X. Pensiliau a Rheolyddion
Aims |
To provide opportunities for children to observe and discuss patterns in a two-way matrix.
For children to solve problems based on implicit algebra.
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Resources |
Vocabulary |
Pencils and Rulers notesheets 1 and 2 – one per pair
Have an enlarged copy of Pencils and Rulers 1 or pre-draw a ‘ready reckoner’ on the board
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lots of, multiply, times, add, in
total, grid, two-way table, charts,
rows, columns
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Organisation |
Near achievement pairs on mixed tables
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Whole class Preparation: (about 5 mins)
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Suppose a pencil costs 10p and a ruler 15p.
Explain that before people used calculators and electric tills, they often used a ‘ready reckoner’ to work out the total price for various combinations of things.
We are going to make up a ‘ready reckoner’ for pencils and rulers.
Look at the ‘ready reckoner’ drawn on the board and point out to the children the numbers of pencils along the bottom and then rulers along the side.
Give out Pencils and Rulers 1.
How could we work out what answer should go in this box? Point to one of the boxes and gradually fill up the bottom two rows of the chart.
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Paired Work: (about 15 mins)
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In pairs, ask the children to fill in the other sections of the table.
After a couple of minutes, stop them and point out the dark section of the sheet that has been destroyed. How could you work out from the other numbers on your sheet, what answers should be in the highest box in the two pencils column? The box which is all by itself?
Work these out with your partner and write down how you worked them out. Then go on to the three questions at the bottom of the notesheet.
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Class sharing |
How did you find out the missing numbers?
Get a range of responses but focus on how the table builds up across the rows (in 10s)and in the columns (by 15s).
Ask what other patterns they see, e.g. diagonals in different directions.
Children may have written 8 x 10 + 7 x 15 for the answer in the top box by looking at the original numbers of pencils and rulers at the bottom and side of the grid.
Question 1 should be straightforward but invite a pair of children to explain how they worked it out.
Question 2 has three entries for 85p and three for 60p. Why are there three answers for 85p and three for 65p?
In answering question 3, encourage the children to describe the upwards movement as ‘up 15’ and the right movement as ‘over 10’. This builds skills and understanding needed for the idea of gradient.
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Paired work :(about 10 mins) |
The charts on Pencils and Rulers 2 are from two different shops where the pencils and rulers cost different amounts.
Work out what the ? represents in each grid.
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Class sharing: (about 10 mins) |
Share strategies for finding the numbers – moving across rows or up columns for the first step. The second step will require the use of informal simultaneous equations (see the discussion in the levels diagram).
Children should reflect on the whole activity – the additive and multiplicative changes within each row and column and the linking of the rows and columns to find a diagonal entry, up and over. They should explain their reasoning and link this with other experiences.
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