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 :-)

Understanding Fractions - Group 7

Today I asked Group 7 (Yr 5/6) to show me fractions in more than one way (multiple representations). This group of students are lacking understanding of the key idea that fractions is about parts of a whole. They are learning that the whole could be five, the whole could be 100, the whole could be a set of objects (discrete), just repeated numbers added together, or even a shape (continuous) but they can all show the same fractional number related to the whole.

In the photos below I asked the students to show me four quarters, this is only the beginning of these lessons so as they progress we will use other materials as well so that they get experience of fractions in many forms.

• The multi-link cubes stuck together are fractions of a shape 
• the groups are fractions in sets 
• The numbers represent equal sharing creating a whole using numbers

Group 7 Problem Solving

Today my group (7) began working on a problem solving task (about workers packing cherries into boxes and cartons) near the end of their session. This group is very enthusiastic about mathematics so we have a lot of cooks in the kitchen as the saying goes. I'm sharing this video to show what is working well and what needs improvement with this group dynamic. 

As you'll see the girls take over quite a bit so we are working on listening to each other more and taking turns with sharing ideas. Jarome tries to share his thinking - that they can use their basic facts to help work out larger numbers when he says, "8 + 4 is 12" in reference to the problem where they are trying to work out 80 + 40, though he is talked over. It also shows how important it is for us as teachers to listen to what our students are muttering or saying within a group especially the quiet ones, as they usually know more than they let on. 

They are very engaged in this task. However, as we continue the task in the next session I will need to go over the expectations of working in a group and also the importance of reading the problem. They have come a long way with their communication in mathematics and they will surely continue to improve :-)

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.

The Egg Couch

Last week I had the chance to interact with other Math Support Teachers from around New Zealand at our Seminar Days. I was shown this fabulous idea of using the below image as a basis for a Mathematical investigation:

Source: Eggs Sofa by Fulvio Bonavia http://adsoftheworld.com/media/print/axn_sonys_satellite_tv_channel_eggs_sofa

Yesterday I used this image for the first time with a class of Year 7 students. Each group were given the image and asked to list down all the things they noticed about the picture i.e.I notice the eggs are in trays, there are lines going down etc. 

Once they had written down the things they noticed, they were then asked to write down any questions under 'I wonder' i.e. I wonder who made this couch? how is it stuck together? etc.

Then we had a quick discussion on the types of questions they had come up with. Are they mathematical, scientific, general?

From here they were asked to make predictions of how many eggs they thought there may be altogether. We discussed what a prediction was and an example of a good prediction versus a silly prediction.

The groups made their predictions and then started thinking about how they would go about completing this investigation i.e. what strategies could we use?

We ran out of time so unfortunately we didn't get any further however, this is great because investigations take time and this helps the students learn perseverance when attempting to solve challenging mathematical problems.

I will be using this same image with my groups, so check back in the next couple of weeks to see how they undertook the challenge : - )

Here are some photo's of room 7 working on the problem:

Here are a couple of photo's of some of my year 7 and 8 students working on their piece of the egg couch:

Is There Another Way?

My students have been working very hard with me ever since they started their groups a few weeks ago. We have spent some time learning what Multiplication looks like as an array, using Animal Strips. I use this material as it is an excellent way for the students to see what the groups are made up of i.e. 9 is made up of 5 and 4, 8 is made up of 5 and 3 etc and it moves them away from counting in ones or skip counting and towards looking at groups of/sets of/lots of instead. While my students are still getting a grasp of their multiplication facts, using this materials enables them to break down the numbers into the Multiplication tables they do already know like 5 and 3 or 5 and 2 and 2 (4). 

This leads them into learning the strategies of Place Value Partitioning (breaking up the tens and ones) and Rounding and Compensating (rounding to the nearest ten and then subtracting the groups you added in the beginning). As my students generally have some knowledge of written strategies like algorithms, we look at those as well, which I teach them can be another tool to check their answer with but it should be one of many ways they know how to solve a problem.

Below are some photos of my year 7 & 8's - Group 8 (Trinaye, Michaela, Katelyn, Aisea, Harrison & Jerome) working with these Animal Strips to develop their understanding of Multiplication and related strategies; followed by a problem solving session where they needed to extract what the questions were asking them and then use their knowledge of these strategies to solve each problem. They were able to demonstrate all three strategies and are becoming much better at identifying the maths question within all the 'words' of each problem.

These questions were taken from the Mathematics Assessment Resource Banks (ARBS) Copyright 2011 Ministry of Education, Wellington, New Zealand.

The first question for example is: 

"Hohepa worked in a store for 16 hours a week for 7 weeks over Christmas. Show how to work out how many hours he worked over that time"

Class Activities from Maths Week

Here are some photos of room 14, 6 & 7 students working together during Maths week to practice measurement through cooking!

Great work everyone, thanks for sharing :-)

Maths Week - Day 5 Problems

Here are your challenges for today:

Junior Problem:

Mr Rangi has lots of rubbish bins to clean. On Monday he cleans three bins. On Tuesday he cleans eight bins. On Wednesday he cleans half the amount of bins that he did on Tuesday. On Thursday he had more to clean. By Friday morning he had cleaned 23 bins altogether. How many bins did he clean on Thursday?

Senior Problem:

Matua Dudley had a whole packet of chocolate biscuits.  He was feeling very hungry, so he ate 4/7 of the biscuits.  That left only 12 chocolate biscuits for Mr Cressey.  How many biscuits were in the packet to begin with?

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Students: Write your answer in the comment section with your name and class.

Teachers: If you have any photos of student thinking please email to Miss Breen 

That is all the problems for this week. Congratulations to all the students and classes who completed the challenges, you are awesome!

Here are some photos from room 17 today:

Maths Week - Day 4 Problems

Here are your challenges for today:

Junior Problem:

Joseph makes a train from a green rod, a yellow rod and a black rod.   Nate makes his train from a green rod, a black rod and a blue rod?  Who has the longest train?  (You will need to use Cuisenaire rods to help you with this problem)

Senior Problem:

Mr Dawson is organising the Year 7&8 ki-o-rahi tournament. Mr Dawson has 156 cones, and each game requires twelve cones.  How many ki-o-rahi games can Mr Dawson run at the same time?

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Students: Write your answer in the comment section with your name and class.

Teachers: If you have any photos of student thinking please email to Miss Breen 

Come back tomorrow for new challenges!

Here is a photo of some student thinking/strategies from room 9 today:

Thinking from room 7 today:

Tina's thinking from room 8:

Maths Week - Day 3 Problems

Here are your challenges for Wednesday:

Junior Problem:

Danielle was given a marble on Tuesday. Then she was given two marbles every day for a long time. When did she get her 11th marble?

Senior Problem:

To make 6 pavlovas, Mr Lysaght needs 10 eggs.  How many eggs does Mr Lysaght need to make 21 pavlovas?

~~~~~~~~~~~~~

Students: Write your answer in the comment section with your name and class.

Teachers: If you have any photos of student thinking please email to Miss Breen 

Come back tomorrow for new challenges!

Here are some photos of room 7's thinking today!

Here is student thinking from room 9:

Maths Week - Day 2 Problems

Here are your challenges for Tuesday:

Junior Problem

Mrs Nathan has baked 5 muffins for her family.  She will decorate each muffin with 3 jelly beans. How many jelly beans will Mrs Nathan need?

Senior Problem

Tori cut her pizza into sixths.  She had 36 onion rings to spread evenly on her pizza slices.  How many onion rings are on 2 sixths of the pizza?

Students: Write your answer in the comments area with your name/class

Teachers: please send photos of student thinking to Miss Breen :-)

Good Luck!

Come back tomorrow for new challenges!

Maths Week - Day 1 Maths Problems

Maths Problems for 2014 Maths Week

JUNIOR PROBLEM - Monday

The bears are planning to go to the woods for a picnic.  There are 2 cars and 10 bears.  Half the bears are to go in each car.  How many bears will be in one car?

Write your answer with your name/class in the comments area :-)

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SENIOR PROBLEM - Monday

Richard withdraws $2389 from the bank in one-hundred dollar notes, ten-dollar notes and one-dollar coins. How many hundred dollar notes does he get?

Write your answer with your name/class in the comments area :-)

GOOD LUCK!

COME BACK TOMORROW FOR NEW CHALLENGES!


Here is a sample of student thinking from room 12 today

New Term - New Groups

It was fantastic to see how much my students progressed in mathematics in terms 1 & 2. Now we are in term 3, I have the priviledge of working with a whole lot of new students.

For the rest of the year I will be working with:

Group 5:          Group 6           Group 7        Group 8       Group 9
Rehia                  Dominic              Hannah           Trinaye          Chevon
Beau                   Jamal                  Anita               Harrison         Shalin
Frank                 Victoria               Jarome           Katelyn           Justin
Serenade           Joshua                Fasia                Michaela         Reece
Jessie                 Hunter                Anna                Jerome
Maraea              Tearani               Wilson             Aisea

Please check back often as I will be updating this blog with videos, photos, activities and their work regularly :-)

The Magic of Fibonacci Numbers

If you read my previous post where I used patterning task cards then this post relates to task card number 6. In that sequence of numbers I was using Fibonacci numbers which can be seen in various shapes and forms in nature. 

Watch this Ted Talks video below where Mathematician Arthur Benjamin presents, "The Magic of Fibonacci Numbers". It's well worth a watch and helps show how amazing numbers and patterns can be : )

The Magic of Fibonacci Numbers

Can You See a Pattern?

Today both year 8 groups were asked to discuss and solve a number of Algebraic patterning tasks. These started off with basic colour patterns and gradually built up to number patterns and representations where they needed to work out the rule/formula/relationship of that sequence.

Patterning is, "critical to the development of mathematical concepts and algebraic thinking and reasoning. General mathematical processes of representation, symbolization, abstraction, generalization and proof rely on initial pattern recognition and application in a variety of shapes, counting, spatial arrays or geometric patterns" (Bobis, Mulligan & al, 2013, pg 54).

As they worked in a group, it allowed every member of the group to contribute. In Group 3, one person made a suggestion of the relationship between the first three numbers in:

1, 2, 3, 5, 8, 13, 21, __

which helped others in the group to explore that idea further which eventually ended with the discovery of the pattern and the answer.

Below is a picture of some of the tasks I used today.

It's All About the Money

Last week my Year 8 groups and my Year 5 group were using play money. Money is good to use because it relates to real life. Everyone of my students has had experience at some point of handling money and buying or selling goods in exchange for money. 

My year 5's role played 'Shop Keepers'. They took turns being a shop keeper and had to sell an item to a customer. The problem for them in this challenge was working out how much change to give back i.e. if a drink cost $3.80 and they were given a $10 note, how much change do they give back? This involves using knowledge of place value, counting, addition, subtraction, parts of a whole (fractions) and decimals

My year 8's were first asked to count the pile of money put in front of them then they were asked to use money to solve addition and subtraction story problems involving money. 

Here are a couple of photos of one of my year 8 groups engaged in using their money to solve a problem.

My year 5 students (below) playing 'Shopkeepers'