Grade 3 Mathematics, Quarter 4, Unit 4.3
Solving Two-Step Problems Overview
Number of instruction days:
6–8
Content to Be Learned Solve two-step problems involving whole numbers using the four operations and represent these problems with equations (with a letter standing for the unknown quantity) Fluently add and subtract within 100 using strategies and algorithms Use mental computation or estimation, including rounding, to assess reasonableness of answers. Solve multistep problems in the conventional order when no parentheses are present (Order of Operations). Apply understanding of place value and properties of operations as strategies in computations. Multiply one-digit whole numbers by multiples of 10–90 using strategies based on place value and properties of operations.
Providence Public Schools
(1 day = 90 minutes)
Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Assign meaning when analyzing problems. Mentally decompose equations during the solution process. Explain relationships between equations and operations. Evaluate reasonableness of various strategies throughout the solution process.
7 Look for and make use of structure. Determine patterns by decomposing equations. Apply properties of operations and place value to solve equations. Analyze the problem for structure and shift perspectives when necessary.
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Grade 3 Mathematics, Quarter 4, Unit 4.3
Solving Two-Step Problems (6–8 days)
Version 5
Essential Questions How can you solve two-step problems using the four operations?
How can the Order of Operations help you to solve multistep problems?
How can you represent the problem with an equation using a letter standing for the unknown quantity?
How can you apply your understanding of place value and/or properties of operations when multiplying and dividing?
Can you explain your most efficient strategy when solving problems involving addition and subtraction through 1000?
How can you explain your strategies when you multiply one-digit whole numbers by multiples of 10–90?
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Solving Two-Step Problems (6–8 days)
Grade 3 Mathematics, Quarter 4, Unit 4.3 Version 5
Standards Common Core State Standards for Mathematical Content Operations and Algebraic Thinking
3.OA
Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.8
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 3 3
This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
Number and Operations in Base Ten
3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.4 4
A range of algorithms may be used.
3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. 3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
Common Core State Standards for Mathematical Practice 1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Providence Public Schools
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Solving Two-Step Problems (6–8 days)
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Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards Prior Learning Students in Grade 2 represented and solved two-step word problems involving addition and subtraction. They fluently added and subtracted within 20. They worked with equal groups of objects to prepare for work with multiplication. Second-grade students also used addition and subtraction to find the total number of objects arranged in rectangular arrays with up to five rows and five columns. They wrote equations to express the total as the sum of equal addends (repeated addition).
Current Learning There are two fluency expectation for grade 3 students: fluently add and subtract within 1000 using strategies and algorithms (3.NBT.2) and fluently multiply and divide within 100. In this unit, students solidify these fluencies, and apply these skills when problem solving two-step word problems using the four operations (3.OA.8). Developing understanding of multiplication and division strategies is a critical area of focus for Grade 3. Students find products and related quotients involving single-digit whole numbers and 10. This is a required fluency for Grade 3. By the end of Grade 3, students conceptually understand and know all products of 2 one-digit numbers from memory (quickly and accurately). Grade 3 students use strategies based in understanding of place value and properties of operations in problem solving. They determine the unknown whole number in a multiplication or division problem. Students solve multiplication and division problems using equal groups, arrays, and measurement. Common multiplication and division situations are detailed in Table 2 in the CCSS Glossary, page 89. D-116
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Solving Two-Step Problems (6–8 days)
Grade 3 Mathematics, Quarter 4, Unit 4.3 Version 5
Although there are three major types of situations, Grade 3 focuses on Equal Groups and Arrays. Students apply properties of operations (commutative, associative, distributive) when multiplying and dividing. They represent problems as equations, solve two-step word problems involving multiplication and division, and multiply one-digit whole numbers by multiples of 10 (10–90).
Future Learning In Grade 4, students will develop fluency with multi-digit multiplication and division using their knowledge of place value to 1,000,000 and the properties of operations. They will continue to calculate mentally and estimate while using the four operations to solve problems. They will use the standard algorithm to fluently add and subtract multi-digit whole numbers. Grade 4 students will continue to build an understanding of the relationship between division and multiplication. In Grade 5, students will finalize fluency with multi-digit addition, subtraction, multiplication, and division.
Additional Findings According to Principles and Standards for School Mathematics, “in grades 3–5, students’ development of number sense should continue, with a focus on multiplication and division. Their understanding of the meanings of these operations should grow deeper as they encounter a range of representations and problem situations, learn about the properties of these operations, and develop fluency in whole number computation.” (p. 149) Principles and Standards in School Mathematics also states that “Students in these grades should also develop computational-estimation strategies for situations that call for an estimate and as a tool for judging the reasonableness of solutions.” (p. 144)
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Grade 3 Mathematics, Quarter 4, Unit 4.3
Solving Two-Step Problems (6–8 days)
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Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students’ attainment of the mathematics within the unit. Solve two-step problems involving whole numbers using the four operations and represent these problems with equations (with a letter standing for the unknown quantity).. Fluently add and subtract within 100 using strategies and algorithms Use mental computation or estimation, including rounding, to assess reasonableness of answers. Solve multistep problems in the conventional order when no parentheses are present (Order of Operations). Apply understanding of place value and properties of operations as strategies in computations. Multiply one-digit whole numbers by multiples of 10–90 using strategies based on place value and properties of operations.
Instruction Learning Objectives Students will be able to: Solve two-step problems using the four operations and represent these problems with equations (with a letter standing for the unknown quantity). Use mental computation or estimation, including rounding, to assess reasonableness of answers. Solve multi-step problems in the conventional order when no parentheses are present (Order of Operations). D-118
Providence Public Schools
Solving Two-Step Problems (6–8 days)
Grade 3 Mathematics, Quarter 4, Unit 4.3 Version 5
Apply understanding of place value and properties of operations as strategies in computations. Multiply one-digit whole numbers by multiples of 10–90 using strategies based on place value and properties of operations. Demonstrate understanding of the concepts and skills learned in this unit.
Resources enVision Math Grade 3, Pearson Education, Inc., 2009 Topic 5 Multiplication Meanings and Facts Teacher Edition Topic 6 Multiplication Fact Strategies: Use Known Facts Teacher Edition Topic 7, Division Meanings Teacher Edition Topic 8 Division Facts Teacher Edition Resource Masters Student Pages Investigations in Numbers, Data, and Space – Grade 3, Pearson Education, Inc., 2008 Implementing Investigations in Grade 3 – Implementation Guide Unit 5 Teacher Edition Equal Groups Teacher Resources Binder Pearson Success Net, www.pearsonsuccessnet.com Investigations (TERC) http://investigations.terc.edu/index.cfm Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment for ample resources to refer to when planning your unit and individual lessons.
Materials 1 inch grid paper, hundreds charts, two-color counters, connecting cubes
Providence Public Schools
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Grade 3 Mathematics, Quarter 4, Unit 4.3
Solving Two-Step Problems (6–8 days)
Version 5
Instructional Considerations Key Vocabulary properties of operations (addition and multiplication)
distributive property inverse property
associative property commutative property
Planning for Effective Instructional Design and Delivery In this unit, students have the opportunity to practice and apply the concepts and skills they have learned to represent and solve multi-step problems and real world situations. Encourage students to use a variety of algorithms and solution paths as they apply what they know about place value and the properties of operations, and support them in sharing their ideas with one another. There are two fluency expectations for grade 3 students: fluently add and subtract within 1000 using strategies and algorithms (3.NBT.2) and fluently multiply and divide within 100. In this unit, students solidify these fluencies, and apply these skills when problem solving two-step word problems using the four operations (3.OA.8). Refer to Table 1 in the CCSS Glossary titled Common addition and subtraction situations. It is imperative that students have experience throughout the year solving all problem types. Refer to Table 2 in the CCSS Glossary. This table provides information on the common multiplication and division situations. It is essential that your students have practice with two situation types presented in the table: Equal Groups and Array/Area situations. The resources identified in this unit have been referenced in prior units. Use those resources that best meet the needs of your students in new and innovative ways. Provide relevant, real world contexts for students to apply and solve a variety of situations. An important component of standard 3.OA.8 is assessing “the reasonableness of an answer” through estimation and other mental strategies. This is a “habit of mind” that needs to be continuously modeled and built into both computation and problem solving. Assessing the reasonableness of an answer can help build critical thinking skills and better number sense. Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction.
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Solving Two-Step Problems (6–8 days)
Grade 3 Mathematics, Quarter 4, Unit 4.3 Version 5
Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources (particularly the Algebra Connections and Teacher Notes sections)
Notes
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