I have been working in Mr. Sontag’s grade 7/8 room for the last few days wrapping up a performance task we had the students complete that had them choose an authentic house floor plan and then create a quote for the cost to floor the house using materials that they had to research online. It covered a variety of concepts:
- calculating the area of a rectangle, triangle, trapezoid and circle by choosing and applying the correct formula
- make metric conversions (and metric to imperial conversions)
- decompose irregular shapes into simpler shapes so the are can be calculated easier
- Estimate costs of materials based on prices you see at different online stores to make comparisons without having to calculate each example you find
- Use mental math calculation strategies when estimating material costs.
- multiply and divide with whole numbers and decimals
- recognize unit rate prices and solve unit rates to calculate cost of materials
- reason and prove that your quote is reasonable, a good product and decent price etc.
The focus of the assessment of learning was on their understanding of area, applying and calculating using the the correct formula, making metric conversions and the math process of reasoning and proving. Mr. Sontag was using the information gathered on the other concepts as diagnostic assessment for further learning, while also seeing assets students may have in those areas or possible gaps in learning.
When I cam into class yesterday one of the students said, “Mr.Stamp why did you make the deck we had to construct onto our house plan be 6 squared kilometres?” Mr. Sontag and I had already went over the student work and had noticed this misconception so it didn’t surprise me that this question came up! Below is picture of the solution from the student who asked the initial question.
The prompt for this part of the task was this: Construct a deck to add onto your floor plan that has an area of about 600 000 squared cm. Then calculate the cost for the floor of the deck into your quote.
You can see in the picture above that this group divided by 100 to get 6000 squared metres. This was a common misconception. Below you can see in this picture of another groups work they did something similar but divided by 1000 and got 600 square metres.
I took this picture after they had used their feedback to figure out their misconception as you can see they did. You can still see the erased part of their original answer which was 600 squared metres. Only two groups didn’t make this conversion error so we knew we needed to maybe do a mini lesson/guided group. Two groups were able to figure out why they had made this error but the rest needed to see the mini lesson.
What is the misconception?
The students who are dividing by 100 are viewing the conversion as a linear measurement conversion instead of an area measurement conversion which is squared. They knew that 1 m is equal to 100 cm so they divided 600 000 squared cm by 100. This was the most common error in the class regarding this conversion.
What did we do about it?
We started the mini lesson with this think-pair-share prompt:
After a solid discussion most still felt that there wasn’t a misconception and that the first option was correct. They couldn’t see why it wouldn’t be divided by 100 since 1m was equal to 100 cm. Then we moved on to this:
We had the students look at the model we built in class and asked these questions: What do you think the big square and the little square represent? (You can see the little 1×1 cm tile above the big square in the pic on the right.) After some think time they were able to say the big one is a square metre and the little one is square centimetre. Then I moved the square centimetre tile into the bottom corner of the square metre and asked: How many of these squared centimetre tiles what it take to fill up the square metre? I must say we had some wild estimates but it didn’t take long for some hands to go up and say 10 000. I asked, “How did you get that?” and one student explained that they used the rectangle area formula and multiplied 100 x 100 because the base and the height would each be 100 cm because 1m is equal to 100 cm.
At this point I think we could see some light bulbs starting to turn on. We now asked these two questions: Now that we know that 1 square metre is equal to 10 000 squared centimetres. How does that change what we are looking for when we asked you to convert 100 000 square cm into square metres? Why can’t we divide 600 000 by 100?
We got some quality responses:
- One student said it changes what we looking for in my mind because I see now we are looking for how many 10 000’s are in 600 000.
- Another student then added in that they think we can’t divide by 100 because we are not looking for a length measurement but an area measurement. So it is squared.
- Just like we saw how many 1 cm by 1 cm tiles fit into the bigger square metre model this would be looking at how many of those big square metres would be in 600 000 square centimetres. And to do that it makes sense to think about how many square centimeters is equal to one square metre and that is 10 000.
Do we feel all the students understand this now, no we don’t but more certainly do. There is more work to be done here with some of these students to help them consolidate this idea. I suggested some more small guided groups or possibly some one on one time.
If you really dig in here this really comes under the umbrella of the big idea of unitizing. Seeing a group of, as a unit. Making any conversions really relies on having built a solid understanding of unitizing. Whether it is seeing 1 m as 100 cm (linear) or in this case 1 square metre as 10 000 square centimeters (area/squared measurement) which is more challenging for students, it comes down to being able to unitize and see the unit you are using and then multiplying/dividing by 10, 100, 1000 etc which is also built on unitizing!
Hope you enjoyed this one!