Prompt 21: Solving Linear Equations

This is a task I gave to a Year 8 mixed attainment class which I think would be a nice pedagogy prompt given the question set below:

Questions:

  1. How do you show balance operations between steps? Is this consistent between department members? Should all department members use a consistent approach?
  2. Do you explicitly reference all variants of linear equations in curriculum documentation? i.e. I can solve a linear equation with unknowns on both sides when one or both unknown terms are negative, or, I can create a linear equation given the solution.
  3. Do you teach only one method for an equation like this? How do you approach different methods? Is this student led or do you explicitly instruct on a range of methods?

Prompt 20: Order of Operations

Which approach do you utilise to help students understand the order of operations?

Some interesting blog posts to back up this post:

  1. Cell 4: Dani Quinn (@danicquinn) – Tried and Tested: GEMS
  2. Cell 2: Colin Foster (@colinfoster77) – Higher Priorities Article in favour of BIDMSA
  3. Cell 5: David Butler (@DavidKButlerUofA) The Operation Tower
  4. Cell 6: Chris McGrane (@ChrisMcGrane84) – Order of Operations Area Model (A task which highlights that BIDMAS is not required).

Prompt 18: Exponential Growth – Comparison with Linear

Additional Fun:

Questions:

  1. Even if simple interest does not appear in the curriculum, would you still utilize it in order to make a direct comparison between linear and exponential growth?
  2. Do you show the entire curve to begin with, or would you build up as in the 3 pictures given in the prompt?

Further examples of exponential growth:

a. My Geogebra Applet on Bacterial Growth.

b. Nrich Task – Modelling an Epidemic (Nice kinaesthetic starter). You could also have an interesting discussion about how gossip could be modelled very simply with exponential growth.

Source: Unknown

c. Grains of Rice on a Chessboard Problem (The video is quite old but I love it).

d. Towers of Hanoi Puzzle (NCTM Applet)

e. Paper Folding TED Talk.

f. Dan Meyer’s Domino SkyScraper.

g. Human Population through Time (exceptional video on population growth shared by @atulruna on twitter)

h. How often do real world phenomena actually follow exponential growth indefinitely?

Source: Unknown

Prompt 17: Understanding the Area of a Circle

Applets:
  1. My applet – Cell 2: Area of a Circle
  2. My applet – Cell 4: Circle Area through Integration
  3. Popular Applet: Anthony Or: Area of Circle

Questions:

  1. In regards to Cell 1, students require Pythagoras’ Theorem to find the area of the inscribed triangle. Would you ever set this as a question after a unit on Pythagoras?
  2. Cell 2, 3 and 4: The parallelogram method may be the most common way of understanding the formula. However, the triangle method may lead more seamlessly into deriving the formula with calculus. Does this affect your decision of which method you show earlier on in school?
  3. The How-before-Why debate is a popular one right now. Would you ensure students knew how to use the formula before promoting understanding or vice-versa? How about if you go backwards from a parallelogram or a triangle into a circle – does drawing on what students already know about areas of shapes change your response to this question?

Prompt 15: Pythagoras 4

This prompt is for those teachers who may not teach beyond Middle School (Year 9/Grade 8) or beyond GCSE (Year 11/Grade 10). I find it’s always pedagogically helpful to know how a concept might be extended beyond the curriculum I teach.

Additional Resources:

  1. Cell 1: See this Gif by Idan Tal (@MagicPi2)

2. A lovely post about the connection between Pythagoras and the Cosine Rule by Andrew Stacey (@mathforge)

3. Cell 4: A little more on time dilation: Time is Relative video

Prompt 14: Nth term of a Linear Sequence

  1. Why do we often only introduce the arithmetic sequence formula post-16?

2. If you only teach one method, which method do you teach and why? Does each method lend itself to another concept in the curriculum?

3. The standard formula for an arithmetic sequence is given in cell 4. However, you can use any term in the formula u_n = u_x + (n – x)d; would you consider teaching this more general form, even though it may not appear in formula booklets?

4. In regards to cell 2, the zeroth term is equivalent to the y-intercept, and the difference is equivalent to the gradient on a co-ordinate grid. How best could you represent or teach this integral connection?

Prompt 12: Equations of Straight Lines

  1. Which method out of cell 1 and cell 3 do you think is most helpful when teaching gradient? Are both important? How can cell 3 be used to enhance understanding of perpendicular lines?
  2. When would you introduce x-intercept form? Before studying the factored form of a quadratic? as an inquiry or investigation during a unit on y-intercept form? How is it helpful in leading into transformation of functions later in school?
  3. Should gradient be reconceptualized as a vertical stretch?
  4. Would you teach the general intercept form or is it simply interesting to know? This catalyses a more general question: How do external tests influence what we teach?