Algebra – Roots of Routes to Algebra

The four roots of algebra are:

  1. Generalised arithmetic
  2. Expressing generality
  3. Possibilities and Constraints
  4. Rearranging and Manipulating

Some tasks to uncover misconceptions:

Activity 1

Invite a class to write down an arithmetic expression such as 13=3*4+1  and then rearrange it without actually performing the operations. Calculators will be most helpful for discussions about whether a rearrangement is correct.




and 3 can be isolated,


²Engage the class or group in discussion about how to write down the rearrangement. You can expect to expose all sorts of weaknesses and confusions about the four operations and their rules!

When a class has caught onto the idea, get pairs of pupils to produce arithmetic all expressions and then challenge other pairs to rearrange them with a particular number isolated on the left-hand side. often it is the case that a number which appears more than once cannot be isolated. Can the class come to any conclusions about when a number can be isolated?

Activity 2

Here are 3 consecutive numbers: 10,11,12

Multiply the first and third numbers: 10 *12=

Now square the middle number: 11*11=

What is the difference between the numbers?

Does this always work for consecutive whole numbers?


(A formal algebraic solution to this problem is:

Let x, x+1,x+2 represent three consecutive numbers.

I want to prove that the difference between the square of the Middle number and the product of the first and third, is one. In symbols, I want (x+1)²-x(x+2)=1

Activity 3

Express algebraically the statement that “There are three feet in a yard.”

One probable response is 3f=1y (f=feet and y=yards). This conveys the sense of the relationship between feet and yards, but it isn’t clear exactly what the letters f and y stand for.

The annotation would be a shorthand rather than an algebraic statement. As the f and y are being seen as units (feet and yards) rather than as generalised numbers.

When the letters are used in a more algebraic sense the appropriate formulation becomes:

3y = 1f (three times the number of yards…)

many children never properly grasp how a letter can be used to represent a generalised number. They tend to think of 2a + 3b as 2 apples plus 3 bananas, using the letters as units of measure rather than things which can take on different values ( twice the number of apples…)


Activity 4

Question a) If a + b = 43, a + b + 2 =

Question b) If e + f = 8, e + f + g =







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NAPLAN and EALD Language and Literacy Levels

NAPLAN writing assessment is a tool to assess student writing in either persuasive or narrative texts. The assessment uses a rubric in a number of specific areas to determine a score.

In both text types the following aspects are analysed:

  • Audience
  • Text structure
  • Cohesion
  • Sentence structure
  • Punctuation
  • Spelling

Additionally when analysing narratives Language devices are assessed, and when persuasive writing is analysed persuasive devices are also monitored.

Comparatively the EALD Language and Literacy Levels divide the analysis of writing into the following areas:

  • Composing learning area texts
  • Text Knowledge
    • Organisational structures
    • Text cohesion- Foregrounding
      • Text and paragraph openers
      • Sentence openers including using passive voice to change what is foregrounded
      • Reference: pronouns, determiners and substitution
  • Grammar Knowledge
    • Sentence structure
      • Simple
      • Compound
      • Complex
    • Punctuation
      • Sentence level
      • Basic punctuation
      • Beyond basic
    • Words and word groups
      • Verbs and verb groups
      • Adverbs, adverb groups/ phrases and prepositional phrases
      • Noun groups/ phrases, including plurals, articles and nominalisation
    • Expressing opinion and point of view
      • Evaluative language
      • Modality
      • Expressing opinion directly
  • Word Knowledge
    • Understanding/ using learning area vocabulary
    • Spelling

NAPLAN is a measurement tool, providing information about where students are in relation to the aspects measured – a summative assessment.
The EALD Language and Literacy Levels are a description of how language skills develop. Therefore they provide much more information to the teacher about where to next – a great teaching and assessment tool; formative and summative assessment.

Although specifically designed to support the development of language and literacy of high needs students, the Levels are a teaching tool to support teachers with all learners. They provide a framework for “high expectations and high support (Mariani 1997)”, enabling a closing of the gap between a student’s language resources and those required for their year level.

Underpinned by a systematic and explicit pedagogy based in:

  • Teaching in advance of language development
  • Stretching students’ knowledge and imagination beyond what they can readily do independently
  • Encouraging the use of strategies such as self-correction and trial and error
  • Customising support for individual learners, including modifying the level of support and the timing of its withdrawal as students move to independence

Implications on practice

Gradual release of responsibility
Zone of proximal development
Explicit teaching – unpacking the demands of the curriculum – How language works, why language choices are made, what the effects of certain choices are and how to use language in powerful ways.
The teaching and learning cycle – 4 key stages
1. Setting the context
2. Modelling and text deconstruction
3. Joint construction
4. Independent construction

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Text Dependent Questions – Grades 6-12

“Classroom discussions allow for the co construction of knowledge. Discussion elevates the act of reading deeply from a private one to a public one.”
Necessary features of discussion:
* sustained dialogue! not just short questioning cycles
* Uptake, such that the teacher poses new questions derived from the comments of students
* Authentic questions that do not always have a single correct answer

“Questions focused on the literal level of meaning of a text will not accomplish the kind of critical thinking we seek from our students. But they are the start of the journey.”

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Planning Leaning Sequences – Kath Murdoch

Ask questions such as:
What are the ways of exploring the concept or concepts that are engaging and relevant to my students?
What understandings, dispositions and skills will students be learning?
What experiences will be used to ensure that students learn?
What evidence will be gathered to inform learning and understanding?
What is a current, real-life issue for this particular cohort of students?

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A New Year, A New School, A New Beginning

Wow, what a difference I have noticed this year.
Eager, appreciative students who want to learn.
Professional teachers, who are used to collaborating and have systems established to help their work.
Welcoming parents.
A calm atmosphere.
High expectations from leadership in a supportive and appreciative environment.
I am looking forward to the next few years.

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Education and Technology – How to Focus on Maximising the Teaching Opportunities

Technology can provide many benefits, including: a wide audience for student work, collaborative opportunities with people all over the world, access to experts, use of tools to deliver a message effectively, and tools to systematise complex processes.

Unfortunately we haven’t got to the stage where technology is available without hiccups yet. Often schools are hindered by lack of access to technology (not having one on one devices), frequent cuts to internet connectivity, popular and useful sites blocked by systems, and lack of skills – teachers and students.

I am committed to finding ways to utilise technology to enhance my teaching practices, not just look for gimmicks that fix a short term issue. To this end I will aim to make use of the following technologies:

  • Blogs – to provide a central storage point for access to links and resources, to provide information to staff, students and parents, to share work, to encourage people to respond to student work, engage a wider audience in conversations
  • Wikis – to provide opportunities  for students  to co-create, collaborate
  • Voki – to engage the audience, direct a specific message in an entertaining way
  • Powtoon – utilise an animation tool to communicate learning powerfully, engage audience (transform learning)
  • Scratch – to teach students coding language
  • Vimeo – to communicate using video
  • Animoto – to encourage students to use visual and auditory elements to communicate a strong message
  • Voice Thread – to use the power of visual, auditory and written messages, combined with people from various parts of the world
  • Diigo – to curate important articles, reference pieces
  • Edmodo – to communicate with students around the world
  • iPads – in the junior primary years – My Story, Explain Everything, Book Creator, Tellagami, QR Reader, Show Me, Skitch, Puppet Pals
  • Twitter – to encourage traffic to blogs, pose questions of my educational Personal Learning Network, access current information and diversity of opinions
  • Skype – to connect with others
  • Fuse – to connect synchronously with teachers in various parts of the world
  • Thinglink?? – gimmicky? or purposeful?

What else would you add to this list?


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