Year Five - Seeing and Connecting Patterns across Representations
Home -Seeing & Connecting Patterns across Representations Map
About this Big Idea
In this Big Idea students explore patterns through multiple representations. Students will understand the importance of finding patterns and how visual representations can facilitate this. This investigation concludes with students using tables and number sentences to describe how patterns grow.
Understanding Goals:
Students will understand:
- that numbers are just one of the many ways mathematicians represent their thinking
- that there are many was to represent and explore patterns in mathematics
- that we explore patterns in mathematics so as to make generalisations
- how mathematics can be beautiful
Background:
Students should be given opportunities to discover and create patterns and to describe, in their own words, relationships contained in those patterns.
Students should be encouraged to use their own words to describe number patterns. Patterns can usually be described in more than one way and it is important for students to hear how other students describe the same pattern. Students' descriptions of number patterns can then become more sophisticated as they experience a variety of ways of describing the same pattern.
Core Content from the Syllabus:
Working Mathematically
Patterns & Algebra
- Continue and create sequences involving whole numbers, fractions and decimals; describe the rule used to create the sequence
- describe the number pattern in a variety of ways and record descriptions using words
- explain why it is useful to describe the rule for a pattern by describing the connection between the 'position in the pattern' and the 'value of the term' (Communicating, Reasoning)
- interpret explanations written by peers and teachers that accurately describe geometric and number patterns (Communicating)
- make generalisations about numbers and number relationships
Language:
- pattern, increase, decrease, missing number, number sentence
Connected to:
Mindset Mathematics Learning Activities
Visualise
Students explore two similar visual growing patterns and develop ways of seeing and describing their growth. They compare the patterns and extend them to see how they diverge. - See page 145
Questions for reflection:
- What are the different ways we can see the growth of each pattern?
- What are the different ways we can describe the growth of each pattern?
- How do numbers help us compare? How do the visuals help us compare?
- What happens to each pattern as you extend it to the 10th or 20th case?
- What were the most useful ways to see and describe the growth of the patterns? Why? What makes you say that?
Play
Students create a pair of growing patterns using shapes, analyze their patterns, and then use them to create an activity for others to explore. Students' patterns become stations for a pattern carnival. - See page 153
Questions for reflection:
- How do you see the patterns growing?
- How are they similar? Different?
- What do you think the 10th [or 20th] case will look like? How many shapes will it take to build the 10th [or 20th] case of each pattern?
- What patterns did you see in the carnival that were most intriguing? Why? What makes you say that?
Investigate
Students investigate how to show a visual growing pattern in a table and on a coordinate graph. Partners create their own patterns using pictures, tables, and graphs, and look for connections between these forms and among their patterns. - See page 159
Questions for reflection:
Part A
- How can we use the graph and table to describe the growth?
- How does growth look different in the picture, table, and graph?
- How do the table and graph change what you see? What might they be useful for?
Part B
- What do the different patterns have in common?
- What makes them different?
- How could we group them to show what they have in common?
- What connections do you see?
- What questions do you have about these patterns? What are you wondering?
- What different things can you learn about a pattern by looking at a picture, at a table, or at a graph? Which do you think gives the most information? Why? What makes you say that?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
