5th Grade - Gateway 1

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

The curriculum is divided into 21 Scopes, and each Scope contains a Standards-Based Assessment used to assess what students have learned throughout the Scope. Examples from Standards-Based Assessments include:

• Scope 3: Read and Write Decimals, Evaluate, Standards-Based Assessment, Question 3, “During a gymnastics meet, Michelle scored eight and seventy-five hundredths on her floor routine. Write her score in numerical form.” (5.NBT.3)

• Scope 9: Add and Subtract Fractions, Evaluate, Standards-Based Assessment, Question 5, “Meghan is painting her bedroom. On Wednesday, she painted \frac{1}{3} of the room. On Thursday, she painted another \frac{3}{7} of the room. How much of the room was painted by the end of the day on Thursday? \frac{16}{21}, \frac{2}{21}, \frac{10}{21}, \frac{4}{10}” (5.NF.1)

• Scope 14: Numerical Expression. Evaluate, Standard-Based Assessment, Question 2, “An expression is shown below. 4\times(2+3) What value is equivalent to this expression? 10, 91, 20, 47.” (5.OA.1)

• Scope 15: Classify Two-Dimensional Figures, Evaluate, Standards-Based Assessment, Question 2, Students see the following shapes: parallelogram, rectangle, trapezoid, and square. “Select all the figures that are rectangles.” (5.G.4)

• Scope 18: Volume in Cubic Units, Evaluate, Standards-Based Assessment, Question 2, “A student used cubes to build a rectangular prism. The edge of each cube was 1 centimeter. The student used 36 cubes to build the first layer. The rectangular prism had a total of five layers. What is the volume of the rectangular prism, in cubic centimeters, that the student built? 41 cubic centimeters, 150 cubic centimeters, 180 cubic centimeters, 31 cubic centimeters.” (5.MD.4)

The materials reviewed for STEMscopes Math Grade 5 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 in Grade 5 as students engage with all CCSSM standards within a consistent daily lesson structure, including Engage, Explore, Explain, Elaborate, and Evaluate. Intervention and Acceleration sections are also included in every lesson. Examples of extensive work to meet the full intent of standards include:

• Scope 7: Multiply Multi-Digit Whole Numbers, Explore 1 and Skill Basics, engages students in extensive work to meet the full intent of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm.) In Explore 1, Standard Algorithm, students connect the area model strategy for three-digit-by-two-digit multiplication with partial products and the standard algorithm. Part 2: The Standard Algorithm, read each problem and set up the equation, use the standard algorithm work mat and digit cards to solve each problem, record your work in the space below. Remnant 1, “Length: 362 feet Width: 27 feet”; Remnant 2, “Length: 436 feet Width: 16 feet”; Remnant 3, “Length: 154 feet Width: 34 feet.” Skill Basics, Problem 1, “Priya has a collection of stickers she keeps in her sticker book. She has collected 48 pages of stickers. There are 28 stickers on each page. How many stickers has Priya collected?” Problem 2, “Miss Mabel’s Pumpkin Farm sells pumpkins. The farm has 26 rows of pumpkins, and there are 32 pumpkins growing in each row. How many pumpkins does Miss Mabel’s Pumpkin Farm have to sell?”

• Scope 13: Divide Unit Fractions, engages students in extensive work to meet the full intent of 5.NF.7a (Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (\frac{1}{3})\div4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (\frac{1}{3})\div=\frac{1}{12} because$$((\frac{1}{12})\times4=\frac{1}{3})$$. Explore 1, “You and two friends are going to share four pizzas at a pizza sampler gathering. You all want to be adventurous and try different toppings on each pizza and have agreed that each person will get an equal portion of each pizza. How many total portions of pizza will each person get?” Candy Sharing Scenario Card 1: “Mack wants to see what it would look like if he divided his 3 candy bars into halves. Make a model with the fraction tiles that would show Mack how many portions he would make when dividing by. Label the number of parts in each candy bar.” Candy Sharing Scenario Card 2: “Mack is still not convinced that if he divides his 3 candy bars by$$\frac{1}{2}$$  he will create more parts. Make a model with the fraction tiles that would show Mack that when you divide a whole number by a fraction, you make smaller parts, but there are actually more of them. Show Mack what would happen if he divided his 3 candy bars by \frac{1}{4}.” Candy Sharing Scenario Card 3: “Show Mack what would happen if he divided his 3 candy bars by \frac{1}{5} .” Explain, Show What You Know Part 1: “Solve each expression. Draw a model and write an equation to represent your thinking. Expression 4\div\frac{1}[5}”

• Scope 14: Numerical Expressions, Explore 1 and 2, engages students in extensive work to meet the full intent of 5.OA.1 (Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.) In Explore 1-Order Matters, Exit Ticket, students create expressions for multi-step problems, evaluate their expression, and explain the process. In the first scenario, students are given a real-life problem. There is a box to show their expression, their process description, how it was evaluated, and their solution. “Create an expression that represents the word problem below, evaluate it, and then describe the process you used to evaluate it. Lilly hired 3 teens to mow her grass and rake the leaves. She paid them $75 to mow and $30 to rake, and the boys split the money equally. Since the boys also swept her sidewalk, she paid each of them an extra $5 as well. Write an expression that shows how much money each teen received. Expression: Process description: Evaluate: Solution:”  In Explore 2-Grouping Symbols, Procedure and Facilitation Points, students work through scenarios with multiple steps, using order of operations and symbols to show any groupings. “1. Distribute Student Journals to each student. Invite the class to read the first problem aloud while students follow along in their Student Journals. a. Catie owns a catering company. She shops at a kitchen supply store. She bought three mixers for $74.25 each and two pans for $26 each. How much did she spend at the kitchen supply store? 2. Allow students two minutes to collaborate with their groups to write a mathematical expression that matches the problem scenario. 3. DOK–1 Write students’ expressions on the board, and discuss any errors as well as similarities between expressions. 4. Remove the parentheses. Ask students to evaluate the expression without the parentheses to see if they get the same value. 5. DOK–1 Discuss. Answers may vary: No. The value is not the same because if we solve from left to right, 74.25\times3=222.75+26=24875\times2=497.5 is not the same as 74.25\times3=222.75+(26\times2)=52, for a total of 222.75+52=274.75. 6. Ask questions such as the following: a. DOK–1 What should be evaluated first? b. DOK–1 What needs to be performed in the second set of parentheses? c. DOK–3 What do you think those parentheses mean? 8. Inform students that they will visit 8 stations. At each station, they will either have to create an expression based on a scenario or they will be given an expression to evaluate. For all problems, they will need to evaluate the expressions given or created. 9. Circulate around the room as student groups work together. Ask the following questions at each station: a. DOK–1 Are parentheses necessary in this scenario? b. DOK–2 How do the two expressions compare? c. DOK–2 Why is the first expression less or greater?”

• Scope 18: Volume in Cubic Units, engages students in extensive work to meet the full intent of  5.MD.3a and 5.MD.3b, (Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.). Explore 1, “The Jolly Elf Hat Factory just got a big order for elf hats at the North Pole, and they have hired you to pack shipping boxes with the elf hats! It is your job to fit as many elf hats into each shipping box as you can. Each centimeter cube contains one elf hat. The boxes will have lids, so the elf-hat boxes cannot go over the top of the shipping box. Give each student a Student Journal. Give each group Shipping Box 1, a centimeter ruler, and centimeter cubes. Ask questions such as the following: DOK–1 How can you pack the boxes so that you can fit as many elf hats into Shipping Box 1 as possible? Explain that students should record how many elf hats fit the length, width, and height of Shipping Box 1 on their Student Journals. Allow students to pack the boxes and write their cube totals. After they have packed their boxes with cubes, students should measure the length, width, and height of the shipping box and the elf-hat boxes using the centimeter ruler. Students should record these measurements on their Student Journals. Ask questions such as the following: DOK–1 What are the dimensions of the elf-hat boxes that you measured? Explain that this is called a cubic centimeter. Have students refer to the cubic foot and cubic yard or meter that was built. Discuss with students how each of these are called cubic units because their length, width, and height are each 1 unit of measure. For this activity, students will be working with cubic centimeters. For the Missing Boxes part of this lesson, students will be using cubic centimeters and cubic inches. DOK–1 How many elf hats could you fit into a shipping box with dimensions of 5 cm long, 4 cm wide, and 3 cm tall? DOK–2 What is the volume of Shipping Box 1? DOK–2 Compare Box 1 and Box 2. Which box do you think will hold more elf hats? …” Part II: Missing Boxes, Tell students the Jolly Elf Hat Factory didn’t get their shipment of new boxes in yet, but they still need to figure out how many elf-hat boxes will fit into each shipping box when they do arrive. The factory knows how many elf-hat boxes will fit in the length, the width, and the height of each box. Questions 3 and 4 are in cubic centimeters, and 5 and 6 are in cubic inches. Be sure to emphasize that students need to be mindful when answering questions, using manipulatives, and labeling correctly… As students are working, monitor and check for understanding. Ask questions such as the following: DOK–2 How did you know how many centimeters the length of Shipping Box 3 was? DOK–1 What are the dimensions of the bottom, or base, of Shipping Box 4? After students have found the volume of Shipping Boxes 3–6, have them meet with another group and review their answers. If there are discrepancies with the answers, have them rebuild the boxes using the centimeter or inch cubes and reach an agreement on the volume of the boxes. …” Exit ticket, “Elf hats for senior elves who have been at the North Pole for many years are fluffier than regular elf hats. They need a special unit-cube elf-hat box that is 1 inch long, 1 inch wide, and 1 inch tall. What is the volume of this unit cube? V=___.” Question 2: “A senior-elf hat shipping box has the following dimensions: L = 4 inches W = 2 inches H = 3 inches How many unit cubes would fit on the bottom layer of a shipping box with these dimensions? What is the volume of the shipping box?” Decide and Defend: “Mrs. Dean is packing up her classroom. She is putting all of her number cubes in the same box. Each number cube is 1 cubic inch. Mrs. Dean has 40 number cubes. Will they all fit in the rectangular prism box?” An image of a rectangular prism packed 6 unit cubes high and packed with a base area of 6 cubes is shown. 

• Scope 20: Graph on a Coordinate Plane, Explore 3-Graphing Real World Problems, Procedure and Facilitation Points, engage students with extensive work to meet the full intent of 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.) “1. Distribute a Student Journal to each student. 2. Read the following scenario: a. You and your business partners own a company that sells shirts. As with any business, there are many tasks that need to be completed for the business to run smoothly and to make money. Today, you and your business partners will be collecting and documenting data on a coordinate plane. It is important that your group works together to help your company run a successful business. 3. Display the Coordinate Plane Anchor Chart to review important vocabulary from past Explores. 4. Invite students to discuss the following questions with their groups before sharing with the class: a. DOK–1 What does the picture on the anchor chart represent? b. DOK–1 What is a coordinate plane? c. DOK–2 How do you think a coordinate plane can help us document our business data? d. DOK–1 What does the vertical number line represent? e. DOK–1 What does the horizontal number line represent? f. DOK–1 What do the numbers (3, 4) represent when using a coordinate plane? g. DOK–1 What is an ordered pair? h. DOK–1 What does the first number represent in an ordered pair? i. DOK–1 What does the second number represent in an ordered pair?. j. DOK–1 What does the point where the x-axis and y-axis meet represent? k. DOK–1 What is the origin? l. DOK–1 How would we plot the ordered pair (3, 4) on our coordinate plane? 5. Explain to the class that they will be using their knowledge of coordinate planes to represent their business scenarios. 6. They will read each station card as a group and then use the coordinate plane and dry-erase markers to represent the data from each scenario. 7. Explain to students that they will need to rotate the job of placing ordered pairs on their coordinate plane. After each student has placed a point on the coordinate plane, they must write the ordered pair and their name on the Who Marked the Spot card. 8. After the group has agreed that the points are representing the ordered pairs correctly, they will then record their data on their Student Journals. 9. Place each group at a station, and monitor student collaboration as they work together as a group. Ask the following questions to assess their understanding: a. DOK–2 After reading the scenario, what do you think the x-coordinate represents? b. DOK–2 After reading the scenario, what do you think the y-coordinate represents? c. DOK–2 What information in the scenario can help you figure out the relationship for the ordered pairs that are being represented on the coordinate plane? d. DOK–1 Is each number line on the coordinate plane counting by the same amount? e. DOK–1 What should you do if your ordered pair doesn’t fall exactly on a line? 10. Before groups can rotate, check each group’s Who Marked the Spot Card to ensure that each student participated in placing ordered pairs on the coordinate plane. Once each group has been quickly checked, then make sure students clean up each station and erase their points and ordered pairs from their coordinate planes and Who Marked the Spot Cards before groups are allowed to rotate. 11. Once the station is cleaned up, allow students to rotate from station to station after an allotted amount of time. 12. After the Explore, invite the class to a Math Chat to share their observations and learning.”

The materials reviewed for STEMscopes Math Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major cluster of each grade.

The instructional materials devote at least 65% of instructional time to the major clusters of the grade:

• The approximate number of scopes devoted to major work of the grade (including assessments and supporting work connected to the major work) is 13 out of 21, approximately 62%.

• The number of lesson days and review days devoted to major work of the grade (including supporting work connected to the major work) is 129 out of 152, approximately 85%.

• The number of instructional days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 145 out of 180, approximately 81%.

An instructional day analysis is most representative of the instructional materials because this comprises the total number of lesson days, all assessment days, and review days. As a result, approximately 81% of the instructional materials focus on the major work of the grade.

The materials reviewed for STEMscopes Math Grade 5 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. Examples of connections include:

• Scope 14: Numerical Expressions, Explore 1–Order Matters connects the supporting work of 5.OA.1 (Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.) to the major work of 5.NF.1 (Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.) Students discover the value of a standard order of operations and how failing to follow the order of operations can change the value of an expression. Scenario–Cookin’ Up Something Sweet, “Arnav followed a recipe that called for several dry ingredients: 4 cups of flour, 2\trac{1}{2} cups of sugar, 2\trac{1}{2} cups of chocolate chips, \frac{1}{2} of a cup of pecans, and \frac{1}{2} of a cup of coconut. How many cups of dry ingredients did the recipe include? ($$4+(2\frac{1}{2}+\frac{1}{2})$$  (4+2)\times(2\frac{1}{2}+\frac{1}{2})  4+(2\times2\frac{1}{2}+\frac{1}{2})  4+2\times(2\frac{1}{2}+\frac{1}{2}) Expression:___, Evaluate:___, Solution Sentence:___” 

• Scope 17: Represent Measurement with Line Plots, Explore, Explore 1–Problem-Solving with Measurement on a Line Plot, Exit Ticket, connects the supporting work of 5.MD.2 (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}). Use operations on fractions for this grade to solve problems involving information presented in line plots.) to the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions.) Students are given a table with the distances run for 10 days and use a line plot and mixed numbers with different denominators to find the distance a person ran. Students “Read the scenario. Draw a line plot to represent the data. Answer the questions. 1 Problem-Solving with Measurement on a Line Plot, Exit Ticket, “Ginger is on the track team at her school. Her coach has asked everyone to keep track of how far they run over a 10-day period. The table below shows Ginger’s data. Day, Distance (miles), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2\frac{1}{2}, 2\frac{1}{4}, 3\frac{1}{2}, 3\frac{1}{8}, 2\frac{1}{4}, 3\frac{1}{2}, 2\frac{1}{2}, 2\frac{1}{4}, 2\frac{1}{4}3\frac{1}{4}. 1. How many total miles did Ginger run on the days she ran less than 3 miles? Write an equation and solve. ___  2. How many total miles did Ginger run on the days she ran more than 3 miles? Write an equation and solve. ___ 3. Use your totals from questions 1 and 2 to find the total distance Ginger ran over a 10-day period. ___”

• Scope 19: Apply Volume Formulas, Explain, Show What You Know–Part 3: Additive Volume of Composite Figures, connects the supporting work of 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them…) to the major work of 5.MD.5c (Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.) Students are given two L-shaped composite figures and need to determine which figure has the greatest volume. Students see a composite figure that is made of a rectangle with dimensions six by three by two connected to a rectangle that is three by two by three.  The other figure is made of two rectangles, two by two by eight and two by two by five.  Students create simple expressions and record the calculations. “Caitie and Libby are five-year-old twins. Their Aunt Jenny is making them special memory boxes in the shape of their initials. Jenny decided to start with Libby’s box because she thought the L shape would be easier. She made two different boxes so Libby could choose her favorite. Libby said she wanted the memory box with the most space since she planned on having lots of memories. Look at the two designs: Which design did Libby choose? Explain your reasoning.”

The materials for STEMscopes Math Grade 5 meet expectations that materials include 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 are sometimes listed for teachers in one or more of the three sections of the materials: Engage, Explore and Explain. Examples of connections include:

• Scope 6: Model the Four Operations with Decimals, Engage, Hook-Slumber Party Shopping, Procedure and Facilitation Points, connects the major work of 5.NBT.A (Understand the place value system) and 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths.) Students find the amount spent using the four operations with decimal numbers. “Part II: Post-Explore, 1. After students have completed the Explore activities for this topic, show the phenomena video again, and repeat the situation. 2. Review the problem and allow students to solve it. 3. Give each pair of students a copy of the Student Handout, a white board or piece of paper, and a dry-erase marker or pencil. 4. Tell the students to use the models on their copy of the Student Handout to help them solve each problem. 5. You will read each scenario below and allow them to work together to solve it. Then students will share with the class before going to the next discussion point. 6. Discuss the following: a. DOK-2 The mom and son want to buy some cake mix and frosting to make cupcakes for the party. They want to make a lot. They buy 3 boxes of cake mix and 4 cans of frosting. How much did they spend? b. DOK-2 They spent $5 on pizza for the slumber party. How many pizzas did they buy? c. DOK-2 They also bought 7 liters of soda for the party. How much did they spend on soda? d. DOK-2 The last item needed for the party is ice cream. They bought 2 gallons of ice cream. How much did they spend on the ice cream? e. DOK-2 How much money did they spend on all the groceries for the party? f. DOK-2 If they paid with $20.00, how much money did they have left?”

• Scope 13: Divide Unit Fractions, Explore 1–Divide Whole Numbers by Unit Fractions, Candy Bar Scenario Cards 1, 2 & 3 connects the Number & Operations–Fractions domain to the Number & Operations in Base Ten domain. “Mack wants to see what it would look like if he divided his 3 candy bars into halves. Make a model with the fraction tiles that would show Mack how many portions he would make when dividing by \frac{1}{2}. Label the number of parts in each candy bar.” Sharing Candy Card #2 “Mack is still not convinced that if he divides his 3 candy bars by \frac{1}{2} he will create more parts. Make a model with the fraction tiles that would show Mack that when you divide a whole number by a fraction, you make smaller parts, but there are actually more of them. Show Mack what would happen if he divided his 3 candy bars by \frac{1}{4}.” Sharing Candy Card #3 “Show Mack what would happen if he divided his 3 candy bars by \frac{1}{5}.”

• Scope 17: Represent Measurement with Line PLots, Evaluate, Skills Quiz, Questions 9-12 connects the Measurement & Data domain to the Operations & Algebraic Thinking domain. A line plot titled “Pounds of Apples” is shown with a line iterated in halves from 1 to 4. “9 - Label the line plot with the fraction increments represented. What fraction increment is represented in this line plot? 10 - What are the measurements of apples picked that are shown in this line plot? 11- How many pounds of apples picked are shown in this line plot? 12 - Mrs. Jones took her kids apple picking in an orchard. Each child picked a certain amount of apples. If Mrs. Jones wanted to redistribute the apples picked evenly by weight amongst her four kids, how many pounds of apples would each child get?”

• Scope 19: Apply Volume Formulas, Explore, Explore 2–Using Three Dimensions to Find Volume, Exit Ticket connects the Measurement & Data domain to the Number & Operations in Base Ten domain. “MeWOW’s Pet Store makes beds for kittens! They have 96 cubic inches of filling left to make one small rectangular kitty bed, but they are not sure how high this will make the bed. Can you help them out?”An image of a kitty bed with the dimensions of 6 inches and 8 inches labeled. “ l= ___ w= ____ h= ___ Show your work. Check your answer by using the two volume formulas. V= ___ V= ___”

The materials reviewed for STEMscopes Math 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.

Prior and future connections are identified within materials in the Home, Content Support, Background Knowledge, as well as Coming Attractions sections. Information can also be found in the Home, Scope Overview, Teacher Guide, Background Knowledge and Future Expectations sections. 

Examples of connections to future grades include:

• Scope 7: Multiply Multi-Digit Whole Numbers, Home, Scope Overview, Teacher Guide, Future Expectations connects 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm) to future learning. “Sixth-grade students will extend their fluency in using standard algorithms for all four mathematical operations when solving problems involving multi-digit decimal numbers.”

• Scope 12: Fractions as Division, Home, Content Support, Coming Attractions connects 5.NF.3 (Interpret a fraction as division of the numerator by the denominator (\frac{1}{b}=a\div b) (a/b = a ➗ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers…) to future learning. “Sixth-grade students will apply and extend previous understandings of multiplication and division to dividing fractions by fractions. They will use their knowledge of (\frac{1}{b}=a\div b) and explore how (\frac{a}{b})\div(\frac{c}{d})=\frac{ad}{bc}. Grade seven students understand that integers can be divided and that the quotient makes a rational number. Eighth graders begin to use rational numbers to approximate irrational numbers. In grade six geometry, students use their knowledge of multiplying fractions to help them find the area of rectangular prisms. Connections are made in grades six and seven as students begin working with ratio concepts and reasoning. In grade six, they explore the concept of unit rates, and in seventh grade, students will find the unit rate of fractions. They use these understandings to apply these principles to real-world contexts.”

• Scope 21: Generate and Graph Numerical Patterns, Home, Content Support, Coming Attractions, connects the work of 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.) to the work of 6.RP.3a (Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.) “In sixth grade, the domain Number and Operations in Base Ten is replaced with the domain Ratios and Proportional Reasoning. Knowledge gained about patterns within base-ten numbers between kindergarten and grade five builds an essential foundation that is used to support algebraic thinking in later grades. In sixth grade, students make tables of equivalent ratios, compare equivalent ratios, find missing values in the tables, and plot pairs of values on the coordinate plane. Sixth-grade students use input/output tables to write linear equations to show the rule-dependent and independent variables. Sixth-grade students create graphs on the coordinate plane, and the points of the graphs are representations for a rule that shows the relationship of output for a given input.”

Examples of connections to prior grades include:

• Scope 3: Read and Write Decimals, Home, Content Support, Background Knowledge, connects 5.NBT.3a (Read and write decimals to thousandths using base-ten numerals, number names, and expanded form…) to previous work. “Kindergarten through third grade built a firm foundation with whole numbers. Second grade partitions different shapes into halves, fourths, and quarters. Third grade develops an understanding of fractions as numbers, being composed of unit fractions. They begin to use fractions to solve problems, understanding that the size of a fractional part is relative to the size of the whole. Fourth grade extends the fractional concept to include fraction equivalence and operations with fractions. As decimals are introduced, students learn decimal notation for fractions with denominators of 10 or 100. Students worked with decimals and fractions interchangeably, to include locating them on a number line. Fifth grade builds on this, and students are expected to read and write decimals to thousandths using different methods.”

• Scope 11: Multiplication Problem Solving Using Fractions, Home, Scope Overview, Teacher Guide, Vertical Alignment, Background Knowledge, connects the work of 1.G.3 (Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of…), 2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc…), 3.NF.1 (Understand a fraction \frac{1}{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction \frac{a}{b} as the quantity formed by a parts of size \frac{1}{b}.) and 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.) to the work of 5.NF.4a (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product \frac{a}{b}\timesq as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b…) “The foundational skills of using visual representations of fractions and operations of fractions begins with earlier work in the domains of number, measurement, and geometry. First- and second-grade students make sense of the base ten system and use fraction language to describe shapes partitioned into equal shares (halves, thirds, and fourths). In third grade, students measure with rulers marked with halves and fourths of an inch. Third-grade students use fraction notation and visual models (tape diagrams, number lines, and area models) to represent and compare fractions. In fourth grade, students solidify an understanding of equivalent fractions, add and subtract fractions and mixed numbers with like denominators, and they multiply fractions by whole numbers. In fifth grade, students add and subtract fractions and mixed numbers with unlike denominators, interpret multiplication as scaling, divide whole numbers by unit fractions, and divide unit fractions by whole numbers.”

• Scope 19: Apply Volume Formulas, Home, Scope Overview, Teacher Guide, Background Knowledge connects 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume) to work done prior to 5th grade. “Competencies in shape composition and decomposition, spatial structuring, and arithmetic form the foundation for understanding volume. In Kindergarten, students begin to develop geometric concepts and spatial reasoning as they identify, describe, and compose 2-D and 3-D shapes. In first grade, students reason about shapes in relation to their attributes. In second grade, students recognize and draw specific types of shapes, such as triangles, quadrilaterals, pentagons, and hexagons. In third grade, students make generalizations about properties that are shared between categories and subcategories of shapes. Third-grade students measure and estimate liquid volume and investigate area by covering two-dimensional spaces. In fourth grade, students more precisely name 2-D shapes by classifying them based on parallelism, perpendicularity, and angle types. Fourth-grade students solve measurement problems and use place value understanding and properties of operations to perform multi-digit arithmetic.”