This summer I’ve had opportunities to review math units that I’ll be teaching in September. I’ve been beefing up the units with formative assessments and intentional questions that focus on math reasoning. One skill that seems to need bolstering every year falls in the category of proportional reasoning. This becomes quite evident when students encounter fractions and rates. Some students may use proportional reasoning,but it’s not necessarily identified as a strategy or communicated using that specific vocabulary.
I also picked up this book over the summer. I read it about a year ago, but I’m finding so many gems in there a second time around. The authors reveal research that indicates teaching proportional reasoning has benefits. The authors also showcase that proportional reasoning is difficult to define, but they can categorize what people can do with this type of reasoning. People that use proportional reasoning understand the relationships that numbers have put together and how they relate individually. They can analyze numbers and look the difference (additive) between them and observe the ratio (multiplicative).
My takeaway from this section of the book comes from the authors’ five reminders. These reminders come in handy when thinking of how to create learning experiences involving proportional reasoning.
1.) Use unit and multiplicative models. Double-down on using the idea of a rates, which can be applied to the idea of a proportion. Specifically, I can think of rate tables to be helpful with this.
2.) Identify proportional and non-proportional comparisons.
3.) Include measurement, prices, graphs, and geometry to show proportions. Proportional reasoning can be found in a variety of contexts.
4.) Solve proportions using different strategies. Focus on reasoning. This may be ignited by planting questions that elicit different ways to solve the problem. Students should be able to compare and discuss what comparisons exist. This can also be addressed through the use of “what’s my rule” tables.
5.) Have students recognize that short-cut methods such as cross-multiplication aren’t helpful in developing reasoning.
Being able to identify proportional reasoning can help teachers emphasize its usefulness. Having in-depth conversations about this type of reasoning has benefits. I realize that this post is heavy on tables and that’s not the only form that proportions take. I do feel as though the tables help students observe the relationships a bit easier since it’s organized. While exploring this topic I came across a MARS activity that I’m planning on using in September.
My third grade class is studying unit prices this week. They investigated in/out tables on Monday and determined how to find missing values. During this process students started to explore strategies to solve function machines. The class transitioned to unit prices on Wednesday. This was more challenging, especially when students needed to find the price per ounce. Being able to round answers and place the decimal in the appropriate place seemed to cause some issues. Students knew what operation to use but had trouble placing the decimal.
On Friday students started off the day with a brief activator involving price per ounce. Students were given three Starbuck cup sizes. They had to find the price per ounce. Students worked in groups and experimented with different strategies to find the right place to put the decimal. The groups checked their reasonableness by multiplying their unit price answer to find the total price. This took around 10-15 minutes. Click here to find the template.
After this activity the class formalized a process to find the unit price of an item. We then moved to the main project for today’s lesson.
Students were given iPads and asked to visit a pizza site. The site was up to the student. Students visited Papa Johns, Pizza Hut, Giordano’s, Dominos and Lou Malnati’s websites. They were asked to customize one large pizza. Students put together their dream pizza and found the final price. In order to be consistent students were told that each large pizza has around nine pieces.
Students took a screen shot of their fake order and saved it to their camera roll. They submitted their screen shots and added a caption that included a number model of the unit price.
I’m reviewing the student screenshots tonight and am finding that students are becoming better at understanding unit rates. Next week we’ll be looking at better buy problems.
My fifth graders are currently studying rates and proportions. Earlier in the week they explored rates by looking at unit prices and solving problems with some type of cross-multiplication strategy. Although they’ve made progress I still feel as some many still need to cement their understanding of a ratio and proportion. So it was time to switch up the instruction model.
I decided to go with using a spreadsheet. In this case, the spreadsheet would be in the form of an Excel document. Each student grabbed a laptop and opened up Excel. The students used Excel earlier in the year so they were familiar with some of the basic functions.
After entering a few text cells, students were asked to put a random number above zero in cells B4 and C4. Then the class discussed what GCD stood for. Most of the students said “greatest common denominator.” That response made sense because that’s heavily emphasized in fourth grade as students add and subtract fractions. In this case, GCD means greatest common divisor. The class then discussed what that meant when comparing two numbers and the helpfulness in finding the GCD when exploring equivalent fractions. The discussion then transitioned from equivalent fractions to finding ratios.
Students entered in the formula =GCD(b4,c4) to find the GCD of the two different numbers. Students observed how the GCD changed as they updated their numbers.
The next part was a bit tricky. I asked the students to write a formula to express the ratio in simplest form. The class used the GCD and trial and error to come up with the ratio formula. Once students wrote the formula and placed it in E4. Students then explored how the ratio changed when their numbers were updated.
The class then reviewed why the formula actually worked. The class discussed that basically the formula took each number and divided it by the GCD of both numbers. What was great was that students were starting to connect the reasoning behind the creation of a ratio. Instead of just cross-multiplying, students are starting to show a deeper understanding of how ratios are constructed and the process used to simplify. The students were able to save and print out their spreadsheets for later review.
The topic of mathematical rates was introduced earlier this week. Personally, I tend to find this unit enjoyable as there are many opportunities to connect the topic outside of the classroom. To introduce the topic my classes go home and find examples of rates in their kitchen’s pantry. The next day the class shares out what they found. This usually leads to an in-depth conversation about rates and patterns. After our conversation I felt as though more examples and experiences were needed.
That evening I found some masking tape in my desk. I decided to create a race path around the classroom. The path varied in width and it purposely had a few sharp turns. A roll of masking tape was used as well as a few proactive comments to the janitorial staff to not pick up the tape overnight. When students came into the room the next day they saw this:
When the students walked into the room they were surprised. A few started to jog around the track and ask questions about the room. Already I was fairly excited as the students were pumped to see what I was up to. I explained to the students that we were going to use the track to discuss rates, patterns and measurement. The class then measured out the track and found that it was 66 feet long. We had a conversation about how this track could be used to emphasize rates. I then introduced the students to the sheet below.
Students were starting to see the big picture of this activity. Students then took turns and quickly walked the course. While they walked I had a few students become referees to make sure that no one stepped outside of the path. I used an online counter and displayed the results as students quickly walked. Once all the students completed their route and wrote down their results the class reviewed how patterns can be developed with rates. Students were able to find the amount of feet traveled per second and then used that information to find how fast they walk one foot. I was finding that students were trying out different mathematical strategies to find a solution. I gave them opportunities to work with each other to find solutions. I asked clarifying questions when needed, but for the most part the students were on track. When the class finished this part of the sheet I gave them the second part.
This portion of the activity was more challenging. Students were able to find the total amount of seconds, but converting the seconds to minutes was a struggle. Many students asked how they could convert 12.9 minutes to minutes and seconds. I was proud to see that students understood that 0.9 doesn’t mean 9 or 90 seconds. This was a great opportunity to explore the concept of converting decimals to actual minutes. The class used different calculations and found that 12.5 would actually be 12 minutes and 30 seconds. As progress was made students started to find a conversion strategy to correctly convert the decimal to seconds.
As a class we shared our results and found patterns and the mean. This activity worked so well that I used it with a few other classes this week. I’m finding that students are developing a better conceptual understanding of rates while participating in a learning experience that I hope they don’t forget too quickly.