Category: hands on manipulatives

Animal Algebra: General Terms of Sequences

Use linking cubes and make these three animals.

algebra animals

Ask students to discuss:

  • How do they see these animals growing?
  • What does the next animal look like? How many cubes will be needed for its {head/body/front leg/hind leg}?

Using these answers, bring students to see the general term for the number of cubes used. I’ve used this picture to help:

animal-expanded

Now do a similar exercise using growing L-shapes, growing T-shapes, growing Z-shapes, growing animals of their creation.

I have used this lesson countless times and it continues to be a favourite. I like that general terms are introduced without the method of making a table of values. I want to avoid reducing geometric sequences to a meaningless sequence of numbers.

Excellent similar ideas are found at visualpatterns.org and this nrich problem about cable bundles. (The nrich problem has sample student work, making it good for teacher workshops, as well.)

Does anyone know where this idea is from? It’s not original to me and the second image in this post is a screenshot from a long lost book. I have also seen that Colin Foster had an idea like this in his (amazingly free) book, Instant Maths Ideas for Key Stage 3 Teachers: Number and Algebra.

a mathematics lesson that worked

Update:

Thanks to Colin, I think the original idea is from Paul Andrews’ book, Linking Cubes and the Learning of Mathematics. It’s available for sale from the ATM and I highly recommend it. (I have just bought a new copy.)

Factors Using Multilink Cubes

My year 8s (twelve year olds) have been learning about multiples and now it was time to talk about factors. Some of them have not got all their times tables memorised, which presents some difficulty for our unit of work on factors, multiples, and primes. So my teaching assistant and I doled out the multilink cubes and I asked them to make rectangles. I wanted to know if any number of cubes could be made into a rectangle shape, fully filled with cubes (no gaps).

2014-09-02 11.53.39

After the students played with this for a while, I started making a list on the board of all the sizes of rectangles they had made. Then I asked, which numbers of cubes can be made into more than one shape of rectangle? Above are two rectangles with 8 cubes and below are three rectangles with 12 cubes.

2014-09-02 11.53.36

 

Next I introduced the idea of a factor by talking about the sizes of the rectangles and after some discussion we listed all the factors of 12 using the rectangles in the picture above.

After that, students made their own lists of all the factors of some numbers of their choosing. My assistant and I went around to their tables, asking if they were noticing anything. Using their results, we made a table on the board of all the factors of the numbers from 1 to 15 and talked about what they noticed.

Top on their list of noticings were: 1 is always a factor of every number and the number itself is always a factor. These were not obvious statements to my students. We discussed why each was true by talking about making a long skinny rectangle of 1 x __ for any number.

Next lesson is going to be about prime numbers, so I hope that their next noticing is that some numbers have a lot more factors than others.

Do you use multilink cubes in your lessons?