Year Six - Reasoning with Proportions
Home - Reasoning with Proportions Map
About this Big Idea
In this Big Idea students explore quantitative literacy by exploring the relative proportion of everyday objects. They visualise the proportions of animal dimensions and make comparisons between animals using proportions to unlock fresh insights. Later they investigate seating arrangements in a restaurant and expand their understanding of the relationship between area and perimeter through a lens of proportion. This Big Idea lays a conceptual foundation for mathematical concepts that students will return to in Senior School.
Understanding Goals:
Students will understand:
- how seeing proportions allows us to better understand relationships and see new patterns
Essential Questions
- What do we mean by 'proportion' and why do mathematicians care about proportions?
- What do proportions help us see? When might a proportion help you see something new?
- How can we visualise proportions?
Background:
Ratios and rates underpin proportional reasoning needed for problem solving and the development of concepts and skills in other aspects of mathematics. This unit is an introduction only and is designed to allow students to develop a fundamental understanding of the concept. Later in their mathematical learning they will return to this concept and explore it in greater depth and technicality. In this unit the emphasis is on visualising proportions and looking for proportion relationships as a way of better understanding of common items e.g. animal dimensions, tables and chairs, area & perimeter.
Core Content from the Syllabus:
Working Mathematically
Multiplication and Division
- select and apply efficient mental and written strategies, and appropriate digital technologies, to solve problems involving multiplication and division with whole numbers
- solve word problems involving multiplication and division, e.g. 'A recipe requires 3 cups of flour for 10 people. How many cups of flour are required for 40 people?'
- use appropriate language to compare quantities, e.g. 'twice as much as', 'half as much as' (Communicating)
- use a table or similar organiser to record methods used to solve problems (Communicating, Problem Solving)
Language:
- proportion, relationship, rate, ratio, unit rate, multiply, multiplied by, product, multiplication, data, unit, divide, division, fraction
Connected to:
Mindset Mathematics Learning Activities
Visualise
Students visualise proportions by exploring a graph of animal jump length in proportion to body length. As they make sense of the data, students explore how proportions change the way they understand the animals the data represents. - See Page 144
Questions for reflection:
- What did you notice in the graph that helped you make sense of it? Did you make mistakes in reading it? What did you learn?
- What observations did you make?
- What is the most surprising thing you noticed? What makes you say that?
- How did the proportion data change the way you thought about the data and comparisons?
- What did you find when you added yourself? How do you compare to other animals?
- What do proportions help us see? When might a proportion help you see something new?
Play
Building on the Animal Jump Graph in the Visualise activity, students play with ways to represent animal data that use proportions and help readers see the animals in a new way. - See page 149
Questions for reflection:
- What surprises you about the animals?
- How do the displays help you see new things?
- How do the proportions change your thoughts about comparisons?
- What do you wonder now?
- Where do you think you would fit on the graphs?
- What connections do you see across graphs?
- What did you have to do to design your displays?
- How did you find your proportions?
- How precise do you think proportion data needs to be to communicate differences accurately and effectively?
- What was challenging and why?
- What do proportions help us see? When might a proportion help you see something new?
Investigate
Students investigate the relationships between seats and tables at a restaurant, developing ways to communicate this relationship using proportions, ratios and unit rates. - See page 156
Questions for reflection:
- What patterns did you notice as you put tables together?
- How can you see those patterns in your table?
- What ways did you develop for expressing the relationships between the number of tables and number of chairs?
- What is the most efficient way of seating people? What makes you say that?
- How are the tables and chairs connected to area and perimeter?
- What does our investigation of tables and chairs tell us about area and perimeter?
- Where do you see proportions in the classroom right now? Draw pictures to help explain your thinking?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 6
NESA - Mathematics K-10 - 2012
