Math Worth Thinking About: Seeing the Beautiful Complexity in Simple Number Patterns
Watch this video to see:
Watch this video to see:
- how students might transition from simpler to more sophisticated ways of thinking about and describing numerical patterns, and
- the surprising and satisfying way in which related sets of numbers are generated.
The CT/Coding Advantage: Abstraction, Automation, and Resilience
Watch this video to see:
- how the combination of tangible abstraction and automation can create dynamic models that provide students with settings for rich investigation, and
- how CT/coding helps students develop greater resilience to academic struggle.
Minds On: Building a Patio
1. Provide students with the following problem: You have decided to build a square patio in your backyard. The patio will be made of square tiles. Use the following instructions to build models that will help you find the information you need for your planning:
Give students an appropriate amount of time to work on the problem, providing support as required. |
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2. Bring the whole class together to debrief the activity. Work together to create a class table of the patio data. The following questions (taken from the lesson Backyard Improvements in The Super Source – Patterns/Functions – Grade 7-8) support student thinking and discussion for this activity:
- What patterns do you see in the data?
- How would you describe the number sequence in the number of blocks added column?
- How would you describe the number sequence in the perimeter column?
- How would you describe the number sequence in the total number of blocks column?
- How many blocks would be needed for the 11th square? the 12th square? the nth square?
- How many blocks would be needed for the 20th square? How do you know?
- What algebraic expressions can you write to express the relationships you found?
Action: Coding the Patio
3. Have students, still in pairs, explore the Backyard DIY Scratch code linked here. As the image to the right shows, it creates models of the square patios as well as tables of values for blocks added, perimeter, and total number of blocks. In addition, the code graphs each set of data. The algebraic expressions for each pattern are written into the code. After students have had sufficient time to explore, facilitate a short class discussion about the Scratch code. Some points of discussion might include:
3. Have students, still in pairs, explore the Backyard DIY Scratch code linked here. As the image to the right shows, it creates models of the square patios as well as tables of values for blocks added, perimeter, and total number of blocks. In addition, the code graphs each set of data. The algebraic expressions for each pattern are written into the code. After students have had sufficient time to explore, facilitate a short class discussion about the Scratch code. Some points of discussion might include:
- The graphs – Students will not have encountered graphs in the unplugged activity. They may require some explanation of how the three lines are related to the models and tables of values. They may notice that one of the lines is not straight. This is a good opportunity to introduce them to the distinction between linear and non-linear patterns.
- The algorithms – This code is quite robust (it has to be in order to create models, tables of values, and graphs). To make it more manageable, three separate blocks have been created (Reset, Create Tile Models, and Add to Lists). The algorithms that create the graphs are remixes of a single code, and they are separated from the main code by broadcast blocks. As a result, what could have been a single, but very large and confusing, code has been presented as a collection of smaller, more accessible parts. Students should be told that this is an example of the computational thinking skill of decomposition. Spend some time looking closely at these smaller parts to see how the individual blocks combine to create the desired functions. Tell them that this kind of thinking, known as algorithmic thinking, is a key CT skill.
- The algebraic expressions – Ask students to find where the algebraic expressions (from their work on the unplugged task) exist in the Scratch code.
Action: The Staircase
4. Provide students with the follow-up problem:
Now that you have built your patio, you need to build a set of stairs from one level of your backyard to another. You plan to use railroad ties for the steps. Use the following instructions to build models that will help you find the information you need for your planning:
4. Provide students with the follow-up problem:
Now that you have built your patio, you need to build a set of stairs from one level of your backyard to another. You plan to use railroad ties for the steps. Use the following instructions to build models that will help you find the information you need for your planning:
- Use colour tiles to represent the side view of the railroad ties. Work with your partner to build models of staircases. Start with the smallest possible staircase (i.e., a single stair), then add onto it to build increasingly larger ones.
- Each time you build a new model, record the following pieces of information:
- The number of railroad ties you added to the previous model to build the current one
- The perimeter of the staircase
- The total number of railroad ties in the staircase
- Use the provided table to organize your data.
- Record data for the first ten staircases. You do not have to build all ten models. If you find patterns in the data, use them to predict the values of higher terms for all three categories.
- Look for relationships between the model number and each of the three data categories (number of railroad ties added, perimeter, and total railroad ties). Think about how you might generalize these three relationships. Write an algebraic expression for each one.
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5. Inform students that the end goal is for them to produce a Scratch code that, like the one for the patio, creates models, tables of values, and graphs for the staircase data. Provide them with several options for their work, including:
- They may begin by working in an unplugged setting, then, at a time most appropriate to them, bring their work into Scratch.
- Their Scratch code may be a complete remix of the provided code, something that is entirely new and different, or a combination of these ideas.
Consolidating the Learning
6. Bring the whole class together to debrief the activity. Work together to create a class table of the staircase data. Invite students to share their observations with the class. The following questions (taken from the lesson Backyard Improvements in The Super Source – Patterns/Functions – Grade 7-8) support student thinking and discussion for this activity:
6. Bring the whole class together to debrief the activity. Work together to create a class table of the staircase data. Invite students to share their observations with the class. The following questions (taken from the lesson Backyard Improvements in The Super Source – Patterns/Functions – Grade 7-8) support student thinking and discussion for this activity:
- What patterns do you see in the data?
- How would you describe the number sequence generated by the entries in the number of railroad ties added column? the perimeter column? the total number of railroad ties column?
- How many railroad ties would be needed for the 11th staircase? the 12th staircase? the nth staircase?
- How many railroad ties would be needed for the 20th staircase? How do you know?
- What algebraic expressions can you write to express the relationships you found?
- What did you notice when you compared the sequences from Part 1 with those from Part 2?
Extensions
7. Once students have completed their codes for the staircase problem, have them apply their new knowledge and skills to different contexts. The Super Source – Patterns/Functions – Grade 7-8 offers a number of lessons that could serve as effective follow-up activities to Backyard DIY. A good candidate for this purpose is called Ripples (the full lesson is linked here). In this lesson, students use various pattern blocks as pebbles that, when tossed into a pond, create specific ripple patterns. Students model these scenarios with physical tiles, collect numerical data, and represent the patterns in various ways. Students could remix the Backyard DIY Scratch code to model these patterns.
8. In Backyard DIY, students investigate square numbers and triangular numbers. Invite them to consider the possibility of sets of numbers based on other polygons. This could lead to an investigation of pentagonal and hexagonal numbers. Once these sets are discovered, students could adapt the Backyard DIY Scratch code to create tables of values, models, and graphs for them.
7. Once students have completed their codes for the staircase problem, have them apply their new knowledge and skills to different contexts. The Super Source – Patterns/Functions – Grade 7-8 offers a number of lessons that could serve as effective follow-up activities to Backyard DIY. A good candidate for this purpose is called Ripples (the full lesson is linked here). In this lesson, students use various pattern blocks as pebbles that, when tossed into a pond, create specific ripple patterns. Students model these scenarios with physical tiles, collect numerical data, and represent the patterns in various ways. Students could remix the Backyard DIY Scratch code to model these patterns.
8. In Backyard DIY, students investigate square numbers and triangular numbers. Invite them to consider the possibility of sets of numbers based on other polygons. This could lead to an investigation of pentagonal and hexagonal numbers. Once these sets are discovered, students could adapt the Backyard DIY Scratch code to create tables of values, models, and graphs for them.
Curriculum Expectations
Backyard DIY addresses the following curriculum expectations:
Backyard DIY addresses the following curriculum expectations:
Acknowledgements
Backyard DIY is partially based on "Backyard Improvements" from The Super Source: Patterns/Functions – Grades 7-8. The original lesson is found here.
Backyard DIY is partially based on "Backyard Improvements" from The Super Source: Patterns/Functions – Grades 7-8. The original lesson is found here.
References
CAST. (2018). Universal design for learning guidelines version 2.2. Retrieved April 11, 2018, from http://udlguidelines.cast.org
Gadanidis, G., Brodie, I., Minniti, L., & Silver, B. (2017, April). Computer coding in the K-8 mathematics curriculum? Retrieved April 06, 2018, from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/computer_coding_k8_mathemathics_system.html
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
[Polygonal numbers as quadratic sequences]. (n.d.). Retrieved August 6, 2018, from https://drchristiansalas.com/2015/01/04/polygonal-numbers-as-quadratic-sequences/
Sengupta-Irving, T. (2016). Doing things: Organizing for agency in mathematical learning. Journal of Mathematical Behavior, 41, 210-218.
The super source: Patterns/functions - Grades 7-8. (2007). Vernon Hills, IL: ETA/Cuisenaire.
Voogt, J., Fisser, P., Good, J., Mishra, P., & Yadav, A. (2015). Computational thinking in compulsory education: Towards an agenda for research and practice. Education and Information Technologies, 20(4), 715-728.
CAST. (2018). Universal design for learning guidelines version 2.2. Retrieved April 11, 2018, from http://udlguidelines.cast.org
Gadanidis, G., Brodie, I., Minniti, L., & Silver, B. (2017, April). Computer coding in the K-8 mathematics curriculum? Retrieved April 06, 2018, from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/computer_coding_k8_mathemathics_system.html
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
[Polygonal numbers as quadratic sequences]. (n.d.). Retrieved August 6, 2018, from https://drchristiansalas.com/2015/01/04/polygonal-numbers-as-quadratic-sequences/
Sengupta-Irving, T. (2016). Doing things: Organizing for agency in mathematical learning. Journal of Mathematical Behavior, 41, 210-218.
The super source: Patterns/functions - Grades 7-8. (2007). Vernon Hills, IL: ETA/Cuisenaire.
Voogt, J., Fisser, P., Good, J., Mishra, P., & Yadav, A. (2015). Computational thinking in compulsory education: Towards an agenda for research and practice. Education and Information Technologies, 20(4), 715-728.