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Percentage Change introduction

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This padlet is an open space for sharing resources. Feel free to explore suitable means to build your resource package and e-portfolio.

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study the transformation of functions and graphs based on a context (open in Evernote)

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There are 3 simulations in PhET related to integers on a number line:

ordering

distance

operations

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Although manipulatives and representations are commonly used in primary mathematics or have been historically considered the reserve of lower prior attainers, they can be equally valuable for revealing mathematical structures at secondary level for all students.

We have produced guidance for some of the most useful representations for Key Stage 3 mathematics. Each guidance document details:

what they are

why they are important

how they might be used.

Click the following links to download the guidance documents.
Please note some of the links to NCETM content in the documents below may not work. You can quickly find where the links point using the search tool on this website. If you cannot find what you're looking for, contact info@ncetm.org.uk.

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Using mathematical representations at KS3

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Guidance for some of the most useful representations for Key Stage 3 mathematics.

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Is it easy to predict how the intersection will move?

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By asking students to compare the areas and reason about the change, they may make sense of algebraic expressions. In this case, expansion of an expression is done with a purpose. The factored and expanded forms are equivalent but carry different meanings.

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Introduce the identity, difference of two squares, through a problem about comparing areas. See the making of this file in YouTube -> http://youtu.be...

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Modelling addition and subtraction of directed numbers as collection of tiles

Try it in Desmos

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modelling addition and subtraction of positive and negative numbers

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Desmos activity: create patterns with graphs

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Desmos activity: parallel lines and angles

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Desmos activity: ratio of lengths and areas

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Geometry Tasks from Internet Archive books

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Geometry Tasks from Internet Archive books

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This is an example of balance scale puzzle made in Polypad. There is a link to more info about building this kind of puzzles and planning a lesson. (more from this page)

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Virtual manipulatives, dynamic geometry, graph plotting, data science and more: explore the ultimate mathematical playground!

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a simple runner model for describing multiplication of directed numbers

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open GeoGebra page: https://www.geogebra.org/m/rzzexnyu

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Perfect square identities by algebra tiles

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Using algebra tiles in teaching and learning perfect square identities

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Lesson Outline and Objective

Algebra tiles are a powerful tool for helping students make the connection between symbolic manipulation and geometrical representation using area models. In this activity, students can explore completing the square using the Polypad Algebra tiles, and get a sense of why the process is called "completing the square".

Warm Up

Optional Review of Area Model of Multiplication

If your students could benefit from a review of the area model of multiplication, consider solving 27 x 15 in three different ways using this Polypad canvas:

https://mathigon.org/polypad/5fbacacc958ef80012f2bd0b

This can also serve as a good review of multiplication of negative numbers. This Polypad canvas shows the solutions:

https://mathigon.org/polypad/45ahtupqCn1aA

Spot the Squares

https://mathigon.org/polypad/5fbad36f5721f80012bb83dc

Students may wish to explore using the Polypad algebra tiles to start off with, but by thinking about the structure of the squares that they make, students may be able to answer some of the questions without needing to create the diagrams. It might be worth sharing as a class any strategies that students used to help them decide whether an expression could be written as a square or not.

Solutions

If your students could benefit from a reminder of equivalent expressions, consider substituting in a value for x in one of the solutions above to show that the two expressions are equal (at least for the chosen value of x). To show this visually on Polypad, you could use the pen tool and the algebra tiles to show this check for :

Main Activity

Share this Polypad canvas and allow students time to work on forming squares with the tiles provided:

https://mathigon.org/polypad/iM2ZSUrmaip5yQ

After giving students some time to work,, discuss the algebraic representations of their findings.

One way to manage filling in the gap is to add pairs of 1 and -1 tiles (which keeps the expression the same), giving:

You can show your students that the +1 and -1 tiles cancel out to make 0's by dragging them on top of each other. This shows that you're not changing the value of the expression on the canvas:

Now, take apart the 0 tiles and move use some of them to create a square:

Given the ease of adding tiles to the canvas, some students may think just adding 7 tiles to the diagram "completes the square." So, they may incorrectly go from the first image below to the second image:

Now give students some time to explore completing the square. Here are some expressions they could try:

The first is chosen to be similar to the examples they have already seen. The second has a negative constant term, and the third introduces a negative coefficient of x so they will need to think carefully about positive and negative numbers. The fourth would require cutting up the algebra tiles as the coefficient of x is odd, so this could be a good focus for discussion at the end of the lesson.

Possible Solutions

As this last example uses fractions, students can't represent it directly using algebra tiles but may wish to annotate using the pen tool to show the parts of their answer:

Support and Extension

A nice extension to this activity is to invite students to think about how they could use completing the square to help them solve quadratic equations. As with all explorations of quadratics, it is worthwhile to encourage students to move between algebraic and graphical representations and inviting them to explain any relationships they notice so that they start to develop a better understanding of how features of an algebraic representation relate to features of the graph.

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Explore our free library of tasks, lesson ideas and puzzles using Polypad and virtual manipulatives.

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Nov 3, 2021 (Unit Circle & Trigonometry)

We studied some examples in GeoGebra and Desmos related to trigonometry and unit circle.

I recommend the teaching strategy explained by W.W. Sawyer.

main recording

see the GeoGebra tasks involved (a section about Unit Circle in a GeoGebraBook of Trigonometry)

some Desmos activities with similar pedagogical design:

Unit Circle Trig (part 1 and part 2)

some questions based on DSE paper 2

another interesting example of circular stitching in Desmos