Introducing the Concept of Ratios
Last week, we had flipped our classroom learning and the children enquired into ratios at home with three levelled options of understanding see: Flipping Classroom for Ratios Enquiries
Part of that enquiry entailed the children creating their own ratio word problems for their peers to solve. Looking at which levelled YouTubes they chose to learn from and the types of questions they wrote served as a pretty good pre-assessment of where each child was at with their conceptual understandings.
Using that data, I formed a small group that showed that ratios were very new to them and whilst other groups where enquiring into ways to measure mass to extend that previous unit, I was able to support this group in in getting clearer key foundation understandings of ratios. I figured that if I can spend some additional time with these children now, then when we as a whole class start our enquiries into ratios they will be better equipped with their understandings.
Using their 'flipped classroom' notes they made (sample):
we brainstormed together what we had found out (see below):
To find out our understandings a bit more, I presented the situation of the number of boys to girls in our class and how we could see this as a ratio. How about the ratio of boys to the entire class?
To extend our understanding a bit further, we added me to the ratio.
This helped us to discover that a ratio doesn't just have to be two numbers, it can be more than two:
How might we use ratios in real life situations?
Why should we learn about ratios?
- We can use them for recipes when we need to make more than what the ingredients say.
- We can use them when we want to compare how much of something I have compared to my friend.
- Sometimes in video games, we can see the ratio energy points of characters in them.
That was pretty good I thought for a group I had pegged as being in the most need of support.
To help us visualise ratios, I presented them with some green and orange cubes.
What is the ratio we see?
If we then had 6 green cubes, how many orange would we have if it was an equivalent ratio?
- Some of us thought 4 and some thought 5 so we discussed strategies we used and which seems more reasonable.
What if we had 8 oranges cubes, how many green would we have at an equivalent ratio?
What is the strategy we use? - multiply both values.
To help them discover some practical ways we might use ratios, we tossed a ball into a bin 10 times and recorded the number of times it went in and the number of times it went out.
We decided our ratio was 6 : 4 ( 6 in, 4 out )
What was the ratio from the number of ins to the total?
= 6 : 10
What was the ratio of the number of misses to the total?
= 4 : 10
If we continued throwing the ball into the bin at the same ratio, how many ‘ins’ would we expect if we threw the ball a total of:
° 100 times? = 60 times
How did we calculate that?
- We multiplied the number of ins by 10 because the the total number of throws was ten.
Oh, I see!
So, what if we threw the ball 50 times and it landed in the bin at an equivalent ratio?
= 30 times
How do we know that?
- Because half of 100 is 50 and so we halved 60.
Oh, that makes sense. So what we are saying is we can make equivalent ratios by multiplying the numbers?
Let's say that we throw the ball 25 times, but this time we want to know how many times we missed the bin at the equivalent ratio?
- Hmmmm........Good question Mr Anshaw!
Thanks. I thought so too. :)
I bit of head scratching and discussion took place as they tested their answers for reasonableness.
- A few different answers were suggested and after hearing them all we heard the strategies we used and that helped us to see what the correct answer was.
What does this make us wonder about ratios?
- I wonder if we can divide the values to also find equivalent ratios.
- I wonder what other ways we use ratios in real life.
- I wonder if ratios always stay this easy or do they get more complicated.
Giving children the opportunity to share wonderings helps them to:
° take ownership of their own learning
° develop curiosity and thus greater engagement in what we are doing
° serves as an interesting formative assessment (What sort of questions are they wondering about and how could we use those to enquire into?)
To help us perhaps discover our wonderings, I asked them what could we do next?
One child suggested we could roll a die, so that is what we did.
We decided we would roll the die 12 times and create a ratio of the number of times it lands on odds or evens.
I think when we ask children to do the thinking like in this situation, we are giving them an opportunity to think creatively and also to think a bit more deeply about the concept. In this case, how might we use ratios in real life situations.
To help them develop different strategies, I introduced a t-bar and we recorded the number of times it landed odds or evens in that. I knew that this would come in handy later in our unit as a useful tool.
We had rolled 8 evens and 4 odds and knew to write this as 8 : 4
This time, they posed the questions for themselves to solve.
- What if we rolled the die 24 times at an equivalent ratio?
I asked why they chose 24 and it was explained how 12 and 24 have an easy relationship.
Makes sense to me. Actually, it makes a lot of sense and helped us to think about number relationships.
- What about 48 rolls?
I predicted we might then suggest a higher number, but one student threw us a bit by suggesting what if we only rolled the die 6 times at the same ratio?
- Interesting question!
It helped us discover a wondering of whether we could divide ratios to find equivalents too.
Another then asked if we could then simplify our ratio?
We figured we could and simplified it to 4 : 2
- Wait! We can simplify it even further to 2 : 1
So are we saying that for every two times we rolled an even, we rolled an odd 1 time?
Oh, I see.....
To help further with that understanding of dividing to find equivalent ratios, we went back to our cubes:
From this, we were able to see how we can indeed divide the values in ratios as well as multiply them to find equivalents.
So, how are we feeling about ratios?
- I like ratios. They are easy and we can do fun things with them.
- I think they are easy to understand especially when I can see them visually with the cubes.
- Can we do more fun ratio activities like this tomorrow?
Absolutely. And that seemed like the best way to end our investigation.......