Kindergarten - Gateway 2

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include: 

• In Unit 2, Lesson 5, Introduction, students engage with K.G.6, compose simple shapes to form larger shapes, as they complete puzzles using geometric shapes. The materials state, “Step 1 says I’m going to pick a puzzle. Step 2 says I need to Decide what shape might fit. T & T: How can I make sure that happens? Strategy 1: Keep trying shapes until one fits the space. Strategy 2: Look at the space you are trying to fill. What shape might fit because of its attributes? Then find the shape that has the same attributes. Remember! You can flip and turn the pattern blocks. What shape do you think would fit? Why did you pick that shape?”
• In Unit 3, Lesson 2, Introduction and Workshop, students engage with K.CC.4, demonstrate understanding of the relationship between numbers and quantities, as they play “Counting Bags/Jars.” Students count the number of pattern blocks in the bag and then show the same amount using cubes. During the Workshop the teacher asks students, “How do you know this is the same amount / how are you showing the same amount?”
• In Unit 5, Lesson 6, Introduction, students engage with K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, as they play a game called Compare. During the Introduction, two cards are drawn (example, 7 and 9) and students are asked to pictorially show which is more or less by drawing circles on their whiteboards. The teacher asks, “How do you know from the picture?” A sample student response might be, “I know because in the picture you can see that there are extra circles in the row of 9 and the row of 7 is missing some.” 
• In Unit 6, Lesson 3, Introduction, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as the students complete a dice game while the teacher checks for understanding with questions leading the students to describe their thinking. Workshop states, “Step 1: Roll 2 cubes; record their amounts. (“roll” a 4 and a 2) Show first cube: How many? 4 (record) Show second cube: How many? 4 (reccord) Show second cube: How many? (give time to count as needed) 2 (record) Lap 2: Conceptual: Which strategies are kids using? What misconceptions are arising? Check for Understanding: How did you solve? Why does that work? How does your equation match what you did?”
• In Unit 8, Lesson 2, Workshop, students engage in K.NBT.1, compose and decompose numbers 11 to 19 into ten ones and some further ones, as students bundle objects into a group of ten and count on to determine the number of objects in a bag. The teacher is given suggestions for guiding the students to develop the concept of teen numbers. The materials state, “What did you notice about the group of ten ones, loose ones and the way we write the numbers?”

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

• In Unit 6, Lesson 10, Exit Slip, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situation, verbal explanations, expression, or equations, as they find the difference between two numbers using manipulatives and represent with an equation. Students are directed, “Use your counters and tens frames to find the difference. Fill in the equation to show what you did.” Students are provided with the digits, 8 and 5, and given a blank equation to fill in. 
• In Unit 7, Lesson 6, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they look at a picture of a rekenrek and find another equation that is equal/the same. Problem 1 states, “Look at the picture and equation in box 1. Use your teddies to write another equation that is equal/the same.” In box 1, there is a picture of a rekenrek with 4 on the top and 1 on the bottom and the matching equation, $$4 + 1 = 5$$. 
• In Unit 8, Practice Workbook G, students engage with K.NBT.1, compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, as they independently draw pictures to show the decomposition of the number 18 into ten ones and 8 more ones. Problem 6 states, “Draw a picture to show 18 as ten ones and some more ones. Write a number sentence to match.”
• In Unit 9, Practice Workbook F, students engage with K.OA.4, by finding the number that makes 10 for any number 1 to 9 by using objects or drawings and recording the answer. Problem 3 states, “Draw circles and write a number to show how many more are needed to make 10.” Students are given 2, 4, 7, 6, 3, 1, 8, 9, and 5.

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The instructional materials develop procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• In Unit 3, Lesson 12, Workshop Worksheet, students engage with K.CC.5, count to answer “How many?” questions about 20 things arranged in a line, a rectangular array, or a circle, as they count 18 pieces of silverware arranged in a rectangular array. The materials state, “Mr. Lohela was having a dinner party. He set out the silverware. How many pieces of silverware did he set out?”
• In Unit 6, Lesson 19, Introduction, students engage with K.OA.5, fluently add and subtract within 5, as they represent a story problem. The materials state, “Step 1: Visualize. Make a mind movie while I read. There were 7 carrot sticks on Hubina’s plate. She ate 3 of them. How many carrot sticks are on her plate? Step 2: Represent and Retell. Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.”
• In Unit 7, Practice Workbook E, Making 3, 4, and 5 finger Combinations, students engage with K.OA.5, fluently add and subtract within 5, as they play a game to develop fluency within 5. The materials state, “The teacher uses different finger flashes and students determine how many fingers are needed to make a target sum.” Once students understand the game, they play with a partner. 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they use fingers to calculate the missing addend. For example, “Activity: Making 3, 4, and 5 Finger Combinations. T: I’ll show you some fingers. I want to make 3. Show me what is needed to make 3. (Show 2 fingers.) S: (Show 1 finger.)”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• In Unit 3, Practice Workbook C, students engage with K.CC.3, write numbers from 2 to 20, as they independently complete a number sequence filling in missing numbers from 10 -15. Problem 5 states, “Fill in the missing numbers, “10, 11, ____, ____, ____, ____.”
• In Unit 6, Lesson 22, Assessment, students engage with K.OA.5, fluently adding and subtracting within 5, as they complete equations. Problem 5 states, “Solve. $$2 + 3 =$$ ___.”
• In Unit 7, Lesson 8, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they independently create equations with a sum of 10. The materials state, “Show all of the ways you could make 10. (You may not need to fill in every equation.)” Blank equations equalling 10 follow the directions. 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently add and subtract within 5, as they independently solve a series of addition problems with a sum of 2-5. Problem 1 states, “$$3 + 2 =$$____”

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that the materials reflect the balance in the standards and help students meet the standards’ rigorous expectations by helping students develop conceptual understanding, procedural skill and fluency, and application. The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

Conceptual understanding

• In Unit 3, Lesson 16, Exit Slip, students engage with K.CC.5, count to answer “how many” questions about as many as 20 things, as they represent a quantity 10 -20 pictorially by using a strategy to keep track of the count. The Exit Slip shows the number 16 with two blank ten frames. Students are expected to draw circles on the ten frame to represent 16.
• In Unit 6, Lesson 11, Exit Ticket, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as they represent and solve subtraction problems while using counters and tens frames. The materials state, “Use your counters and tens frames to solve. $$7 - 3 =$$____ and $$9 - 5 =$$ _____” 
• In Unit 9, Practice Workbook F, students engage with K.OA.3, decomposing numbers less than or equal to 10 into pairs in more than one way, as they draw pictures to show more than one way to make each number. Problem 4 states, “Draw a picture to show 2 ways to make each number. 6; 4; 7; 6; 3; 1; 8; 9; 5.”

Procedural skills (K-8) and fluency (K-6)

• In Unit 6, Lesson 5, Exit Slip, students engage with K.OA.5, fluently add and subtract within 5, as they are given an image with two numbers to add them together, and a spot for an equation. The materials state, “Cube 1 (6) Cube 2 (3) Equation _____ + _____= _____.” 
• In Unit 7, Practice Workbook E, Shake and Spill, students engage with K.OA.5, fluently add and subtract within 5, as they spill five two-sided counters in a cup to find combinations of 5. The materials state, “The students determine how many of each color is showing and record the sum using drawings or equations. The students should ‘shake and spill’ several times to show different pairs of numbers that sum to 5.” 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they solve put together and take apart problems with the result unknown within 5. Problem 8 states, “$$2 + 2 =$$ ___.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The student materials prompt students to both construct viable arguments and analyze the arguments of others even though mathematical dialogue is mainly between the teacher and individual students.

Examples of constructing viable arguments include: 

• In Unit 3, Lesson 26, Introduction, students transition from counting by ones to counting by tens and discuss the transition. T&T states, “How could I figure out how many? Counting by ones took a LONG time. Is there a way we can use our tens frames without having to count every dot?” 
• In Unit 4 Assessment, students compare the relative capacity of two items and explain how they know. Item 3 states, “(Give the student the cup and the basket). Ask, which item holds less? Use the words holds more/holds less. How did you figure that out?/How did you know that?” 
• In Unit 5, Lesson 3, Introduction, every pair of students have five yellow cubes and eight green cubes. The materials state, “Step 3 says Compare: We have gotten really good at telling which is more, but what if I ask Which is LESS? T&T: Figure it out with your partner- which color is less? How do you and your partner know?” Example response states, “Strategy 1:1: Matching 1:1: I know because I matched the cubes and some ___ cubes didn’t have a partner.”
• In Unit 6, Lesson 8, Introduction, students justify their thinking while solving a story problem using a strategy chosen by them. The materials state, “Step 2: Represent and Retell. Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.” 

Examples of analyzing the arguments of others include:

• In Unit 3, Lesson 12, Share/Discussion, students determine and write how many objects in a set (10-20 objects) by using a strategy to keep track. The teacher asks 2-3 students to share their work/strategies. The materials state, “How did ____ count? How did ____ count? What is the same about these strategies? What is different? Why do they both work?” 
• In Unit 6, Lesson 6, Workshop, Problem 1 states, “Gia had 4 apples in her basket. She picked 4 more and put them in her basket. How many apples does she have now?” Share/Discussion states, “Facilitate a discussion around a major misconception. Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR: 2-3 students share their work/strategies: How did ____ represent? How did ____ represent? How do both of these strategies work to represent the story?”
• In Unit 7, Lesson 5, Share/Discussion, students are engaged in a class discussion sharing their strategies after finding all of the possible combinations of 7. The materials state, “2 - 3 students share their work/strategies involving compensation: How did ______ find another solution? How did ______ find another solution? How/why do both of these strategies work?”
• In Unit 8, Lesson 8, Lesson Task states, “Introduction: Tanya has ten beads and five beads. Marie has ten beads and three beads. Tanya says she has more beads than Marie. Is Tanya correct? Show and tell how you know.”

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

• In Unit 2, Lesson 6, Introduction, Step 2 states, “What shape do you think would fit? Why did you pick that shape? SMS: It had a side that was the same length as the side of the puzzle (or some other attribute reference). If kids are struggling to articulate this idea, have a student come up and try it and the teacher can narrate the ideas to give kids the language - ‘Oh I see Danny is lining up the side of square with the side of the picture. Noticing the side helped here!’”
• In Unit 3, Lesson 18, Introduction, teachers are provided guidance in helping students to construct viable arguments demonstrating how to find the total of three numbers rolled on dot cubes. The materials state, “Step 2 is for us to find out the total. When we did this before, we only had to figure out the total for two dot cubes. T&T: How would we do it for three dot cubes? SMS: It’s the same! It’s just more dots, so I can count them all. If a student says this, have a student come up and demonstrate touching each dot and counting and a student showing the amounts on fingers and counting all. SMS: I can just see (subitize) and say the number on one dot and count on from there. Have a student demonstrate.”
• In Unit 4, Lesson 4, Share/Discussion, during Workshop, students are picking two objects and determining which is heavier or lighter using either a balance or hefting. The materials state, “Facilitate a discussion around a major misconception (i.e. an object that is longer/taller doesn’t always have to be heavier). Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR, 2-3 students share their work/strategies: How did ___ compare their objects? How did ___ compare their objects? What is the same about these strategies? What is different? Why do both work?”
• In Unit 5, Lesson 3, Mid-Workshop Interruption, students determine which number is more and which is less by building towers or matching one to one. The materials state, “If $$>\frac{2}{3}$$ of students are successful, ask students to describe the relationship between 2 towers (green 8 and blue 3) in a turn and talk. Hunt for a student who says one tower is more and another who says the other tower is less. Share their answers and ask who is right; students should see that both students are right- the green tower is more and the blue tower is less. Discuss how this is true; students should articulate that they are opposites and that if one tower is more the other will always be less and vice versa. Challenge students to circle the amount that is more as well one the recording sheets moving forward. If $$>\frac{2}{3}$$ of students are successful, call students back together to clarify expectations through a misconception protocol or role play.”
• In Unit 7, Lesson 1, Introduction, students name and record (with equations) various ways to decompose the totals 4 and 5. During a demonstration, the teacher tosses 3 red chips and 1 yellow chip. The materials state, “Step 2: says what do you see? TT: What do you see? I see 3 red counters and 1 yellow counter and 4 counters altogether. (Record guiding questions on VA.) -The purpose of this question is to get students to generate the numbers they will use in their number sentences; feel free to use the questions below to help: - If students say, ‘I see 4counters/chips,’ ask, ‘What colors do you see?’ -If students say, ‘I see red and yellow,’ ask, ‘How many red do you see? How many yellow?’ -If students say, ‘If students say, ‘I see red and yellow,’ but don’t notice the total, ask, ‘How many do you see altogether/How many does that make altogether?’”
• In Unit 7, Lesson 11, Introduction, Play Again and Check for Understanding, teachers are instructed to pose a fictional problem for the students to analyze. The materials state, “Rather than playing a full game, pose this problem: Mr. Lynch was playing the game, and he drew a 3, so he recorded like this: $$3 +$$ ____ $$= 10$$ (show). Then when he went to show that many on his tens frame, he realized that he didn’t have any! They had all been cut in half for an art project. So he used just the top half, like this (Show the top row of the tens frame with 3 counters on it.). Then, he said, ‘How many to ten?’ and he counted the empty squares. 2! He wrote $$3 + 2 = 10$$. EV: Does this work? No. TT: Why not? That is how many to 5, not ten. The tens frame works because it has 10 squares in all, so if we show how many we have we can count the empty squares to figure out how many to ten. There are not 10 squares in all.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials also use precise and accurate terminology and definitions when describing mathematics, and support students in using them. 

Examples of explicit instruction on the use of mathematical language include:

• In Unit 2, Overview, Identify The Narrative states, “Students are formally introduced to solid, 3D shapes in the lesson titled, ‘Stack/Roll/Slide.’ In the lesson, students observe solid shapes and begin to develop the vocabulary needed to describe their attributes, including vertices, faces, and edges. They begin to think about how these attributes distinguish them from 2D shapes.”
• In Unit 4, Lesson 1, Introduction, Introduce the Math states, “Today we’ll figure out how long (kinesthetic: make arms wide horizontally) or tall (kinesthetic: make arms wide vertically) things are by comparing two objects. That’s called length. CR: Length is (how long or how tall things are). Do again with kinesthetic movements. Show Measurable Attributes VA. When we are talking about how long or how tall things are, they can be LONGER (longer-choral response and motion: start with hands together and move apart) or SHORTER (shorter-choral response and motion: start with hands apart and move together).”
• In Unit 6, Lesson 1, Introduction, provides students with explicit instruction on the meaning of the + and = as they learn to write addition equations. Introduce the math states, “First, let’s learn the 2 math symbols we will be using today. This sign is called ‘plus’. It looks like the letter t. It means put together (put two hands together, interlocking fingers). We read it as ‘and’. (point to it 2 times and have kids say, ‘and’ and put their fingers together). This sign is called ‘equals’. It means is. (point to it 2 times and have kids say ‘is’ or ‘makes’.)”
• In Unit 7, Lesson 11, Introduction, students learn the name of the tool, tens frame. The materials state, “Today we are going to work with a special tool: The tens frame. (show tens frame) Why is this called a tens frame? What is special about it? It has 10 squares. Count them with me. (count the ten squares together) Today, we are going to play a game called ‘How many to Ten,’ let’s see how we can use our tens frames to help us today!”

Examples of the materials using precise and accurate terminology and definitions:

• In Unit 4, Lesson 7, Introduction, students are prompted to use accurate language to compare the capacity of two containers. The materials state, “Which one held more rice? The _______ held more rice than the _______. I know because it held _____ scoops of rice and the _____ held ______ scoops of rice. Be sure to prompt for accurate comparative language.”
• In Unit 5, Lesson 1, Assessment and Criteria for Success, students are expected to use the terms more, greater, the same, and equal to describe sets of objects. Questions are provided for teachers to support students in the use of these terms. The materials state, “Teachers should circulate during workshop to gather data on student mastery. All students should be able to use the words, ‘more,’ ’greater,’ ‘the same,’ and ‘equal’ to describe their sets. Teachers should ask: 1. Which is more/has a greater amount of cubes? How do you know? 2. How can you describe this tower? (pointing to a tower that is more). 3. How can you describe these towers? (showing two towers that are the same).”
• In Unit 6, Lesson 2, Introduction, Introduce the Math, the teacher adds a symbol to the visual aid to assist in students’ understanding. The materials state, “The symbol for addition is the ‘+’ sign that you all already know how to use. (add to picture part of va). It means we put the red and green together. (Draw the equals sign and the red and green together).”  
• In Unit 7, Overview, Identify the Narrative states, “They conclude that there are many different ways to make a total. They record using addition equations, with a continued emphasis on the language ‘and’ for ‘+’ and ‘makes’ or ‘is’ for ‘=.’ (If they have 3 red and 2 yellow, they record ‘$$3 + 2 = 5$$’ or ‘$$5 = 3 + 2$$’ and say ‘3 and 2 is five’ or ‘3 and 2 makes 5’ or ‘5 is 3 and 2.’) Teachers intentionally use this language when writing and reading equations and expressions in order to build deep conceptual understanding of the symbols.”