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. 

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols.

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

This task allows students to relate addition and subtraction problems to money in a context that introduces the concept of scarcity.

In this task students are asked to write an equation to solve a real-world problem. There are two natural approaches to this task. In the first approach, students have to notice that even though there is one variable, namely the number of firefighters, it is used in two different places. In the other approach, students can find the total cost per firefighter and then write the equation.

This task is the first in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

This task provides a context for some of the questions asked in "6.NS Multiples and Common Multiples." A scaffolded version of this task could be adapted into a teaching task that could help motivate the need for the concept of a common multiple.

This series of 5 word problems lead up to the final problem. Most students should be able to answer the first two questions without too much difficulty. The decimal numbers may cause some students trouble, but if they make a drawing of the road that the girls are riding on, and their positions at the different times, it may help. The third question has a bit of a challenge in that students won't land on the exact meeting time by making a table with distance values every hour. The fourth question addresses a useful concept for problems involving objects moving at different speeds which may be new to sixth grade students.

VA SOL 3.17a     The student will create equations to represent equivalent mathematical relationships.This task allows students to use a numberl ine to solve a word problem while learning how to write an equation  for addition.  This remix includes manipulatives as a way to support student learning.

While students need to be able to write sentences describing ratio relationships, they also need to see and use the appropriate symbolic notation for ratios. If this is used as a teaching problem, the teacher could ask for the sentences as shown, and then segue into teaching the notation. It is a good idea to ask students to write it both ways (as shown in the solution) at some point as well.

The first of these word problems is a multiplication problem involving equal-sized groups. The next two reflect the two related division problems, namely, "How many groups?" and "How many in each group?"

The purpose of this task is to show three problems that are set in the same kind of context, but the first is a straightforward multiplication problem while the other two are the corresponding "How many groups?" and "How many in each group?" division problems.

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).