Assignment 1 – Open Tasks, Learning for All
Number Sense and Numeration
In 1970, the population of Canada was estimated at 21.32 million people. In 2010, the population was estimated to be 34.01 million.
If Canada continues to grow at the same rate as this data predicts, estimate which decade the population will exceed 50 million?
This problem encompasses the essence of an open math task, providing multiple starting points and a variety of strategies and methods to reach a final estimate. Given that the problem focuses on the strategy of proportional relationships and less about a final, concrete answer, students can explore their thinking in a variety of creative ways. Additionally, it connects with the overall and specific expectations listed in a number of ways, such as estimation, computation, and reasonableness/efficiency. Students could approach this problem by finding the population quantities per year, per decade, or another measurement of time. Also, students have a wide margin of error to work within since the question is asking for a decade rather than a specific year. This, in turn, adds to the openness of the question and the overall differentiated framework of this math problem.
University of Waterloo. "Problem of the Week." The Centre for Education in Mathematics and Computing (CEMC) - Web Resources.
Using an object of your choice as a guide, predict the total perimeter of the classroom.
Extension: Using your perimeter measurements, calculate the area of the classroom.
This question asks the students to complete a specific task, but within the framework of student choice and multiple strategies. Students are asked to utilize a manipulative of their choice, which creates multiple entry points into solving the task. Through the use of manipulatives in this problem, students are called upon to use proportional reasoning to calculate the perimeter of the classroom (i.e., 1 object = 10 cm). By having the students move around the classroom and work together with their peers, they are experiencing embodied learning while satisfying the importance of learning skills, such as collaboration, communication, and creativity (in their choice of manipulative).
This problem provides a great framework for an open discussion and consolidation following the task. Within this discussion, students are able to compare strategies with classmates, talk about what they decided to do first, and use appropriate and relevant math terms to communicate their understanding of the task. This problem also provides an unique opportunity to analyze what units of measurement each student chose to use while calculating the perimeter of the classroom (mm, cm, dm, m).
Geometry and Spatial Sense
Toys “R” Us are designing their new line of fidget spinners using the various shapes seen below:
Using what you know about polygons and their various features/characteristics, list the similarities and differences of the shapes. Consider such things as: number of sides, types of angles, type of polygon, etc.
This problem at its very essence is an open task, available for students of every ability to answer. The variety of shapes give the students multiple entry points when starting the problem; students could focus on the triangles first, compare the same-coloured polygons, work from top to bottom, etc. There is no specified minimum requirement or limit on the number of similarities or differences that the students could list. This creates a tiered question, allowing high-performing students to challenge themselves by listing many different similarities and differences, while also providing lower-performing students with enough choice that they could easily find their first response to the problem.
Manipulatives could easily be used during this task, which would satisfy the tactile and bodily-kinesthetic learners, as opposed to relying on the visual component of the task. This task also provides a great opportunity for a consolidating math talk, in which students could use geometrical terms and language (as outlined throughout the curriculum) and explore themes found within other strands (i.e., side length).
Ginger Snaps Clip Art. “Primary Geometric Shapes Set.”
Patterning and Algebra
“I am a number pattern with the number 60 as my 6th term.”
Using a table of values to show your thinking, write as many possible pattern rules as you can that make this statement true.
This task is open in a number a different ways. Firstly, the task is open for students to create different types of patterns: growing, shrinking, and repeating. This provides the educator with a good idea of which types of patterns the students are most comfortable creating and which areas the students could develop further skills (formative assessment). The task is also open in that the quantity and difficulty of the patterns created is solely determined by the student. This differentiates the task for all levels of students. For example, students on IEPs could count by simple numbers (1s, 2s, 3s…) to complete their number pattern, while other students could create two-rule patterns (+4, -2). As with many other open tasks, there is a great potential for a consolidation discussion with peers about the strategies they used when approaching this problem. The strategies shared could also connect to curriculum expectations, especially when student use the patterning terminology used throughout the unit.
Data Management and Probability
Khadeeja opened a box of Smarties. She noticed that some of the Smarties were red, some were purple, and some were brown. There were 8 Smarties in the box all together.
What could the possible probabilities be if Khadeeja will most likely pull out a red Smartie first? Use images and other strategies to support your thinking.
Extension: If Khadeeja pulled out a purple Smartie first, how would this affect the probability of her pulling out a red Smartie next?
This challenging open task asks students to consider a number of different components prior to making a final answers. Students must take into account the colours other than red (purple and brown) and what effect they may have on the total quantity of red Smarties. The problem provides students with other important information regarding the quantity of the red Smarties, such as it will most likely be pulled first. Students must understand what “most likely” means (larger probability when compared to the other colours) and take this into account when finding possible probabilities. This, in turn, has students think about total number of each colour and the affect that this has on the probability of it being pulled.
This task calls upon students to use images and other strategies to support their thinking. Therefore, students are free to approach the problem in a number of ways, such as drawing, using fractions, displaying results in charts, and running trials with coloured manipulative, among other strategies. With the information provided and the ability to answer the question in any chosen way, students of all levels are able to access the question and demonstrate their learning. For students with math IEPs, a number of factors could be differentiated, such as the total number of Smarties in the box, the total number of colours, and the number of possible probabilities requested.
Small, Marian. Making Math Meaningful. “Opening-Up Strategies.”
Ginger Snaps Clip Art. “Primary Geometric Shapes Set.” N.p., 30 July 2012. Web. 10 July 2017.
Ontario Teachers’ Federation. Lesson Plan. “Parallel and Open Task Problem-Solving Math Bank.”
Small, Marian. Making Math Meaningful. “Opening-Up Strategies.”
University of Waterloo. "Problem of the Week." The Centre for Education in Mathematics and Computing (CEMC) - Web Resources. University of
Waterloo, n.d. Web. 10 July 2017. <http://www.cemc.uwaterloo.ca/resources/potw.php>.