Kindergarten - Gateway 1

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Above-grade-level assessment questions are present but could be modified or omitted without a significant impact on the underlying structure of the materials.

The series is divided into nine units, and each unit contains a Unit Assessment available online in the Unit Overview document and can also be printed for students. The Unit Assessments contain written and interview questions. Some units contain suggestions for use of Post-Unit Assessment questions as Pre-Unit Assessment questions. Teachers are directed to adjust instruction according to the Pre-Assessment results. 

Examples of assessment questions aligned to grade-level standards include:

• Unit 2, Geometry Interview Questions, Task 1, “Put a triangle above a square and ask, ‘Where is the triangle in relation to the square?’” (K.G.1)

• Unit 5, Counting & Comparing Unit 5 Assessment, Question 2, “Look at the slices of pizza, (picture of five slices of pizza). Circle the group of ice cream cones that has more than the slices of pizza, (picture of one group of eight ice cream cones and another picture of six ice cream cones).” (K.CC.6)

• Unit 6, Counting Unit 6 Assessment, Question 9, “There were 10 cupcakes on the table. Jamaine ate 4 cupcakes. How many cupcakes are on the table now?” (K.OA.2)

• Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 3, “Draw a picture and write a number sentence to show 17 as tens and ones.” (K.NBT.1)

There are examples of above-grade-level assessment questions. In Unit 8, four of the seven questions assess above-grade-level content. The Guide to Implementing AF Math: Grade K, “Teachers should remove these items or use them for extension purposes only.” For example:

• Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 4, “How many tens are in the number 37? A. 3 B. 37 C. 7 D. 73.” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 

• Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 5, “Keisha drew sticks and dots to show how many blocks she had. How many blocks does Keisha have? A. 34 B. 7 C. 43 D. 44.” A picture representing 43 is between the question and answers. The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 

• Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 6, “Gloria wants to draw a picture to represent the number 26 as tens and ones. What could she draw to show the number 26 as tens and ones?” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 

• Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 7, “Use sticks and dots to show 42.” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value.

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

Each unit consists of lessons that are broken into four components: Introduction, Workshop/ Discussion, Independent Practice, and Exit Ticket. In addition to lessons, there are Math Stories “to enable students to make connections, identify and practice representation and calculation strategies, and develop deep conceptual understanding through the introduction of a specific story problem type in a clear and focused fashion with deliberate questioning and independent work time,” and Math Practice (Practice Workbook) for students “to build procedural skill and fluency.” Examples include:

• Unit 2, Lesson 4, Introduction and Workshop Resources, students describe objects in the environment using names of shapes and describe the relative positions of these objects using terms (K.G.1). Students play a game called “Where is my Shape?” where they pick a card that tells them to draw a shape above, below, beside, in front of, behind, and next to a picture of a Ninja Turtle. The full intent of the standard is met as all relevant positional words are included in this lesson. Practice Workbook B includes five problems that address K.G.1.

• Unit 2, Lesson 6, Exit Ticket, students engage with K.G.6 as they compose 2D shapes out of other 2D shapes by noticing their attributes. Students are provided with eleven exit tickets to independently practice the standard. “Jordan filled his hexagon using all triangles yesterday. Today he started using triangles but then ran out. How can he finish his puzzle? Circle the shape that would fit.” Students are given a picture of a hexagon with three triangles inside forming a trapezoid. They are then provided with three multiple choice items with shape pictures: A. trapezoid B. square C. oval.

• Unit 3, Lesson 1, Introduction and Workshop, students engage with K.CC.3 as they count objects and represent them with the written numerals 0-20. This is one of fifty lessons addressing this standard. As a result, the full intent of the standard is met for all students as this lesson focuses on students counting objects up to 15 and writing the corresponding numeral, prior lessons focused on up to five objects, then up to ten objects, and future lessons go up to 20 objects. Students are provided with extensive work on K.CC.3 through introduction activities, games, workshop time, and independent practice time. Lesson 1, Introduce the Math, “We are now experts at counting and writing numbers up to 10! Over the next few days we are going to be working with bigger numbers, up to 15. Let’s count to 15 together. Today, we are going to keep practicing counting by playing Counting Bags/Jars.” In the game Counting Jar, students are directed to count objects and write the corresponding numeral. The directions for the game are to “1. Take 1 bag; 2. How many? (move and count/organize and count); 3. Record.”

• Unit 6, students engage with K.OA.2 in 22 lessons. Lesson 6, Exit Slip, students solve addition and subtraction word problems. “Freddy had 6 books on his bookshelf at home. He went to the library and got 4 more books. How many books does he have now?”

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

• The approximate number of units devoted to major work of the grade, including assessments and supporting work connected to the major work, is 6 out of 9, which is approximately 67%.

• The number of lessons devoted to major work of the grade, including assessments and supporting work connected to the major work, is approximately 124 out of 163, which is approximately 76%. 

• The instructional block includes a math lesson, math stories, and math practice components. The non-major component minutes were deducted from the total instructional minutes resulting in 9,420 major work minutes out of 13,855 total instructional minutes. As a result of dividing the major work minutes by the total minutes, approximately 68% of the materials focus on major work of the grade. 

A minute-level analysis is most representative of the materials because the minutes consider all components included during math instructional time. As a result, approximately 68% of the materials focus on major work of the grade.

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The publishers identify connections between supporting content and major work within the lesson plan in the “Standards in Lesson” section, as well as in the Guide to Implementing AF Math: Grade K. Additional connections exist within the materials, although not always stated by the publisher. In addition, the publisher identifies the CCSSM clusters at the top of each lesson plan. However, in some cases, supporting clusters are misidentified as major clusters. For example, in Unit 2, Lesson 3, the publisher incorrectly identifies the Geometry Clusters as the major work of the grade. Examples of the connections between supporting work and major work include:

• Unit 1, Lesson 5, Introduction, students engage with the supporting work of K.MD.3, classify objects into categories, and the major work of K.CC.5, count to answer “how many” questions. In Step 4, students sort pattern blocks by shape, then answer, “How Many? We need to figure out how many are in each group.”

• Unit 2, Lesson 3, Exit Ticket, students engage with the supporting work of K.G.2, correctly name shapes, K.G.5, model shapes in the world by building shapes from components, and the major work of K.CC.5, count to answer “how many” questions. Students count sides and vertices and build them using geoboards. Problem 3 states, “Circle the shape that has 4 corners.” Students are shown pictures of a hexagon, a circle, a triangle, and a square. 

• Unit 2, Lesson 12, Understand: Introduce the Problem, students engage with the supporting work of K.G.4, analyze and compare two-dimensional shapes, and the major work of K.OA.2, solve addition word problems. Students are asked to visualize the shapes being mentioned. The teacher says, “Get ready to make a mind movie! Close your eyes and turn on your ears!” The teacher poses the problem by reading it 2-3 times, “Noah has three shapes. Noah has one square. Noah has one rectangle. Noah has one triangle. Noah counts all the corners of each shape. How many corners does Noah count all together? Show and tell how you know.” After the problem is read, students create a drawing of the three shapes based on their knowledge of their attributes. They then count the corners to add them, and represent the addition with an equation. In this lesson K.G.4 is the only standard identified, not K.OA.2.

• Unit 4, Lesson 5, Introduction, students engage with supporting work of K.MD.2, compare two objects to see which holds “more of”/”less of” the attribute, and the major work of K.CC.6, count to determine which group holds more. Students play a game called “Which holds more?” where they compare two objects and determine which holds more scoops of rice. The teacher asks, “How can we figure out which object has a larger capacity or holds more?” Students might say, “We can put in scoops of rice and count each to compare.” 

• Unit 9, Lesson 5, Workshop engages with the supporting work of K.MD.3, classify objects into categories, count the number of objects in each category, and sort the categories by count; and the major work of K.OA.3, compose and decompose numbers less than or equal to 10 in more than one way while finding multiple combinations of 10 pink and blue beads to make groups of 10. Workshop worksheet, “Introduction: Linda has pink beads and blue beads. Linda has some bags. Linda wants to put 10 beads in each bag. Some must be pink and some must be blue. How many different ways can Linda put pink beads and blue beads in bags? Show all of your mathematical thinking.”

• Kindergarten Practice Workbook A, students engage with the supporting work of K.MD.3, classify objects into given categories, and the major work of K.CC.5, count the number of objects to answer “How many?” Problem 1, “Color each group of 3.” The directions are followed by a picture of three rectangles, five triangles, three circles, and four squares.

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

The publisher identifies the CCSSM Clusters at the top of each lesson plan. However, in some cases, supporting clusters are misidentified as major clusters. For example, in Unit 2, Lesson 8, the publisher incorrectly identifies the Geometry clusters as the major work of the grade. Examples of connections include:

• Unit 3, Lesson 24, Exit Ticket, students engage with K.CC.B, count to tell the number of objects, and K.NBT.A, work with numbers 11-19 to gain foundations for place value, as they count a number represented in two ten frames. Problem 1 shows two tens frames with ten and six ones, “How many are there?”

• Unit 2, Lesson 8, Workshop, Student Workshop Worksheet students engage with K.G.A, identify and describe shapes, and K.G.B, analyze, compare, create, and compose shapes, as they match the faces of 3D solids to 2D shapes. “Makkelle had a can of soup. She wanted to put a label on it that would cover the whole top face. What shape would the label be? A. (insert a picture of a rectangle) B. (insert a picture of a circle) C. (insert a picture of a triangle).” Students are provided with a cylinder to represent the can of soup.

• Unit 8, Lesson 7, Workshop, Exit Ticket, students engage with K.NBT.A, work with numbers 11-19 to gain foundations for place value, and K.OA.A, understand addition as putting together and adding to, and understand subtraction as taking apart and taking from, as they make drawings to decompose teen numbers into ten ones, and some more ones. Problem 1, “Write a number sentence to show the number 13 as a group of ten ones and some more ones.”

• Practice Workbook B, students engage with K.G.A, identify and describe shapes, and, although not stated, K.G.B, analyze, compare, create, and compose shapes, as they draw shapes in relation to one another. Problem 2, “Look at the star. Draw a circle below the star. Draw a triangle above the star. Draw a rectangle next to the star.”

The materials reviewed for Achievement First Mathematics Kindergarten meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

 The Unit Overview supports the progression of Kindergarten standards by explicitly stating connections between prior grades and current grade level work. Each Unit Overview contains an Identify the Narrative component that identifies connections to what students learned before entering school and concepts previously learned in Kindergarten. 

Each Unit Overview also contains an Identify Desired Results: Identify the Standards section that makes connections to supporting standards learned prior to the unit. In addition, some lessons make connections to previous grade-level learning in the Narrative section. Examples include: 

• Unit 2, Geometry Unit Overview, Identify the Narrative, “Coming into this unit, students use the informal language of their everyday world to name and describe flat shapes (rectangle, triangle, square, circle, hexagon) without yet using mathematical concepts and the vocabulary of geometry. At this stage, a figure is a square because it looks like a book; another figure is a circle because it is round like the wheel of a car. Students make these observations without explicitly thinking about the attributes or properties of squares and circles.”

• Unit 3, Counting Unit Overview, Identify the Narrative, “Up to this point in K, students have worked intensively within 10. They have counted sets of objects and pictures and written numerals up to 10. This unit will help students build on their knowledge of numbers within 10 and extend it to larger quantities.”

• Unit 4, Measurement Unit Overview, Identify the Narrative, “After two units of counting (students can now count groups and record numbers to 20), and one unit of Geometry where students observed, analyzed, composed, decomposed and classified objects by shape, students now compare and analyze length, weight, capacity. This unit supports students’ understanding of amounts and their developing number sense.”

• Unit 5, Counting and Comparing Unit Overview, Identify The Narrative, “Lessons 10-12 ask students to write out hundreds charts to 100. This relates back to the pattern between and within the decades that students discovered in Unit 3. The kindergarten standard is to orally count to 100, while writing numbers to 100 is a first grade standard. In Kindergarten students must be able to orally count to 100 and write numerals up to 20. When doing the hundreds chart activities, point students toward appropriate resources in order to master the writing of numbers to 100 while allowing ample time for students to practice counting their strips and hundreds charts aloud.”

• Unit 6, Addition and Subtraction Unit Overview, Identify Desired Results: Identify the Standards, K.OA.1, K.OA.2, and K.OA.3 (all are addition and subtraction standards) are identified as the standards to be learned in Unit 6. The previous kindergarten standards identified as foundational are counting standards, K.CC.1, K.CC.2, K.CC.4, K.CC.5, K.CC.6, and K.CC.7. 

• Unit 8, Two-Digit Numbers Unit Overview, Identify Desired Results: Identify the Standards, K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones) is identified as one of the standards to be learned in Unit 8. The previous kindergarten standards identified as foundational are the counting standards (K.CC.1, K.CC.2, K.CC.3, K.CC.4) and K.OA.1 (Represent addition and subtraction). 

The Unit Overview documents contain an Identify the Narrative component that looks ahead to content taught in future grades. In addition, the Linking section includes connections taught in future grades, units, or lessons. Evidence of future grade-level work supporting the progressions in the Standards is identified. Examples include: 

• Unit 1, Sorting and Counting Unit Overview, Identify The Narrative, Linking, ”Looking ahead to the remainder of kindergarten, students will continue the counting sequence beyond 10 and up to 100. They will use the counting skills developed in this unit to develop strategies for addition and subtraction and to compose and decompose numbers within ten (K.OA.3) and into tens and ones, beginning with teen numbers (K.NBT.1) and then with all two-digit numbers (1.NBT.2). They will use their understanding of counting to compare sets (K.CC.6) and their place value understanding to compare two digit numbers. (1.NBT.3)”

• Unit 2, Geometry Unit Overview, Identify The Narrative, Linking, “Most importantly, it will help them access more complex geometrical standards in first grade in regard to distinguishing between defining attributes versus non-defining attributes, composing two-dimensional shapes to create a composite shape, creating new shapes from the composite shapes, and partitioning circles and rectangles. In second grade, students will need to draw shapes based on a given set of attributes; in third grade, students will focus on quadrilaterals and understand that a quadrilateral can also be categorized in a number of different ways; in fourth grade, students focus on points, line, ray, and parallel versus perpendicular lines.”

• Unit 5, Counting and Comparing Unit Overview, Identify the Narrative, Linking, “Students expand the counting sequence beyond 100 in first grade and begin to relate the way we say and write numbers to place value understanding.”

• Unit 8, Two-Digit Numbers Unit Overview, Identify The Narrative, Linking, “In first grade, students will use their understanding of place value to represent two-digit numbers in expanded notation and begin to add and subtract two-digit numbers.” 

• Unit 9, Exemplars Unit Overview, Identify the Narrative, Linking, “In second grade, students must understand the place value of three-digit numbers and use that understanding to add, subtract, and compare three-digit numbers, as they did in first grade with two-digit numbers. They will work to add and subtract within 1000 and fluently add and subtract within 100. In third grade, scholars are expected to fluently add and subtract within 1000 using place value strategies. Finally, in 4th grade, scholars are introduced to the standard algorithm for addition and subtraction.” A structural overview is under the Linking narrative. The image shows the progression of addition and subtraction through 4th grade.