GROWING PATTERNS

(c) 2021 George Gadanidis

“The ideas in this workshop were used over 10 years ago as a low floor, high ceiling way of teaching the content of growing patterns in a rich math context in grades 1-2. I have used these same ideas to teach about linear relations in grade 9.”

George Gadanidis

MINDSET

Grades 1-2 children singing parent comments on the math they shared at home.


ABOUT THE GROWING PATTERNS WORKSHOP

  • For students (enrichment; Covid catch-up), teachers (professional learning; online teaching resource) & parents (math therapy)
  • Self-serve workshops: pick and choose activities to complete
  • Or, for digital certificate: complete and submit tasks (get PDF) to receive a digital certificate
MENUTOPICS COVERED
A. Growing patterns
B. Plotting growing patterns
C. Creating growing patterns
D. Algebraic representations
E. Mathematician interview
F. Modelling with code
— growing patterns
— steepness / slope / linear relations
— how mathematicians think
— history of mathematics
— coding extensions

A. GROWING PATTERNS
Helpful materials:
  • linking blocks of different colours
  • crayons / markers / pencils
  • PDF handout for sections A.1 – A.4
  • PDF handout for sections A.5 – A.7

A.1. GROWING PATTERN #1

What is missing in the table below?

A.2. GROWING PATTERN #2

What is missing in the table below?

A.3. GROWING PATTERN #3

What is missing in the table below?

A.4. COMPARING GROWING PATTERNS

Describe how the 3 growing patterns are similar or different.


A.5. GROWING PATTERN #4

What is missing in the table below?

A.6. GROWING PATTERN #5

What is missing in the table below?

A.7. GROWING PATTERN #6

What is missing in the table below?

A.8. GROWING PATTERN #7

What is missing in the table below? Draw or build a growing pattern to match the pattern in the table.

A.9. GROWING PATTERN #8

Create a new growing pattern in the table below. Draw or build a matching pattern.

A.10. PATTERNS IN THE GROWING PATTERNS

Study all of the growing patterns above. Discuss with a partner:

  • How do the blue numbers/blocks change in each of the patterns?
    • What is similar?
    • What is different?
    • What is interesting?
  • How do the red numbers/blocks change in each of the patterns?
    • What is similar?
    • What is different?
    • What is interesting?
  • How do the gray numbers, after the equal sign (=), change in each of the patterns?
    • What is similar?
    • What is different?
    • What is interesting?

A.11. WHAT DID YOU LEARN?

What did you learn about the role/meaning of each of the following in the growing patterns?

  • blue numbers / blocks
  • red numbers / blocks
  • gray numbers / blocks

What is interesting or surprising about what you learned?

Work with a partner to script how you might share what you learned in a way that others might also find it interesting or surprising.


B. PLOTTING PATTERNS
Helpful resources:

B.1. BAR GRAPHS OF GROWING PATTERNS

Graph each of the 3 patterns below on inch grid chart paper using colour dabbers, or use colour crayons and the  Square Grid Handout (PDF).

a)

b)

c)

B.2. WHICH ONE IS STEEPER?

Graph each of the 3 patterns below on inch grid chart paper using colour dabbers, or use colour crayons and the  Square Grid Handout (PDF).

a) Imagine the bar graphs as staircases, as shown below. Which staircase would be most difficult to climb? Why?

b) Why are some graphs steeper than others? Is it because of the blue blocks, the red blocks, or because of both?

c) HINT: Imagine graphing only the blue blocks, as shown below. Does the steepness change if the red blocks are removed?

d) Do you think the following statement is true? If not, can you draw two bar graphs to prove that this statement is false?

“The tallest bar graph is always the steepest.”

B.3. TABLES OF VALUES & BAR GRAPHS

Complete each table of values. Build the first 3 stages of the each pattern using connecting cubes. Then, create a bar graph to represent each table of values.

a)

b)

c)

B.4. practice

1. Represent each pattern as a bar graph.

a)

b)

c)

2. Complete the table. Draw a bar graph to represent the pattern.

3. Create a new growing pattern in the table below. Draw a matching bar graph.

B.5. WHAT DID YOU LEARN?

What did you learn about the graphs of growing patterns?

  • about steepness
  • about the roles of red and blue blocks

What is interesting or surprising about what you learned?


C. CREATING GROWING PATTERNS
Helpful materials:

C.1. 2 COLOUR PATTERNS

So far, we have created growing patterns using 2 colours, such as the ones below. In these patterns, we had one colour where the blocks grew in number, and another colour where the blocks stayed the same. For example, in the second pattern below, the blue blocks grow 2, 4, 6, 8 and so on, and the red blocks stay at 2.

We have also noticed that when we plot these patterns as bar graphs, some patterns create steeper bar graphs than others.

a.  Create a new growing pattern using 2 colours, where one colour grows and the other is constant.

  1. Build the pattern using connecting cubes.
  2. Complete the table below for your pattern.
  3. Draw a bar graph of your pattern.
  4. Use words to describe your pattern.

b. Create a different growing pattern using 2 colours, where one colour grows and the other is constant, so that its bar graph will be steeper.

  1. Build the pattern using connecting cubes.
  2. Complete the table below for your pattern.
  3. Draw a bar graph of your pattern.
  4. Use words to describe your pattern.

c. Create a different growing pattern using 2 colours, where one colour grows and the other is constant, so that its bar graph will be less steep.

  1. Build the pattern using connecting cubes.
  2. Complete the table below for your pattern.
  3. Draw a bar graph of your pattern.
  4. Use words to describe your pattern.

d. Create a shrinking pattern using 2 colours, where one colour shrinks and the other is constant.

  1. Complete the table below for your pattern.
  2. Draw a bar graph of your pattern.
  3. Use words to describe your pattern.

C.2. 3 COLOUR PATTERNS

Let’s use 3 colours to build growing patterns.

a. Here’s a growing pattern with 3 colours, where only 1 colour is growing.

  • complete the table below for the pattern
  • draw a bar graph of the pattern
  • use words to describe the pattern

C.3. WHAT DID YOU LEARN?

What did you learn about building growing patterns?

  • about what changes
  • about what stays the same

What is interesting or surprising about what you learned?


D. ALGEBRAIC REPRESENTATIONS

D.1. MATCHING ALGEBRAIC EXPRESSIONS

Which pattern matches which algebraic expression?

NOTE:

  • x is a variable that represents the stage number
  • y is a variable that represents the number of blocks at each stage.
1. This image has an empty alt attribute; its file name is train-base-2x1-1.pngy = 3x + 3
2. This image has an empty alt attribute; its file name is train-base-2x2-1.jpegy = x + 1
3. This image has an empty alt attribute; its file name is train-base-3x3-1.jpegy = 2x + 2

D.2. BUILDING PATTERNS FROM ALGEBRAIC EXPRESSIONS

Draw a growing pattern of red and blue blocks for each algebraic expression.

NOTE: Use x for the stage number and y for the number of blocks at each stage.

1. This image has an empty alt attribute; its file name is train-base.pngy = x + 3
2. This image has an empty alt attribute; its file name is train-base.pngy = 3x + 1
3. This image has an empty alt attribute; its file name is train-base.pngy = 5x + 2

D.3. WHAT DID YOU LEARN?

What did you learn about the algebraic representations of growing patterns?

  • about variables?
  • about constants?

What is interesting or surprising about what you learned?


E. MATHEMATICIAN INTERVIEW

The series of videos below are of applied mathematician Dr. Lindi Wahl (Western University) completing, discussing and extending the the above activities.

E.1. GROWING PATTERN # 1

‘I study patterns in biology. All mathematicians study patterns. And this is a beautiful one.

E.2. GROWING PATTERN # 2

Lindi builds and graphs another growing pattern.

E.3. CONSTANTS & VARIABLES IN MY WORK

‘I think one of the more interesting things we can see here is that there are parts of the pattern that change and parts that don’t change.’

‘Here we have it separated really beautifully, because we can see the red dots don’t change and the blue dots do change.’

‘And that’s actually an important part of what I do in my work.’

E.4. SLOPE

‘If I put the pointer through all the top red dots I’ve got a perfect straight line there, and it’s very steep.’

E.5. HOW BACTERIA GROW

‘In my research I study bacteria and viruses and bacteria and viruses don’t generally grow in patterns like these. They grow by dividing in two. … We have a curvy shape, so it’s a completely different pattern.’

E.6. USING SYMBOLS

‘We’ve represented these patterns in our little blocks, and we’ve also done it on the page (using bar graphs), and the way I would do it as a mathematician is to use symbols.’

E.7. BEAUTIFUL EXPRESSIONS

‘One thing that is beautiful and exciting for me as a mathematician is this formula right here for example: 2 to the exponent x.’

‘And the neat thing about it is that we’ve done this representation with the blocks and with the dots, but we can sum it all up, we can encapsulate it, in just these two little symbols.’

‘These two symbols here capture everything about this entire pattern.’

E.8. “SEEING AS”

Jan Zwicky

The Canadian musician, philosopher and poet Jan Zwicky says in her book Wisdom & Metaphor (2003) that “geometrical representations” such as those shown below attract our attention and say: “look at things like this” (p. 38).

They also lead to “understanding” through “seeing as” (p. 3).

What do you “see as” in the images below? What do you understand?

E.8. WHAT DID YOU LEARN?

What did you learn from the interview with Dr. Lindi Wahl?

  • about her work?
  • about constants and variables?
  • about different representations of math patterns?
  • about different growing patterns?
  • about how a mathematician thinks?

What is interesting or surprising about what you learned?


F. WITH CODE

Computer code can dynamically model math concepts and relationships.

F.1. TALK MATH TO YOUR COMPUTER

F.2. A CODING SIMULATION

Please see the Growing Patterns coding simulation at https://researchideas.ca/mathncode/sims-growpatt.html.

  • Mouse over the values and change them using the arrows that appear?
  • How is this similar to what you did in the previous sections above? How is it different?
Different representations of “instructions”

Notice that the code can also be seen as a flowchart and as a list on instructions using text (also called pseudocode).

  • Which view do you prefer?
  • Why?
Different representations of the growing patterns?

Also notice that the growing pattern can be seen as a bar graph, a table of values and as numbers on a hundred grid.

  • Which representation do you find most appealing?
  • Are the instructions above also representations of the growing patterns?

F.3. WHAT DID YOU LEARN?

What did you learn about coding and growing patterns?

What is interesting or surprising about what you learned?