Grade 3: Unit 3.OA.A.1-4 Represent and Solve Problems Involving Multiplication & Division

Overview: The overview statement is intended to provide a summary of major themes in this unit.

In this unit, students move from solving problems about single objects to solving problems involving groups of objects. In these problems there are a specific number of groups with the same number of objects in each group. Students use groupable manipulatives, pictures, words, numbers, and equations to represent their thinking and solutions. Students develop their understanding of the meanings of multiplication and division through activities and problems involving equal-sized groups, arrays, and area models. Students will determine the unknown in each multiplication or division situation. The unknown can be located in any of the three positions in an equation in either a multiplication or division problem that represents the situation. Students use the relationship between multiplication and division to employ efficient strategies for solving problems.

Teacher Notes: The information in this component provides additional insights which will help the educator in the planning process for the unit.

  • Review the Progressions for Grades K-5 Counting and Cardinality; K-5 Operations and Algebraic Thinking at to see the development of the understanding of multiplication and division as stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.
  • As Table 2 on page 89 of the Common Core State Standards indicates, there is more than one way to solve a problem. It is VERY important to help students see that one student could use multiplication to solve a problem while another might use division, and a third might use a comparison or number sense. It is very important to expose students to all of the approaches modeled in Table 2, Page 89, CCSS, to have them discuss what they know from reading or hearing a problem and what they need to find. Students can then approach the problem in a way that makes sense to them and see if it is effective and leads to a clear solution. See the following site to access this table:
  • Table 3 on page 90 of the Common Core State Standards shares the various Properties of Operations that are vital to this unit. See access them.
  • Students should engage in well-chosen, purposeful, problem-based tasks. A good mathematics problem can be defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method (Hiebert et al., 1997). A good mathematics problem will have multiple entry points and require students to make sense of the mathematics. It should also foster the development of efficient computations strategies as well as require justifications or explanations for answers and methods.
  • Choose problems carefully for students. For example, determine if you wish to focus on using doubling and halving in multiplication, or on using landmark numbers. Specific types of problems typically elicit certain strategies.
  • Focusing on ‘Key Words’ limits a child’s ability to successfully solve problems since it locks them into one and only one approach, which is not necessarily the best for that problem, and possibly not even correct.
  • Classroom discussion, “think-alouds”, and recording students’ ideas as they share them during group discussion are integral in developing algebraic thinking as well as building on students’ computational skills. It is important to record a student’s method for solving a problem both horizontally and vertically.
  • The vocabulary words that students should learn to use with increasing precision with this cluster are: operation, multiply, divide, factor, product, quotient, divisor, dividend, equation, unknown, strategies, reasonableness, mental computation, estimation, rounding, patterns, and properties or rules about how numbers work. Students need not use the formal terms Commutative Property, Associative Property, or Distributive Property when dealing with these properties of operations.
  • 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. The Order of Operations applies, which states that with no parentheses or exponents, you complete multiplication and/or division in the order shown followed by addition and/or subtraction in the order shown.
  • Variables can be used in three different contexts: as unknowns, as changing quantities, and in generalizations of patterns.
  • Multiplicative compare situations are more complex than equal groups and arrays, and must be carefully distinguished from additive

compare problems. Multiplicative comparison first entersthe Standards at Grade 4.OA.A.1. For more information on multiplicative compare problems, see the Grade 4 section of the progressions document noted above.

  • Note that in the PARCC Model Content Frameworks ( the following statements are made:
  • Students must begin work with multiplication and division (3.OA) at or near the very start of the year to allow time for understanding and fluency to develop.
  • Note that area models for products are an important part of this process (3.MD.C.7). Hence, work on concepts of area (3.MD.C.5–6) should likely begin at or near the start of the year as well.

Enduring Understandings: Enduring understandingsgo beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

  • A mathematical statement that uses an equal sign to show that two quantities are equivalent is called an equation.
  • Equations can be used to model problem situations.
  • Operations model relationships between numbers and/or quantities.
  • Addition, subtraction, multiplication, and division operate under the same properties of operations in algebra as they do in arithmetic.
  • The relationships among the operations and their properties promote computational fluency.
  • Students use mathematical reasoning and number models to manipulate practical applications and to solve problems.
  • Through the properties of numbers we understand the relationships of various mathematical functions.
  • Flexible methods of computation involve grouping numbers in strategic ways.
  • Computation involves decomposing and composing numbers using a variety of approaches.

Essential Questions: A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

  • Why do I need mathematical operations?
  • How is thinking algebraically different from thinking arithmetically?
  • How do the properties contribute to algebraic understanding?
  • What is meant by equality?
  • What do I know from the information shared in the problem? What do I need to find?
  • How do I know which computational method (mental math, estimation, paper and pencil, and calculator) to use?
  • How do you solve problems using multiplication or division in real world situations?
  • What are some strategies for solving to find the unknowns in equations?
  • How do you estimate answers using rounding to the greatest place?
  • How can you decide that your calculation is reasonable?
  • What are different models of and models for multiplication and division?
  • What questions can be answered using multiplication and division?
  • What are efficient methods for finding products and quotients?
  • How do multiplication and division relate to each other?
  • How are parentheses used in numeric expressions?
  • What computation tools are best suited to which circumstances?
  • How do I record a solution to a multiplication or division problem in an equation?

Content Emphasis by Cluster in Grade 3:According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key:

Major Clusters

Supporting Clusters

○Additional Clusters

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

Understand the properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Number and operations in Base Ten

○Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations – Fractions

Develop understanding of fractions as numbers.

Measurement and Data

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Represent and interpret data.

Geometric measurement: understand concepts of area and relate area to multiplication and addition.

  • Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometry

Reason with shapes and their attributes.

Focus Standards: (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):

According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators should give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

  • 3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
  • 3.OA.C.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, now from memory all products of two one-digit numbers.
  • 3.MD.C.7Relate area to the operations of multiplication and addition.
  • 3.MD.C.7aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
  • 3.MD.C.7b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
  • 3.MD.C.7c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning.
  • 7.MD.C.7d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

PossibleStudent Outcomes: The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeply into the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

The student will:

  • Solve word problems using drawings and equations involving multiplication and division within 100.
  • Write equations to represent word problems involving multiplication and division.
  • Identify the value of the unknown in multiplication or division equations.
  • Solve multiplication and division problems in which the unknown is in any of the three possible positions.
  • Model the solution of multiplication or division word problems using equal groups, arrays, drawings, or equations.
  • Identify the errors in another student’s work which yields an incorrect response.
  • Become engaged in problem solving that is about thinking and reasoning.
  • Collaborate with peers in an environment that encourages student interaction and conversation that will lead to mathematical discourse about multiplication and division.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

  • The Common Core Standards Writing Team (May 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at:

Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.

  • Key Advances from Previous Grades:

Students in Kindergarten:

  • Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
  • Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects, drawings, and mental math to represent the problem.
  • Decomposenumbers less than or equal to 10 into pairs in more than one way, e.g., by using objects, drawings, and mental math and then record each decomposition by a drawing or writing anequation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
  • When given any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects, drawings, and mental math and then record the answer with a drawing or equation.
  • Fluently add and subtract within 5. (Students in Kindergarten work with addition and subtraction to 10 but must be fluent up to 5.)

Students in Grade 1:

  • Use addition and subtraction within 20 to solve word problems.
  • Solve word problems that call for the addition of three whole numbers whose sum is less than or equal to 20.
  • Apply properties of operations as strategies to add and subtract.
  • Understand subtraction as an unknown-addend problem.
  • Relate counting to addition and subtraction.
  • Fluently add and subtract within 20 by the end of grade 1.
  • Understand the meaning of the equal sign.
  • Determine if equations involving addition and subtraction are true or false.
  • Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.

Students in Grade 2:

  • Add and subtract within 1000.
  • Fluently add and subtract within 100.
  • Use addition and subtraction to solve one- and two-step word problems.
  • Add up to four two-digit numbers.
  • Explain why addition and subtraction strategies work.
  • Work with equal groups of objects to gain foundations for multiplication.
  • Additional Mathematics:

Students in Grade 4:

○Use the four operations with whole numbers to solve problems.

○Work with factors and multiples.

○Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers.

○Find whole-number quotients and remainders with up to four-digit dividends, and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. They illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Students in Grade 5 and beyond:

○Write and interpret numerical expressions, using parentheses, brackets, or braces and evaluate these expressions.

○Perform operations with multi-digit whole numbers and with decimals to hundredths.

○Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

○Compute fluently with multi-digit numbers and find common factors and multiples.

○Apply and extend previous understandings of numbers to the system of rational numbers.

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections tograde-level standards from outside the cluster.

Over-Arching
Standards / Supporting Standards
within the Cluster / Instructional Connections outside the Cluster
3.OA.A.1: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7. / 3.OA.B.5: Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 Is known, then 4 x 6 = 24 is also known (Commutative property of multiplication). 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30 (Associative property of multiplication). Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 (Distributive property).
3.OA.A.2: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.A.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. / 3.OA.C.7:Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.OA.D.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.MD.C.7:Relate area to the operations of multiplication and addition.
3.MD.C.7a:Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7b: Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7c: Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning.
7.MD.C.7d: Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations: 8 x ? = 48, 5 = ?÷ 3, 6 x 6 = ?. / 3.OA.B.6: Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.