## Plans for action research into children’s use of algebra to describe growing patterns

Over the course of the next couple of weeks in school, I intend to begin my work with the children. I haven’t really finished my reading yet, but I have a solid foundation to begin from – also I have the time available to me in school over these two weeks and after that it will be increasingly difficult to be able to do these sorts of things.

**Session 1**: Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding. These may not be accurate at this stage – but that’s ok (and the whole point of the sessions…)

**Fir Tree investigation**: uses pattern block triangles to create growing fir trees. Children must extend the pattern, complete the table of values and describe the 10th tree.

**Session 2**: use ideas from algebraic infants. Modelling how a family of dogs can be constructed. Legs, shoulders, bum, body, head – with only legs and body growing, Children need to show each stage in a table for the dog. They then create their own animal, growing it in the same way. Multilink cubes needed.

**Session 3**: **Tables & Chairs investigation** challenges students to find a rule to describe the relationship between the number of small square restaurant tables placed together in a line and the number of diners that can be seated at the larger table if only one person sits on each side. Can model with shapes if needed (squares and circles). As before, children make a table to shapes used and differences to aid thinking.

**Session 4**: Part 1: **Ships Ahoy**. Children look at a simple pattern of horn blows and predict future patterns.

Part 2: **Rockets**. Children look at a pattern for building rockets. They are challenged to see how many parts will be needed for the 50^{th} stage. Whiteboard/paper drawings probably best – stickers might be helpful though?!

**Session 5**: **Task Cards**. Children choose a task card then construct it using squares or triangles. Encourage them to look at the patterns and decide how it is growing. Create and complete a table to show the growth pattern. Use the tiles to make the next two shapes in the pattern. Be prepared to explain how the pattern is growing. If there is time, choose another task card. Cards attached at the end.

Session 5 – Challenge Cards (PDF)

**Session 6**: Same as session 1. Can the children identify the growing pattern and express the rules algebraically?

Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. This pattern must be similar but not the same as session 1. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding.

**Hexagon dragon investigation**: dragons made from equal numbers of hexagons and triangles, each new term adding on more. Requires children to extend the pattern, create an input/output table to describe the growing pattern, then draw and/or describe the 10th dragon in words. An extra challenge asks students to generate a rule for this pattern so that Miguel can figure out how many blocks he will need to build a dragon of any size.

Each of these sessions will be completed during the children’s Numeracy time. I am likely to choose 4 or 5 children to work with from across ability groups and a mix of genders. I will take copies of their written work and record conversations to help me analyse their progress and thinking over the course of the sessions – I feel that this will give me the data I need to go further. Session 6 is a repeat of session 1 in order to set a baseline and see the progress, if any, the children have made in between.

Clearly, this is such a small scale research project that results can’t be read into too much. However, it is a beginning for my school to look at how we can use algebra in wider contexts. I have been careful to choose similar tasks throughout the sessions as I will only have at most half an hour to complete the tasks with them.