4th Grade - Gateway 2

The instructional materials for Achievement First Mathematics Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. Examples include: 

• Unit 4, Lesson 8, students develop conceptual understanding of 4.NBT.5, as they use place value blocks to help them solve multi-digit multiplication problems. In Problem of the Day, “Problem: A video store display shelf has DVDs stacked in 3 rows. There are 246 videos in each row. How many videos can the shelves hold? TT: How can we represent this problem with an equation? We can write 246 videos x 3 rows = K total videos. Add to VA. Why does that work? It works because this is a problem about equal groups. In this problem, we have 3 groups—the rows—with 246 DVDs in each. We need to figure out the total number of DVDs. We can do this with multiplication. Today we’re going to solve 2, 3, and 4-digit multiplication with place value blocks. Work with your partner to solve this equation with place value blocks.”

• Unit 6, Lesson 6, students develop conceptual understanding of 4.NF.1, as they use tape diagrams and number lines to find equivalent fractions. In Workshop, Problem 2, “Markette is using a number line to figure out how many sixths are equal to \frac{2}{3}. She tells her partner, “We should partition each interval on our number line into 3 new parts, because 3+3 is 6. Is Markette’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.”

• Unit 7, Lesson 6, students develop conceptual understanding of 4.NF.7, compare two decimals to the hundredths by reasoning about their size. In the Workshop, Problem 1, students use visual models and number lines to support their reasoning about comparisons. “For each problem, shade each decimal amount on the given grids and plot them on the number line. Then use those models to compare the decimals using <, > or =0.2__$$0.19$$.” 

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

• Unit 2, Lesson 4, students demonstrate conceptual understanding of 4.NBT.2, as they use a provided place value chart and their knowledge of place value to determine the reasonableness of a provided answer. Independent Practice, Problem 2, “Kate used the place value chart to write the number below in standard form. Is Kate’s work correct? Explain why or why not on the lines below.”

• Unit 6, Lesson 1, students demonstrate conceptual understanding of 4.NF.3, as they draw a visual model and write an equation to solve a problem. Independent Practice, Problem 1, “Terrell is keeping track of his running for the week. Draw a visual model and write an addition equation to model Terrell’s running plan. How far will he have to run at the end of the week?” 

• Unit 10, Lesson 3, students demonstrate conceptual understanding of 4.MD.5, as they use manipulatives to find the measure of a given angle. In the Exit Ticket, Problem 3, “Using pattern blocks, how can you find the measure of the angle below? Use pictures, words and numbers to show how you found your answer.”

The materials for Achievement First Mathematics Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

Math Practice is intended to “build procedural skill and fluency” and occurs four days a week for 10 minutes. There are eight Practice Workbooks in Achievement First Mathematics, Grade 3. One workbook, C, contains resources to support the procedural skill and fluency standard 4:NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm. In the Guide To Implementing Achievement First Mathematics Grade 4, teachers are provided with guidance for which workbook to use based on the unit of instruction. Examples include:

• Practice Workbook C, Problem 1, students solve subtraction problems. “Find the difference. 51,348 and 22,122. Use the standard algorithm to solve.” (4.NBT.4)

• Practice Workbook C, Problem 6, students solve subtraction problems. “Use a strategy that makes sense to you to solve. 59,637 – 34,721 = .” (4.NBT.4)

• Practice Workbook C, Problem 9, students practice subtraction. “$$56432 - 33224 =$$ _____.“ (4.NBT.4)

Cumulative Reviews are intended to “facilitate the making of connections and build fluency or solidify understandings of the skills and concepts students have acquired throughout the week to strategically revisit concepts, mostly focused on major work of the grade.” Cumulative Review occurs every Friday for 20 minutes. Examples include:

• Unit 4, Cumulative Review 4.5, Problem 4, students solve subtraction problems. “Find the difference. Show your work. 6,241 - 1,366 = _____.” (4.NBT.4) 

• Unit 4, Cumulative Review 4.5, Problem 2, students solve addition and subtraction problems using the standard algorithm. “Camden Yard sold 5,864 tickets and 2,549 students tickets to last Friday's Baltimore Orioles baseball game. How many total tickets were sold for last Friday’s game?”(4.NBT.4)

• Unit 6, Cumulative Review 6.4, Problem 5, students solve a subtraction problem. “Find the difference. 2,301 - 1,976 = ___” (4.NBT.4)

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications, especially during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Guide to Implementing AF, Math Stories, October, students engage with 4.OA.3 as they solve a multi-step word problem posed with whole numbers and having whole-number answers using the four operations in a non-routine format. Sample problem 13, “Jamal and Sarah are playing a game with 5 counters. On each person’s turn they can take either 1or 2 counters from the pile. The player with the last turn loses. If Jamal starts the game and takes 1 counter away, what are two possible outcomes for the game?”

• Unit 7, Lesson 11, Problem of the Day, students engage in a routine problem with 4.NF.5 as they apply their understanding of fractions. "Victoria finds a multicolored quilt that exactly matches the colors in her bedroom. Victoria is so excited that she phones her mom to tell her about the quilt. This is what Victoria tells her mom: The quilt is a rectangle with one hundred squares, \frac{36}{100} of the quilt is made of red and yellow squares \frac{5}{10} of the quilt is blue squares, \frac{14}{100} of the quilt is green squares. Victoria's mom is very excited about the new quilt. She asks Victoria what total fraction of the quilt is made of blue and green squares. What fraction should Victoria tell her mom is the total fraction of the quilt made of blue and green squares? Show all your mathematical thinking.”

• Unit 9, Guide to Implementing AF, Math Stories, May, students engage with 4.NF.1 as they apply the use of a visual fraction model to generate equivalent fractions and solve a non-routine problem. "Akilah draws two rectangles of the same size, and divides them into a different number of total parts. In the first rectangle, she colors in 4 parts. In the second rectangle, she colors in 5 parts. The colored areas are equal. What fractions could she have divided her rectangles up into?”

• Unit 10, Guide to Implementing AF, Math Stories, June, students engage with 4.MD.2 as they use the four operations to solve a routine word problem involving money. Sample Problem 1, “There are 3,418 students at Brookside Elementary and 2,192 students at La PLaya Elementary. All of the students are going on a field trip to the Natural history museum, where tickets for children are $3 each. The schools have a budget of $20,000 to spend on field trips. How much money will they have left over?”

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

• Unit 1, Lesson 11, Independent Practice, students engage with 4.OA.2 as they solve a routine word problem involving multiplicative comparisons. Problem 2, “Kenny is 56 years old. His sister is 7 years old. How many times younger is Kenny’s sister than him?”

• Unit 2, Cumulative Review 2.2, Problem 4, students engage in non-routine application of 4.OA.4 as they use their knowledge of factor pairs to solve a problem in more than one way. "Yvette is making bracelets for her friends. Each bracelet will have an equal number of charms. She has 24 charms and she wants each bracelet to have at least 2 charms, but no more than 8 charms. Part A: Which is NOT a way that Yvette can make her bracelets? a) 8 bracelets with 3 charms on each bracelet. b) 6 bracelets with 4 charms on each bracelet. c) 4 bracelets with 6 charms on each bracelet. d) 4 bracelets with 8 charms on each bracelet. Part B: Are there any other ways that work for Yvette to make her bracelets? Show your work below.”

• Unit 6, Lesson 22, Independent Practice, students engage with 4.NF.3 as they solve routine word problems involving addition and subtraction of fractions. Problem 1, “A cabinet has shelves that are 11\frac{1}{4} inches tall. Mike stacked a speaker that is 4\frac{3}{4} inches tall on top of a DVD player that is 3\frac{2}{4} inches tall. How much space is left between the objects and the top of the shelf?”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations in that 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.

The instructional materials include opportunities for students to independently demonstrate the three aspects of rigor. Examples include:

• Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding of factors by determining whether there are additional factors for a number within 100. Problem 7, “Marco and Desiree made 56 cookies for a bake sale. They will put an equal amount of cookies into bags. Marco and Desiree want to put more than 2 cookies but fewer than 10 cookies into each bag. Desiree says that they can only put 7 cookies into 8 bags or 8 cookies into 7 bags. Marco thinks there are more ways to put an equal number of cookies into bags. Who is right? Why are they right?” (4.OA.4)

• Practice Workbook D, students develop procedural skill and fluency as they multiply whole numbers. Problem 14, “Calculate the product of 64 × 35.”  (4.NBT.5) 

• Unit 6, Lesson 23, Exit Ticket, students apply their understanding of fraction multiplication as they solve word problems. “Edwin uses \frac{3}{4} of a teaspoon of baking powder for each batch of muffins he makes. He needs to make 3 batches for his Cub Scout meeting and 4 batches for his study group. How many teaspoons of baking powder will Edwin need?” (4.NF.4) 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Unit 7, Lesson 11, Independent Practice, students apply their conceptual understanding of adding decimals to solve a real-world problem. Part 1, “Mrs. Evans, the physical education teacher, is forming relay teams to help raise money for cancer research. There must be two students on each relay team. To determine the teams, Ms. Evans uses the students’ practice times from the last physical education class. Ms. Evans wants the teams to be as evenly matched as possible so they have a fair chance to win the race. What would the best combination of students be for each of the relay teams? Show all your mathematical thinking.” (4.NF.5, 4.NF.7) 

• Unit 5, Lesson 7, Exit Ticket, students apply their conceptual understanding of multiplication to solve a two-step word problem using tape diagrams and equations. Problem 1, “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. (A × 2) + 4,892 = 6,392.” (4.OA.3) 

• Unit 8, Lesson 4, Independent Practices, students apply their conceptual understanding of place value to solve a problem involving the value of coins. Problem 6, “Which is more, 68 dimes or 679 pennies? Prove with a place value chart and then explain on the lines below.” (4.MD.2)

The materials reviewed for Achievement First Mathematics Grade 4 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 Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• Unit 3, Lesson 3, Independent Work Question 8, students engage with MP1 as they make sense of a problem involving multi-digit addition of whole numbers. “Milos’s family is keeping track of their steps each week. So far his sister has walked 15,678 steps, his father has walked 123,098 steps, and his mother has walked 435,607 steps. If Milos has walked twice as many steps as his sister, how many total steps has the family walked altogether?”

• The Unit 6 Overview outlines the intentional development of MP1. “Students apply the meaning of area and perimeter in order to interpret word problems in which area and perimeter are implicitly stated. Students practice division and multiplication strategies in the context of word problems. Calculations can be tedious and long, and students must continue to persevere through many steps in order to solve problems. Students employ a variety of problem solving skills in order to solve conversion problems. They must use ratios and calculations, but also determine which operation to use to convert.”

• Unit 5, Lesson 4, “How will embedded MPs support and deepen the learning?”, teachers are provided with explanations of connections between content and practices, “Students continue to practice SMP 1 as they plan to represent and solve multi-step problems by identifying all of the values and relationships between the values in the word problem, paying close attention to the questions asked in the word problem.”

• Unit 8, Lesson 6, Workshop Problem 3, students engage with MP1 as they work through multi-step word problems that require them to apply the concept of elapsed time. “Ms. Johnson has 20 minute meetings with students during the school day. She has a five-minute break between meetings. She does not have a break before her first meeting or after her last meeting. If she starts meetings at 7:45am, and has 4 meetings scheduled, what time will she be finished?”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 2 Overview outlines the intentional development of MP2. “When students start to work with numbers in greater place values such as the hundred thousands and ten thousands, they use abstract reasoning to understand quantitative meanings. Since there are no place value blocks big enough to show a hundred thousand and they often can’t draw quantities this large, they must apply patterns of the place value system to logically understand the magnitude of larger numbers. SMP 2 is developed in lessons 2, 3, 5, & 9-11.”

• Unit 2, Lesson 3, Independent Practice Question 5, students engage with MP2 as they read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. “Lee and Gary visited South Korea. They exchanged their dollars for South Korean bills. Lee received 5 thousand dollar bills, 6 hundred dollar bills, 9 ten dollar bills, and 5 one dollar bills. What was Lee’s total amount of money in standard, written, and expanded form?”

• The Unit 6 Overview describes development of MP2. “The true understanding of the fundamental meaning of a fraction is abstract reasoning. Students must learn to take an abstract representation of two numbers (a numerator and a denominator) and give it a new meaning referring to a part of a whole – a value less than one. Through visual models and many examples, students should begin to understand that a fraction is a quantity in itself that has a position on a number line. This is extremely abstract quantitative reasoning. Students reason abstractly when they compare fractions of different wholes. The idea that a fraction can have a different value based on the size of its whole, but the same whole is implied when it is not specified, is a challenging abstract concept for students to grasp.”

The materials reviewed for Achievement First Mathematics Grade 4 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.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 16, Workshop, students critique the reasoning of others and construct a viable argument as they evaluate the estimation strategies used by other students to determine who is correct. Problem 2, “Patricia said the best way to estimate the solution to 8,421 - 462 is to round each number to the nearest hundred. Matthew said the best way to estimate is to round each number to the nearest thousand. Who is correct? Explain your answer.” 

• Unit 7, Lesson 4, Exit Ticket, Problem 3, “Patrice is measuring the rainfall for December. On Monday there was 0.09 of an inch of rainfall. On Tuesday there was 0.9 of an inch of rainfall. Patrice tells his sister that it rained the same amount on Monday and Tuesday. Tell whether or not Patrice is correct on the lines below.”

• Unit 7, Lesson 8, Discussion, teachers are provided with guidance and questions to engage students in critiquing the reasoning of others. “Last year I had a scholar who told me that when you compare decimals it's like the opposite of comparing whole numbers. What do you think they meant by that? How are comparing whole numbers and comparing decimals similar?”

• Implementation Guide, Unit 7, Math Stories, February/March, students are provided with an opportunity to share their math thinking as they solve problems involving fractions. Problem 12, “Yanira ran \frac{3}{4} miles each day for 6 days. How many miles did she run over the course of 6 days?” Teachers are provided with three specific protocols to assist them in helping students represent and/or solve the problem, including sentence stems, for example: “First I put ____ because the story ____. Then I put ____ because in the story ____. Finally, I put ____ because in the story/we need to figure out ____.”

• Unit 10, Lesson 4, Exit Ticket, students construct an argument and critique the reasoning of others based on their knowledge of shapes. Problem 3, “Carlos is helping his brother with homework. He tells his brother that if you want to draw an obtuse angle, you should always use the bottom set of degrees on the protractor arc, and if you want to draw an acute angle you should always use the top set. Is Carlos’s reasoning accurate? Explain why or why not on the lines below.”

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Math Stories Guide, Promoting Reasoning through the Standards for Mathematical Practice, MP4, “Math Stories help elementary students develop the tools that will be essential to modeling with mathematics. In early elementary, students become familiar with how representations like equations, manipulatives, and drawings can represent real-life situations.” Within the K-4 Math Stories Representations and Solutions Agenda, students are given time to represent, retell, and solve the problem on their own.

• The Unit 1 Overview describes the intentional development of MP4. “Students model real-world mathematical situations using equations and tables with patterns. Students use tables, pictures, and mathematical formulas to solve problems involving patterns, multiplicative comparisons, and to determine factors, and classify numbers as prime or composite. SMP4 is developed in lessons 2 and 5 - 9.”

• The Unit 4 Overview provides guidance for connecting MP4 with area, perimeter, and solving conversion problems. “Students interpret word problems referring to area and perimeter and represent the information. When students solve conversion problems, they use many different types of models to determine how to convert, or show how they converted. They use the ratio to draw appropriate pictures, create tables, and write equations that use mathematics to model how to convert from one unit of measurement to another. Students also use benchmarks to understand units of measurement, which is a way of modeling a mathematical concept with real-world objects.”

• Unit 6, Lesson 18, Pose the Problem, students subtract mixed numbers by using fraction tiles and drawings. “Moira ordered 6 pizzas for the Student Council meeting. At the end of the meeting there were 2\frac{3}{4} pizzas left. How many pizzas did the student council eat during the meeting?”

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 2 Overview describes the intentional development of MP5. “Students choose between many methods when working with place value. In almost every aspect of this unit (place value relationships, expanding, reading, writing, comparing and rounding numbers, and non-standard partitioning) students have a variety of tools that could assist them. They can use place value charts, place value blocks, pictures of dots in each place value, pictures of place value blocks, organized lists, etc. to solve these types of problems. They must determine which tools are most effective for certain tasks.”

• Unit 6, Lesson 6, Exit Ticket Question 2, students use various tools, models and representations to show the meaning of fractions and different ways of showing fractional quantities. “Delilah is using a number line to figure out how many eighths are equal to \frac{3}{4}. She tells her partner, ‘We should partition each interval on our number line into 4 new parts, because 4 + 4 is 8.’ Is Delilah’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.”

• Unit 9, Lesson 4, Independent Practice Question 6, students use square corners and rulers to determine types of lines and angles. “Can a triangle have two right angles? Explain and draw an example to prove your thinking.”

At times, the materials are inconsistent. The Unit and Lesson Overview narratives describe explicit connections between the MPs and content, but lessons do not always align to the stated purpose. 

The materials do not provide students with opportunities or guidance to identify and use relevant external mathematical tools and resources, such as digital content located on a website.

The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 2, Lesson 1, Exit Ticket, students solve problems requiring them to demonstrate an understanding of the term expanded form. Problem 3, “What two hundreds is seven hundred twelve between? Write seven hundred twelve in standard form. Write seven hundred twelve in expanded form.” 

• Unit 8, Lesson 2, Independent Practice Question 1 (Bachelor Level), students use precision to solve problems related to volume and capacity. “The capacity of each pitcher in the teacher work room is 3 quarts. Right now, each pitcher contains 1 quart 3 cups of liquid. If there are 3 pitchers in the room, how much more total liquid can the pitchers hold?”

• In the Unit 9 Overview, “Students attend to precision when naming and identifying lines, angles and triangles based on names using points.”

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• Unit 4, Lesson 15, Try One More, teachers are provided with instructions to explicitly teach the term remainder. “We do have 1 leftover in this problem. This is called our remainder. A remainder is the amount leftover after dividing a number when one number does not divide evenly into another number. What was our answer before the remainder and why?” Students might say, “114 because we have 1 hundred + 1 ten + 4 ones.” The teacher replies, “Yes. Now our answer becomes 114 R1 because our answer is 114 with a remainder of 1.”

• Unit 6 Lesson 1, teachers are provided with guidance in reviewing vocabulary related to fractions. The introduction, “Before you get started, let’s review some key fraction vocabulary. In a fraction, what does the denominator tell us? The denominator tells us the total number of parts in a whole. In a fraction, what does the numerator tell us? The numerator tells us the total number of parts being referred to.”

• Unit 9, Lesson 4, the introduction provides teachers with guidance in introducing terminology related to triangles through a series of questions. “We have learned about different angle types, which will help us in our work today. What are the different types of angles? Today you will use the different types of angles to help you classify triangles!” Students work to observe and note information about triangles. The teacher prompts, “What did you observe about triangles? Triangles are classified by two names, kind of like how you have a first name and a last name. One name tells us about their angles, and one name tells us about their angles, and one name tells us about their sides.”

The materials reviewed for Achievement First Mathematics Grade 4 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 grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 5, Independent Practice, students solve problems by looking for structures based on place value. Problem 4, “Tiana drew 12 hundreds blocks on her paper. How many tens is that equal to? a. 1,200  b. 120 c. 12 d. 12,000.” 

• Unit 5, Lesson 4, Independent Practice, Question 2, students look for structure as they solve problems involving parts adding up to a whole by using tape diagrams to represent the situations. “Malia is keeping track of the subway riders on Saturday. At the first stop, some people got on the train. At the second stop, three times more people got on the train than at the first stop. At the last stop some people got off the train. How many people are on the train now?”

• Unit 8, Lesson 5, Exit Ticket, Question 2, students solve word problems involving adding enough of a smaller unit in order to regroup in the context of money amounts and determining change. “Meiling needed $5.35 to buy a ticket to a show. In her wallet, she found 2 dollar bills, 11 dimes, and 5 pennies. How much more money does Meiling need to buy the ticket?”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 5, Lesson 8, Exit Ticket, Question 1, students look for regularity in repeated reasoning as they solve problems by interpreting and labelling a representation such as a tape diagram. “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. (A\times2)+4,892=6,392.”

• Unit 6, Lesson 16, Independent Practice, Question 2, students see regularity in regrouping as they add and subtract mixed numbers. “Khalia and Jermaine are in a pie-eating contest. After 5 minutes, Khalia ate 3\frac{3}{4} pies and Jermaine at 4\frac{3}{4} . How much total pie did they consume altogether?”

• Unit 10, Lesson 2, Independent Practice, students look for repeated calculations as they solve a problem involving a circle divided into angles with given measurements. Problem 3, “Joanne cut a round pizza into equal wedges with angles measuring 30 degrees. How many pieces of pizza does she have?”