Fractions and Play

I found myself planning another pizza fraction problem when it dawned on me that kids don't just eat pizza. Fractions play a part in many different aspects of our lives. So instead of coming up with another pizza or biscuit sharing activity, I decided to bring in water with cups and bottles, different fake foods, paper, lego, and blocks and let the kids explore.

I asked my students to show me their understanding of 'a whole'.

I wanted them to show me different examples of 'a whole'. e.g A whole piece of paper, a whole bunch of bananas, a whole cup of water.

I also asked them to show me how different parts can make a whole. 1/2 + 1/2 = 1 whole, 5 bananas make the whole bunch, 5 cups of water fill the whole water bottle etc.

I basically wanted them to show me their understanding of the central idea.

Fractions are a ways of representing whole-part relationships.

Students playing with making different fraction amounts.

Cutting up a whole piece of paper into different parts.

Creating a whole soccer team, two whole teams, half of the field, talking about half time.

Students playing with creating a whole box of bagels, making a whole ham burger, a whole much of bananas.

What fraction of the food is meat?

Creating different fraction amount with cups of water. 

This student was showing me that 1/2 + 1/2 = 1 whole. She then asked what if we use 'x' (multiplication)? This really intrigued her and she went to ask another student.  He shrugged! I wrote this on the board to reflect on in our next lesson. A hard concept to understand (I must admit I had to google it!), but it reinforces what can happen when you let kids inquire and explore in maths.

Introducing Fractions

Today I introduced our new central idea for our Proportion and Ratio Unit. 

First, I asked kids to read and discuss what they thought were the KEY words from within the CI. Then, on post it notes, I got my students to write down what they knew about fractions. 

Some kids could recall prior knowledge from previous Grade 1. 

Other students were open enough to admit that they don't know... yet. So pleasing to see growth mindset developing in the way they think about maths.

I then showed them this video I made using Lego Movie as a provocation. I wanted to see if this would give me more insight into the students prior knowledge. 

I asked them to answer these questions:

1. What are the problems the gorilla and the monkey face?

2. How would they solve the problems?

There responses to this showed me that some students already have a good understanding about the concept of equal sharing and knowledge of using vocabulary such as half or thirds to represent whole -part amounts.

Most could write the symbols, but some had difficulty using fractions to represent the correct amount. e.g. 3/1 is half a cake.

Interestingly, some students who said that they didn't know anything about fractions, were able to solve this problem.

This always reminds me of the importance of formative assessment and getting students to share their initial understandings .


              

                      

Posing "Challenging" Problems

Posing problems is a appropriate title to this post as they tend to pose me problems trying to write them.

My school was lucky enough to recently have Mignon Weckert talk during a maths job a like. She is passionate about how important posing challenging maths problems are.

So off I went...

The process I went through is based on readings and advice from people who are leaders in inquiry maths and with more knowledge than myself, most notably Mignon.

First, I started with a Central Idea:

Number operations can be used to solve problems in a variety of ways

and with the idea that...
Multiplication and division are effective and efficient ways to solve problems.

From my formative assessment (see previous post) I found that...

• Some students are skip counting
• Some students are using repeated addition
• Some students are using multiplication facts
• Some students are using materials
• Some students are answering it ease!
• Some students have no idea (they told me so!)
• Some students counted all. 

Great 7 different "groups"! 
NOTE: I don't group students any more. The benefits of students working in mixed groups are just too great. I am just aware of where each student is at in their learning.

Now, my first thought was, "How do I pose questions that cater to all of these groups?" 

The advice I was given was..."pose problems that challenge them all!"

Great, but easier said than done!

So, I looked at my "groups" and focused on the central idea. I wanted my student to realise that multiplication and division are efficient and effective methods of solving problems. 

Problem Number 1:

So more by chance than good planning. My class were going on a field trip. I was going to divide them into groups, but I thought... well why don't you let them do it. Now with 21 students the problem immediately presented a challenge as we needed to get into groups of 5. It was also difficult as it dealt with division even before we explore multiplication. I decided to give it a go and see how it went. 

Reflection:

• It was challenging for most, due to the 21 
• To extend those who needed it.  I said each group should have even boys and girls (knowing we have more boys than girls).
• I had rich conversations with my strugglers. We used materials, talked about skip counting or using basic facts.  One student even said that multiplication makes it easier BINGO!
• Some kids drew pictures others used materials.
• I challenged the higher, some even said that it was tricky.
• Some said 4 groups of 5, that didn't work. So we then talked about 5 groups of 4, with 1 more remaining. Commutative properties! 

 Problem Number 2:

I gave students 4 different sets of arrays. Instead of using easy amounts, I made arrays that went over groups of 10, for example 15 groups of 2 or 2 groups of 15. The reason for this was to challenge students to experiment with breaking groups ups (division), combining the use of repeated addition or using skip counting to help solve the problem. The problem was simply to find the totals of each array and show me how.

Reflection:

• Posing harder problems opened up a range of strategies.
• Students worked in groups and learnt from listening to other strategies.
• Students had the flexibility to solve problems in different ways using different operations.
• Students made connections between all 4 operations. 
• It didn't really challenge those higher working students. This is something that I still need to work on. 

My big take away so far is that... that when you make questions challenging, you bring out questions, challenge understandings and encourage discussions because you are making the students think! 

Problem Number 3.

I went to hang out my washing I had 19 pieces of clothing to hang up and 35 pegs. If each piece of clothing needs two pegs do I have enough?

Reflection:

• We discovered the link between addition, subtraction, multiplication and division. i.e. split 19 into 10 and 9 or round up to 20 to help us solve 19 x 2.
• Kids who tried 35 divided by 19 were stumped. Then one said lets use multiplication 19 x 2 as it is easier to solve.
• When working with larger numbers such as 19, it allows you to challenge those who need it and to also support the others. I could work with students on 9 x 2 by creating arrays, but still allowing them to solve the problem. 

Some more questions I have come up...

I have between 15 and 23 apples. How many different ways could I divide them into equal bags?

50 / 5 + 10 = 20 write a story to explain what happens here?

3 X 5 = 30/2 - true or false why?

Use 23 linking cubes to show your understanding the 2, 3, 5 times tables.

Create your own stories using the 2,3,5 times tables.

9 x 0 = 0 Why?

I will continue to add to this post as I move through the unit. If you have any questions that have worked in your class I would love to hear them. 

Connecting Maths with our UOI!

Today I showed students this puzzle to share at our parents maths afternoon. I then posed the idea that we make it... they soon latched on and were truly engaged!

"Let's do it now!" they said. I then thought this links nicely with our UOI central idea: Many systems are involved in getting things to our table. 

Teacher questions

How can we organise this puzzle?

What systems will we need to create?

What is the maths involved in this process?

A puzzle was a great system to organise and a natural link with our unit.

We got out blocks, talked about how we are going to organise it., chose leaders, reflected on the process and discussed what's involved in putting things together.

The end result... a great math puzzle! 

Breaking Down the Central Idea

I broke up the central idea for our Addition and Subtraction unit. Students were asked to add their ideas to the meaning of main concepts such as 'number' 'operation' 'solve' and 'problems'.

I was impressed with how much thought the students put in. 

We had just finished our Number and Place Value Unit so I was really pleased with the reflection comments about number.