process did you and your partner just show? [Student answers should reflect understanding that the solution is getting bigger.]” K.OA.1 (Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.)

The materials reviewed for Snappet 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. 

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 1 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 teacher's support and independently throughout the units. Per Snappet Learning phases math, "MP1: Make sense of problems and persevere in solving them. Found in almost every math problem across the board. It means that students must understand the problem, figure out how to solve it, and work until it is finished. Standards encourage students to work with their current knowledge bank and apply the skills they already have while evaluating themselves in problem-solving. This standard is easily tested using problems with a tougher skill level than already mastered. While students work through more difficult problems, they focus on solving them instead of just getting to the correct answer." Examples include:

• Unit 1: Numbers to 5, Lesson 1.4, Math practices, Exercise 4c, “This slide allows students to apply MP 1 (Make sense of problems and persevere in solving them) as they think about the relationship between 0 and 1. Ask: Is there more than 1 bird in the tree or less than 1 bird in the tree? [less] Ask: What does that tell you about the number 0? [It is less than 1.]” The exercise states, “How does 0 compare to 1?”.

• Unit 2: Numbers to 10, Lesson 2.5, Math practices, Exercise 4a, “Exercise 4 gives students practice with MP 1 (Make sense of problems and persevere in solving them) as they work to describe what you are trying to determine. Ask: What is the question asking you to do? [Decide whether there are more hammers or saws.]” The exercise shows images of hammers and saws and states, “Which has more?”

• Unit 7: Addition and Subtraction Strategies, Lesson 7.9, Math practices, Exercise 4a, “Exercise 4 is designed to provide students with practice applying MP 1 (Make sense of problems and persevere in solving them) as students plan a solution pathway instead of jumping to a solution by selecting a method to add or subtract using mental math. Alert students that there will be more than one answer. Ask: What must you pay close attention to in this exercise? [Sample answer: whether the expressions are addition or subtraction] Call on students to share their ideas and responses to the question. [Sample answers: I know when zero is added to or subtracted from a number, the number does not change.; I can calculate them in my head.; I can use my fingers.]” The exercise states, “Which equal 2? How do you plan to solve this problem? $1+1$; $1+1$; $1+1$; $1+1$; $1+1$; $1+1$.”

MP 2 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 teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP2: Reason abstractly and quantitatively. When trying to problem solve, it is important that students understand there are multiple ways to break apart the problem in order to find the solution. Using symbols, pictures or other representations to describe the different sections of the problem will allow students to use context skills rather than standard algorithms.” Examples include:

• Unit 1: Numbers to 5, Lesson 1.6, Math practices, Exercise 4a, “The intent of Exercise 4 is to allow students to practice MP 2 (Reason abstractly and quantitatively) as they make sense of quantities and their relationships. Say: Draw a set of marbles that is equal to the set of marbles above. [Students should draw 4 marbles.]” The exercise shows four marbles and states, “Draw a set of marbles that is equal to the set of marbles above.”

• Unit 6: Understand Subtraction Within 10, Lesson 6.5, Math practices, Exercise 4c, “Exercise 4 give students a chance to practice MP 2 (Reason abstractly and quantitatively) as they make sense of quantities and their relationships to represent “take apart” subtraction problems.” “In this problem, students are provided a word problem and an image. Tell students to think about the previous problems they have done like this to help them. Call on a student to share their explanation. [Sample answer: I need to find how many lollipops are not yellow. In other words, how many of the 10 lollipops are either blue, red, or orange.] Ask: What expression did you write to represent this subtraction problem? [$$10-2$$]” The exercise states, “There are 10 lollipops. 2 are yellow. How many are other colors? Explain what you need to find. Write the missing numbers. - ”

• Unit 7: Addition and Subtraction Strategies, Lesson 7.6, Math practices, Exercise 4a, “Exercise 4 is designed to give students some experience applying MP 2 (Reason abstractly and quantitatively) as students make sense of quantities and their relationships when composing 10. Ask: What are two ways you can tell how many balls are already in the pool? [Sample answer: count them; look for the given addend] Ask: How many total balls should there be in the pool? [10] Call on a student to respond to the question. [Sample answer: Find what number, when added to 7, has a sum of 10.]” The exercise shows seven balls and states, “How many more balls make 10? $7+_=10$” Students drag 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10 to complete the equation. “What are you trying to find?”

The materials reviewed for Snappet 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 Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 3 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students construct viable arguments and critique the reasoning of others as they work with the support of the teacher and independently throughout the units. Per Snappet Learning phases math, “MP3: Construct viable arguments and critique the reasoning of others. This standard is aimed at creating a common mathematical language that can be used to discuss and explain math as well as support or object to others’ work. Math vocabulary is easily integrated into daily lesson plans for students to be able to communicate effectively. “Talk moves” are important in developing and building communication skills and can include such simple tasks as restating a fellow classmate’s reasoning or even supporting their own reason for agreeing or disagreeing. Prompting students to participate further in class mathematical discussions will help build student communication skills. Examples include:

• Unit 2: Numbers to 10, Lesson 2.6, Math practices, Exercise 4d, “It also allows students to engage in MP 3 (Construct viable arguments and critique the reasoning of others) as students defend their selection of numbers. Have students choose their numbers. Then have their partner check the solution. If the partner thinks the solution is incorrect, have them explain why.” The exercise states, “Write numbers that can go in the boxes. 10” “Less” is above the first box, “more” is above the “10.”

• Unit 6: Understand Subtraction Within 10, Lesson 6.4, Math practices, Exercise 4a, “Exercise 4 provides students practice with MP 3 (Construct viable arguments and critique the reasoning of others) as they ask themselves clarifying questions to solve “take from” word problems. Ask: What does the 7 represent? [the cookie that is eaten] Ask: What does this missing number represent? [the number of cookies the baker has left]” The exercise states, “The baker has 7 cookies. You eat 1 of them. Drag to show what you ate. Write the missing number. $7-1=$__ What does the missing number show?”

• Unit 9: Geometry, Lesson 9.4, Math practices, Exercise 4c, “The purpose of Exercise 4 is to give students a chance to use MP 3 (Construct viable arguments and critique the reasoning of others) as they use assumptions and definitions to name three-dimensional shapes.” “Ask: Which shape did you select? [middle or second] Call on a student to share their answer to the question. [Sample answer: Pyramids always have triangles on their sides.] Ask: What is the first shape, or the shape on the left? [cone] Ask: what is the third shape, or the shape on the right? [cube]” The exercise shows a cone, pyramid, and cube. “Which is a pyramid? What makes a pyramid?”

The materials reviewed for Snappet Math Kindergarten meet expectations for supporting the intentional development of MP6: Attend to precision and to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the Mathematical Practice Standards.

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 6 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 and the specialized language of mathematics as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP6: Attend to precision. Math, like other subjects, involves precision and exact answers. When speaking and problem-solving in math exactness and attention to detail are important because a misstep or inaccurate answer in math can be translated to affect greater problem-solving in the real world.” Examples include:

• Unit 1: Numbers to 5, Lesson 1.4, Math practices, Exercise 4b, students “practice MP 6 (Attend to precision) as they communicate precisely about the number 0.” “Ask: What are some other ways to say how many birds are in the tree? [Sample answer: There are no birds in the tree.]”  The exercise states, “What is another way to say how many birds are in the tree?”

• Unit 2: Numbers to 10, Lesson 2.7, Math practices, Exercise 4b, students “practice MP 6 (Attend to precision) as they make sure they accurately record the number of objects.” “Ask: How can you make sure you write the correct number? (I should count the circles twice to make sure I counted correctly.) Check that students correctly write the number 7.” The exercise states, “How many? Write the number.”

• Unit 5: Understand Addition Within 10, Lesson 5.2, Math practices, Lesson 4b, students “apply MP 6 (Attend to precision) as they learn the meaning of the plus sign and how to construct addition expressions to represent a model. Students will also recognize the order does not matter in addition (Commutative Property of Addition).” “Students will also recognize the order does not matter in addition (Commutative Property of Addition).” “This problem asks students to select the addition sentence that represents what is taking place in the picture. Ask: What is taking place in this picture? [Sample answer: A boy is placing a book on a table that already has 8 books on it.] Call on a student to share their response to the question and have them explain. [$$8+1$$; Sample answer: The same number of books (9) is represented if the addition sentence is $1+8$ or $8+1$.]” The exercise states, “$$1+8$$; $1+6$; $1+4$. Drag the addition sentence to the box that matches the picture. What is another addition sentence that matches?”

• Unit 8: Measurement and Data, Lesson 8.3, Math practices, Exercise 4b, students “apply MP 6 (Attend to precision) as they use clear mathematical language when they describe the lengths of objects.” “In this problem students compare the width of two sofas and select the one that is wider. Ask: Which sofa did you select? [purple sofa] Call on a student to share their explanation for how they knew which sofa was wider by eliciting a non-formal definition of width. [Sample answers: The purple sofa has less space on either of its sides than the red sofa. The purple sofa is longer from the left side to the right side.]” The exercise states, “Which is wider? Explain how you know.”

The materials reviewed for Snappet 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. 

The Mathematical Practice Standards are identified in the Course Overview/Unit Pacing Guide, Teacher Guide, Unit Overviews, and Lesson Overviews. Each lesson has a Math Practices tab that provides 3-5 structured exercises supporting the intentional development of each Math Practice throughout the year. 

MP 7 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 use structure as they work with the teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP7: Look for and use structure. When students can identify different strategies for problem-solving, they can use many different skills to determine the answer. Identifying similar patterns in mathematics can be used to solve problems that are out of their learning comfort zone. Repeated reasoning helps bring structure to more complex problems that might be able to be solved using multiple tools when the problem is broken apart into separate parts.” Examples include:

• Unit 1: Numbers to 5, Lesson 1.6, Math practices, Exercise 4b, “Students also apply MP 7 (Look for and make use of structure) as they think about the meanings of more and less. Say: Draw a set of books that has more than the number of green books above. Ask: What words could describe the number of books someone could draw? [Sample answers: more, greater, three, four]” The exercise states, “Draw a set of books that has more than the number of green books above.” Two books are shown.

• Unit 3: Numbers to 20, Lesson 3.9, Math practices, Exercise 4c, “The purpose of Exercise 4 is to focus on MP 7 (Look for and make use of structure) as students count to 20 starting from any number.” “Ask: What are the next three numbers? [14, 15, 16] Have a student come to the board and write these numerals, as some students may be struggling to write numerals. Call on a student to respond to the question. [Sample answer: I just counted 12, 13, 14, 15, 16 like I would if I was counting objects.]” The exercise states, “Write the next three numbers. 12, 13, ___. How did you know what numbers to write?”

• Unit 9: Geometry, Lesson 9.13, Math practices, Exercise 4a, “This exercise provides students with the opportunity to apply MP 7 (Look for and make use of structure) as they use smaller shapes to make larger given shapes. Provide students with a set of tangrams. When they are finished making the figure, have them compare their figure with a partner to be sure they look the same. Call on a student to share their answer to the question. [Sample answer: whale]” The exercise states, “Make this figure using tangrams. What do you think it is?”

MP 8 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 teacher's support and independently throughout the units. Per Snappet Learning phases math, “MP8: Look for and express regularity in repeated reasoning. In mathematics, it is easy to forget the big picture while working on the details of the problem. For students to understand how a problem can be applied to other problems, they should work on applying their mathematical reasoning to various situations and problems. If a student can solve one problem the way it was taught, it is important that they also can relay that problem-solving technique to other problems.” Examples include:

• Unit 3: Numbers to 20, Lesson 3.11, Math practices, Exercise 4a, “Exercise 4 provides students with the opportunity to apply MP 8 (Look for and express regularity in repeated reasoning) as they evaluate the reasonableness of their results when using patterns to order numbers. Ask: Did anyone compare the numerals above the boxes? Call on a student that raises their hand and ask: How did you decide? [Sample answer: The numbers both have one 10. I can look at the 1s. The number with more 1s is more.] Ask: Did anyone compare the picture models? Call on a student that raises their hand and ask: How did you decide? [Sample answer: I looked at the number of pencils in the bottom row.]” The exercise shows twelve pencils and fourteen pencils. “What number of pencils is more. Tap that box. How did you decide?”

• Unit 8: Measurement and Data, Lesson 8.5, Math practices, Exercise 4b, “Exercise 4 gives students the chance to apply MP 8 (Look for and express regularity in repeated reasoning) as they look for generalizations when using attributes of objects to sort them.” “Ask: How does this problem want you to sort the items? [Sample answer: into two categories, people and fish] Ask: How many people are there? [4] Ask: How many fish are there? [5]” Students sort people and fish into the boxes labeled “people” and “fish.”

• Unit 9: Geometry, Lesson 9.5, Math practices, Exercise 4a, “Exercise 4 provides students the chance to practice MP 8 (Look for and express regularity in repeated reasoning) as they use the attributes to describe and compare squares and rectangles. In this problem, students will analyze a picture frame and identify whether it is a square or a rectangle. Call on a student to say the name of the shape pictured. [rectangle] Call on a student to share their answer to the question. [Sample answer: I know it is a rectangle because all four of its sides are not the same size.]” The exercise states, “What shape is the picture frame? square; rectangle. “How can you tell?”

The materials reviewed for Snappet Math Kindergarten meet expectations for providing teacher guidance with useful annotations and suggestions for enacting the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development. The Documentation section of the materials provides comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• Snappet Teacher Manual, 3.1 Teacher Guide, “A Teacher Guide is available for every instructional lesson in Snappet, both digitally and on paper. The Teacher Guide contains the lesson overview, math content standards alignments, materials needed, vocabulary, EL/SEL strategies, common errors, and step-by-step support for teaching the lesson. Consistent design: The Teacher Guide, like the lesson itself, always has the same structure and is, therefore, easy and clear to follow. From the Teacher Guide, the teacher can access the learning path for every learning objective with constant visibility into the class's progress.  Full support: The learning phases explained in the teacher manual are also visible while teaching the lesson in the digital environment. This gives the teacher the support they need not only while planning their lessons but also while teaching their lessons. Easy to print: The teacher manual is easy to print by course or by lesson. Each downloadable and printable Teacher Guide is customized with the most up-to-date information about the progress and skill development for each student.”

• Instructional videos include 1-2 minute videos showing how to use the software, 5-minute videos of the classroom condensed to show each segment of the lesson, and full lesson videos. 

• Grade K-Pacing Guide provides the number of weeks to spend on each Unit as well as a Materials list for each Unit.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Unit/Lesson Overviews and Teacher Tips provide useful annotations. Examples include:

• Grade K-Unit Overviews, Unit 7 Overview: Addition and Subtraction Strategies, Understanding the Math, “Numbers can be decomposed and composed. Decomposing a number is breaking it into two or more parts that when added will equal the number. Composing a number is adding other numbers to make a greater or equal number. Decomposing and composing numbers represent addition and subtraction fact families that can be used to mentally compute both operations.”

• Unit 3: Numbers to 20, Lesson 3.10, Instruction & guided practice, Exercise 1c, Teacher Tip, “(SEL) Give a number card to each student. If you have more than 21 students, have the others distribute the rubber bands. Place a piece of string on the floor. Each student with a number should get an equal number of rubber bands for their card.Have students place their number and the rubber bands in order from least to greatest. Ask: Does number 5 have more or fewer rubber bands than number 17? Number 3? As we go up inandorder, do the numbers become greater or fewer? Ask the student with 20 to step away, saying 20 as they leave. Repeat with 19, 18 all the way back to 0. Ask: Are there more or fewer rubber bands as you down from 0?”

• Unit 6: Understand Subtraction Within 10, Lesson 6.1, Instruction & guided practice, Exercise 1e, Teacher Tip, “Ask: What do the Xs on the marbles mean? [They are the marbles I give away.] What does the marbles without the X mean? [That is the number of marbles that I left.]”

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

Snappet Math provides explanations for current grade-level concepts within the Understanding the Math and Learning Progressions components of the Unit Overviews. Prior, current, and future standards are connected within the Lesson Overview of each lesson. Additionally, each Lesson Overview includes Expanding Content Knowledge and Application Beyond Kindergarten, which provides explanations and examples of more complex grade-level concepts and concepts beyond the current course. Examples include:

• Unit Overviews, Unit 1: Numbers to 5, Learning Progression, “In this grade level, students will count, write, and compare numbers from 0 through 5. They will count numbers through 5. They will recognize and read the numerals from 0 through 5. They will learn the terms more, less, and equal that will allow them to compare the numbers through 5. They will write the numerals from 0 through 5. In future grade levels, students will count numbers through 120. They will read, write, and pronounce all the two-digit numbers and the numbers through 120. They will locate numbers on a number line (1.NBT.A.1). They will count to 100 by using and making groups (1.NBT.B.2.a). They will compare two-digit numbers by using a number line and by using place value. They will order numbers to 100 (1.NBT.B.3). They will model and write three-digit numbers (2.NBT.A.3) They will count forward and backward to 1,000 (2.NBT.A.2). They will compare and order numbers to 1,000 (2.NBT.A.4).”

• Unit 3: Numbers to 20, Lesson 3.2, Lesson Overview, “In prior lessons, students have counted numbers to 10 to find how many. (K.CC.B.4, K.CC.B.5); composed and decomposed a number from 11 to 19. (K.NBT.A.1). In this lesson, students will represent 11–15 as a ten and a number of ones. (K.NBT.A.1); decompose a number from 11 to 15 as a ten and a number of ones. (K.NBT.A.1). In future lessons, students will count to 20 by making groups. (K.CC.B.4); read and write numbers 11–20 (K.CC.A.3).”

• Unit Overviews, Unit 5: Understand Addition Within 10, Understanding the Math, “Addition is the joining of two or more groups. Addition can be thought of as a shortcut for counting. The lulus sign (+) is used to denote addition. Learning basic addition facts is critical for not only learning how to add greater numbers, but also subtract and eventually multiply. Two forms of addition word problems are “add to” and “put together.” “Add to” word problems involve adding some to an existing number. “Put together” word problems involve joining two groups.” 

• Unit 6: Understand Subtraction Within 10, Lesson 6.6, Lesson Overview, Expanding Content Knowledge and Application Beyond Kindergarten, “Introduction to Algebraic Concepts: “Take apart” subtraction word problems lay the groundwork for early algebraic thinking. As students advance, they will use these foundational skills to solve for unknowns in equations, like $x - 3 = 4$. Understanding how to break down numbers and problems in kindergarten paves the way for algebraic manipulation and problem-solving in later years.”

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

Correlation information is present throughout the grade level and can be found in the Pacing Guide, Unit Overviews, and each Lesson Overview. Explanations of the role of the specific grade-level mathematics in the context of the series can be found in each Lesson Overview under The Specific Role of the Standard in the Overall Series. Examples include:

• The Pacing Guide provides a table separated by unit and includes columns identifying previous, grade-level, and future skills. The skills are grouped by standard and are linked to identify lesson(s) standard alignment.  

• Unit Overviews identify the standards addressed in each unit and a lesson standard alignment. The Unit Overviews also include a learning progression that links current standards to previous and future standards for each unit.  

• Unit 3: Numbers to 20, Lesson 3.2, Lesson Overview, Mathematical Content Standards, “K.CC.A.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).”

• Unit 9: Geometry, Lesson 9.3, Lesson Overview, Specific Role of the Standard in the Overall Series, referring to K.G.A.2, “Bridging Concrete and Abstract Understanding: This standard acts as a bridge between the concrete understanding of shapes in the real world and abstract concepts they represent in mathematics. By learning to name shapes, students begin to see the abstract qualities of these figures, such as the number of sides or angles, which are fundamental concepts in higher-level geometry. This ability to shift from seeing shapes as mere physical objects to understanding their geometric properties is a critical skill in the mathematical development of students.”

The materials reviewed for Snappet Math Kindergarten meet expectations for providing explanations of the instructional approaches of the program and identification of research-based strategies. Instructional approaches of the program are described within the Teaching using the Snappet Method document. The four lesson components— Instruction and Guided Practice, Extend Learning using Math Practices, Independent and Adaptive Practice, and Small Group Instruction, are described. Examples include:

• Instruction and guided practice, “The lesson design for instruction follows the CRA approach to teaching: Concrete, Representational, Abstract. The exercises begin with Activate Prior Knowledge exercises which are designed to be used as real-time feedback opportunities during the introduction of the new lesson. This is followed by Student Discovery where manipulatives, games, or activities will be introduced to prepare students minds and bodies for new learning. These activities are followed by instruction slides that provide opportunities for students to think out loud, think pair share, co-craft questions, and talk about the new concept in a variety of ways. Instruction is followed by Guided Practice exercises where students can try it on their own while being supported by the teacher. The Guided Practice exercises also give the teacher the opportunity to identify if students are ready to begin practicing independently and to identify any common errors that might be occurring. Following Instruction and Guided Practice, teachers can go deeper into the mathematics by introducing the Math Practices exercises.”

• Extend Learning using Math Practices, “Teachers will utilize the exercises available in Math Practices to go deeper in the complexity of student learning. These exercises are designed to be non-routine, open-ended, and an extension of the discussions that occurred during the lesson. Often, these exercises will extend beyond the Student Discovery activities. It is recommended to group students into groups of 2 (K-2) or 3 (3-5) to encourage students to discuss their thinking and give evidence for their reasoning.”

• Independent and Adaptive Practice, “Students continue their learning of the concepts during independent practice. Independent Practice exercises are written at grade level and act as a “diagnostic assessment” to determine the appropriate level of Adaptive Practice. Adaptive practice offers 5 levels of difficulty that are defined by the quintile measures. Level 3 is considered grade-level proficient. Quality is the goal over quantity. It is recommended that only 1-3 sets (10-30 questions) of adaptive practice exercises be completed in any one practice session. Once students have reached their target goals and attained their desired level, they should either practice on a different concept or finish practice for the day.”

• Small Group Instruction, “Every lesson includes a Small Group Instruction intervention lesson for students that are struggling with the concept. This becomes evident when students are not able to progress during adaptive practice. Student initials will appear in yellow and will be identified as being “stuck” on their progress towards their target goals. It is recommended to provide reteaching to these students in a small group setting using the exercises in the small group instruction section. These exercises are scaffolded to provide support for struggling students. Once you have completed this lesson with students and they have demonstrated understanding using the guided practice exercises in the small group lesson, you can continue to monitor the students progress by having them continue to practice adaptively on the lesson.”

Research-based strategies within the program are cited and described in the Snappet Teacher Manual within Research-based strategies. Snappet Math states, “The Snappet Math curriculum integrates a series of rigorously research-based instructional approaches and strategies explicitly designed to facilitate effective K-5 mathematics education. Informed by eminent educational researchers and institutions, including the National Council of Teachers of Mathematics (NCTM) and the Institute of Education Sciences (IES), the key strategies are as follows:...” Examples include: 

• Concrete-Pictorial-Abstract (CPA) Approach, “This method involves the sequential use of concrete materials, pictorial representations, and abstract symbols to ensure thorough understanding (Bruner, 1966). Snappet's curriculum employs and explicitly references the CPA approach in the lesson phases ‘Apply in a concrete pictorial representation’ and ‘Apply in an abstract representation.’”

• Problem-Solving Instruction, “Snappet encourages students to engage with real-world problems, enhancing the relevance and application of mathematical concepts and procedures (Jonassen, 2000). Guidance is provided on various problem-solving strategies (Polya, 1945) in both instruction & guided practice and during independent practice.”

• Formative Assessment, Feedback, and Error Correction, “Regular assessments help to understand a student's learning progress, provide opportunities to give feedback, and adjust instruction (Black & Wiliam, 1998). Feedback is one of the most powerful influences on learning and achievement (Hattie, 2003), and correcting common errors has been identified as a factor that positively influences student achievement (Smith & Geller, 2004). Due to Snappet’s elaborate and immediate feedback system, every activity serves as a formative assessment. During instruction and guided practice, student responses appear on the Interactive Whiteboard in real-time for all students and the most common errors made by the students are summarized and highlighted. This feedback allows teachers to identify and correct common errors quickly, promoting student understanding and success. For every lesson and standard, both the teacher and students get continuous feedback on the current performance and progress. The immediate and actionable feedback, along with prompt error correction, is integral to promoting student achievement and progress in the Snappet Math curriculum.”

• Direct Instruction, “Direct instruction is a major factor in contributing to student achievement (Rosenshine, 2012). This involves clear, concise teaching where the teacher models what is to be learned and provides guided practice with immediate feedback. The Snappet Math curriculum incorporates this approach, with teachers provided with detailed lesson plans, strategies for explicitly teaching concepts, and resources for modeling mathematical thinking. The interactive nature of Snappet also allows for real-time guided practice and these exercises are explicitly referenced in every lesson with the guided practice icon ( ), aligning with the principles of direct instruction.”

The materials reviewed for Snappet Math Kindergarten meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The program provides a Material List, and specific lessons include a Materials heading needed to support instructional activities within the Lesson Overview. Examples include:

• Grade K-Material List, “The list below includes materials used in the Kindergarten Snappet Math course, excluding printed materials and templates. The quantities reflect the approximate amount of each material that is needed for one class. More detailed information about the materials needed for each lesson can be found in the Lesson Overview.” A table lists the Materials, Unit(s), and Approximate Quantity Needed, “Counters (markers); 1, 2, 3, 5, 6, 7; 20 per student.”

• Unit 1: Numbers to 5, Lesson 1.10, Lesson Overview, Materials, “Per pair: 1 set of number/ word cards with numbers 3–5, Per student: pencil and paper.”

• Unit 5: Understand Addition within 10, Lesson 5.1, Lesson Overview, Materials, “Per Pair: 15 markers/counters.”