After reading Jo Boaler’s book Mathematical Mindsets and collaborating with other educators, I was determined to change what I do and STICK with it! As we came back from Christmas Break, I started my students off with a task called Growing Rectangles from Boaler’s book. With the help of the Principles to Actions math chat on Twitter #NCTMp2A, I was able to really narrow down the learning goal for the lesson…
Students create rectangles and through applying a scale factor of (k), discover the area of the new rectangle is k² larger than the original.
The first thing students had to do was create 5 rectangles with an area of 20 cm². They quickly realized there were only three options with whole numbers. Using focused questions, we discussed other options.
Once students had their rectangles, they had to enlarge them by a scale factor of 2. Without even knowing what scale factor means, I asked them to predict what they thought it meant based on the context. They were able to define it themselves.
Then we played a little Notice & Wonder. Many of them noticed that the area was 4 times as large as their original. I asked them, “Does this always happen?” To test this theory, they proceeded to choose a rectangle with an area other than 20cm² to see if it had the same pattern, and we continued to Notice and Wonder. Two responses are below. Students discussed their ideas and noticed in the second example to, “take the 2’s that you multiplied with you add them and get 4 which is what the area is multiplied by.” Knowing that adding was a mistake, instead of addressing it at the moment and taking over, I asked the students to test this theory. Do we add the scale factor from both sides or do we multiply? (my handwriting in blue came after question 5).
To test the theory of adding vs. multiplying they did this…
Selecting the work here was very important. I wanted to select something that wasn’t too hard for them to make connections with. So I chose groups who created a rectangle with an original area that was a multiple of 10 before they scaled them by 3, 4, and 5. When I put the student work up on the left, many students immediately noticed that the area was no longer being multiplied by 4. They realized here that when applying a scale factor of 3, the area of the enlarged rectangle was 9 times as big, a scale factor of 4 created a rectangle with an area that was 16 times as big and a scale factor of 5 created a rectangle with an area that was 25 times as big. At this point, I drew them back to the responses to #4 and asked, “So what does that have to do with this?” They immediately said that 3×3=9, 4×4=16 and 5×5=25 so we don’t add the sides we multiply them (this is when I changed it in blue on the slide above). Then I asked them, “Does this always work? How do we know?” After displaying the student work on the right with rectangles that had an original area of 20cm², they realized that this always worked which led to…
My writing in blue was a record of student debating in class. They eventually came to the understanding that applying a scale factor of k will create a rectangle with an area that is k². I was really pleased with how successful this was as I planned on linking the idea of scale factors to scale drawings and proportionality.
The next task was taken from CPM. The task was a problem I did last year for the first time. I made the same mistake that my students did so I knew it would be a task worth doing. First, as a class we did an entry level scale drawing problem which included Notice & Wonder, many table and partner talks and coming to a consensus about the values the kids were assigning to the problem and whether or not they made sense.
The scale drawing at right shows the first floor of a house. The actual dimensions of the garage are 20 feet by 25 feet. All angles are right angles. What are the actual length and width of the living room (in feet)?
Next they completed the following as a task in their groups…
Hank is planning his vegetable garden. He has created the scale drawing at right. The actual area for the tomatoes will be 12 feet by 9 feet. All angles are right angles. If the horizontal length of the zucchini plot shown in the diagram is 1 5/8, what is the area of the real vegetable garden in square feet?
We spent a lot of time Noticing and Wondering on this one. The more I used that strategy, the more I see how the basic “notices” turn into more “math specific notices” which definitely help focus their thinking.
As they were noticing, I recorded their “noticings” on the board. Eventually, they figured out that the scale was 1 in. for every 6 ft. They also noticed that tomatoes are really a fruit, not a vegetable, so I had to re-define that part of the question to having them find the area of the entire garden. After “noticing” they wonder the questions I’m going to ask them anyway!
I gave them some time to get started. I walked around and kept seeing blank papers. They had done a great job noticing things so it wasn’t making any sense to me why they weren’t writing anything down. I secretly started freaking out inside because I wasn’t sure about my next instructional move. As I was freaking out, I was reminded of a session at CMC that I attended presented by Matt Vaudrey. I remember him asking , “What questions do you have?” So I stopped class and said to them, “It seems like some of you are struggling. What questions do you have?” They asked…
- Do we keep the fractions in fraction form?
- How do we convert inches to feet?
- Do we need to pair up gardens and add them together?
- How many feet is 1.625 inches?
- How many feet is 3.125 inches?
- How do I know if 3.125 inches is equal to 18.75ft?
We table and partner talked with these questions above (the first three were from a different class than the last three). I’m glad they asked how to convert inches to feet. In response to that question I asked them, “Well how is this related to the growing rectangles problem? How are they similar? How are they different?” They instantly saw that if 1 in. was equal to 6 ft., then the scale factor would be 6, which led them to the idea of multiplication which led them to knowing how to turn inches to feet. They were confusing the fact that 1 foot has 12 inches with the scale in the problem. I also noticed that many of them were getting hung up on computation which was getting in the way, so I allowed them to use calculators to eliminate that struggle. Here is what they came up with and how I sequenced it…
This first table had minor computation errors. What the students noticed here is that they took the scale factor of 1 in. = 2 ft. and changed it to .5 in. = 3 ft. so they could easily work with converting the decimal part of 6.25. I was impressed with their method as they were already thinking about proportionality. The solution of 37.5 ft.² came next in the sequence (the typed info on the slides came after students made connections through table/partner talks). This helped them find the error in the previous one. We spent most of the time comparing 37.5 ft.² with 225 ft.². Which one makes sense? How do we know? I wish I would have video taped this lesson because I can’t even remember all of the questions I was asking them to get them where they needed to be. One student did say, “if the tomato garden is 12 ft. by 9 ft., then the area is 108 ft.² so we know that the 37.5 ft.² can’t be the right answer.” Through asking, “What does this/that mean?” (I feel like I ask this a lot) and having the kids continue discussing to find meaning in the numbers, they realized one group found the area in inches and then tried to scale it to feet, while the other scaled the sides to feet first, then found the area. Some seem confused because the first strategy made sense to a lot of them (which was also the mistake I initially made when I solved this problem). So I posed the question, “What if this were a growing rectangle? How would that change things?” To which a student replied drawing the image below.
They hadn’t really made the connections but were almost there. So I went back to the slide with the work above and posed a new question, “How are 37.5, 225, 6.25 and 36 related?” Sad that the bell rang, I asked that they reflect on the question at home for HW and be ready to share the next day. I was pretty pleased with the responses (please forgive their grammar, they have the right mathematical ideas).
Because of the growing rectangle activity, they realized that the area of the vegetable garden in feet had to be 36 times as large as the original. If the original area is 6.25 in.² then 6.25 × 36 = 225.
If 6.25 × 6 = 37.5, the area of the new rectangle was only 6 times the original, therefore 37.5 × 6 again would give them 225, an area 36 times as large.
Then to check their understanding I posed the question, “So where are the 6’s in 18.75 × 12?” I thought I would stump them because there are no 6’s in the problem when you look at it. I didn’t stump them at all. They knew that the 18.75 and the 12 had already been scaled from inches to feet so (3.125 in. × 6 ft.)(2 in. × 6 ft.) = 225 ft.².
I stressed out about how long it took to complete these activities, but I knew the learning was worth it. Fewer tasks with depth are more meaningful than multiple lessons that “practice” procedures and don’t develop understanding. I went home feeling like THIS is why I teach EVERY DAY!
Here is the handout I gave my students. 4-15 Task
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