2.6 - Estimating and Determining Sums and Differences

The student will

a) estimate sums and differences;

b) determine sums and differences, using various methods

c) create and solve single-step and two-step practical problems involving addition and subtraction.

• So that I can estimate when we guess a number as closely as possible instead of finding an exact answer
  
• So that I can be used to quickly check the reasonableness of a sum
  
• So that  we can estimate the number of candies in a jar
  
• So that we can estimate the cost of our groceries so that we do not spend too much money
  
• So that I can round numbers and use the estimates to help me add. This can help me when I am buying things at the store. I can figure out if I have enough money to get what I need
  
• So that I can understand that solving a mathematical problem is similar to solving problems I will face in life
• So that I can understand that sometimes I have to work through more than one step in order to solve a problem

• So that I can learn to think through a problem and come to a solution

• Addition and subtraction should be taught concurrently in order to develop understanding of the inverse relationship.
  
• Grade two students should begin to explore the properties of addition as strategies for solving addition and subtraction problems using a variety of representations, including manipulatives and diagrams.
  
• 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 commutative property of addition states that changing the order of the addends does not affect the sum (e.g., 4 + 3 = 3 + 4).
• 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 associative property of addition states that the sum stays the same when the grouping of addends is changed (e.g., 4 + (6 + 7) = (4 + 6) + 7).
• An equation (number sentence) is a mathematical statement representing two expressions that  are equivalent. It consists of two expressions, one on each side of an 'equal' symbol (e.g., 5 + 3 = 8,  8 = 5 + 3, and 4 + 3 = 9 - 2).  An equation can be represented using a balance scale, with equal amounts on each side (e.g., 3 + 5 = 6 + 2).
  
• Rounding is one strategy used to estimate.
  
• Estimation skills are valuable, time-saving tools particularly in practical situations when exact answers are not required or needed.
  
• Estimation can be used to check the reasonableness of the sum or difference when an exact answer is required.
  
• Problem solving means engaging in a task for which a solution or a method of solution is not known in advance. Solving problems using data and graphs offers one way to connect mathematics to practical situations.
  
• The problem-solving process is enhanced when students:
  
• create their own story problems; and
• model word problems, using manipulatives, drawings, or acting out the problem.
• The least number of steps necessary to solve a single-step problem is one.
  
• Using concrete materials (e.g., base-10 blocks, connecting cubes, beans and cups, etc.) to explore, model and stimulate discussion about a variety of problem situations helps students understand regrouping and enables them to move from the concrete to the abstract. Regrouping is used in addition and subtraction algorithms.
  
• Conceptual understanding begins with concrete and contextual experiences. Next, students must make connections that serve as a bridge to the symbolic. Student-created representations, such as drawings, diagrams, tally marks, graphs, or written comments are strategies that help students make these connections.
• 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 addition and subtraction.  Problem type examples are included in the following chart:
  

  Join (Result Unknown)	Join (Change Unknown)	Join (Start Unknown)
  Sue had 28 pencils. Alex gave her 14 more pencils. How many pencils does Sue have all together?	Sue had 28 pencils. Alex gave her some more pencils. Now Sue has 42 pencils. How many did Alex give her?	Sue had some pencils. Alex gave her 14 more. Now Sue has 42 pencils. How many pencils did Sue have to start with?
  Separate (Result Unknown)	Separate (Change Unknown)	Separate (Start Unknown)
  Brooke had 35 marbles. She gave 19 marbles to Joe. How many marbles does Brooke have now?	Brooke had 35 marbles. She gave some to Joe. She had 16 marbles left. How many marbles did Brooke give to Joe?	Brooke had marbles. She gave 19 to Joe. Now she has 16 marbles left. How many marbles did Brooke start with?
  Part-Part-Whole (Whole Unknown)	Part-Part-Whole (One Part Unknown)	Part-Part-Whole (Both Parts Unknown)
  The teacher has 20 red markers and 25 blue markers. How many markers does he have?	The teacher has 45 markers. Twenty of the markers are red, and the rest are blue. How many blue markers does he have?	The teacher has a tub of red and blue markers. She has 45 markers in all. How many markers could be red? How many could be blue?
  Compare (Difference Unknown)	Compare (Bigger Unknown)	Compare (Smaller Unknown)
  Ryan has 20 books and Chris has 9 books. How many more books does Ryan have than Chris? Ryan has 20 books. Chris has 9 books. How many fewer books does Chris have than Ryan?	Chris has 9 books. Ryan has 11 more books than Chris. How many books does Ryan have? Chris has 11 fewer books than Ryan. Chris has 9 books. How many books does Ryan have?	Ryan has 11 more books than Chris. Ryan has 20 books. How many books does Chris have? Chris has 11 fewer books than Ryan. Ryan has 20 books. How many books does Chris have?

  
  
• Strategies for adding and subtracting two-digit numbers can include, but are not limited to, using concrete objects, a hundred chart, number line, and invented strategies.
  
• Mental computation helps build number sense in students.  Strategies for mentally adding or subtracting two-digit numbers should be student-invented strategies.  Some of these strategies may include:
  
  
• The terms used in addition are
       23    → addend
    + 46    → addend
       69    → sum
• The terms often used in subtraction are
       98    →  minuend
    – 41    → subtrahend
       57    → difference
• At this level, students do not need to use the terms addend, minuend, or subtrahend for addition and subtraction as shown above.
• 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.

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

• Estimate the sum of two whole numbers whose sum is 99 or less and recognize whether the estimation is reasonable (e.g., 27 + 41 is about 70, because 27 is about 30 and 41 is about 40, and 30 + 40 is 70). (a)
  
• Estimate the difference between two whole numbers each 99 or less and recognize whether the estimate is reasonable. (a)
  
• Determine the sum of two whole numbers whose sum is 99 or less, using various methods. (b)
  
• Determine the difference of two whole numbers each 99 or less, using various methods. (b)
• Create and solve single-step practical problems involving addition or subtraction. (c)
• Create and solve two-step practical problems involving addition, subtraction, or both addition and subtraction. (c)
  
  

2nd Math Vocabulary