As well as teaching year 4 children, I also set regular puzzles for more able children in years 5 and 6. I will post examples of activities I have used and web pages I have found which have helped generate ideas.
www.nrich.maths.org is always the first place I look for ideas. Here is an example of a puzzle I found there, and how I adapted it to take the learning further.
- Fitted is an activity that appeared in the key stage 3 section in November 2011. I slightly adapted it because the class teacher was working on area and perimeter in class.
“Nine squares with side lengths1,4,7,8,9,10,14,15 and18 cm can be fitted together with no gaps and no overlaps, to form a rectangle. What is the perimeter of the rectangle?”
In the next session I used www.mathspickle.com because it has a terrific video section that takes the same idea a bit further, but allows you to explore balanced equations in a practical context.
Although the website has worsheets for the activities, I rearranged them to save paper and to concentrate on area and perimeter (click on the picture).
We stuck with the idea of multiplying numbers together to find products, but used a calculator to explore products from consecutive numbers and square numbers. I have this activity in the context of area and in the context of mountains. I have attached resources for both.
The final activity we looked at linked to square numbers also came from nrich. I posted the recording sheet under multiplication activities. With the more able children we concentrated on multiplication using the grid method and using algebra to find the reason why the difference increases in square numbers.
Sticking with an algebra theme we are now investigating Fibonacci sequences. I found these resources.
(http://www.teachfind.com/national-strategies/fibs-and-truths) and adapted them to produce these.
Next week we are going to revisit problems introduced in year 4. The strategy taught then was based on a guess and test approach. We will be refining this to using an algebraic approach to solve them more efficiently.
Leaf Pyramid Algebra
See the rainforest post for a picture, or click here
to download resources.
We investigated the possible totals that could be made when the numbers on the bottom of the pyramid were consecutive. When solving the three layer pyramid, children noticed that the numbers in the top row were all multiples of 4, and that the number was always four times the middle number on the bottom row.
I then challenged them to find the bottom four numbers if the top number was 100. This led to some children dividing the top number by 4. We then tried to complete the pyramid using algebraic terms with the bottom row being: n, n+1, n+2 and discovered the top number was 4n+4. If the top number = 100, then 4n+4=100. The children were able to manipulate this to calculate that n was 24. They then tried to complete the pyramid given different top numbers, before moving onto 4 layer pyramids.
Adding consecutive numbers (the Guass way).
Sticking with consecutive numbers…
Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a very famous German mathematician. One well known story about him happened after the he misbehaved in primary school. His teacher, J.G. Büttner, gave him a task: add the numbers from 1 to 100. The young Gauss produced the correct answer within seconds, to the astonishment of his teacher. Can you find a quick way of adding all the numbers that he might have used?
Solving this puzzle with children reminded me of other consecutive number puzzles we have solved over the years. I always teach the children the strategy:
- draw loops to join first and last number and find total.
- repeat for second and second last numbers. what do you notice about the ‘loop’ total?
- repeat, joing all numbers in sequence and record all ‘loop’ totals.
- Record information systematically in table with the following headings:
last number : how many loops : loop total : overall total
- explain the relationships in the table, how many loops is always 1/2 the last number, the loop total is always 1 more than the last number, the overall total is always the product of loops x loop total. This leads the children the algebraic formula.
Handshakes (link to NGFL CYMRU interactive)
PS there are other brilliant interactive introductions to interesting investigations on this website; I particularly like the tower of hanoi puzzle and the leapfrog puzzle, which like the last two puzzles require the children to use a table to collect data, then analyse data to find patterns before expressing relationships algebraically.
Totalling consecutive numbers (taken from the Nrich website)
I like to do this activity early on in year 4 to introduce the idea of adding sequences of numbers by linking them together as described above. Although it takes determination and perseverance, the more able childrenfind the solution that you cannot make the powers of 2 (1, 2, 4, 8, 16, 32, 64). I then challenge them to use these powers of 2 to make different totals, which can link to Egyptian Multiplication and to this puzzle from the blue problem solving book.