Addition and subtraction with whole numbers

Purpose

This unit explores situations that involve addition and subtraction of whole numbers. Students are expected to choose among a range of strategies, based on their understanding of place value.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
Specific Learning Outcomes
  • Choose appropriately among strategies such as using standard place value, using tidy numbers with compensation, and equal adjustments to solve addition and subtraction problems with whole numbers.
  • Choose correct operations to solve joining, separating, and difference problems with whole numbers.
  • Record solutions using equations, algorithms, and calculator inputs.
Description of Mathematics

There are three types of situation to which addition and subtraction are applied.

  1. Joining parts to form a whole
    Example: You have 35 toy cars and are given 47 more. How many cars do you have?
    Equation: 35 + 47 = 82
  2. Separating parts from a whole
    Example: You have 82 toy cars and give 47 away. How many cars do you have?
    Equation: 82 - 47 = 35
  3. Comparing parts or comparing whole.
    Example: You have 82 toy cars, and your friend has 35 toy cars. How many more cars do you have than her?
    Equation: 35 + □ = 82 or 82 – 35 = □.

The role of the unknown changes the difficulty of the problem and the operation needed to solve it.

  1. Joining change unknown
    Example: You have 35 toy cars and get some more cars. Now you have 82 cars. How many cars did you get?
    Equation: 35 + □ = 82
  2. Separating start unknown
    Example: You have some toy cars and give 47 cars to your friend. Now you have 35 cars. How many cars did you have to start with?
    Equation: □ - 47 = 82
  3. Comparing start unknown
    Example: You have 82 toy cars.  That is 47 more cars than your friend. How many cars does your friend have?
    Equation: □ + 47 = 82

Notice that the same equation can represent a range of different situations.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Provide place value materials so that students can anticipate actions and justify their solutions. Use materials like place value blocks, bundled iceblock sticks, and beaNZ in canisters that are proportional, e.g. ten ones comprise the tens unit. Physical representation diminishes the degree of abstraction required and helps students see what is occurring with the quantities that numbers represent.
  • Connect symbols and mathematical vocabulary, especially the symbols for addition and subtraction (+, -) and for equality (=). Explicitly model the correct use of equations and algorithms and discuss the meaning of the symbols in context.
  • Alter the complexity of the numbers that are used. More places, e.g. three digits or two digits, tends to increase complexity though much depends on the renaming that is required. Addition tends to be easier than subtraction though that depends on the nature of the numbers.
  • Encourage students to collaborate in small groups and to share, and justify, their ideas.
  • Use technology, especially calculators, in predictive, pattern-based ways to estimate sums and differences, e.g. Is the answer to 83 – 47 closer to 20, 30, 40, or 50? How do you know? Allow use of calculators where you want students to focus more on the process of getting a reasonable answer than on practising calculation skills.

The context used for this unit include occupations, finance, and collections of objects. Adjust the contexts to everyday situations that your students are likely to be interested in, and relfect their cultural identities. Addition and subtraction problems can easily be framed in term of objects that are meaningful to your students, such as natural objects (e.g. pine cones), people in their kura or whānau, or points earned in games or other competitions. Encourage students to be creative by accepting a variety of strategies from others and asking students to create their own problems for others to solve, in contexts that are meaningful.

Required Resource Materials

 

Activity

Session One

Begin each session in this unit with an addition/subtraction grid. This session gives you detail about how to introduce the activity. Grids for the other sessions are provided in PowerPoint One. This activity and discussion of how students found the answers should take no more than ten minutes.

  1. Show Slide One of PowerPoint One.
    grid
    Discuss how to complete the grid. The answers to the additions go in the body of the table. For example, 7 + 6 = 13 so 13 can be placed in the second cell in the third row.
    Some of the addends (numbers to add) are missing.
    How will you workout what numbers they are?
    Students might recognise that 2 goes in the bottom of the left-hand column because 7 + 2 = 9.
  2. Let students complete the grid. Some students enjoy the challenge of bettering their time from day to day, but others may not. Students can record their time each day if that motivates them.
    Use the animation of Slide One (mouse clicks) to show the answers and how those answers can be determined from the information that is given.
  3. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 3 on pages 56-58 and select problems that are appropriate for your students. Make sure that you have physical materials that represent the place value structure of whole numbers, such as Place Value Blocks, beaNZ in canisters, iceblock sticks in bundles, or Place Value People.  Money is not a proportional representation, and one cent coins are no longer used. Use problem stories other than those involving money, if possible.
    Represent each calculation using both the physical materials and empty number lines. For example, 56 + 37 = 93 can be represented as:
    image            image
    56 equals 5 tens, and 6 ones                     Adding 30 gives 86, 8 tens, and 6 ones
    image           image
    Adding 7 gives 8 tens, and 13 ones           10 ones make 1 ten so the answer equals 93
    numberline

Session Two

  1. Use Slide Two of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 4 on pages 58-60 and select problems that are appropriate for your students. The key idea is about using tidy numbers with compensation to solve addition and subtraction problems. Use proportional place value materials rather than money. Change the contexts of the problems to be about items that are of interest to your students.
  3. Pay particular attention to the inverse compensation needed for subtraction.
    For example, consider 72 – 38 =  (in an appropriate story context).
    With place value blocks the problem might be represented as:
    image          image
    72 equals 7 tens, and 2 ones                    Subtract 40 (4 tens) leaves 3 tens, and 2 ones (32)
    image
    Compensating for removing 40 instead of 38 gives 3 tens and 4 ones (34)
    The same operation on an empty number line looks like this:
    number line
  4. Pose problems with three-digit whole numbers as well, such as:
    There are 725 trout in Lake Kahurangi at the start of the fishing season.
    297 of the trout are caught.
    How many trout are left?
  5. Let students practise from Copymaster One.

Session Three

  1. Use Slide Three of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 5 on pages 61 and select problems that are appropriate for your students. The key idea is about using the inverse relationship between addition and subtraction, particularly solving subtraction problems by adding on. Use proportional place value materials rather than money. Change the contexts of the problems to be about items that are of interest to your students.
  3. It is important that students understand why it is possible to solve a problem like 95 – 68 = □ by solving 68 + = 95, as well as developing the procedural fluency in doing so.
    image               image
    95 equals 9 tens and 5 ones                        If 68 was subtracted there would be ? left
    image
    That means 68 + ? = 95
    In empty number line form the problem is represented like this:
    image

    number line
  4. Let students practise from Copymaster Two.

Session Four

  1. Use Slide Four of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 7 on pages 63 and select problems that are appropriate for your students. The key idea is about students making sensible strategic choices about how to solve addition and subtraction problems. Encourage students to use mental and written methods. Only use proportional place value materials when students need additional support. Change the contexts of the problems to be about items that are of interest to your students.
  3. Discuss the criteria for deciding which method/s are the best for certain calculations. Encourage students to make up problems that each strategy (place value, tidy numbers, inverse thinking) would be useful for.They might conclude:
    • Place value works on any numbers but is more difficult where renaming is required.
    • Tidy number strategies work best when the one or both numbers are close to a tidy number (usually a decade or century).
    • Inverse thinking strategies work on any numbers but are especially useful if renaming is needed for subtraction.
  4. Equal adjustment is another strategy given in Key Idea 6 on Page 62 of Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. The addition form of equal adjustment is conceptually easy, but the subtraction is not. Choose whether, or not, to offer those strategies to your students but do not invest too much time in doing so. Here are examples:
    • Addition 48 + 34 =  
      image       
                            image
      48 add 34                                     has the same sum as 40 + 32 = 72
      number line
      number line
    • Subtraction 81 -58 =
      image          image
      The difference between 81 and 58 is the same as the difference between 83 and 60.
      83-60 = 23.
      number line

      number line
  5. To practice applying addition and subtraction strategies ask students to work on these Figure It Out activities:

Session Five

  1. Use Slide Five of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 8 on pages 64 and select problems that are appropriate for your students. The key idea involves use of standard written forms for addition and subtraction. Students should also use calculators as another way to find answers to addition and subtraction problems. Choose problems from the given examples but use the simplest written form for addition rather than the expanded form given in the lesson.
  3. Use proportional place value materials when developing the written forms so students understand what is occurring with the quantities. Alongside learning for understanding give students sufficient practice to develop fluency. Here are examples:
    • Addition
      image
    • Subtraction (using decomposition)
      image
  4. Let students practise from Copymaster Three.
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Level Three