Using CAS to allow students to investigate unit rates opens up a world of options due to the device recognizing words as variables. Teachers can actually enter one cup of sugar makes 2 cookies in fraction form on a TI-Nspire CX CAS and get the results shown below.
Asking questions that promote deep thinking about the topic gets students into the discussion.
- Why are these true?
- What stays the same?
- What changes?
- What other ratios would work?
- What other ratios would NOT work?
Asking the right questions is key to starting effective discussions, but what then? Multiple representations help students visually see what the math is doing which in turn leads to better understanding and skill mastery. Look at the following situation:
Joe can mow 7 lawns in 4 hours. How many lawns can he mow in 3 days?
Now look how the CAS can investigate this problem using Numerical form, a table, and a graph.
This allows students to see relationships between “doing the math” and “using the math.” See how long it takes the students to realize that the data is in hours but the problem is in days. The number 3 is nowhere in the data which will generate questions from students. This opens up a broad range of “teachable moments” which is what teachers love to see. The students want you to explain instead of you begging them to listen. Now let’s throw in some geometry.
This introduces using the unit rates for unit conversion. Although unit rates are a 6th grade skill, students can see how solving equations will play a part even though they haven’t mastered that particular skill. Ask the students to investigate how the device got that answer. You may never have to teach “cross multiply” again. 🙂
As I think about important skills for students, percentages stand out to me as an important skill for both elementary and middle school students that is not understood completely. The students memorize the algorithms we set before them but never really reach the mathematical understanding of why and how we use them. Using the TI-Nspire CX CAS handheld and some teacher preparation, students can delve into the why’s and how’s of percentages. For example, start the lesson by putting the following slide on the board or send to student handhelds.
Ask students the following questions:
- What stays the same?
- What changes?
- Why do you think the last one is false?
Using a Quick Poll in the TI Navigator system or a cooperative learning strategy, facilitate student discussions on their answers. The students can use their devices to investigate other percentages to see if their theories hold true. Teachers can use the activity Solving Percent Problems from the TI Math Nspired website as a follow up activity or intro activity for the lesson on using percents to solve problems. As the unit progresses other investigations and discussion starters could look like this:
Ask questions like, “What do you notice about the numbers?”, “What is the relationship between the numbers?”, “What conclusion can you make based on this pattern?”. Also show some other percentages such as the slide below asking similar questions.
Of course as in any good unit of study, opportunities for practice and hands-on applications are needed throughout the unit to master the skill but getting students motivated to understand the math is the first step. Investigations such as these will allow students to delve deeper into the math instead of skimming the surface with algorithms only.
When I was first introduced to a CAS device, I thought it was a great tool for my AP Calculus class. The more I used CAS activities I realized it would be a great tool for all math. I believe student investigations are key in deeper understanding of mathematics. CAS devices such as the TI-Nspire CX handheld and iPad app give students the opportunity to investigate the why’s and how’s of math. If I write the steps on the board, the students only half believe. If I give them technology and pose a “dilemma” such as why does ½ + ⅓ = ⅚, then the real learning begins. I created this blog to share my experiences with and thoughts about using CAS in all grade levels.