5th Grade - Gateway 1

The materials reviewed for Everyday Mathematics 4, Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Summative Interim Assessments include Beginning-of-Year, Mid-Year, and End-of-Year. Unit Assessments found at the end of each unit assess the standards of focus for the unit. Open Response Assessments found at the end of odd-numbered units provide tasks addressing one or more content standards. Cumulative Assessments found at the end of even-numbered units include items addressing standards from prior units.

Materials assess grade-level standards. Examples include:

• Unit 1 Assessment, Item 7, “How many cubes would it take to fill this prism? ____ cubes; What is the volume of this prism? ____ cubic units.” A rectangular prism partially filled with cubes is shown. (5.MD.3, 5.MD.4)

• Unit 2 Assessment, Item 3, “a. Jesse collects cans for recycling. When he has 1,500 cans, the recycling center will pick them up from his house. Jesse has 120 bags with about 35 cans in each bag. Should he call the recycling center to arrange a pick-up? Explain how you know. b. Did you have to find an exact answer to solve Problem 3a? Explain why or why not.” (5.NBT.5)

• Unit 4 Assessment, Item 19, “Gina is donating money to her neighborhood food pantry. Her aunt agreed to donate two dollars more than Gina donates. The table below shows some of the possible amounts of money they may donate. a. Write the data in the table above as ordered pairs. b. Plot the ordered pairs as points and use a straightedge to connect them.” A table of values is shown. (5.G.2) 

• End-Of-Year Assessment, Item 12, “Reed’s class is painting a giant chessboard on the playground. A chessboard consists of 64 squares arranged in 8 rows and 8 columns. His class is making each square \frac{1}{3} m by \frac{1}{3} m. a. What will be the length and width of the chessboard in meters? Show your work. b. What will be the area of the completed chessboard? Show your work. Give your answer in square meters. c. How could you use the number of squares on the chessboard to find the area of the chessboard in square meters?” (5.NF.4b)

The materials reviewed for Everyday Mathematics 4, Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Instructional materials engage all students in extensive work with grade-level problems. Each lesson provides opportunities during Warm Up, Focus Activities, and Practice. Examples include:

• Lesson 1-10, Visualizing Volume Units, Focus: Converting Volume Units, Math Journal 1, Problem 3, students convert between cubic units, ”a. How many cubic inches do you think are in a cubic foot? b. How many inches are in a foot? c. How many square inches are in a square foot? How did you find your answer? d. How many cubic inches are in a cubic foot? How did you find your answer?” Lesson 2-6, Application: Unit Conversions, Teacher’s Lesson Guide, Focus, Converting Miles to Feet, students convert miles to feet using a provided chart, “Display a two-column table labeled Miles and Feet. Fill in the numbers 1-5 in the Miles Column and complete the first two rows of the Feet column. How many feet of fencing would the rangers need for a 3-mile section of path?” The Table Chart shows Miles 1, 2, 3, 4, 5 and Feet 5,280, 10,560, ?, ?, ?. Students engage in extensive work with grade-level problems for 5.MD.1, “Convert like measurement units within a given measurement system.”

• Lesson 2-12, Strategies for Choosing Partial Quotients, Focus: Using Partial-Quotients Division with Lists of Multiples, Math Journal 1, students use lists of multiples to find partial quotients and use area models, “For Problems 1-4, make an estimate. Then use partial-quotients division to solve. Show your work. You make a list of multiples on Math Masters, page TA10 to help you. 1. 1,647 / 28 ? Estimate: ___ Answer: ___. 2. 4,319 / 42 ? Estimate: ___ Answer: ___. 3. 2,628 / 36 ? Estimate: ___ Answer: ___. 4. 9,236 / 41 ? Estimate: ___ Answer: ___. 5. Paul drew the area model at the right for his solution to Problem 1. What partial quotients did he use to solve the problem?” Lesson 4-4, Comparing and Ordering Decimals, Practice, Interpreting Real-World Remainders, Math Journal 1, students interpret remainders, “Solve each number story. Show your work. Explain what you decided to do with the remainder. 1. Bre earned 189 tickets playing different games at the fair. If each prize cost 15 tickets, how many prizes can Bre get? Number Model: ___. Quotient: ___. Answer: Bre can get ___ prizes. Circle what you did with the remainder. Ignored it, Reported it as a fraction, Rounded the quotient up. Why? ___.” Students engage in extensive work with grade-level problems for 5.NBT.6, “Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.”

• Lesson 5-9, Understanding an Algorithm for Fraction Multiplication, Focus: Multiplying Fractions, Math Journal 2, students practicing using a fraction multiplication algorithm, “1. In your own words, describe a method for multiplying fractions discovered in class. Use the fraction multiplication algorithm described above to solve Problems 2-7. 2. \frac{1}{2}\star\frac{3}{6}. 3. \frac{2}{3}\star\frac{1}{4}. 4. \frac{3}{5}\star\frac{1}{6}. 5. \frac{3}{4}\star\frac{3}{8}. 6. \frac{2}{5}\star\frac{4}{10}. 7. \frac{7}{9}\star\frac{2}{12}. For Problems 9 and 10, write a number model. Then solve. 9. Sheila had \frac{3}{4} pound of blueberries. She used \frac{1}{3} of them in a fruit salad. How many pounds of blueberries did she use? Number model: ___. Answer: ___ pound. 10. the mirror in a dollhouse is \frac{2}{4} inch wide and \frac{3}{4} inch tall. What is the area of the mirror in square inches? Number model: ___. Answer: ___ square inch.” Students engage in extensive work with grade-level problems for 5.NF.4, Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 5 standards through a consistent lesson structure. According to the Teacher’s Lesson Guide, Problem-based Instruction “Everyday Mathematics builds problem-solving into every lesson. Problem-solving is in everything they do. Warm-up Activity- Lessons begin with a quick, scaffolded Mental Math and Fluency exercise. Daily Routines - Reinforce and apply concepts and skills with daily activities. Math Message - Engage in high cognitive demand problem-solving activities that encourage productive struggle. Focus Activities - Introduce new content with group problem-solving activities and classroom discussion. Summarize - Discuss and make connections to themes of the focus activity. Practice Activities - Lessons end with a spiraled review of content from past lessons.” Examples of meeting the full intent include:

• Lesson 1-9, Two Formulas for Volume, Focus: Find a Second Volume Formula, Practice: Math Journal 1, students use multiplication to solve for volume, “Use a formula to find the volume of each prism. Record the formula you used. 1. A rectangular prism is shown with 8 cubes in length, 5 cubes for width, and 4 cubes to show height. Volume: ____. Formula: ____.” Lesson 1-11, Volume Exploration, Focus, Estimating Volumes of Musical Instrument Cases, Math Journal 1, students find the volume of figures composed of rectangular prisms by adding the volumes of prisms, “In Problems 1-3, use the mathematical models to estimate the volume of the instrument cases. 1. Trombone case. The volume of the trombone case is about ___ in^3.” The measurements are 8 in., 4 in., 10 in., 9 in., 4 in., and 31 in. Try This, Problem 4, “Asher needs to take the xylophone, the trombone, and the French horn with him to a band concert. His trunk has 13 cubic feet of cargo space. Can he fit all three cases in his trunk? Explain how you know.” Students engage in the full intent of 5.MD.5, Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.”

• Lesson 4-1, Decimal Place Value, Reading and Writing Decimals, Practice: Math Journal 1, students read and write decimals to thousands using place value, “Write the following decimals in words. Use the place-value chart on journal page 112 to help you. 1. 0.67, 2. 3.8, 3. 3.622, 4. 0.804. Write each decimal using numbers. Record them on the place-value chart on page 112. Then write the value of 4 in each decimal. 5 a. four and eight tenths. b. 4 is worth ___. 6 a. forty-eight hundredths. b. 4 is worth. 7. a. forty-eight thousandths. b. 4 is worth. 8. a. six and four hundred eight thousandths. b. 4 is worth.” Lesson 4, Comparing and Ordering Decimals, Practice, Math Masters, students practice comparing decimals to thousandths, “Darryl and Charity are playing Decimal Top-It. Their record sheet is shown below. 1. Compare their decimals for each round and write >, <, or = in the middle column. Use the place-value chart above to help you. Round 1, Player 1- Darryl 0.378 ___ Player 2- Charity 0.860. Problem 3a. Put Darryl’s decimals in order from least to greatest. 0.378, 0.9, 0.804, 0.547, and 0.72. 3b. Put Charity’s decimals in order from least to greatest. 0.860, 0.59, 0.92, 0.6, and 0.098.” Students engage in the full intent of 5.NBT.3, “Read, write, and compare decimals to thousandths.” 

• Lesson 7-5, A Hierarchy of Triangles, Focus: Making a Triangle Hierarchy, Practice: Math Journal 2, students define a triangle hierarchy to classify triangle cards, “On the left, write the categories and subcategories from the triangle hierarchy you created in class. Use the hierarchy to classify your triangle cards. When you are finished, glue or tape the cards in place.” Students have Triangles, Isosceles triangles, and Equilateral triangles as their triangle hierarchy. Problem 4, “a. All ___ have three sides and three angles. b. All ___ have at least two sides of the same length. c. All ___ have a line of symmetry.” Lesson 7-6, A Hierarchy of Quadrilaterals, Practice, students use the quadrilateral hierarchy to answer questions. Math Masters, “1. Fill in the blanks. a. All ___ are ___, but not all ____ are ____. b. All ____ are ____, but not all ___ are ___. c. All ___ are ____ but not all ____ are ___. 2. a. All parallelograms have two pairs of parallel sides. Does this mean that all rectangles have two pairs of parallel sides? Explain how you can tell by looking at the hierarchy. b. All trapezoids have at least one pair of parallel sides. Explain how you can tell by looking at the hierarchy.” Students engage in the full intent of 5.G.4, “Classify two-dimensional figures in a hierarchy based on properties.”

The materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade. 

• There are 8 instructional units, of which 6 units address major work of the grade or supporting work connected to major work of the grade, approximately 75%.

• There are 113 lessons, of which 88.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 78%.

• In total, there are 170 days of instruction (113 lessons, 37 flex days, and 20 days for assessment), of which 95.5 days address major work of the grade or supporting work connected to the major work of the grade, approximately 56%. 

• Within the 37 Flex days, the percentage of major work or supporting work connected to major work could not be calculated because the materials suggested list of differentiated activities do not include explicit instructions. Therefore, it cannot be determined if all students would be working on major work of the grade.

A lesson analysis is most representative of the instructional materials. As a result, approximately 78% of the instructional materials focus on the major work of the grade.

The materials reviewed for Everyday Mathematics 4 Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Digital materials’ Main Menu links to the “Spiral Tracker” which provides a view of how the standards spiral throughout the curriculum. The Lesson Landing Page contains a Standards section noting standards covered by the lesson. Teacher Edition contains “Correlation to the Standards for Mathematics” listing all grade-level standards and correlating lessons. Examples include:

• Lesson 2-7, U.S, Traditional Multiplication, Part 3, Focus: Estimating and Multipying, students write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them (5.OA.2) and fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5). Teachers guide students through this problem, “Display the work shown to the right. Ask: Do you think 384 is the correct answer for 64 x 15? How do you know? Interpret the expression 64 x 15 without evaluating it. 64 x 15 is equivalent to a number that is 15 times as large as 64. Ten times as much as 64 is 640, so that means that 64 x 15 is greater than 640. Since 384 is less than 640, it cannot be the correct answer.” 

• Lesson 3-8, Renaming Fractions and Mixed Numbers, Math Journal 1, students write and interpret numerical expressions (5.OA.A) and apply and extend previous understandings of multiplication and division (5.NF.B). Students use an equal-grouping interpretation of division to explore connections between fractions and division. Problem 1 states, “Write a division expression to model each story, then solve. You can use fraction circles or draw pictures to help. Olivia is running a 3-mile relay race with 3 friends. If the 4 of them each run the same distance, how many miles will each person run?”

• Lesson 6-4, Line Plots, Focus: Creating and Interpreting a Line Plot with Fractional Pencil-Length Data, students make a line plot to display a data set of measurements in fractions of a unit (5.MD.2) and add and subtract fractions with unlike denominators (including mixed numbers) (5.NF.1). Students measure the length of their pencils to the nearest 1/4 inch and collect data about the lengths of their peers’ pencils. Then students use this data to make a line plot and answer questions about the data. Teacher prompt states, “What is the difference in length between the longest pencil and the shortest pencil?” 

• Lesson 8-1, Planning an Athletic Center, Focus: Finding Areas of Playing Surfaces, Math Journal 2, students convert like measurements within a given measurement system (5.MD.1) and fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5). Problem 2 states, “The town has 4 acres of land to use for an athletic center. The land is a rectangle with a length of 160 yards and a width of 121 yards. The town wants the athletic center to have a variety of sports playing surfaces. You have been asked to help decide which playing surfaces should be included and how the surfaces should be arranged. Explain how you and your group created your plan.”

The materials reviewed for Everyday Mathematics 5 Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

The Teacher Edition contains a Focus section in each Section Organizer identifying major and supporting clusters covered. There are connections from supporting work to supporting work and major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Lesson 4-11, Addition and Subtraction of Decimals with Hundredths Grid, Math Journal 1, Problems 1 and 2, students use grids to solve decimal addition and subtraction problems, “Shade the grid in one color to show the first addend. Shade more of the grid in a second color to show the second addend. Write the sum to complete the number sentence.” 1. “$$0.6+0.22=$$ ____? 2. 0.18+0.35= ____?” This connects the major work of 5.NBT.A, “Understand the place value system” to the major work of 5.NBT.B, “Perform operations with multi-digit whole numbers and with decimals to hundredths.”

• Lesson 5-10, Identifying and Visualizing Patterns, Math Journal 2, Problem 2, students compare rules from different problems and determine how the rules affect the graph, “a. A rule is given at the top of each column in the table below. Use the rules to fill in the columns. In (x) Rule: +6: 0, ?, ?, ?, ?. Out (y) Rule +2: 0, ?, ?, ?, ?. b. What rule relates each in number to its corresponding out number? 3. Think about the rules you used to fill in the in and out columns. Why did the rule you found in Part b make sense? d. Write the numbers from the table as ordered pairs. Then graph them. Draw a line to connect the points.” This connects the supporting work of 5.OA.B, “Analyze patterns and relationships” to the supporting work of 5.G.A, “Graph points on the coordinate plane to solve real-world and mathematical problems.”

• Lesson 6-2, Playing Exponent Ball: Focus: Exponent Ball, students learn to play a game to multiply and divide decimals by powers of 10, and compare decimals and whole numbers. students learn the game. Directions from the game include, “If the card is black, multiply your starting number by the power of 10. If the card is red or blue, divide your starting number by the power of 10. Write an expression to show how to multiply or divide your number. Then find the value of the expression.” This connects the major work of 5.NBT.A “Understand the place value system” to the major work of 5.NBT.B, “Perform operations with multi-digit whole numbers with decimals to hundredths.”

• Lesson 7-11, Rules, Tables, and Graphs, Part 1, Focus: Displaying Data on a Graph, students relate operations with fractions to operations with whole numbers. Teachers use the following example with students, “$$4\frac{2}{3}\star7=4\star7+\frac{2}{3}\star7$$.” This connects the major work of 5.NBT.B, “Perform operations with multi-digit whole numbers” to the major work of 5.NF.B, “Apply and extend previous understandings of multiplication and division.”

• Lesson 8-1, Planning an Athletic Center, Math Message, students perform operations with multi-digit whole numbers and apply and extend previous understandings of multiplication and division to multiply and divide fractions. Students convert measurements to find the area of an Olympic beach volleyball court, “The dimensions of an official Olympic volleyball court are 52 feet 6 inches by 26 feet 3 inches. Find the area of the court in square feet.” This connects the major work of 5.NBT.B, “Perform operations with multi-digit whole numbers and with decimals to hundredths” to the major work of 5.NF.B, “Apply and extend previous understandings of multiplication and division to multiply and divide fractions.”

• Lesson 8-2, Applying the Rectangle Method for Area, Math Journal 2, Problems 3-6, students create a table, rule, and graph to model a situation, “Write the numbers from the table in Problem 1 as ordered pairs. Remember to use parentheses and a comma in each ordered pair. Then graph the ordered pairs on the coordinate grid. Draw a line to connect the points. 4. Use the table. How much will 3 pounds of supplies cost? 5. Use the rule from 2a. How much will 10 pounds of apples cost? 6. Use the graph. How much will 6 pounds of apples cost?” This connects the supporting work of 5.OA.B, “Analyze patterns and relationships” to the supporting work of 5.G.A, “Graph points on the coordinate plane to solve real-world and mathematical problems.”

The materials reviewed for Everyday Mathematics 4 Grade 5 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.

Materials relate grade-level concepts to prior knowledge from earlier grades. Each Section Organizer contains a Coherence section with “Links to the Past” containing information about how focus standards developed in prior units and grades. Examples include:

• Unit 1, Area and Volume, Teacher’s Lesson Guide, Links to the Past, “5.OA.1: In Grade 3, students inserted parentheses into number sentences and solved number sentences containing parentheses.” 

• Unit 5, Operations with Fractions, Teacher’s Lesson Guide, Links to the Past,”5.NF.5, 5.NF.5a: In Unit 4, students made conjectures about how an image on a coordinate grid would change based on multiplying one or more of the coordinates. In Grade 4, students interpreted multiplication equations as comparisons.” 

• Unit 7, Multiplication of Mixed Numbers; Geometry; Graphs, Teacher’s Lesson Guide, Links to the Past, “5.NF.4, 5.NF.4b: In Unit 1, students used informal strategies to find areas of rectangles with fractional side lengths. In Unit 5, students represented products as rectangular areas as they learned procedures for fraction multiplication. In Grade 4, students applied the area formula for rectangles to solve problems.” 

Materials relate grade-level concepts to future work. Each Section Organizer contains a Coherence section with “Links to the Future” containing information about how focus standards lay the foundation for future lessons. Examples include:

• Unit 2, Whole Number Place Value and Operations, Teacher’s Lesson Guide, Links to the Future, “5.OA.1: In Unit 7, students will use grouping symbols in an expression to model how to solve a multistep problem about gauging reaction time. In Grade 6, students will evaluate expressions and perform operations according to the Order of Operations.”

• Unit 6, Investigations in Measurement: Decimal Multiplication and Division, Teacher’s Lesson Guide, Links to the Future, “5.NBT.2: In Unit 8, students will multiply and divide numbers by powers of 10 to help them solve rich, real-world problems. In Grade 6, students will write and evaluate numerical expressions with whole-number exponents.”

• Unit 8, Application of Measurement, Computation, and Graphing, Teacher’s Lesson Guide, Links to the Future, “5.MD.1: In Grade 6, students will use ratio reasoning to convert measurement units.”

Instructional materials contain content from future grades in some lessons that is not clearly identified. Examples include:

• Lesson 4-12, Decimal Addition Algorithms, Math Masters, Decimal Addition Algorithms, Focus, Extending Whole-Number Addition Algorithms to Decimals, “Students practice using decimal addition algorithms and use estimates to check the reasonableness of their answers (5.NBT.7).” Problem 3, “Find the page in your Student Reference Book that shows how to use your algorithm to add decimals. Write the page number below. Read the example. Then use your algorithm to solve 2.965+7.47. Record your work in the space at the right.” Since one of the numbers includes the thousandths place, this problem aligns to 6.NS.3, fluently adding, subtracting, multiplying, and dividing multi-digit decimals using the standard algorithm for each operation.

• Lesson 4-13, Decimal Subtraction Algorithms, Math Masters, Decimal Subtraction Algorithms, Focus, Extending Whole-Number Subtraction Algorithms to Decimals, “Students practice using decimal subtraction algorithms and use estimates to check the reasonableness of their answers (5.NBT.7).” Problem 2, “Find the page in your Student Reference Book that shows how to use your algorithm to subtract decimals. Write the page number below. Read the example. Then use your algorithm to solve 9.48-7.291. Record your work in the space at the right.” Since one of the numbers includes the thousandths place, this problem aligns to 6.NS.3, fluently adding, subtracting, multiplying, and dividing multi-digit decimals using the standard algorithm for each operation.

• Lesson 8-3, Planning an Aquarium, Focus: Choosing a Fish Tank, Math Journal 2, “Students use area and volume guidelines to choose a tank (5.MD.5).” The image provided shows two fish tanks, and students find the volume of each fish tank. Fish Tank B provides fractional side lengths. Problem 1, “Choose the fish tank that you want for your aquarium. Fish Tank B: Side lengths include 16 in., 6\frac{1}{2} in., 10 in., 10\frac{1}{2} in., 8 in., 20 in.” Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism aligns to 6.G.2.