In Unit 2, students will build on their work on multi-digit multiplication and division from Grade 4 as well as their understanding of the structure of the base-ten system in Unit 1 to finalize fluency with multi-digit multiplication and extend multi-digit division to include two-digit divisors.

In Grade 4, students attained fluency with multi-digit addition and subtraction (4.NBT.4), a necessary skill for computing sums and differences in the standard algorithm for multiplication and division, respectively. Students also multiplied a whole number of up to four digits by a one-digit whole number, as well as two two-digit numbers (4.NBT.5). By the end of Grade 4, students can compute those products using the standard algorithm, but “reason repeatedly about the connection between math drawings and written numerical work, help[ing] them come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (Progressions for the CCSSM, “Number and Operation in Base Ten, K-5", p. 14). Students also find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (4.NBT.6). Similar to multiplication, by the end of Grade 4, students can compute these quotients using the standard algorithm alongside other strategies and representations so that the algorithms are meaningful rather than rote.

Unit 2 of Grade 5 begins with writing, evaluating, and interpreting simple numerical expressions (5.OA.1, 5.OA.2). This serves both to review basic multiplication and division facts, which supports major content later on in the unit, and as a way to record calculations that will grow increasingly complex as the unit progresses. Then, students solidify the standard algorithm for multiplication with the computational cases from Grade 4 before extending its use to larger and larger factors (5.NBT.5). Next, students follow a similar progression with division, first computing quotients involving cases from Grade 4 using a variety of strategies and then extending those methods to computations involving two-digit divisors. Note, however, that unlike multiplication, fluency with the standard division algorithm is not expected until Grade 6 (6.NS.2). Throughout the unit, students “learn to use [the] structure [of base-ten numbers] and the properties of operations to reduce computing a multi-digit…product or quotient to a collection of single-digit computations in different base-ten units” (MP.7) (Progressions for the CCSSM, “Number and Operation in Base Ten, K-5", p. 4). Further, “repeated reasoning (MP.8) that draws on the uniformity of the base-ten system is a part of this process” (Progressions for the CCSSM, “Number and Operation in Base Ten, K-5", p. 4).

In Unit 3, students will explore volume of three-dimensional shapes (5.MD.3—5), connecting it to the operations of multiplication and addition (5.NBT.5, 4.NBT.4). They also use their understanding that they gradually built in prior grade levels to classify shapes in a hierarchy, seeing that attributes of shapes in one category belong to shapes in all subcategories of that category (5.G.3—4).

This unit builds off of students’ well-established understanding of geometry and geometric measurement. Similar to students’ work with area, students develop an understanding of volume as an attribute of solid figures (5.MD.3) and measure it by counting unit cubes (5.MD.4). Students then connect volume to the operation of multiplication of length, width, and height or of the area of the base and the height and to the operation of addition to find composite area (5.MD.5). Throughout Topic A, students have an opportunity to use appropriate tools strategically (MP.5) and make use of structure of three-dimensional figures (MP.7) to draw conclusions about how to find the volume of a figure.

Students then move on to classifying shapes into categories and see that attributes belonging to shapes in one category are shared by all subcategories of that category (5.G.3). This allows students to create a hierarchy of shapes over the course of many days (5.G.4). Throughout this topic, students use appropriate tools strategically (MP.5) to verify various attributes of shapes including their angle measure and presence of parallel or perpendicular lines, as well as attend to precision in their use of language when referring to geometric figures (MP.6). They also look for and make use of structure to construct a hierarchy based on properties (MP.7).

In the fourth unit for Grade 5, students extend their computational work to include fractions and decimals, adding and subtracting numbers in those forms in this unit before moving to multiplication and division in subsequent units.

Unit 4 starts with a refresher on work in Grade 4, starting with generating equivalent fractions and adding and subtracting fractions with like terms. While students are expected to already have these skills, they help to remind students that one can only add and subtract quantities with like units, as well as remind students of how to regroup with fractions. Then, students move toward adding and subtracting fractions with unlike denominators. They start with computing without regrouping, then progress to regrouping with small mixed numbers between 1 and 2, and then to regrouping with mixed numbers. Throughout this progression, students also progress from using more concrete and visual strategies to find a common denominator, such as constructing area models or number lines, toward more abstract ones like multiplying the two denominators together and using that product as the common denominator (5.NF.1). Then, students use this general method in more advanced contexts, including adding and subtracting more than two fractions, assessing the reasonableness of their answers using estimation and number sense (MP.1), and solving one-, two-, and multi-step word problems (5.NF.2), (MP.4). Then, the unit shifts its focus toward decimals, relying on their work in Grade 4 of adding and subtracting decimal fractions and their deep understanding that one can only add like units, including tenths and hundredths as those units, to add and subtract decimals (5.NBT.7). They use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, relating the strategy to a written method and explain the reasoning used (MP.1). Students then apply this skill to the context of word problems to close out the unit (MP.4).

In Grade 5 Unit 5, students continue their exploration with fraction operations, deepening their understanding of fraction multiplication from Grade 4 and introducing them to fraction division.

Students began learning about fractions very early, as described in the Unit 4 Unit Summary. However, students’ exposure to fraction multiplication only began in Grade 4, when they learned to multiply a fraction by a whole number, interpreting this as repeated addition. For example, 4×2/3 is thought of as 4 copies of 2 thirds. This understanding is reliant on an understanding of multiplication as equal groups (3.OA.1). In Grade 4, however, students also developed an understanding of multiplicative comparison (4.OA.1), which will be of particular importance to the new ways in which students will interpret fraction multiplication in this unit.

The unit begins with students developing a new understanding of fractions as division. In the past, they’ve thought of fractions as equal-sized partitions of wholes, but here they develop an understanding of a fraction as an operation itself and represent division problems as fractions (5.NF.3). Students now see that remainders can be interpreted in yet another way, namely divided by the divisor to result in a mixed-number quotient. Then, students develop a new understanding of fraction multiplication as fractional parts of a set of a certain size (5.NF.4), which is a new interpretation of multiplicative comparison. Students use this understanding to develop general methods to multiply fractions by whole numbers and fractions, including mixed numbers. Throughout this work, students develop an understanding of multiplication as scaling (5.NF.5), “an important opportunity for students to reason abstractly” (MP.2) as the Progressions notes (Progressions for the Common Core State Standards in Mathematics, Number and Operations - Fractions, 3-5, p. 14). Then, students explore division of a unit fraction by a whole number and a whole number by a unit fraction (5.NF.7), preparing students to divide with fractions in all cases in Grade 6 (6.NS.1). Then, students also solve myriad word problems, seeing the strategies they used to solve word problems with whole numbers still apply but that special attention should be paid to the whole being discussed (5.NF.6, MP.4), as well as write and solve expressions involving fractions as a way to support the major work (5.OA.1, 5.OA.2). Finally, students make line plots to display a data set of measurements in fractions of a unit and solve problems involving information presented in line plots (5.MD.2), a supporting cluster standard that supports the major work of this and the past unit of using all four operations with fractions (5.NF).

In Unit 6, students use their procedural knowledge of multiplication and division with whole numbers, combined with their newly acquired understanding of multiplication and division with fractions, to multiply and divide with decimals, reasoning about the placement of the decimal point. They then apply this to the context of word problems, including those involving measurement conversion.

This unit starts with multiplying a decimal by a single-digit whole number, then multiplying a decimal by a multi-digit whole number, and finally multiplying a decimal by another decimal. Then, students progress to dividing a decimal by a single-digit whole number, then dividing a decimal by a two-digit whole number, and finally solving cases involving decimal divisors. Throughout these topics, students use the same methods to compute decimal products and quotients as they did for whole-number products and quotients, but they must reason about the placement of the decimal point. It is only in the last lesson of each topic that students generalize the pattern of the placement of the decimal point. The various lines of reasoning, and their advantages and disadvantages, can be read on pages 19 and 20 of the NBT Progression linked in the “Unit-Specific Intellectual Preparation” section. Students also solve myriad word problems as well as write and solve expressions involving decimals as a way to support the major work (5.OA.1, 5.OA.2). Finally, the unit closes with students learning to convert among different-sized customary measurement units within a given measurement system and solve word problems that use those conversions (5.MD.1), which extends the work from Grade 4 of converting from a larger unit of measurement to a smaller one in Grade 4 (4.MD.1—2). As noted in the Progressions, “this is an excellent opportunity to reinforce notions of place value for whole numbers and decimals, and the connection between fractions and decimals (e.g., meters can be expressed as 2.5 meters or 250 centimeters)” (GM Progression, p. 26), as well as computations with these types of numbers (5.NBT.7, 5.NF), thus connecting the work of unit conversion with major work of the grade.

Reasoning about the placement of the decimal point affords students many opportunities to engage in mathematical practice, such as constructing viable arguments and critiquing the reasoning of others (MP.3) and looking for and expressing regularity in repeated reasoning (MP.8). For example, “students can summarize the results of their reasoning as specific numerical patterns and then as one general overall pattern such as ‘the number of decimal places in the product is the sum of the number of decimal places in each factor’” (NBT Progression, p. 20).

In Unit 7, the final unit of the year for Grade 5, students are introduced to the coordinate plane and use it to represent the location of objects in space, as well as to represent patterns and real-world situations.

Students have coordinated numbers and distance before, namely with number lines. Students were introduced to number lines with whole-number intervals in Grade 2 and used them to solve addition and subtraction problems, helping to make the connection between quantity and distance (2.MD.5—6). Then in Grade 3, students made number lines with fractional intervals, using them to understand the idea of equivalence and comparison of fractions, again connecting this to the idea of distance (3.NF.2). For example, two fractions that were at the same point on a number line were equivalent, while a fraction that was further from 0 than another was greater. Then, in Grade 4, students learned to add, subtract, and multiply fractions in simple cases using the number line as a representation, and they extended it to all cases, including in simple cases involving fraction division, throughout Grade 5 (5.NF.1—7). Students’ preparation for this unit is also connected to their extensive pattern work, beginning in Kindergarten with patterns in counting sequences (K.CC.4c) and extending through Grade 4 work with generating and analyzing a number or shape pattern given its rule (4.OA.3).

Thus, students start the unit thinking about the number line as a way to represent distance in one dimension and then see the usefulness of a perpendicular line segment to define distance in a second dimension, allowing any point in two-dimensional space to be located easily and precisely (MP.6). After a lot of practice identifying the coordinates of points as well as plotting points given their coordinates with coordinate grids of various intervals and scales, students begin to draw lines and figures on a coordinate grid, noticing simple patterns in their coordinates. Then, after students have grown comfortable with the coordinate plane as a way to represent two-dimensional space, they represent real-world and mathematical situations, as well as two numerical patterns, by graphing their coordinates. This visual representation allows for a rich interpretation of these contexts (MP.2, MP.4).

 The purpose of this lesson is to help students develop their understanding of algebra using the patterns of multiplcation that occur on a hundreds board.  This is an effective strategy that will help students develop an understanding of multiples which is very crucial in 4th grade and above.  It also reinforces mutiplication as repeated addition.  This is also an effective strategy to use with English Language Learners, Special Educational Students and struggling learners because it provides a hands-on, interactive format for the whole class to see and participate.This lesson is also very engaging and it promotes most of the 5 C's: critical thinking, creative thinking, collaboration, communication, and citizenship.Use the following sites to look at 2 products that I have used from EAI and Didax.https://www.eaieducation.com/Product/504038/Make_a_Splash%e2%84%a2_120_Mat_Floor_Game.aspx (see image)https://www.didax.com/hip-hoppin-hundreds-mat.html 

Dan Meyer has created many mathematics lessons centered around storytelling, along with a methodology for using these with students. Before you do any of his lessons, please see the explanation at his blog: https://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/ To see all the lessons in a Google Docs spreadsheet, go to https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNN
C6Z4/edit#gid=0

The lessons are all CC BY, and you may edit these to fit your own students’ needs. The links from #GoOpenVA go to his own website because the structure of the lessons (including pertinent videos) is integral to the delivery of the lessons. Some lessons are stored as downloadable zip files, and these are noted as part of the link.

After completing a rough draft of a teacher-directed writing assignment, students will work together as de-bugging buddies to engage in the peer editing process using a rubric to document the mistakes that their peer has made. Students will fill out the rubric with the appropriate scale for the writing they are editing.

To actively engage students in recognizing common errors in writing such as capitalization, punctuation, sentence structure, and spelling.

A course in Graphic Arts Technology provides students with an understanding of the processes and systems common to careers in publishing, printing, and other forms of media distribution. Representative topics include graphic design concepts; art and copy preparation; image generation and editing; desktop publishing; on-demand publishing; school yearbook and magazine layout; advertising and promotion; printing technology; binding and finishing; and screen printing.Students will be committed to lifelong learning as they grow individually, participate in groups, think analytically, create artistic products, and contribute to production of a major project. Students will learn illustration design software such as Adobe Illustrator, photo editing software such as Adobe Photoshop, and page layout software such as Adobe InDesign to create projects that will be printed in traditional and digital formats.

Students will organize various stem items, make observations regarding those items such as their color, size, etc and then graph their observations.

Students will construct and analyze data from a Skittle bag in bar graphs.

Students will go around the school (assign a few kids to one class, then data will come from many rooms) and record data on the various options kids have to play on at recess. Using the data collected they will create a graph from the tally chart they surveyed with. As a class they will compare their graphs and then discuss what they predict kids will play on for that afternoon using the data. **The teacher will need to ask for assistance from various other classroom teachers to interview or survey their students.

Analyzing data displayed in graphsMathematics Instructional Plans (MIPs) help teachers align instruction with the Mathematics Standards of Learning (SOL) by providing examples of how the knowledge, skills and processes found in the SOL and curriculum framework can be presented to students in the classroom.

In this lesson, the students will learn what a graph is and why we use graphs. We will look at an example of an object and picture graph, discuss the parts of a graph, and then create a class graph. By the end of the lesson, students will work to create their own graph.

Graphing Linear EquationsMathematics Instructional Plans (MIPs) help teachers align instruction with the Mathematics Standards of Learning (SOL) by providing examples of how the knowledge, skills and processes found in the SOL and curriculum framework can be presented to students in the classroom.