Math - 2018-19

4.4 - Addition, Subtraction, Multiplication, Division, Estimation, Multi-Step Practical Problems

The student will

a)  demonstrate fluency with multiplication facts through 12 × 12, and the corresponding division facts;*

b)  estimate and determine sums, differences, and products of whole numbers;*

c)  estimate and determine quotients of whole numbers, with and without remainders;* and

d)  create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication, and single-step practical problems involving division with whole numbers.

*On the state assessment, items measuring this objective are assessed without the use of a calculator.



Adopted: 2016

BIG IDEAS

  • So that I can use word problems to apply math concepts to real life situations.

  • So that I can solve word problems about favorite football teams, classmates, shopping trips, and daily activities.

  • I can use word problems to help me become a great math thinker!

UNDERSTANDING THE STANDARD

  • Computational fluency is the ability to think flexibly in order to choose appropriate strategies to solve problems accurately and efficiently.

  • The development of computational fluency relies on quick access to number facts.  There are patterns and relationships that exist in the facts.  These relationships can be used to learn and retain the facts.

  • A certain amount of practice is necessary to develop fluency with computational strategies; however, the practice must be motivating and systematic if students are to develop fluency in computation, whether mental, with manipulative materials, or with paper and pencil.

  • In grade three, students developed an understanding of the meanings of multiplication and division of whole numbers through activities and practical problems involving equal-sized groups, arrays, and length models.  In addition, grade three students have worked on fluency of facts for 0, 1, 2, 5, and 10.

  • Three models used to develop an understanding of multiplication include: 
    • The equal-sets or equal-groups model lends itself to sorting a variety of concrete objects into equal groups and reinforces the concept of multiplication as a way to find the total number of items in a collection of groups, with the same amount in each group, and the total number of items can be found by repeated addition or skip counting.
                           
    • The array model, consisting of rows and columns (e.g., three rows of four columns for a 3-by- 4 array), helps build an understanding of the commutative property.
                           
    • The length model (e.g., a number line) also reinforces repeated addition or skip counting.
                            
  • The number line model can be used to solve a multiplication problem such as 3 x 6. This is represented on the number line by three jumps of six or six jumps of three, depending on the context of the problem. 
                                 
  • The number line model can be used to solve a division problem such as 6 ÷ 3 and is represented on the number line by noting how many jumps of three go from 6 to 0.

  • In order to develop and use strategies to learn the multiplication facts through the twelves table, students should use concrete materials, a hundreds chart, and mental mathematics.  Strategies to learn the multiplication facts include an understanding of multiples, properties of zero and one as factors, commutative property, and related facts. Investigating arithmetic operations with whole numbers helps students learn about the different properties of arithmetic relationships. These relationships remain true regardless of the whole numbers.

  • Grade four students should explore and apply the properties of addition and multiplication as strategies for solving addition, subtraction, multiplication, and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols).
  • The properties of the operations are “rules” about how numbers work and how they relate to one another.  Students at this level do not need to use the formal terms for these properties but should utilize these properties to further develop flexibility and fluency in solving problems.  The following properties are most appropriate for exploration at this level:
    • The identity property of addition states that if zero is added to a given number, the sum is the same as the given number.  The identity property of multiplication states that if a given number is multiplied by one, the product is the same as the given number.
    • The commutative property of addition states that changing the order of the addends does not affect the sum (e.g., 24 + 136 = 136 + 24).  Similarly, the commutative property of multiplication states that changing the order of the factors does not affect the product (e.g., 12 x 43 = 43 x 12).
    • The associative property of addition states that the sum stays the same when the grouping of addends is changed (e.g., 15 + (35 + 16) = (15 + 35) + 16).  The associative property of multiplication states that the product stays the same when the grouping of factors is changed [e.g., 16 x (40 x 5) = (16  x 40)  x 5].
    • The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.  Several examples are shown below:

      •  3(9) = 3(5 + 4)
      •  3(5 + 4) = (3 × 5) + (3 × 4)
      •  5 × (3 + 7) = (5 × 3) + (5 × 7)
      •  (2 × 3) + (2 × 5) = 2 × (3 + 5) 
      •  9 × 23
      • 9(20+3)
      • 180 + 27
      • 207
      • 34 x 8
  • Addition is the combining of quantities; it uses the following terms:
                                addend    →      45,623
                                addend    →   + 37,846
                                sum          →     83,469
  • Subtraction is the inverse of addition; it yields the difference between two numbers and uses the following terms:
                                 minuend      →       45,698
                                subtrahend   →    – 32,741   
                                
    difference     →       12,957
  • The terms associated with multiplication are
    •                      factor    →     376
                           factor    →    x 23
                          product  →   8,648

  • In multiplication, one factor represents the number of equal groups and the other factor represents the number in or size of each group.  The product is the total number in all of the groups. 
  • Multiplication can also refer to a multiplicative comparison, such as: “Gwen has six times as many stickers as Phillip”.  Both situations should be modeled with manipulatives.
  • Models of multiplication may include repeated addition and collections of like sets, partial products, and area or array models.
  • Division is the operation of making equal groups or shares.  When the original amount and the number of shares are known, divide to determine the size of each share.  When the original amount and the size of each share are known, divide to determine the number of shares.  Both situations may be modeled with base-ten manipulatives.
  • Division is the inverse of multiplication. Terms used in division are dividend, divisor, and quotient.

    dividend ÷ divisor = quotient

                           quotient
           divisor  ) dividend

                   dividend 
                    divisor      =  quotient

  • Students benefit from experiences with various methods of division, such as repeated subtraction and partial quotients.
  • Estimation can be used to determine the approximation for and then to verify the reasonableness of sums, differences, products, and quotients of whole numbers. An estimate is a number that lies within a range of the exact solution, and the estimation strategy used in a particular problem determines how close the number is to the exact solution. An estimate tells about how much or about how many.

  • Strategies such as rounding up or down, front-end, and compatible numbers may be used to estimate sums, differences, products, and quotients of whole numbers.

  • The least number of steps necessary to solve a single-step problem is one.

  • The problem-solving process is enhanced when students create and solve their own practical problems and model problems using manipulatives and drawings.

  •  In problem solving, emphasis should be placed on thinking and reasoning rather than on key words.  Focusing on key words such as in all, altogether, difference, etc.,encourages students to perform a particular operation rather than make sense of the context of the problem.  A key-word focus prepares students to solve a limited set of problems and often leads to incorrect solutions as well as challenges in upcoming grades and courses.

  • Extensive research has been undertaken over the last several decades regarding different problem types.  Many of these studies have been published in professional mathematics education publications using different labels and terminology to describe the varied problem types.

  • Students should experience a variety of problem types related to multiplication and division.  Some examples are included in the following chart:


  • Students need exposure to various types of practical problems in which they must interpret the quotient and remainder based on the context.  The chart below includes one example of each type of problem.

ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

  • Demonstrate fluency with multiplication through 12 × 12, and the corresponding division facts. (a)
  • Estimate whole number sums, differences, products, and quotients, with and without context. (b, c)
  • Apply strategies, including place value and the properties of addition to determine the sum or difference of two whole numbers, each 999,999 or less. (b)
  • Apply strategies, including place value and the properties of multiplication and/or addition, to determine the product of two whole numbers when both factors have two digits or fewer. (b)
  • Apply strategies, including place value and the properties of multiplication and/or addition, to determine the quotient of two whole numbers, given a one-digit divisor and a two- or three-digit dividend, with and without remainders. (c)
  • Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than. (b, c)
  • Create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication with whole numbers. (d) 
  • Create and solve single-step practical problems involving division with whole numbers. (d)
  • Use the context in which a practical problem is situated to interpret the quotient and remainder. (d)

KEY VOCABULARY

Updated: Aug 22, 2018