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A study of number properties

Keywords:
Achievement Objectives:

Achievement Objective: NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
AO elaboration and other teaching resources

Purpose: 

The purpose of this unit is to develop the students’ deeper understanding of the way numbers behave, to enable them to use everyday language to make a general statement about these behaviours, and to understand the symbolic representation of these ‘properties’ of numbers and operations.

Specific Learning Outcomes: 
  • Recognise that there are number properties and that these describe the behavior of number operations.
  • Understand that a generalisation of an important idea can be expressed using letters (variables).
  • Describe the commutative property of addition and the commutative property of multiplication, name and apply these.
  • Describe the associative property of addition and the associative property of multiplication, name and apply these.
  • Describe the distributive property of multiplication over addition, and name and apply this property.
  • Recognise how number properties are ‘useful’ in their own mathematics.
Description of mathematics: 

As students solve problems using the operations of addition, subtraction, multiplication and division, they come to recognise, informally, some of the behaviours that numbers always exhibit when number operations are applied to them. They sense that this is ‘just the way numbers work’. Even though students may not be challenged to articulate generalisations which that state number properties, these behaviours become familiar. As problem-solvers, the students ‘draw’ on these properties, using relational thinking strategies, albeit in an unconscious way.

For example, a student uses a place value strategy to solve a problem such as:
There are three aliens. Each has 27 teeth. How many teeth are there altogether?
The student explains, “I said three times two is six, so three times twenty is sixty and I said three times seven is twenty one. Then I added sixty and twenty-one together, so it’s eighty-one teeth.” They are applying the distributive property of multiplication over addition. The student knows that 3 x 27 or 3 x (20 + 7) = 3 x 20 + 3 x 7 or a x (b + c) = (a x b) + (a x c).

Typically, classroom teaching and discussion is more focused on ‘finding answers’ to computations, in this case, 81, and sharing the strategies for working these out (explaining what they did), rather than being focused on identifying the number properties involved.

Number properties govern how operations behave and relate to one another, and they are essential for computation. Because algebra is the area of mathematics that uses letters and symbols to represent numbers and the relationship between them, students can be enabled, through general statements such as ‘a x (b + c) = (a x b) + (a x c)’, to see clearly the structure and nature of these number properties. It is useful therefore to have students pause to recognise and reflect more formally on ‘the way numbers work’ and in so doing to be gently introduced to variables (as yet unknown amounts), before forming and solving simple linear equations.

Generalisations can be expressed in a number of ways. Students should firstly use their own words to describe what they see happening, for example: “I just know that when you multiply, you can break one of the numbers apart, multiply the numbers separately, and then put them back together again.” The student can then be guided to connect these ideas to the elegantly simple algebraic notation that express these same ideas using letters as variables, and to recognise that is property is true for any (real) numbers. The ability to apply, recognise and understand these number properties, is foundational to ongoing success in algebra as well as arithmetic, and in mathematics in general.

The purpose of these lessons is to make the students fully aware of how the numbers are behaving. Naming each number property and writing and exploring a property using letters, is a way of helping students recognise and understand these. However, having students remember the names for each number property is not a focus of these lessons.

Links to the Number Framework
Advanced Additive/Early Multiplicative
Advanced Multiplicative

Required Resource Materials: 
Plastic pegs
Coloured counters
Activity: 

The context for this unit of work is Enviroschools projects. A fictitious school, Kiwi School, and its projects, are used here. If possible, explore these lessons in a similar way, using the practical context of your own school projects.

Session 1

SLOs:

  • Recognise that there are number properties and that these describe the behavior of number operations.
  • Describe in words the additive and multiplicative identities, recognise and explain the representations of these properties, and understand they have formal names because they are important.
  • Describe in words the additive inverse and understand its representations.
  • Understand that a generalisation of an important idea can be expressed using letters (variables).

Activity 1

Set the scene for this unit of work.
Begin the session by asking, ‘What happens when you drop a ball.’
(Answer, ‘It falls to the ground.’)
Ask, ‘Why?’
(Answer, ‘Gravity.’)
Ask several more questions about natural phenomena (such as, ‘What happens when we boil water? Answer. ‘It evaporates.’ etc.
Explain that scientists have investigated and continue to investigate the way in which things in nature behave, and they give these things names (like ‘gravity’).
Explain that mathematicians have investigated and continue to investigate the way numbers and number operations in mathematics behave. These behaviours are called number properties and these have names too.
Write number properties on the class chart.
Tell the students that in the next three lessons they will be working in a slightly different way with maths ideas that they ‘already know’ because they have been using these, perhaps without noticing, ever since they started solving problems.

Activity 2

  1. Introduce Kiwi School. It is an Enviroschool and the students like to recycle their waste materials. If possible they throw nothing away.
  2. Make pencils and paper available to each student. Display Attachment 1 and read this together.
    Pose this task:
    Look carefully at these statements and write an equation to show what happened in each classroom. Write about one thing that you notice.
    Have students each complete their equations and then share them with a partner.
  3. On the class chart, have students record an equation for each classroom in Kiwi School.
    Room 1: 14 – 14 = 0
    Room 2: 18 – 18 = 0
    Room 3: 21 + 0 – 0 – 21 = 0
    Room 4: 5 – 5 = 0
    Room 5: 20 – 20 = 0
    Room 6: 21 + 5 – 5 = 21
    Room 7: 0 + 24 – 24 = 0
    Have the students explain what they notice and record selected ideas. Require students to reference their observations to specific Room numbers.
    For example: “When you subtract a number from itself you end up with zero.” (Rooms 1, 2, 3, 4, 5)
    “When you add zero to, or subtract zero from a number, that number doesn’t change.” (Room 3.)
    “When you add a number then take that number away again it’s like you add nothing, or zero.” (Rooms 6 and 7).
    “If you start with zero and add nothing you still have zero.”(Room 7)
    Tell the students that they have noticed and described some really important number and operation ‘behaviours’. These are number properties.
    Highlight the fact that all of the examples involve zero.
  4. Write a large zero on the class chart and add the title, What we know about zero.
    Brainstorm together and record the students ideas about zero. These are likely to include the ideas above, and others such as:
    “It’s like nothing.”
    “If you add it with another number, you just get the same number.”
    “You can kind of ignore zero, unless it’s at the end of a number, like 20, then it’s really important.”
    “When you plus or minus zero you could just leave it out because it makes no difference.”
    Record these ideas inside the large zero. Use this as a class poster.
  5. Return to the classroom equations. Highlight Rooms 1, 2, 4, and 5. Have a student describe again what is happening. “When you subtract a number from itself you end up with zero.”
    Ask, “Is there a really simple way we can show this?” Accept ideas.
    Write on the class chart a – a = 0
    Ask, “What is ‘a’?” Listen to explanations then agree and recorda stands for any number.” Write the words variable and pronumeral, and explain that both are names for a letter that is used to represent a number. Make it clear that ‘a’ isn’t short for anything and it isn’t the first letter of anything. Any letter could be used here. For example, we could write, y – y = 0.
  6. Explain that, because this is a really important idea in mathematics, it has a special name. They don’t have to remember just now.
    (+) a – a = 0 is known as the additive inverse. Discuss the word ‘inverse’ to highlight what is happening.
  7. Return to the other equations, which have not been highlighted and together record similar equations, using variables.
    Room 3: 21 + 0 – 0 – 21 = 0           a + 0 – 0 – a = 0
    Room 6: 21 + 5 – 5 = 21           a + b – b = 0
    Room 7: 0 + 24 – 24 = 0           0 + b – b = 0
    Highlight Room 6.
    Write a + b – b = a + 0 = a Ask, “Is this true? Discuss this with a partner.”
    Say, “Let’s find the zero.” Highlighting a + b – b = a + 0 = a
    Make the connection between a – a = 0 in Step 6 above.
    Write on the class chart a + 0 = a
    Explain that, because this is a really important idea in mathematics, it has a special name.
    a + 0 = a or 0 + a = a is known as the additive identity. Discuss the name to highlight that an identity is an ‘exact sameness’ and it means that ‘a’ stays just the same.
  8. Provide student pairs with a number line and some counters.
    Write again on the class chart: a – a = 0 and a + 0 = a
          
    Have both students in the pair model both equations using a piece of equipment, choosing different numbers for ‘a’. Have some students model this again for the class. (These are simple to model, but doing so is helpful.)
    Write on the class chart:
    Important number properties that we often use
    a – a = 0 (additive inverse)
    a + 0 = a (additive identity)
    Write in:                            (multiplicative identity)
    Have students discuss what this might be and how it can be written.
    Model, discuss and agree a x 1 = a or 1 x a = a means that ‘a’ keeps its identity.
    Write the equations in place on the class chart.
  9. Have students complete Attachment 2 (Exploring zero).
    Have students buddy mark their equations then conclude the lesson by having them share their summary comments.

Session 2

SLOs:

  • Describe in words the commutative property of addition and the commutative property of multiplication, and name these properties.
  • Describe in words the associative property of addition and the associative property of multiplication, and name these properties.
  • Describe in words the distributive property of multiplication over addition, and name this property.

Activity 1

  1. Refer to Session 1. Focus on the sustainability work that is happening in Kiwi School.
    Display Attachment 3 (Kiwi School planting plan) and pose this scenario:
    Kiwi School students planned and have planted green, red and yellow flaxes and grasses in an area near their adventure playground. Here is their plan.
    Set the task:
    Look at the plan (Attachment 3). In pairs, have them discuss what they see and write the many ways they know for working out how many of each kind of plant there are. Have students pair-share their equations, discussing and recording any different calculations.
  2. Have students record their equations on the class chart. Ensure that the equations below are included in their recording and explained with reference to the planting plan.
    Yellow: 4 x 3 = 12 and 3 x 4 = 12, 2 x 12 = 24
    2 x 4 x 3 = 24
    Green: 2 x 14 = 28 and 14 x 2 = 28
    Red: 2 x 13 = 26 and 13 x 2 = 26
    Green and Red: 2 x (14 + 13)
    So 2 x 27 = 54
    Altogether: (28 + 26) + 24 = 78 and 28 + (26 + 24) = 78
  3. Distribute a copy of Attachment 4 to each student and discuss the tasks. Emphasise that they are to look at what is happening with the numbers and the number operations in each example, and write a description of this in words in the space provided. They should write other equations to which this description applies. These may be from the planting plan, or may be unrelated equations. They should leave the final column blank, unless they know or have their own ideas.
  4. Have students pair-share their work.
  5. Discuss student ideas as a class. Explain that what they have been describing are number properties that mathematicians have found are true for all (real) numbers, which are ‘points on an infinitely long number line’. We accept number properties as true and we do not have to prove them.
    Explain that mathematicians have given names to these behaviours and properties. Ask if anyone knows the names of these properties, recording ideas.
    Write on the class chart:
    The associative property of addition. 
    The associative property of multiplication. 
    The commutative property of addition.
    The commutative property of multiplication. 
    The distributive property of multiplication over addition.

    Read these together. Discuss the root words of associate, commute and distribute. How are the definitions of the properties related to the meanings of the root words?
    Have students discuss in pairs which property name might go with which set of equations and why.
    Discuss and confirm these. (Refer to the teacher answer sheet.) Have students enter the correct number property names in the final column of Attachment 4.
    Have students now look carefully across their completed page.

Activity 2

  1. Write these equations on the class chart.
    Have students discuss each one and record the number that each box represents. As the value for each unknown is ‘found’, talk about how they know this, using the language of number properties.
    ☐ + 235 = 235 + 17
    25 x ☐ = 7 x 25
    (125 + 16) + ☐ = 125 + 16 + 17
    (15 x ☐) x 10 = 15 x (2 x 10)
    8 x (19 + 3) = (☐ x 19) + (8 x 3)
  2. In the class group, have students work in pairs to check the value of the expressions on both sides of the equation.
    Record these beside each equation on the class chart.
    For example: (15 x 2) x 10 = 15 x (2 x 10)       300 = 300
  3. Conclude the lesson by listing key ideas that students have learned.

Session 3

SLOs:

  • Recognise representations, using variables, of the commutative property of addition and the commutative property of multiplication and write equations to show these properties in a practical context.
  • Recognise the representations, using variables, of the associative property of addition and the associative property of multiplication, and write equations to show these properties in a practical context.
  • Recognise representations, using variables, of the distributive property of multiplication over addition, and write equations to show this property in a practical context.
  • Recognise how number properties are ‘useful’ in their own mathematics.

Activity 1

  1. Restate the focus of this series of lessons: To look more closely at the behaviours of numbers and operations so that we understand what is happening as we are solving problems. Refer to Attachment 3 and Attachment 4 from Session 2.
  2. On the class chart, write these equations, using variables, to represent the generalisations of the number properties.
    a + b = b + a
    a x (b x c) = (a x b) x c
    a x (b + c) = a x b + a x c
    a x b = b x a
    (a + b) + c = a + ( b + c )
    Read them together and have students describe what they see.
    Make Attachment 5, paper and pencils available to each student.
    Have student pairs match the equations above, with the number properties they have described and named. Have each student write these in pencil in the third column of their own copy of Attachment 5.
  3. Check these together as a class, having students read the words of each property and checking the symbolic representation.

Activity 2

  1. Pose this scenario:
    The Kiwi School Board of Trustees paid $635 for the plants. Two kinds of the coloured plants cost $8.50 per plant. The third kind of plant cost $7.50 each.
    Pose the task: Work in pairs to find and record on your own paper.
    A. The unit and the total price of each kind of plant: red, green and yellow.
    (Answer: 26 red plants @ $8.50 each, + 28 green plants @ $7.50 each + 24 yellow plants @ $8.50 each = $635)
    B. Write equations, in column one of Attachment 5, to illustrate each of the number properties, using some of the numbers and calculations from part A.
  2. Have students pair-share their results.
  3. As a class share results, and, as property examples are shared, have students read through the descriptions of the properties. Check that they ‘match’.
  4. Discuss and list reasons why recognising these number properties is useful. Accept all suggestions. Highlight that, by using a range of the number properties in this way, calculations can be checked for accuracy. Also note that the commutative (turn around) property helps us to know and learn number facts quickly.

Activity 3

  1. Pose this problem to conclude the session.
    18 children in Room 5 at Kiwi School and the 21 children in Room 6 to plant 2 plants each. Are there enough plants?
    Have each student solve the problem and demonstrate four of the five number properties, by adding these equations to those already in column one of Attachment 5.
  2. Share and discuss results.

Activty 4

Conclude this session and unit by reflecting on student learning about number properties. Revisit the key ideas recorded on the class chart throughout the sessions.
Have students understand that we continue to apply these number properties when we solve more complex equations.

AttachmentSize
NumberPropertiescm1.pdf41.78 KB
NumberPropertiescm2.pdf39.27 KB
NumberPropertiescm3.pdf503.75 KB
NumberPropertiescm4.pdf32.75 KB
NumberPropertiescm5.pdf40.19 KB