4th Grade - Gateway 1

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. 

The curriculum is divided into nine units, and each unit contains a written End-of-Unit Assessment for individual student completion. The Unit 9 Assessment is an End-of-Course Assessment and includes problems from across the grade. Examples from End-of-Unit Assessments include: 

• Unit 1, Factors and Multiples, End-of-Unit Assessment, Problem 1, “a. Is 27 a prime number or a composite number? Explain or show your reasoning. b. Is 29 a prime number or a composite number? Explain or show your reasoning.” (4.OA.4)

• Unit 3, Addition and Subtraction of Fractions, End-of-Unit Assessment, Problem 4, “Jada needs 2 pounds of walnuts for a trail mix. She has 3 packages of walnuts that each weigh \frac{3}{4} pound. Does Jada have enough walnuts to make the trail mix? Explain or show your reasoning.” (4.MD.2, 4.NF.4c)

• Unit 5, Multiplicative Comparison and Measurement, End-of-Unit Assessment, Problem 1, “There are 93 students in the cafeteria. There are 3 times as many students in the cafeteria as there are students on the playground. a. Write a multiplication equation that represents the situation. b. How many students are on the playground? Explain or show your reasoning.” (4.OA.1, 4.OA.2)

• Unit 7, Angles and Angle Measurement, End-of-Unit Assessment, Problem 6, “Use a protractor to complete the following: a. Draw a ray that makes a 25 degree angle with the given ray. b. Draw a ray that makes a 60 degree angle with the given ray. c. What is the size of the angle made by the two rays you drew? Explain how you know.” One ray is provided in the problem. (4.MD.6, 4.MD.7)

• Unit 9, Putting It All Together, End-of-Course Assessment and Resources, Problem 1, “Select the number where the value of 6 is 1,000 times the value of the 6 in 463. a. 643, b. 6,118, c. 63,479, d. 627,385.” (4.NBT.1)

The instructional materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The instructional materials provide extensive work in Grade 4 as students engage with all CCSSM standards within a consistent daily lesson structure. Per the Grade 4 Course Guide, “A typical lesson has four phases: a Warm-up, one or more instructional activities, the lesson synthesis, a Cool-down.” Examples of extensive work include:

• Unit 2, Fraction Equivalence and Comparison, Section B, Lessons 7, 10, and 11 engage students in extensive work with 4.NF.1 (Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions). Lesson 7, Equivalent Fractions, Activity 2, Launch students find equivalent fractions for fractions given numerically, “Groups of 2. ‘Work with a partner on this activity. One person is partner A and the other is B. Your task is to find two equivalent fractions for each fraction listed under A or B, and then convince your partner that your fractions are equivalent.’” Lesson 10, Use Multiples to Find Equivalent Fractions, Activity 1, Student Work Time, students use visual representations to generate equivalent fractions, “‘Think quietly for a couple of minutes about what Elena did and how it relates to Andre’s number lines.’ 1–2 minutes: quiet think time for the first problem. 3–4 minutes: partner discussion on the first problem. Pause for a brief whole-class discussion. Invite students to share their ideas about Elena’s work and how it is related to Andre’s number lines. 4–5 minutes: independent work time for the last problem. Monitor for students who find equivalent fractions for \frac{1}{8} by multiplying time a factor other than 2, 3 or 4.” Student Facing, “Elena thought of another way to find equivalent fractions. She wrote: ‘\frac{1}{5} is multiplied by \frac{2}{2}, \frac{3}{3}, \frac{4}{4}, \frac{5}{5}, and \frac{10}{10}.’ 1. Analyze Elena’s work. Then, discuss with a partner: a. How are Elena’s equations related to Andre’s number lines? (The equivalent fractions are displayed on a number line.) b. How might Elena find other fractions that are equivalent to \frac{1}{5}? Show a couple of examples. 2. Use Elena’s strategy to find five fractions that are equivalent to \frac{1}{8}. Use number lines to check your thinking, if they help.” Lesson 11, Use Factors to Find Equivalent Fractions, Activity 2, Student Work Time, students generate equivalent fractions by applying the numerical strategies they learned, “‘Work on the activity independently. Then, share your responses with your partner and check each other’s work.’ 8–10 minutes: independent work time. 3–5 minutes: partner discussion.” Student Facing, “Find at least two fractions that are equivalent to each fraction. Show your reasoning. a. \frac{16}{8} b. \frac{40}{10} c. \frac{7}{6} d. \frac{90}{100} e. \frac{5}{4}.”

• Unit 4, From Hundredths to Hundred-thousands, Section B, Lesson 11, Section C, Lesson 13, and Section D, Lesson 21 engage students in the extensive work with 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons). Lesson 11, Large Numbers on a Number Line, Activity 1, Student Work Time, students use their understanding of place value and the relative position of numbers within 1,000,000 to partition and place numbers on a number line. Student Facing, “a. Locate and label each number on the number line.  347 (a number line labeled 300 at one end and 400 at the opposite end is located under the problem), 3,470 (a numberline label 3000 at one end and 4000 at the opposite end is located under the problem), 34,700 (a numberline label 30,000 at one end and 40,000 at the opposite end is located under the problem), 347,000 (a numberline label 300,000 at one end and 400,000 at the opposite end is located under the problem). b. Locate and label each number on the number line. 347 (a number line labeled 340 at one end and 350 at the opposite end is located under the problem), 3,470 (a number line labeled 3400 at one end and 3500 at the opposite end is located under the problem), 34,700 (a number line labeled 34,000 at one end and 35,000 at the opposite end is located under the problem), 347,000 (a number line labeled 340,000 at one end and 350,000 at the opposite end is located under the problem). c. What do you notice about the location of these numbers on the number lines? Make two observations and discuss them with your partner.” Activity 2, Student Work Time, students place a set of numbers that are each ten times as much the one before it on the same number line. Student Facing, “Your teacher will assign a number for you to locate on the given number line. A. 347 B. 3470 C. 34,700 D. 347,000 a. Decide where your assigned number will fall on this number line. Explain your reasoning. b. Work with your group to label the tick marks and agree on where each of the numbers should be placed.” Number 1 has a number line with endpoints 0 and 400,000 labeled. Number 2 has a number line with 0 and 400,000 labeled and three tick marks on the line to be labeled. Lesson 13, Order Multi-digit Numbers, Cool-down, students use their place value understanding to order numbers. Student Facing, ”Order the following numbers from least to greatest 94,942; 9,042; 279,104; 9,420; 59,000; 500,492; 279,099.” Lesson 21, Zeros in the Standard Algorithm, Warm-up: Which One Doesn’t Belong: Numbers with 0, 2, and 5, students analyze and compare features of multi-digit numbers, “Groups of 2. Display numbers. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’” Student Facing, “Which one doesn’t belong? A. 2,050 B. 2,055 C. 205.2 D. 20,005.”

• Unit 5, Multiplicative Comparison and Measurement, Section A, Lessons 2, 3, and 5 engage students in extensive work of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). Lesson 2, Interpret Representations of Multiplicative Comparisons, Activity 1, Student Work Time, students analyze and describe how images and diagrams can show “n times as many.” Student Facing, students see pictures of connecting cubes (6 cubes and 2 cubes), “a. Jada has 4 times as many cubes as Kiran. Draw a diagram to represent the situation. b. Diego has 5 times as many cubes as Kiran. Draw a diagram to represent the situation. c. Lin has 6 times as many cubes as Kiran. How many cubes does Lin have? Explain or show your reasoning.” Lesson 3, Solve Multiplicative Comparison Problems, Activity 2, Student Work Time, students make sense of and represent multiplicative comparison problems in which a factor is unknown. Student facing, “1. Clare donated 48 books. Clare donated 6 times as many books as Andre. a. Draw a diagram to represent the situation. b. How many books did Andre donate? Explain your reasoning. 2. Han says he can figure out the number of books Andre donated using division. Tyler says we have to use multiplication because it says ‘times as many’. a. Do you agree with Han or Tyler? Explain your reasoning. b. Write an equation to represent Tyler’s thinking. c. Write an equation to represent Han’s thinking. 3. Elena donated 9 times as many books as Diego. Elena donated 81 books. Use multiplication or division to find the number of books Diego donated.” Lesson 5, One- and Two-step Comparison Problems, Activity 1, Student Work Time, students solve contextualized problems using multiplicative comparison. Student Facing, “For this year’s book fair, a school ordered 16 science books and 6 times as many picture books. Last year, the school ordered 4 times as many picture books and 4 times as many science books than they did this year. a. How many picture books were ordered this year? b. How many picture books were ordered last year? c. How many more science experiment books were ordered last year than this year?”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 4 standards through a consistent lesson structure. According to the Grade 4 Course Guide, “The first event in every lesson is a Warm-up. Every Warm-up is an Activity Narrative. The Warm-up invites all students to engage in the mathematics of the lesson… After the Warm-up, lessons consist of a sequence of one to three instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class… After the activities for the day, students should take time to synthesize what they have learned. This portion of class should take 5-10 minutes before students start working on the Cool-down…The Cool-down task is to be given to students at the end of the lesson. Students are meant to work on the Cool-down for about 5 minutes independently and turn it in. The Cool-down serves as a brief formative assessment to determine whether students understood the lesson.”Examples of meeting the full intent include:

• Unit 5, Multiplicative Comparison and Measurement, Section C, Lesson 16 and Unit 6, Multiplying and Dividing Multi-digit Numbers, Section D, Lesson 22 engage students in the full intent of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems). Unit 5, Lesson 16, Compare Perimeters of Rectangles, Cool-down, students reason about the perimeter of rectangles. Student Facing, “1. Rectangle Y has a perimeter of 20 inches. Name a possible pair of side lengths it could have. 2. Rectangle Z has a perimeter of 180 inches. Complete this statement: a. The perimeter of rectangle Z is ___ times the perimeter of rectangle Y. b. If the length of rectangle Z is 70 inches, how many inches is its width? Explain or show your reasoning. Draw a diagram if it is helpful.” Unit 6, Lesson 22, Problems About Perimeter and Area, Activity 2, Student Work Time, students perform operations with multi-digit numbers to solve situations about perimeter and area. Student Facing, “A classroom is getting new carpet and baseboards. Tyler and a couple of friends are helping to take measurements. Here is a sketch of the classroom and the measurements they recorded. For each question, show your reasoning. a. How many feet of baseboard will they need to replace in the classroom? How many inches is that? b. 1,200 inches of baseboard material was delivered. Is that enough? c. How many square feet of carpet will be needed to cover the floor area?” A composite figure is included with measurement labels for each side. 

• Unit 3, Extending Operations to Fractions, Section B, Lessons 8 and 9 engage students with the full intent of 4.NF.3a (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole). Lesson 8, Addition of Fractions, Activity 2, Student Work Time, students use number lines to represent addition of two fractions and to find the value of the sum. Student Facing, “1. Use a number line to represent each addition expression and to find its value. a. \frac{5}{8}+\frac{2}{8}, b. \frac{1}{8}+\frac{9}{8}, c. \frac{11}{8}+\frac{9}{8}, d. 2\frac{1}{8}+\frac{4}{8}. 2. Priya says the sum of 1\frac{2}[5} and \frac{4}{5} is 1\frac{6}{5}. Kiran says the sum is \frac{11}{5}. Tyler says it is 2\frac{1}{5}. Do you agree with any of them? Explain or show your reasoning. Use one or more number lines if you find them helpful.” Lesson 9, Differences of Fractions, Cool-down, students use number lines to represent subtraction of a fractions with the same denominator, including mixed numbers. Student Facing, “Use a number line to represent each difference and to find its value. a. \frac{12}{5}-\frac{4}{5}. b. 2\frac{1}{5}-\frac{7}{5}.”

• Unit 7, Angles and Angle Measurement, Section B, Lessons 7, 8, and 11 engage students in the full intent of 4.MD.5a (An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through \frac{1}{360} of a circle is called a "one-degree angle," and can be used to measure angles). Lesson 7, The Size of Angles on a Clock, Activity 2, Student Work Time, students use the clock as a tool for reasoning and for talking about “how much” of a turn. Student Facing, “1. Here are some angles formed by the two hands of a clock. In each pair of angles, which angle is larger? Explain or show your reasoning. a. 5:00, 3:00, b. 1:15, 1:20, c. 2:50, 11:20, d. 8:58, 9:35. 2. How large is this angle? Describe its size in as many ways as you can.” A clock shows 12:20. Lesson 8, The Size of Angles in Degrees, Activity 1, Student Work Time, students compare angles on clocks and use degrees as a unit of measure. Student facing, “A ray that turns all the way around its endpoint and back to its starting place has made a full turn. We say that the ray has turned 360 degrees. 1. How many degrees has the ray turned from where it started? (part a shows a 180 degree angle, b shows a 90 degree angle and c shows a 270 degree angle) 2. Sketch two angles: a. an angle where a ray has turned 50\degree b. an angle where a ray has turned 130\degree.” Lesson 11, Use a Protractor to Draw Angles, Warm-up: Estimation Exploration: Long Hand and Short Hand, Student Work Time, students estimate the measure of an angle on a clock face using what they have learned about angles. Student Facing, “How many degrees is the angle formed by the long hand and the short hand of the clock? Make an estimate that is: too high, just right, too low.” An unlabeled clock face shows an angle that is about 3:40 for reference.

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65% of instructional time to the major clusters of the 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, approximately 67%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 117 out of 158, approximately 74%. The total number of lessons devoted to major work of the grade include: 109 lessons plus 8 assessments for a total of 117 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 117 out of 155, approximately 75%.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 74% of the instructional materials focus on major work of the grade.

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers on a document titled “Lessons and Standards” found within the Course Guide tab for each unit. Connections are also listed on a document titled “Scope and Sequence”. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Section B, Lesson 13, Activity 2, Student Work Time, connects the supporting work of 4.MD.4 (Make a line plot to display a data set of measurements in fractions of a unit [\frac{1}{2}, \frac{1}{4}, \frac{1}{8}]. Solve problems involving addition and subtraction of fractions by using information presented in line plots) to the major work of 4.NF.3d (Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators). Students create a line plot using measurements to the nearest \frac{1}{4} and \frac{1}{8} inch and use their understanding of fraction equivalence to plot and partition the horizontal axis. Student Facing states, “1. Andre’s class measured the length of some colored pencils to the nearest \frac{1}{4} inch. The data are shown here: 1\frac{3}{4}, 2\frac{1}{4}, 5\frac{1}{4}, 5\frac{1}{4}, 4\frac{2}{4}, 4\frac{2}{4}, 6\frac{1}{4}, 6\frac{3}{4}, 6\frac{3}{4}, 6\frac{3}{4} a. Plot the colored- pencil data on the line plot. b. Which colored-pencil length is the most common in the data set? c. Write 2 new questions that could be answered using the line plot data. 2. Next, Andre’s class measured their colored pencils to the nearest \frac{1}{8} inch. The data are shown here: 1\frac{6}{8}, 2\frac{2}{8}, 5\frac{2}{8}, 5\frac{4}{8}, 4\frac{4}{8}, 4\frac{4}{8}, 6\frac{6}{8}, 6\frac{6}{8}, 6\frac{6}{8}, 6\frac{4}{8} a. Plot the colored-pencil data on the line plot. b. Which colored-pencil length is the most common in the line plot? c. Why did some colored-pencil lengths change on this line plot? d. What is the difference between the length of the longest colored pencil and the shortest colored pencil? Show your reasoning.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 17, Section C, Activity 2, Student Work Time, connects the supporting work of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems) to the major work of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison). Students consolidate their learning from the past few units to solve problems about length measurements in a mathematical context. Student Facing states, “Your teacher has posted six quadrilaterals around the room. Each one has a missing side length or a missing perimeter. a. Choose two diagrams—one with a missing length and another with a missing perimeter. Make sure that all six shapes will be visited by at least one person in your group. Find the missing values. Show your reasoning and remember to include the units. b. Discuss your responses with your group until everyone agrees on the missing measurements for all six figures. c. Answer one of the following questions. Explain or show your reasoning. 1. The perimeter of B is how many times the perimeter of D? 2. The perimeter of one figure is 1,000 times that of another figure. Which are the two figures? 3. The perimeter of F is how many times the perimeter of B?”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers,Section A, Lesson 4, Activity 2, Student Work Time, connects the supporting work of 4.OA.5 (Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself) to the major work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). In the activity, students continue to analyze patterns in numbers and use them to look at the relationship between the multiples of 99 and multiples of 100. Student Facing states, “Andre’s class did a choral count by 99. Here are the first six numbers they said. a. Study the list of numbers. Make at least 3 observations about features of the pattern.” The list of numbers shows, “counting by 99: 99, 198, 297, 396, 495, 594.” Students then answer, “b. Extend the list with the next four multiples of 99. Be prepared to discuss how you know what numbers to write. c. Why do you think the digits in the numbers change the way they do?”

The instructional materials for Open Up Resources K–5 Math 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. 

Materials are coherent and consistent with the Standards. These connections can be listed for teachers in one or more of the four phases of a typical lesson:  instructional activities, lesson synthesis, or Cool-down. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Section C, Lesson 15, Activity 1, Student Work Time, connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on a whole number). Students reason about problems that involve combining or removing fractional amounts with different denominators in the context of stacking playing bricks. Student Facing states, “Priya, Kiran, and Lin are using large playing bricks to make towers. Here are the heights of their towers so far: Priya: 21\frac{1}{4} inches, Kiran: 32\frac{3}{8} inches, Lin: 55\frac{1}{2} inches. For each question, show your reasoning. 1. How much taller is Lin’s tower compared to: a. Priya’s tower? b. Kiran’s tower? 2. They are playing in a room that is 109 inches tall. Priya says that if they combine their towers to make a super tall tower, it would be too tall for the room, and they’ll have to remove one brick.  Do you agree with Priya? Explain your reasoning.”

• Unit 5, Multiplicative Comparison and Measurement, Section C, Lesson 15, Activity 1, Student Work Time, connects the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers) to the major work of 4.OA.A (Use the four operations with whole numbers to solve problems). Students analyze length measurements listed in a chart, perform multiplication, and convert distances involving fractional amounts in order to compare them. Student Facing states, “Six students were throwing frisbees on field day. Here is some information about each person’s first throw. Elena’s frisbee went 3 times as far as Clare’s did. Andre’s frisbee went 4 times as far as Tyler’s did. a. Complete the table with Elena and Tyler’s distances. Explain or show your reasoning. b. Who are the top 3 throwers for that round? Find out by listing the students and their distances in feet and in order, from longest to shortest.” Values in the table show: Han 17 yards, Lin 51\frac{1}{2} feet, Clare 21\frac{1}{3} feet, Andre 22 yards 2 feet, and Elena and Tyler are blank.

• Unit 7, Angles and Angle Measurement, Section A, Lesson 5, Activity 1, Student Work Time and Activity Synthesis, connects the supporting work of 4.G.A (Draw and identify lines and angles, and classify shapes by properties of their lines and angles) to the supporting work of 4.MD.C (Geometric measurement: Understand concepts of angle and measure angles). Students work with a partner to replicate images of angles as they use the vocabulary they have learned to describe figures. In Student Work Time, Student Facing states, “1. Work with a partner in this activity. Choose a role: A or B. Sit back to back, or use a divider to keep one person from seeing the other person’s work. Partner A: Your teacher will give you a card. Don’t show it to your partner. Describe both images on the card - as clearly and precisely as possible—so that your partner can draw the same images. Partner B: Your partner will describe two images. Listen carefully to the descriptions. Create the drawings as described. Follow the instructions as closely as possible. 1. When done, compare the drawings to the original images. Discuss: Which parts were accurate? Which were off? How could the descriptions be improved so the drawing could be more accurate? 2. Switch roles and repeat the exercise. Compare the drawings to the original images afterwards. 2. If you have time: Request two new cards from your teacher (one card at a time). Take turns describing and drawing the geometric figure on each card.” Activity Synthesis states, “‘How are the two drawings on each card the same?’ (They each have 2 rays. The rays start at the same point. One ray is pointing in the same direction in both drawings.) ‘How are they different?’ (The rays are pointing in different directions on some cards. The rays are farther apart in some cards.) ‘How did you describe what you saw? What terms did you use to help you describe the directions of the rays?’ (We tried to explain by describing the hands on a clock. We tried using words like north, south, east, and west. We described them in relation to vertical and horizontal.) As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations. Remind students to borrow language from the display as needed. ‘Did anyone use the term “angle?” Did anyone measure something or use measurements? The figures that you drew are angles. An angle is a figure that is made up of two rays that share the same endpoint. The point where the two rays meet is called the vertex of the angle.’”

• Unit 9, Putting It All Together, Section C, Lesson 9, Activity 1, Launch and Student Work Time, connects the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic) to the major work of 4.OA.A (Use the four operations with whole numbers to solve problems). Students analyze a situation and solutions in order to think about what questions were asked. The Launch states, “Groups of 2. ‘Have you ever gone on a long hike? What is the longest distance you ever traveled just by walking? Let’s look at the work a student did to answer questions about two men who set world records for traveling by walking.’” In Student Work time, Student Facing states, “George Meegan walked 19,019 miles between 1977 and 1983. He finished at age 31. He wore out 12 pairs of hiking boots. Jean Beliveau walked 46,900 miles between 2000 and 2011 and finished at age 56. Here are the responses Kiran gave to answer some questions about the situation. Write the question that Kiran might be answering. In the last row, write a new question about the situation and show the answer, along with your reasoning.”

The instructional materials reviewed for Open Up Resources K–5 Math 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. 

Prior and Future connections are identified within materials in the Course Guide, Scope and Sequence Section, within the Dependency Diagrams which are shown in Unit Dependency Diagram, and Section Dependency Diagram. An arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section. While future connections are all embedded within the Scope and Sequence, descriptions of prior connections are also found within the Preparation tab for specific lessons and within the notes for specific parts of lessons. 

Examples of connections to future grades include:

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section B, Lesson 11, Preparation connects the work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to work with multiplying multi-digit whole numbers using the standard algorithm in 5.NBT.5. Lesson Narrative states, “This lesson extends students’ analysis to include the standard algorithm for multiplication of multi-digit numbers. In grade 4, the standards focus on understanding place value and how it is represented in different methods for finding products. The work here serves to build the groundwork for making sense of the standard algorithm in grade 5, so students are not expected to use the standard algorithm at this time.”

• Unit 8, Properties of Two-dimensional Shapes, Section A, Lesson 3, Activity 2 connects 4.G.2 (Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.) to work with classifying two-dimensional figures in 5.G.B. Narrative states, “Students are not expected to recognize that the attributes of one shape may make it a subset of another shape (for example, that squares are rectangles, or that rectangles are parallelograms). They may begin to question these ideas, but the work to understand the hierarchy of shapes will take place formally in grade 5. During the synthesis, highlight how sides and angles can help us define and distinguish various two-dimensional shapes.”

• Unit 9, Putting It All Together, Section A, Lesson 2, Preparation connects 4.NF.A (Extend understanding of fraction equivalence and ordering) and 4.NF.B (Build fractions from unit fractions) to work adding and subtracting fractions with unlike denominators in 5.NF.1. Lesson Narrative states, “In this lesson, students apply what they know about equivalence and addition and subtraction of fractions to solve problems. Throughout the lesson, students have opportunities to reason quantitatively and abstractly as they connect their representations, including equations, to the situations (MP2) and to compare their reasoning with others' (MP3). The work of this lesson helps prepare students for adding and subtracting with unlike denominators in grade 5. If students need additional support with the concepts in this lesson, refer back to Unit 3, Section B in the curriculum materials.”

Examples of connections to prior knowledge include:

• Unit 1, Factors and Multiples, Section A, Lesson 1, Preparation connects 4.OA.4 (Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite) to work with concepts of area from 3.MD.7a. Lesson Narrative states, “In grade 3, students learned how to find the area of a rectangle by tiling and found that multiplying the side lengths yields the same result. The purpose of this lesson is for students to apply their understanding of area and multiplication to build rectangles and find their area. As students consider the areas of rectangles with a given side length, they explore the idea of multiples. Students learn that a multiple of a number is the result of multiplying that whole number by another.”

• Grade 4 Course Guide, Scope and Sequence, Unit 2, Fraction Equivalence and Comparison, Unit Learning Goals connects 4.NF.A (Extend understanding of fraction equivalence and ordering) to work with unit fractions from Grade 3. Narrative states, “In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \frac{1}{b} results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line. Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100….As the unit progresses, students use equivalent fractions and benchmarks such as \frac{1}{2} and 1 to reason about the relative location of fractions on a number line, and to compare and order fractions.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section B, Lesson 5, Activity 1 connects 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to work with concepts of multiplication from Grade 3. Narrative states, “In this activity, students build on grade 3 work with arrays to consider how to find the total number in an array without counting by 1. Students are not asked to find the answer, but instead share their strategies for doing so. This allows teachers to observe how students make sense of multiplying larger numbers.”