4th Grade - Gateway 1

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 4 are divided into nine units, and each unit contains a written End-of-Unit Assessment. Additionally, the Unit 9 Assessment is an End-of-Course Assessment, and it includes problems from the entire grade level. Examples of End-of-Unit Assessments include:

• Unit 2, Fraction Equivalence and Comparison, End-of-Unit Assessment, Problem 4, “List three different fractions that are equivalent to \frac{4}{5}. Explain or show your reasoning.” (4.NF.1)

• Unit 3, Extending Operations to Fractions, End of Unit Assessment, Problem 7, “Find the value of each expression. 1. \frac{7}{100}+\frac{8}{100}+\frac{1}{10} 2. \frac{3}{10}+\frac{17}{10}+\frac{2}{10} 3. \frac{14}{100}+\frac{5}{10}+\frac{26}{100}.” (4.NF.5)

• 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 2, “Select all correct statements. A.There are 360 one-degree angles in a circle. B.There are 180 one-degree angles in a circle. C.There are 90 one-degree angles in a right angle. D.There are 180 one-degree angles in a right angle. E.There are 4 right angles in a circle.” (4.MD.5, 4.MD.7)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials provide extensive work with and opportunities for students to engage in the full intent of Grade 4 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 4, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 12, Ways to Compare Fractions, engages students with extensive work with 4.NF.2 (Extend understanding of fraction equivalence and ordering). Activity 1: The Greatest of Them All, students compare two fractions that share the same denominator or the same numerator. Student Task Statements, “Here are 25 fractions in a table. For each question, be prepared to explain your reasoning. 1. Identify the greatest fraction in each column (A, B, C, D, and E). 2. Identify the greatest fraction in each row (1, 2, 3, 4, and 5). 3. Which fraction is the greatest fraction in the entire table?’ The table includes the following fractions: \frac{2}{3}, \frac{2}{5}, \frac{2}{10}, \frac{2}{12}, \frac{2}{100}, \frac{4}{3}, \frac{4}{5}, \frac{4}{10}, \frac{4}{12}, \frac{4}{100}, \frac{7}{3}, \frac{7}{5}, \frac{7}{10}, \frac{7}{12}, \frac{7}{100}, \frac{11}{3}, \frac{11}{5}, \frac{11}{10}, \frac{11}{12}, \frac{11}{100}, \frac{26}{3}, \frac{26}{5}, \frac{26}{10}, \frac{26}{12}, \frac{26}{100}.” Cool-down: Pick the Greater Fraction, Student Task Statements, students use strategies to compare and order common fractions. “In each pair of fractions, which fraction is greater? Explain or show your reasoning. \frac{5}{12} and \frac{5}{8}, \frac{11}{10} and \frac{18}{100}, \frac{6}{10} and \frac{7}{12}.”

• Unit 5, Multiplicative Comparison and Measurement, Lessons 1, 2, and 3 engage students with extensive work with grade-level problems from 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 1, Times as Many, Activity 2, Times As Many, students extend the idea of representing “twice as many” to represent “4, 6, and 8 times as many. “Draw a picture to show the number of cubes the students have in each situation. Be prepared to explain your reasoning.’ Problem 1, ‘Andre has the following cubes and Han has 4 times as many.’ (image of 5 linking cubes with the name “Andre” next to it, and the name “Han” with empty space next to it.)” Lesson 2, Interpret Representations of Multiplicative Comparison, Warm-Up: How Many Do You See: Times As Many, students use grouping strategies to describe the images they see. “‘How many do you see and how do you see them?’ Flash the image. (Images- 3 connected blue rectangles horizontal; 6 connected rectangles horizontal, 3 blue, 3 orange; 12 connected rectangles horizontal, 3 blue, 3 orange, 3 blue, 3 orange).” Lesson 3, Solve Multiplicative Comparison Problems, Activity 1: A Book Drive, students rely on the relationship between multiplication and division to solve multiplicative comparison problems. “‘This diagram shows the books Lin and Diego donated for the school book drive (diagram shows a discrete tape diagram to represent Lin’s books (16) and Diego’s books (4)).’ Problem 1, ‘Lin donated 16 books. Diego donated 4 books. How many times as many books did Lin donate as Diego did? Explain or show your reasoning. Use the diagram if it is helpful.’”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 20, Interpret Remainders in Division Situations, engages students with extensive work with grade-level problems with 4.OA.3 (Use the four operations with whole numbers to solve problems). Activity 2: Save for a Garden, students solve word problems with remainders. Student Task Statements, “1. A school needs $1,270 to build a garden. After saving the same amount each month for 8 months, the school is still short by $6. How much did they save each month? Explain or show your reasoning. 2. Choose one of the following division expressions. 711\div3, 3128\div8. a. Write a situation to represent the expression. b. Find the value of the quotient. Show your reasoning. c. What does the value of the quotient represent in your situation?”

Examples of full intent include:

• Unit 3, Extending Operations to Fractions, Lessons 8, Addition of Fractions, and Lesson 9, Differences of Fractions, 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). In Lesson 8, Addition of Fractions, Activity 2, What is the Sum?, Student Task Statements, students use number lines to represent addition of two fractions and to find the value of the sum. “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}.” In Lesson 9, Differences of Fractions, Cool-down: Differences of Fifths, students use number lines to represent subtraction of a fraction by another fraction with the same denominator including a mixed number and by a whole number. “Use a number line to represent each difference and to find its value. 1. \frac{12}{5}-\frac{4}{5}. 2.  2\frac{1}{5}-\frac{7}{5}.” 

• Unit 4, From Hundredths to Hundred-thousands, Lessons 16, Round Numbers, and Lesson 17, Apply Rounding,  provides the opportunity for students to engage with the full intent of standard 4.NBT.3 (Use place value understanding to round multi-digit whole numbers to any place). Lesson 16, Round Numbers, Activity 1: Round to What, students connect the idea of “nearest multiple” to rounding. “Noah says that 489,231 can be rounded to 500,000. Priya says that it can be rounded to 490,000. 1. Explain or show why both Noah and Priya are correct. Use a number line if it helps. 2. Describe all the numbers that round to 500,000 when rounded to the nearest hundred-thousand. 3. Describe all the numbers that round to 490,000 when rounded to the nearest ten-thousand. 4. Name two other numbers that can also be rounded to both 500,000 and 490,000.” Lesson 17, Apply Rounding, Activity 1: Apart in the Air, students make sense of a situation and decide how to round the quantities. Problem 2, “Planes flying over the same area need to stay at least 1,000 feet apart in altitude. Mai said that one way to tell if planes are too close is to round each plane's altitude to the nearest thousand. Do you agree that this is a reliable strategy? In the last column, round each altitude to the nearest thousand. Use the rounded values to explain why or why not. A 2 column table included with labels “plane” in column 1 and “altitude (feet)” in column 2. The following altitudes are listed: WN11-35,625; SK51-28,999; VT35-15,450; BQ 64-36,000; AL16-31,000; AB25-35,175; CL48-16,600; WN90-30,775; NM44-30,245”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 16, Compare Perimeters of Rectangles, and Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 22, Problems about Perimeter and Area, 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, Rectangles Y and Z, students reason about the perimeter of rectangles. “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.” In Unit 6 Lesson 22, Problems About Perimeter and Area, Activity 2: Replace the Classroom Carpet, Student Task Statements,  students apply the formulas for area and perimeter. “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. 1. How many feet of baseboard will they need to replace in the classroom? How many inches is that? 2. 1,200 inches of baseboard material was delivered. Is that enough? 3. How many square feet of carpet will be needed to cover the floor area?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to the major work of the grade (including assessments and supporting work connected to 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 major work) is 121 lessons out of 158 lessons, approximately 77%. The total number of lessons include 113 lessons plus 8 assessments for a total of 121 lessons. 

• The number of days devoted to major work of the grade (including assessments and supporting work connected to major work) is 119 days out of 167 days, approximately 71%.

The lesson-level analysis is the most representative of the instructional materials, as the lessons include major work, supporting work connected to major work, and assessments in each unit.  As a result, approximately 77% of the instructional materials focus on major work of the grade.

The materials reviewed for Imagine Learning Illustrative Mathematics 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 with supporting standards/clusters connected to the major standards/clusters of the grade. These connections are listed for teachers on a document titled, “Pacing Guide and Dependency Diagram” found on the Course Guide tab for each Unit. Teacher Notes also provide the explicit standards listed within the lessons. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Lesson 14, Problems About Fractional Measurement Data, Activity 2, Larger Shoes, Anyone? 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) with the major work of 4.NF.3 (Understand a fraction \frac{1}{b} with a>1 as a sum of fractions \frac{1}{b}). Student Task Statement Problem 2, “If Han’s shoe length now is 9\frac{1}{8} inches, what was his shoe length in third grade?”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 10, Multi-step Measurement Problems, Cool-down: Hydration Here and There, connects the supporting work of 4.MD.2, (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, adn problems that require expression measurements given in a larger unit in terms of a smaller unit), to the major work of 4.OA.3, (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted). Students use conversions to solve multi-step problems. Student Task Statements, “Halfway through a soccer game, Han drank 210 mL of water. At the end of the game, he drank 4 times as much as he did at halftime. Did Han drink more or less than 1 L of water in total? Explain or show your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 12, Solve Problems Involving Multiplication, Cool-down, Leap Year, connects the supporting work of 4.MD.2, (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expression measurements given in a larger unit in terms of a smaller unit), to the major work of 4.OA.3, (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted). Students use multiplication strategies to solve problems. Student Task Statements, “In a leap year, the month of February has 29 days. How many hours are in that month? Show your reasoning.”

The materials reviewed for Imagine Learning Illustrative 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. 

Materials are coherent and consistent with the Standards. Examples of connections between major work to major work and/or supporting work to supporting work throughout the materials, when appropriate, include:

• Unit 3, Extending Operations to Fractions, Lesson 15, An Assortment of Fractions, Cool-down, 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 Whole Numbers). Students add two sets of fractions and decide which set is greatest: “Which stack of foam blocks is taller: Two \frac{1}{3}-foot blocks and one \frac{1}{6}-foot block, or One \frac{1}{2}-foot block and two \frac{1}{6}-foot blocks?”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 24, Assess the Reasonableness of Solutions connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). In Activity 2, Languages in Philadelphia and Chicago, students use place value understanding with the four operations to solve problems. Problem 2, “What is the difference between the number of people who speak only English and those who speak another language? Show how you know (table has two columns; first column- language and second column- number of people in Philadelphia: English only-1,224,539; Spanish-127,352; Other Indo-European-6,750; Asian-364).”

• Unit 7, Angles and Angle Measurements, Lesson 5, What is an Angle?, Activity 3: Discover Angles 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) as students identify and sketch angles. Student Task Statements, “Here are two figures. 1. Find 2–3 angles in each figure. Draw pairs of rays to show the angles. 2. Sketch a part of your classroom that has 2–3 angles. Draw pairs of rays to show the angles.” An image shows the number 7 and letter K.

• Unit 9, Putting It All Together, Lesson 12, Number Talk, Activity 1: Related Numbers, Related Expressions connects the major work 4.NBT.A (Generalize place value understanding for multi-digit whole numbers) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). Students decompose, rearrange, and regroup numbers or make use of structure to find the value of expressions. Student Task Statements, “1. Here are two addition expressions. Think of at least two different ways to find the value of each sum mentally. a. 15+29 b. 30+58. 2. Here are three subtraction expressions. Think of at least two different ways to find the value of each difference mentally. a. 91-11. b 91-16.  c. 391-86. 3. Can you write a fourth subtraction expression that uses the same strategy you used to find the value of the other differences?”

The materials reviewed for Imagine Learning Illustrative 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. 

The Section Dependency Chart explores the Unit sections relating to future grades. The Section Dependency Chart states, “arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” 

Examples of connections to future grades include:

• Unit 5, Multiplicative Comparison and Measurement, Lesson 11, Pounds and Ounces, Activity 2, Party Prep, connects 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit…) and 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding) to the work in grade 5. “In the first activity, students learned that one pound is 16 times as heavy as 1 ounce. Here they apply this knowledge to convert quantities into ounces and to solve multi-step problems. The quantities include a fractional number of pounds and one expressed in a combination of pounds and ounces. As in earlier lessons in which they encountered a fractional amount of a unit of measurement, students are not expected to find the number of ounces in \frac{1}{2} pound by writing \frac{1}{2}\times16. Instead, they can reason about half of a quantity using their understanding of fractions and by dividing an amount by 2. Those who do write \frac{1}{2}\times16 represent the situation correctly, but this reasoning and the related operation will be developed in grade 5.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 3, Lesson Preparation connects 4.G.1 (Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures), 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 in grade 5. About this lesson, “In this lesson, students identify and sort quadrilaterals based on their angles and sides, including whether their sides are parallel. Students are introduced to the term parallelogram to describe quadrilaterals with two pairs of parallel sides, but they are not expected to use this term throughout the unit. In grade 5, students will continue the work of classifying polygons using these categories.”

• Unit 9, Putting It All Together, Lesson 2, Lesson Preparation connects 4.NF.A (Extend Understanding Of Fraction Equivalence And Ordering) to work in grade 5. About this lesson, “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.”

Examples of connections to prior knowledge include:

• Unit 1, Factors and Multiples, Lesson 1, Multiples of a Number, Activity 1: Build Rectangles and Find Area 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…) to the work in grade 3, “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 (3.MD.7).”

• Unit 3, Fraction Operations to Fractions, Lesson 1, About this lesson, connects 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.) with work done in grade 3. “In grade 3, students represented multiplication of whole numbers with arrays, equal-group drawings, area diagrams, and expressions. In an earlier unit, students used diagrams to represent and compare fractions. In this unit, they extend their understanding of multiplication to include equal groups of unit fractions while using familiar representations to support their thinking.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 16, Round Numbers, About this lesson, connects 4.NBT.3 (Use place value understanding to round multi-digit whole numbers to any place.) to work done in grade 3. “In grade 3, students rounded whole numbers to the nearest 10 and 100. In previous lessons, they worked to find the closest multiples of powers of 10. Here, students build on this work to round whole numbers to the nearest 1,000, 10,000, and 100,000. Students revisit the convention of rounding up when a number is exactly halfway between two consecutive multiples of a power of 10.”