This video is part of the Continue to Know with WHRO TV series. Watch Victoria Valenciana Brannen teach about creating and solving single-step practical problems involving addition, subtraction, and multiplication of decimals.

In this lesson, students will be introduced to the idea of decomposition. Specifically the lesson caters to math word problems, but could be easily modified to any subject (as found in the modificaitons section of the lesson plan). Students will engage with each other and the vocab to work through an easy process to decompose word problems into manaeagable pieces as a strategy to solve. All activities are low prep and can be modified to your needs. This can be a stand alone lesson or expanded by using Part 2 and Part 3 to deepen understanding through coding activities. 

This lesson expands upon the ideas of decomposition by using GameChangineer to incorporate commands to create a coded mini game from decomposing word problems. Students will assist the teacher in this guided lesson on how to create commands and use the website before engaing independently in Part 3. Activities are low prep with modifications included, but do require organized planning to implement effectively. If you have not done a lesson on decomposition, it is suggested you use Part 1 to help student's gain the necessary understandings of the processes used in this lesson. 

This is the final part of an extended lesson on decomposition. Students will create a word problem to decompose and then use GameChangineer to create a mini game that is reflective of the word problem and its solution. Students will be using the plan, design, and review process thourhgout their creations. A rubric and self reflection tool for the final products are included. Activities are low prep with modifications included, but do require organized planning to implement effectively. If you have not done a lesson on decomposition, it is suggested you use Part 1 to help student's gain the necessary understandings of the processes used in this lesson. If you have not done a lesson on writing commands and using GameChangineer, it is suggested you use Part 2 before implementing this independent activity. 

Unit 2 opens students’ eyes to some of the most important content students will learn in Grade 3—multiplication and division. In this unit, “students begin developing these concepts by working with numbers with which they are more familiar, such as 2s, 5s, and 10s, in addition to numbers that are easily skip counted, such as 3s and 4s,” allowing the cognitive demand to be on the concepts of multiplication and division themselves rather than the numbers (CCSS Toolbox, Sequenced Units for the Common Core State Standards in Mathematics Grade 3). Then in Unit 3, students will work on the more challenging units of 0, 1, 6–9, and multiples of 10.

In Grade 2, students learned to count objects in arrays using repeated addition (2.OA.4) to gain a foundation to multiplication. They’ve also done extensive work on one- and two-step word problems involving addition and subtraction, having mastered all of the problem types that involve those operations (2.OA.1). Thus, students have developed a strong problem-solving disposition and have the foundational content necessary to launch right into multiplication and division in this unit.

At the start of this unit, students gain an understanding of multiplication and division in the context of equal group and array problems in Topic A. To keep the focus on the conceptual understanding of multiplication and division (3.OA.1, 3.OA.2), Topic A does not discuss specific strategies to solve, and thus students may count all objects (a Level 1 strategy) or remember their skip-counting and repeated addition (Level 2 strategies) from Grade 2 to find the product. In Topics B and C, however, the focus turns to developing more efficient strategies for solving multiplication and division, including skip-counting and repeated addition (Level 2 strategies) as well as “just knowing” the facts, which works toward the goal that “by the end of grade 3, [students] know from memory all products of two single-digit numbers and related division facts” (3.OA.7). As the Operations and Algebraic Thinking Progression states, “mastering this material and reaching fluency in single-digit multiplications and related division may be quite time consuming because there are no general strategies for multiplying or dividing all single-digit numbers as there are for addition or subtraction” (OA Progression, p. 22). Thus, because “there are many patterns and strategies dependent upon specific numbers,” they first work with factors of 2, 5, and 10 in Topic B, since they learned these skip-counting sequences in Grade 2. Then in Topic C, they work with the new factors of 3 and 4. Only then, when students have developed more familiarity with these factors, will students solve more complex and/or abstract problems with them, including determining the unknown whole number in a multiplication or division equation relating three whole numbers (3.OA.4) and solving two-step word problems using all four operations (3.OA.3, 3.OA.8), assessing the reasonableness of their answers for a variety of problem types in Topic D.

Throughout the unit, students engage in a variety of mathematical practices. The unit pays particular attention to reasoning abstractly and quantitatively, as students come to understand the meaning of multiplication and division and the abstract symbols used to represent them (MP.2). Further, students model with mathematics with these new operations, solving one- and two-step equations using them (MP.4).

Unit 3 extends the study of factors from 2, 3, 4, 5, and 10, which students explored in Unit 2, to include all units from 0 to 10, as well as multiples of 10 within 100. To work with these more challenging units, students will rely on skip-counting (a Level 2 strategy) and converting to an easier problem (a Level 3 strategy dependent on the properties of operations). They then will apply their understanding of all four operations to two-step word problems as well as arithmetic patterns. Finally, the unit culminates with a focus on categorical data, where students draw and solve problems involving scaled picture graphs and scaled bar graphs, a nice application of the major work of multiplication and division.

Topic A begins by reminding students of the commutative property they learned in Unit 2, as well as introducing them to the distributive and associative properties, upon which they will rely for many of the strategies they learn for the larger factors. In order to be able to use these properties, they need to understand how to compute with a factor of 1, which they explore along with 0, as well as understand how to use parentheses. They’ll then explore the factors of 6, 7, 8, and 9 in Topics B and C. Because of the increased difficulty of these facts, students will rely on both skip-counting (a Level 2 strategy) as well as converting to an easier problem (a Level 3 strategy). Converting to an easier problem is dependent on the properties of operations (e.g., to find 6 x 7, think of 5 x 7 and add a group of 7 is dependent on the distributive property). Thus, students will work with the properties extensively throughout the unit, with their understanding of them and notation related to them growing more complex and abstract throughout the unit. In Topic D, students will multiply one-digit numbers by multiples of 10 and by two-digit numbers using the associative property. Then, students solve two-step word problems involving all four operations, assessing the reasonableness of their answer, and identify arithmetic patterns and explain them using the properties of operations. Finally, students explore picture graphs in which each picture represents more than one object and bar graphs where the scale on the axis is more than 1, a key development from Grade 2 (3.MD.3). As the Progressions note, “these developments connect with the emphasis on multiplication in this grade” (MD Progression, p. 7). Students also solve one- and two-step word problems related to the data in these plots, relying on the extensive work students have done with word problems throughout the year. Thus, this supporting cluster standard nicely enhances the major work they’ve been working on throughout this and the previous unit.

In Unit 3, students deepen their understanding of multiplication and division, including their properties. “Mathematically proficient students at the elementary grades use structures such as…the properties of operations…to solve problems” (MP.7) (Standards for Mathematical Practice: Commentary and Elaborations for K–5, p. 9). Students use the properties of operations to convert computations to an easier problem (a Level 3 strategy), as well as construct and critique the reasoning of others regarding the properties of operations (MP.3). Lastly, students model with mathematics with these new operations, solving one- and two-step equations using them (MP.4).

In the first unit for Grade 4, students extend their work with whole numbers and use this generalized understanding of the place value system in the context of comparing numbers, rounding them, and adding and subtracting them.

Students understanding of the base ten system begins as early as Kindergarten, when students learn to decompose teen numbers as ten ones and some ones (K.NBT.1). This understanding continues to develop in Grade 1, when students learn that ten is a unit and therefore decompose teen numbers into one ten (as opposed to ten ones) and some ones and learn that the decade numbers can be referred to as some tens (1.NBT.1). Students also start to compare two-digit numbers (1.NBT.2) and add and subtract within 100 based on place value (1.NBT.3—5). In second grade, students generalize the place value system even further, understanding one hundred as a unit (2.NBT.1) and comparing, adding, and subtracting numbers within 1,000 (2.NBT.2—9). In Grade 3, place value (NBT) standards are additional cluster content, but they still spend time fluently adding and subtracting within 1,000 and rounding three-digit numbers to the nearest 10 and 100 (3.NBT.1—2).

Thus, because students did not focus heavily on place value in Grade 3, Unit 1 begins with where things left off in Grade 2 of understanding numbers within 1,000. Students get a sense of the magnitude of each place value by visually representing the place values they are already familiar with and building from there. Once students have a visual and conceptual sense of the “ten times greater” property, they are able to articulate why a digit in any place represents 10 times as much as it represents in the place to its right (4.NBT.1). Next, students write multi-digit numbers in various forms and compare them (4.NBT.2). Comparison leads directly into rounding, where Grade 4 students learn to round to any place value (4.NBT.3). Next, students use the standard algorithms for addition and subtraction with multi-digit numbers (4.NBT.4) and apply their algorithmic knowledge to solve word problems. The unit culminates with multi-step word problems involving addition and subtraction, using a letter to represent the unknown quantity, then using rounding to assess the reasonableness of their answer (4.OA.3), allowing for students to connect content across different clusters and domains (4.NBT.A, 4.NBT.B, and 4.OA.B).

Throughout the unit, students will repeatedly look for and make use of structure, specifically the structure of the place value system (MP.7). Students develop an understanding that a digit in any place represents 10 times as much as it represents in the place to its right, then apply that understanding of structure to compare, round, and add and subtract multi-digit whole numbers

In Grade 4 Unit 2, students multiply up to four-digit numbers by one-digit numbers, relying on their understanding of place value and properties of operations, as well as visual models like an area model, to solve.

As a foundation for their multi-year work with multiplication and division, students in Grade 2 learned to partition a rectangle into rows and columns and write a repeated addition sentence to determine the total. They also skip-counted by 5s, 10s, and 100s. Then, in Grade 3, students developed a conceptual understanding of multiplication and division in relation to equal groups, arrays, and area. They developed a variety of strategies to build toward fluency with multiplication and division within 100 and applied that knowledge to the context of one- and two-step problems using the four operations.

To begin the unit, students extend their understanding of multiplication situations that they learned in Grade 3 to include multiplicative comparison using the words “times as many.” Next, to continue to refresh students’ work in Grades 2 and 3 on skip-counting and basic multiplication facts and extend it further to values they have not yet worked with, students investigate factors and multiples within 100, as well as prime and composite numbers (4.OA.4). Thus, this supporting cluster content serves as a foundation for the major work with multiplication and division with larger quantities. Tangentially, it will also support the major work in Unit 5 to recognize and generate equivalent fractions. Then, students move into two-digit by one-digit, three-digit by one-digit, four-digit by one-digit, and two-digit by two-digit multiplication, using the area model, partial products, and finally the standard algorithm, making connections between all representations as they go. The use of the area model serves to help students conceptually understand multiplication and as a connection to their work with area and perimeter (4.MD.3), a supporting cluster standard. Finally, with a full understanding of all multiplication cases, they then apply their new multiplication skills to solve multi-step word problems using multiplication, addition, and subtraction, including cases involving multiplicative comparison (4.NBT.5, 4.OA.3, 4.MD.3), allowing for many opportunities to connect content across multiple domains.

This unit affords lots of opportunities to deepen students’ mathematical practices. For example, “when students decompose numbers into sums of multiples of base-ten units to multiply them, they are seeing and making use of structure (MP.7). Students “reason repeatedly (MP.8) about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (NBT Progression, p. 14). Lastly, as students solve multi-step word problems involving addition, subtraction, and multiplication, they are modeling with mathematics (MP.4).

Students’ work in this unit will prepare them for fluency with the multiplication algorithm in Grade 5 (5.NBT.5). Students also learn about new applications of multiplication in future grades, including scaling quantities up and down in Grade 5 (5.NF.5), all the way up to rates and slopes in the middle grades (6.RP, 7.RP). Every subsequent grade level depends on the understanding of multiplication and its algorithm, making this unit an important one for students in Grade 4.

In this unit, students explore the concept of multi-digit division and its applications, such as interpreting a remainder in division word problems and using division to determine the nth term in a repeating shape pattern.

Students developed a foundational understanding of division in Grade 3, when they came to understand division in relation to equal groups, arrays, and area. They developed a variety of strategies to build towards fluency with division within 100, and they applied that knowledge to the context of one- and two-step problems using the four operations. Students also came to understand the distributive property, which underpins the standard algorithm for division.

Just as at the beginning of the previous unit when students expanded their understanding of multiplication beyond Grade 3 understanding to include multiplicative comparison word problems, this unit starts off with the added complexity of division problems with remainders (4.OA.3). This is likely familiar to students from their own real-world experiences of trying to split quantities evenly, and thus the focus is on interpretation of those remainders in the context of various problems. Next, students focus on extending their procedural skill with division to include up to four-digit dividends with one-digit divisors (4.NBT.6), representing these cases with base ten blocks, the area model, partial quotients, and finally the standard algorithm, making connections between all representations as they go. The use of the area model serves to help students conceptually understand division, and as a connection to their work with area and perimeter (4.MD.3), a supporting cluster standard. Lastly, armed with a deep understanding of all four operations spanned over the last three units, students solve multi-step problems involving addition, subtraction, multiplication, and division, including their new problem situations such as multiplicative comparison and interpreting remainders (4.OA.3). They also explore number and shape patterns, using the four operations to draw conclusions about them (4.OA.5).

Throughout the unit, students are engaging with the mathematical practices in various ways. For example, students are seeing and making use of structure (MP.7) as they “decompos[e] the dividend into like base-ten units and find the quotient unit by unit” (NBT Progressions, p. 16). Further, "by reasoning repeatedly (MP.8) about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (NBT Progression, p. 14). Lastly, as students solve multi-step word problems involving addition, subtraction, and multiplication, they are modeling with mathematics (MP.4).

While students are encouraged throughout the unit to use models when appropriate to solve problems, their in-depth experience with the place value system and multiple conceptual models and exposure to the division algorithms prepares them for extending these models to two-digit divisors in Grade 5 (5.NBT.6) and to fluency with the division algorithm in Grade 6 (6.NS.2). Every subsequent grade level depends on the understanding of multi-digit division and its algorithms, making this unit an important one for students in Grade 4.

In this unit, students begin their work with operating with fractions by understanding them as a sum of unit fractions or a product of a whole number and a unit fraction. Students will then add fractions with like denominators and multiply a whole number by any fraction. Students will apply this knowledge to word problems and line plots.

In Grade 3, students developed their understanding of the meaning of fractions, especially using the number line to make sense of fractions as numbers themselves. They also did some rudimentary work with equivalent fractions and comparison of fractions. In Grade 4 Unit 5, they deepened this understanding of equivalence and comparison, learning the fundamental property that “multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction” (NF Progression, p. 6).

Thus, in this unit, armed with a deep understanding of fractions and their value, students start to operate on them for the first time. The unit is structured so that students build their understanding of fraction operations gradually, first working with the simplest case where the total is a fraction less than 1, then the case where the total is a fraction between 1 and 2 (to understand regrouping when operating in simple cases), and finally the case where the total is a fraction greater than 2. With each of these numerical cases, they first develop an understanding of non-unit fractions as sums and multiples of unit fractions. Next, they learn to add and subtract fractions. And finally, they apply these understandings to complex cases, such as word problems or fraction addition involving fractions where one denominator is a divisor of the other, which helps prepare students for similar work with decimal fractions in Unit 7. After working with all three numerical cases in the context of fraction addition and subtraction, they work with fraction multiplication, learning strategies for multiplying a whole number by a fraction and a mixed number and using those skills in the context of word problems. Finally, students apply this unit’s work to the context of line plots. Students will solve problems by using information presented in line plots, requiring them to use their recently acquired skills of fraction addition, subtraction, and even multiplication, creating a contextual way for this supporting cluster content to support the major work of the grade. The unit provides lots of opportunity for students to reason abstractly and quantitatively (MP.2) and construct viable arguments and critique the reasoning of others (MP.3).

In this unit, students relate multiplication, particularly their understanding of “times as many” developed throughout Grade 4, to the conversion of measurement units. Students will also solve word problems involving measurement and measurement conversion.

In previous grades, students have worked with many of the metric and customary units (2.MD.1—6, 3.MD.1—2). They’ve noticed the relationship between some units to help them understand various measurement benchmarks but have not yet done any unit conversions. Not only does this unit build on measurement work from previous grades, but it also relies on myriad skills and understanding developed throughout Grade 4. As the Progressions state, “relating units within the metric system is another opportunity to think about place value” (Progressions for the Common Core State Standards in Mathematics, K-5 Geometric Measurement, p. 20). Further, “students also combine competencies from different domains as they solve measurement problems using all four arithmetic operations, addition, subtraction, multiplication, and division” (Progressions for the Common Core State Standards in Mathematics, K-5 Geometric Measurement, p. 20). Lastly, as directly stated in the language of 4.MD.2, students will convert units and solve problems involving fraction and decimal numbers. Thus, while the unit focuses on supporting cluster standards, their instruction enhances much of the major work of the grade, including place value (4.NBT.A), arithmetic operations (4.NBT.B), word problems (4.OA.A), and fractions (including decimal fractions) (4.NF).

As NCTM’s position statement on the metric system states, “Students need to develop an understanding of metric units and their relationships, as well as fluency in applying the metric system to real-world situations. Because some non-metric units of measure are common in particular contexts, students need to develop familiarity with multiple systems of measure, including metric and customary systems and their relationships" (The Metric System, NCTM). Thus, students will explore both metric and customary systems of measurement, starting with unit conversions from larger to smaller metric units in Topic A, and then similarly with customary units in Topic B. Then, in Topic C, students deal with more complex cases of fractional and decimal unit conversions. At the end of each topic, students apply their new learning in the context of solving multi-step word problems involving unit conversions.

The unit provides rich opportunities for students to engage with the mathematical practice standards. As the Progressions state, “relating units within the traditional system provides an opportunity to engage in mathematical practices, especially ‘look for and make use of structure’ (MP.7) and ‘look for and express regularity in repeated reasoning’ (MP.8)” (Progressions for the Common Core State Standards in Mathematics, K-5 Geometric Measurement, p. 20). Further, when students solve word problems that involve unit conversions, they "may use tape or number line diagrams for solving such problems (MP.1)" (Progressions for the Common Core State Standards in Mathematics, K-5 Geometric Measurement, p. 20).

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).

This video is part of the Learn and Grow with WHRO TV series. Watch Dr. Deborah Fuge teach about solving word problems.

The following word problems are designed for upper elementary students to identify correct operation(s) in order to solve paractical word problems.

This video is part of the Learn and Grow with WHRO TV series. Watch Meghan Dmytriw teach about how to solve multi-step word problems.

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