Exploring Patterns ... extending thinking

 

Patterns are EVERYWHERE! Some suggest that the study of mathematics is the study of patterns ... identifying them, categorizing them, generalizing them.

Five universal generalizations are true about patterns:

• patterns have segments that are repeated
• patterns allow for prediction
• patterns have an internal order
• patterns may have symmetry
• patterns are everywhere

In fact, patterns make a great curriculum organizer for interdisciplinary work since patterns are evident in language, science, history, music, art, and more!

Here are a few ideas for exploring patterns in math in introductory ways ... 

1). Read children's books!  I love to connect with students using a read aloud ... talking through a picture book before getting started with math.  AND yes!  It works well even in middle and high school!

Some of these are more sophisticated than others ... and the last one (bottom right corner) is a coloring book.

2)  Explore polygonal or figurate numbers.  I have a task I've used with students ... you may find it helpful.  "Explore Numbers in Shapes."

3). Check out the activities, explorations in YouCubed ... there are more than a dozen pattern activities ready to implement in your classroom.

4). Visual Patterns is an excellent site ... set up to use regularly as a key activity in your math program.

5). Mathigon has a great series of lessons, interactive, free ... ready to use to enrich students' understanding of patterns.

6) NRich is another excellent source for problem solving, stretching students' thinking and exploring topics.  They have numerous pattern activities at all levels ... perfect for differentiating!

I am sure there are other great resources!  What sources to you recommend for exploring patterns with students?

Growing Shapes - Polygonal Numbers #MTBoS30 - 16

We've been talking about patterns in class.  Today we explored "growing shapes."  We started with this one from Day 5, Week of Inspirational Math, YouCubed:

I asked students what they noticed ... and how they saw the shape growing. I gave them a few minutes to share with their partners.

Students shared numerous ideas ... they talked about columns, rows, colors, triangular shape, symmetry, and more.

Then I asked students to name the pattern. And that's where they had quite a bit of fun! Here are some of the suggestions ...

Space Invaders
Pyramid Effect
Hourglass
Growing Tree
Ant Pile
Power Pyramids
Cloudy with a Chance of Falling Blocks
Boomerang
Propagator
Quadratic Cascading
Stairs Squared
Roof Building
Tetris Mountain

Partners then explored typical figurate numbers, triangular, square, rectangular, and pentagonal numbers.  They discussed the patterns they saw and connections between the patterns.

Students are looking for the answers to these questions, "What 3-digit number is both triangular and pentagonal? What 4-digit number is both square and pentagonal? Can you find a number that is triangular, square, AND pentagonal?"  I overhead a couple of students talking about writing a program to find their answers.  I'm looking forward to our next class.

Pattern Explorations #MTBoS30 - 13

We started our last unit today.  We have completed all of our required standards.

The district curriculum guide suggests that we delve into sequences and series.  Since there are no required standards, I have the opportunity to structure more exploratory activities in the unit. Students will study the typical sequences and series concepts/skills in their next math course (precalculus).

I borrowed from Jo Boaler's YouCubed Week of Inspirational Math, Henri Picciotto's work, and an NCTM Student Explorations (What Shapes Do You See, Jan 2012) to blend together five exploratory lessons.

1. Number Patterns
2. Growing Shapes
3. Patterns in a Triangle
4. Staircase Sums
5. Averages and Sums

As we process students' explorations each day, we will address these concepts:

• The difference between an arithmetic and geometric sequence
• The difference between a sequence and a series
• The difference between convergent and divergent series

We will leave the formulas and details for next year's unit.

Today was day 1 ... and it was a lot of fun!  I wish I had audio recording or pictures but I don't.

First students completed a short exam review learning check with partners.  I loved hearing the buzz in the room as partners convinced one another how to find the solutions asked.  (We are reviewing a small set of questions in each class for the next 2 weeks).

Then I gave a brief introduction to our new unit.  I gave them copies of the visual numbers from Boaler's lessons.  Students got excited when they realized prime numbers were represented by a circle in the visual number display.  Other students were stymied at first - one said, "I just see a bunch of little circles."  I overhead students arguing over how to represent the number 36.

They did a great job with the consecutive number sums.  They caught on quickly when they were allowed to add negative numbers and zero to the patterns.

Only the hundreds' chart patterns slowed them down a bit but it was hilarious to watch light bulbs ignite as they realized the patterns.  Then they argued how to write those patterns algebraically.

I am pumped to see what happens in the future lessons!  Love that we get to wrap up the year with explorations!

#MakeOverMonday Week 2 Checkerboard Borders

Dan Meyer published his second #MakeOverMonday textbook revision challenge tonight - Checkerboard Borders.  Check it out here!

The problem states:  In preparation for back to school, the school administration has planned to replace the tile in the cafeteria. They would like to have a checkerboard pattern of tiles two rows wide as a surround for the tables and serving carts.

I wonder ... when a contractor gets a tiling job, does he use math to determine the number of tiles needed?  What planning does he do before laying out a pattern?

I would give students just the context in italics above ... with some thinking time ... and then proceed:

1.  What questions come to mind?

2.  In teams, go to our 9th grade cafeteria and collect information that will be helpful to you.

3.  In teams, use the graph paper and colored pencils provided to create a diagram that fits the problem description.

4.  Determine how many colored tiles will be needed to create the pattern your team drew.

5.  Suppose the school administration changes its mind and now wants to tile the cafeteria in the 1100 building instead of the 9th grade cafeteria.  Create a generalization or rule that will work on any size cafeteria.  Does it matter if the cafeteria floor is square or rectangular?  Explain your thinking.

6.  The cafeteria manager looks at your diagram and comments that his serving equipment will need more than 2 rows of checkerboard pattern.  He requests 4 rows.  How does that change the rule you created for the contractor?

7.  Create a mini poster with your floor design, calculations, and generalizations.

I made two changes.  First, I asked students what they need to know from the problem situation to encourage their own thinking, curiosity, and problem solving ideas.  Second, I took away the square diagram, choosing instead to use real data.

How would you modify the problem?