4th Grade - Gateway 1

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Each unit contains a Post-Assessment which is a summative assessment based on the standards designated in that unit. 

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

• Unit 1, Post-Assessment, Item 5, “The cost of buying a movie is 4 times the cost of renting a movie. It costs $30 to buy a movie. Write two equations that can be used to determine the cost, r, or renting a movie.” (4.OA.1)  

• Unit 4, Post-Assessment, Item 16, “Divide. 7,285 ÷ 4. Answer choices include:   A. 1,801, B. 1,801 R1, C. 1,821, D. 1,821 R1.” (4.OA.3)

• Unit 9, Post-Assessment, Item 15, “Explain why a square is also a rectangle and a rhombus.” (4.G.2 )

• Unit 10, Post-Assessment, Item 13, “A circular pizza was cut into 5 equal slices from the vertex at the center of the pizza. 3 of the slices of pizza get eaten. What is the measurement of the angle formed at the vertex of the slices that are left?” (4.MD.5) 

Reviewers noted that in the Achievement First Mathematics Grade 4 materials, there was not a Unit 2 Overview therefore an assessment was not available to be reviewed. 

Examples of above-grade-level assessments or assessment items which can be omitted or modified:

• Unit 6, Post-Assessment, Item 2, “For each of the following sums, decide which ones are equal to \frac{22}{17}. For the sums that are equal to \frac{22}{17}, circle YES. For the sums that are not equal to \frac{22}{17}, circle NO. ...; YES or NO \frac{1}{17}+\frac{1}{17}+\frac{9}{17}+\frac{3}{17}.” (4.NF.2, expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.)

• Unit 6, Post-Assessment, Item 14, “Each day Milo reads 18of his new book. Which number sentence best represents the fractions of his book that Milo has read after 7 days? Answer choices include: A. \frac{1}{8}×\frac{1}{7}=\frac{1}{56},  B. \frac{1}{8}×\frac{1}{7}=\frac{7}{8}, C.$$\frac{1}{8}×7=\frac{1}{56}$$, D.$$\frac{1}{8}×7=\frac{7}{8}$$.” (4.NF.4, students are expected to “solve word problems involving multiplication of a fraction by a whole number.” Answer choices A and B do not meet the criteria outlined by the standard.) 

Achievement First Mathematics Grade 4 has assessments linked to external resources in some Unit Overviews; however there is no clear delineation as to whether the assessment is used for formative, interim, cumulative or summative purposes.

The materials reviewed for Achievement First Mathematics Grade 4 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 3, Lessons 2 through 9, students fluently add and subtract multi-digit whole numbers using the standard algorithm, as they solve more than 100 problems in independent practice opportunities (Independent Practice, Exit Ticket, and Practice Workbook) and explain their use of the algorithm (4.NBT.5). Lesson 6, Independent Practice, Problem 5, “$$50,019-12,877$$” and Problem 6, “In the last problem, what place values did you need to regroup and how did you do it? Explain on the lines below.”

• Unit 5, Lessons 1 through 10, students solve multi-step word problems posed with whole numbers and having whole number answers using the four operations, including problems in which the remainders must be interpreted, and represent these problems using equations with a letter standard for the unknown quantity (4.OA.3). There are 83 Independent Practice problems and 15 Exit Tickets that require students to solve multi-step word problems. Lesson 6, Independent Practice, Problem 4, “Mia represented the above problem like this: (437 pies x 9 wards) x 14 = Total pie Pieces,” and asks students, “Is this representation reasonable? Tell why or why not on the lines below.” While there are 98 opportunities for students to engage with this standard, there are less than 10 opportunities for students to represent problems with a letter standard for the unknown quantity.

Unit 10, Lesson 4, students engage with 4.MD.6, measure angles in whole number degrees using a protractor, as they solve problems requiring them to use a protractor to accurately identify the angle measurement. Exit Ticket 1, “For numbers 1-2, use your protractor to create an angle of the given size. Be sure to check if your drawn angle matches the type of angle indicated by the measurement. 1. 67\degree.”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 114 out of 125, which is approximately 91%.

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 121 out of 132, which is approximately 92%. 

• The instructional minutes were calculated by taking the number of minutes devoted to the major work of the grade (10,425) and dividing it by the total number of instructional minutes (11,475), which is approximately 91%. 

A minute-level analysis is most representative of the materials because the units and lessons do not include all of the components included in the math instructional time. The instructional block includes a math lesson, math stories, and math practice components. As a result, approximately 91% of the materials focus on major work of the grade.

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

There are opportunities in which supporting standards/clusters are used to support major work of the grade and are connected to the major standards/clusters of the grade. Examples include:

• Unit 4, Lesson 25, Independent Practice, Problem 2, “Leonard bought 4 liters of orange juice. How many milliliters of juice does he have?” This problem connects the major work of 4.NBT.5, multiply a whole number of up to four digits by a one-digit whole number, to the supporting work of 4.MD.A, as students solve a problem involving measurement conversion from a larger unit to a smaller unit.  

• Unit 8, Lesson 7, Independent Practice, Problem 2, “Julio starts school at 7:45 am and finishes school at 3:30 pm. He has 25 minutes of recess, 32 minutes of lunch, and he has a 46 minute free period in the afternoon. The rest of the time he is in classes. How many hours and minutes does Julio spend in class?” This problem connects the major work standard 4.OA.3 and the supporting work standard 4.MD.2, as students solve multi-step word problems involving intervals of time.  

• Unit 10, Lesson 2, Independent Practice, Problem 2, “Megan has a very large round table. In order for her to seat her guests, she divided it into 10 equal sections. What is the angle measure of each section of the table?” This problem connects the major work of 4.NF.3 and the supporting work of 4.MD.5, as students find the measurement of angles when given a fraction of a circle.

The materials reviewed for Achievement First Mathematics Grade 4 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. Examples include:

• Unit 1, Assessment, students connect the work of 4.OA.B, gain familiarity with factors and multiples, to 4.OA.C, generate and analyze patterns, as they complete a pattern involving multiples. Item 1, “Alfonzo applies numbers on the back of football jerseys. Below are the first five numbers he applies. If the pattern continues, what are the next three numbers he will apply? 9, 18, 27, 36, 45, ___,___,____ a. 54, 63, 72 b. 54, 63, 71 c. 63, 64, 72 d. 63, 72, 8.”

• Unit 5, Lesson 9, students connect the work of 4.OA.A, use the four operations with whole numbers, to solve problems to 4.NBT.B, use place value understanding and properties of operations to perform multi-digit arithmetic, as students use estimation strategies and a visual model to solve multi-step problems. Exit Ticket, Problem 1, “Katie and her sister are saving up money to build a tree house. Each month they have saved 46 each from their allowance. In the last month, they each did some extra jobs in order to get the total amount they needed for the house. They spend 13 months saving and an extra month doing more jobs. If the cost of the tree house was 1299, how much money did they earn in the last extra month? Represent, estimate and solve.” 

• Unit 7, Assessment, students connect the work of 4.NF.A, extend understanding of fraction equivalence, and ordering to 4.NF.C, understand decimal notation for fractions, and compare decimal fractions, as students locate and label points on a number line. Item 19, “Locate and label the following points on the number line below: \frac{130}{10},13.21, 13\frac{12}{100}.”

The materials reviewed for Achievement First Mathematics Grade 4 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. 

Each unit has a Unit Overview and a section labeled “Identify Desired Results” where the standards for the unit are provided as well as a correlating section “Previous Grade Level Standards/Previously Taught & Related Standards” where prior grade-level standards are identified. Examples include:

• Unit 3, Overview, Identify Desired Results: Identify the Standards, identifies 3.NBT.2 under Previous Grade Level Standards/Previously Taught and Related Standards for 4.NBT.5. In Enduring Understandings - What it Looks Like in This Unit, a connection is made between the addition and subtraction work to place value skills in prior grades. “In previous grades, students used place value blocks and pictures of place value blocks to add and subtract numbers. Place value relationships help them regroup. When they need to take away more than they have of a certain place value, they regroup one of a greater place value to ten of that place value.” 

• Unit 6, Unit Overview, Identify Desired results: Identify the Standards identifies 4.NF.1 as being addressed in this unit and 3.NF.1 as Previous Grade Level Standards/ Previously Taught & Related Standards connections. In Identify the Narrative, a description is provided, “In third grade they recognized equivalent fractions using visual models and number lines in ‘special cases’ such as \frac{2}{4}=\frac{1}{2} and \frac{1}{3}=\frac{2}{6}. In fourth grade, they use visual models of equivalent fractions to understand how to use the identity property to find equivalent fractions.”

The materials develop according to the grade-by-grade progressions in the Standards. Content from future grades are clearly identified and are related to grade-level work within each Unit Overview. Each Unit Overview contains a narrative that includes a “Linking” section that describes in detail the progression of the standards within the unit. Examples include:

• Unit 3, Unit Overview, Linking, “Later in the year students will add and subtract mixed units of measurement which will again call upon regrouping concepts -- in this case, from one unit of measurement to another.” An additional reference is made to fifth grade, “When students move to fifth grade, they will continue to solve multi-step word problems with all four operations, so they will be relying on their abilities to add and subtract with the standard algorithm.”

• Unit 6, Unit Overview, Identify the Narrative, refers to prior work students engaged in with fractions. “The fourth grade unit on fractions combines students’ prior knowledge on fractions, the meaning of operations, logical reasoning and new learning experiences to elaborate their understanding of fractions and allow them to operate with fractions.”

• Unit 10, Unit Overview, Linking, “Angle measurements will not come up as a formal part of the math curriculum again until seventh grade. Although there is a large gap in time between this unit and seventh grade, the seventh grade standards rely heavily on students’ skill and knowledge from this unit in fourth grade.”