Recognizing and Describing Patterns in Math

In this post, I will present two patterns for you to look at with the hopes that you will try to dissect them and be able to answer a few questions.  Are you ready?  Here is the first one that I call Consecutive Number Series.

What counting pattern do you see in this sequence?  How would you describe the sequence of numbers that are being added?  What pattern do you see in the answers?  Can you figure out the pattern for 8 × 8 and 9 × 9?  Notice this pattern made a triangle.  Do you know what kind it is?

My next pattern I call The Eights, and you will readily see why. It, too, forms a triangle, but a different kind. Do you recognize this triangle as isosceles?  If you take a ruler, you will find that the base is 4" while the sides are both 3".

What do you think will happen if we take this same pattern and add a 0?  Notice that this pattern does not begin with adding an eight.  Can you figure out why?

I use this type of patterns with my remedial math college students because I consider it important to do some problem solving while recognizing and describing patterns.  After all, problem solving is a part of life.  It doesn't occur in a vacuum. Because students must reason about some specific content, I think patterns are a great place to begin.  Problem solving also helps students to make connections to other parts of mathematics and find some relevance to what they are learning.  And did you know, that problem solvers are typically better test takers?  So take these patterns, and create some of your own questions for your students.  Use them in a journal or as a small group activity.  But whatever you do, have fun learning and discovering patterns in math.

Investigating Math Number Patterns

Mathematics is the science of patterns.  In this post, I want to target some mathematical problems in which we investigate developing patterns.

In the first example below, you will notice we begin by multiplying one by one; then 11 by 11, and so forth. Each time we multiply, the number of digits in the multiplier and the multiplicand increases. Do you see the pattern that progresses in the answer (product)? Notice how this multiplication pattern forms a triangle? Can you figure out what kind of triangle this would be if we added a "peak" or a row at the top?

Here is another interesting pattern. In this one, instead of multiplying by 1, then 11, then 111, the answer (product) looks like the multiplier in the pattern above. Do you notice anything else significant?

Yes, we are multiplying by 9 each time. Now look at the number being added, and count the number of ones you see in each answer. Surprised? Isn’t it amazing how math is ordered, methodical and precise? Maybe that is one reason I love to teach it! Encourage your students to look for and make sense of math patterns and structure in order to deepen their mathematical understanding and retain what they learn.

Summer Fun Using Shaving Cream

Again, I am going to deviate from the subject of math and offer a fun summer activity I do with my grandchildren. It involves a can of shaving cream, cleaning rags and lawn furniture that has set out all winter.

Were you aware that there are many unusual ways to use shaving cream besides using it for shaving? Did you know that you could...

1) Clean jewelry with it? Spray it on your jewelry and use a soft “old” toothbrush to get off the grime. Rinse with water.

2) Give chrome faucets a brilliant shine? Apply the shaving cream to a sponge and rub it on the faucet. Then wipe it off with a damp cloth.

3) Easily remove paint from your hands? Rub the shaving cream onto your hands; then rinse it off with soap and water.

4) Remove carpet stains? Blot the soiled area with a damp sponge and then spray on the shaving cream. Wipe clean with a damp sponge and let the area dry. It will also work on various clothes stains.

5) Clean vinyl lawn furniture? Spray the lawn furniture with the shaving cream and wipe the grubby areas with a damp rag. Rinse when finished.

Item #5 is what I do each summer. Our lawn furniture sets out over the winter on our patio and even though it is covered, it is filthy when summer comes. I always go to the store and purchase the cheapest shaving cream I can find. (Here, Barbasol sells for about $.89 a can. Depending on the number of grandchildren coming over, determines how many cans I purchase. This year, it was three.)  No matter their age, this is one activity that they all look forward to because it is messy!

I write the child’s name on their can of shaving cream and then assign them a piece of furniture to clean. When everyone is done scrubbing and wiping, we get out the garden hose to spray off the remaining shaving cream, and frequently we end up spraying each other.

But what happens to the leftover shaving cream? I think the pictures say it all!

A Short Tutorial on What is Growth Mindset and Fixed Mindset

At my community college, I teach a course entitled Conquering College because many of the students entering college are not prepared. They lack study skills as well as the soft skills of being on time, regularly doing homework, turning in assignments - on time, etc. This class is now required for every student who tests into developmental math and/or reading.  It has three purposes:

1. To enable students to learn and use Advancement Via Individual Determination (AVID) strategies necessary for persistence and success at the college level, 
2. To develop a learning plan based upon personal abilities and goals, and 
3. To become more self-reliant in fulfilling academic goals.

During the sixteen weeks, we spend time focusing on the the growth and fixed mindset. If you haven't heard of the fixed mindset and the growth mindset, I'll summarize it this way - it is our attitudes, thoughts and beliefs about something.

Study the two charts below. In which category do you fit? Where would you place many of your students?

Carol Dweck has done a great deal of research on Growth Mindset. There is a ten minute video on You Tube which is well worth watching and sharing with your fellow teachers. It is called How To Help Every Child Fulfill Their Potential. (click on title) If for no other reason, watch it to discover what 15 years of praising children for their intelligence has done to our students. It is definitely eye opening!