Fractions of sets

This term I am focusing on Fractions, Decimals and Percentages with my students. The students are started to see the connection and relationship between these three areas i.e. that 0.2 is the same as 20% which is the same as 1/5. They are also learning how to simplify fractions and about equivalency at the same time.

So on Monday we started using paper strips to work out the fraction of a certain number of chocolate bars each person would get.

Example:
Miss Breen has 4 Chocolate bars left over at the end of the school term. She wants to share them out with the 5 children that were the best helpers that week. How much chocolate will each child get?

The paper strips helped the students to think about the chocolate bars as wholes before they broke them up into fractions and shared them out. We had a few tries, much trial and error before they began to realise that it was a good idea to start by breaking each whole paper strip into the number of people from each problem.

Once the students understood that each problem related to parts of a whole we moved onto the fractional equations. The students were still free to draw pictures but they began to use multiplication and division as a way to work out their answers.

Here are some photos from this week, of the students from all my groups working on these types of problems.

p.s. I let my kids draw on the tables with Whiteboard markers, it makes for a great working space :-)

Subitising and Quick Images

One of the many things I have learned while being a Maths Support Teacher is the value of Subitising and Quick Images. Now if you aren't sure what subitising means, here's a definition:

Subitising

We often recognise the number associated with a particular pattern straight away, even before we have had time to "count" the items.

The process of immediately recognising how many items are in a small group is called subitising. This name comes from the Italian word subito, which means "immediately" or "right now". When playing a game with dice we normally recognise the number of dots immediately.

By "chunking" information, subitising contributes to early forms of grouping. The process of subitising can also be used with seeing parts in the whole.

If you look at the dot pattern for five you become aware of seeing it also as four and one, or three and two. Domino patterns capture this idea of subitising playing an advanced-organising role. People can see each side of the domino as four individual dots and as "one four". They see the domino as composed of two groups of four and as "one eight".

The notion of units within units is important for separating and combining numbers as well as for multiplication, division and measurement. Subitising contributes to part-whole (sometimes called part-part-whole) number relations.

Interpreting number in terms of part-whole relationships makes it possible for children to think about a number as being made up of other numbers.

Reference: 

State of New South Wales, Department of Education and Training. (2009). Count Me in Too: 

          Subitising.  Retrieved from http://www.curriculumsupport.education.nsw.gov.au/

          countmein/learning_framework_subitising.html

I use subitising quick images as a warm up with every group of students I work with. I created my own set of cards so that I would be able to include more unstructured patterns. By having unstructured and structured dots on one card this leads students to think multiplicatively and decompose numbers (combine dots in various ways) to make groups.

Here is a photo of some of the cards I made:

2d and 3d Geometric shapes can also be used in quick images. This helps promote the use of mathematics vocabulary and also helps them to recognise properties of shapes. With 3d images you can ask students to build them with blocks which helps them to think about parts of the picture/image that they can't see.

Geometric Quick Images to use can be found on the below link:

Click Here

Here are some photos of Group 6 using multi-link cubes to make 3d images of cube arrangements.

How to use quick images:

The students are told that they are going to be shown a card with black dots or shapes or cubes (depending on what you are showing them) on it for three seconds. Their job is to try to work out how many dots are on the page or to describe what the picture was made up of. Then they are shown the card for 3 seconds. After some wait time they are shown the card for another 3 seconds to check their answer. They are using both perceptual and conceptual subitising in this example. The teacher then elicits responses by asking questions such as: 

What did you see? How did you know? How did you count them/add them? 

Below is a short exemplar video of two of my students discussing what they saw after viewing the card in the top right corner of the above photo.  

NB: My students are very familiar with quick images so are now much better at identifying groups of dots quickly. However, at the start of the intervention they would identify individual dots rather than groups, count in ones or skip count and draw the patterns in order to get the answer. Their use of mathematics vocabulary and the concept of 'groups of' (Multiplication) and using equations which you will hear in the video, has been developed over time.