Columns, Rows & Fractions Years 2 - 8

Summary This is not a one off lesson. It will have very little learning value that way. The early part of the activity develops vulgar fraction concepts - the meaning of a fraction and equivalent fractions are the main focus. The activity then develops naturally into a sequence on operations with fractions. The rectangular array of gaps is better seen as a model which can be used each day for a short while over a period of time. The starting point, and therefore the fractions investigated, changes each time, within a familiar structure. These features make the activity suitable for threading. Materials One Poly Plug per pair or one each Drinking straws, pencils or other thin 'dividers'

Special Note

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. In this case the fresh context is an activity titled Rectangle Fractions and its partner Rectangle Fractions Game. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

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These girls placed their straws like this. They divided their whole into three equal parts. Because the whole rectangle is now divided into 3 equal parts we can call each of these parts one third. That's fraction language. We know the whole, we know the whole is divided into parts. We know that the parts are equal. Now count with me by thirds using the girls' board ... one third ... two thirds ... three thirds. Do we know another name for the total of the three thirds? Continue this type of discussion with each different whole rectangle in the class. Record some of the equations which develop in the discussion. For example the picture above could show: 3 x one third = one whole one third + one third + one third = one whole one third + one third = one whole - one third When introducing children to fractions I deliberately write equations in words until the class tells me there are symbols we can use ... and they can explain them to me.

Now the girls are telling me they can show fourths, or quarters, on the same whole. Who can explain to me why using the straws this way shows the whole divided into fourths? Look for an answer which relates the three components of all fraction language: We know what the whole is. The whole is separated into parts. The parts are equal. OK, now I want you to find all the ways your whole can be made into parts with the straws. Each time you find one I want you to draw it and write all the fractions stories you can about that picture.

This boy tells us: I can show fourths and thirds and twelfths. And my blue ones are nine twelfths or three quarters. If I take some straws out I can show halves too. There are lots of sums I can write about all these. Continuing exploration, discussion and recording in this way over a period of time leads naturally into operations on fractions. The key points to develop are: That any rectangle can be separated into equal rows, equal columns and equal 'cells'. That each of these divisions can be described by fraction language. That many collections of plugs within the whole will have alternative fraction names.

Examples from Classrooms

A journal from Winkie Primary School, South Australia, showing records of work as Rows, Columns & Fractions was threaded through the curriculum. Drawings have been recorded on cutouts from Poly Plug Paper. The teacher, Nicholas Dale, has written about the success of threading in Threading Works.

These comments were written by Belinda Rayment, St. Francis of Assisi, Calwell, when involved in a Working Like A Mathematician series of workshops for the Canberra Goulburn Catholic Education Office. The course ran for six days in three sets of two days. These notes were made as a report to the group during Days 3 & 4. Fraction Fury We had fractions and decimals programmed for Year 4 so we started using some of the resources. We have photographed some of the students' work and we continue to thread the activities and gradually introduce new ones. We began with 'Fraction Rectangles - Poly Plug' as part of an introduction to fractions. It was an easy way to see fractions, to introduce and use the language of fractions (it was particularly helpful with denominator and numerator), to manipulate fractions and to find equivalent fractions. We also used the task 'Making Fractions 2' (photo below - enter the name in the search box at the top of this page) as another way to teach and share - fractions as part of a whole. We are looking forward to using Fraction Game. As you can see by the books they found it FUN. (I love the way they record all their work and ideas - it makes continual assessment easier and you can see their thought processes. I also like the way they share ideas and peer teach.) We will be moving on to decimals, so I did trial using the calculator to show a fraction. Only a couple of students worked out how to show a fraction on the calculator but once a couple did, word spread quickly and they found this fascinating. We will return to this. I am looking forward to using 'Fractions to Decimals on a Rope' (from Maths300), Calculator Slido and Move Around with decimals. My class is familiar with Move Around so extending this to decimals should be easier. All these activities correspond with the Australian Curriculum for Year 4: Investigate equivalent fractions used in contexts. (ACMNA077) Count by quarters, halves and thirds, including with mixed numerals. Locate and represent these fractions on a number line. (ACMNA078) Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation. (ACMNA079) Fractions in Action See slides of the way the activity was used in Nichola Brandon's Year 4 class in Chapter 4 of the story Fractions in Action.

Recording shows fractions clearly as part of a whole and equivalences such as two tenths = one fifth (above) and five fifteenths = one third (below).

Adding & Subtracting Fractions: Rectangle Model

As time passes, given a fraction addition, for example, two fifths + one third: the challenge is to create the whole rectangle which could show fifths and thirds simultaneously.

Then decide which is the fifths (rows or columns) and which is the thirds (rows or columns). Next place two columns of plugs to represent the two fifths.

Then add in one row of plugs to represent the one third. If some of the plugs have to be placed on top of ones that are already there, so be it.

Now rearrange the plugs so they are all in an empty space and not on top of each other. It's true that the total can't be calculated in either fifths or thirds now, but all rectangles do have another fraction that can be referenced - the equal 'cells' as above - in this case fifteenths. So: two fifths + one third = eleven fifteenths

Of course it doesn't matter where the 'on top' ones are placed, the result is still 11 fifteenths. Some children may want to rearrange them all to make neat rows; some may want to show the 'on top' ones as blue.

Again, learning is assisted by using this activity for small amounts of time often - that is, threading. The children begin to visualise the operation before actually doing it. After an initial period of practice, some teachers encourage this visualisation by asking the students to imagine the answer before making it.

Extensions

1. If you are a Maths300 member explore Lesson 77, Rectangle Fractions. This begins with rectangular wholes in the classroom such as window panes, develops further with Poly Plug, extends into graph paper rectangles and is supported by a great piece of software. It also has two excellent Classroom Contributions - one from Australia that includes a development of the rectangle model for multiplying fractions and one from USA confirming that high level failure students in Year 9 finally found success with fraction addition using the Columns, Rows & Fractions model.
2. See Fraction Multiplication: A Model to extend this model to multiplication.

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