June Problem to Discuss

Submitted by Brooke Istas on Wed, 06/07/2017 - 08:19
  • 3130 Views
  • 15 Comments

Look at this picture, there is one mountain and then there are two mountains.

Post your prediction and then do the work and post again your response.

  • How many sticks are needed to build a mountain range with 5 peaks, 8 peaks, and 10 peaks?  Remember post your prediction first, then do the work and post again  what your solution is.  Be sure to explain how you arrived at your answer.
  • I will have a follow up question or questions as the weeks progress.

Let's Discuss!

Comments

Hi all,

What a great visual pattern! I’m looking forward to trying this with students using toothpicks or popsicle sticks.

A few us have been collecting visual patterns for teaching functions and algebra. You can access a collection of worksheets at CollectEdNY.org, including the popsicle stick mountain pattern which I added this morning. If you like visual patterns, you should also check out the amazing visualpatterns.org, which has more than 200 patterns teachers have submitted.

I have my predictions. Should I start sharing them?

Best,

Eric

Eric, 

Thanks for sharing the link to the CollectEdNY.org and for the visual pattern's website!  And yes, start sharing!  If we get finished with this problem perhaps we can work on another.

Thanks!

Brooke

Brooke,

I didn't see this discussion until after I already had students work on this pattern, albeit the wording of the problem was presented a little differently. I used the template at CollectEdNY which presents the mountain ranges as Figure #1, Figure #2, and so on. There were two different explicit rules that students came up with:

4x + 2 = y

4(x-1) + 6 = y

An interesting question that students can ponder when looking at the pattern this way is: What would Figure #0 look like?

-Patricia

Dear Heroes,

My guesses:

  • 5 Peaks - 22 sticks
  • 8 Peaks - 34 sticks
  • 10 Peaks - 42 sticks

I'll do some work and see what I come up with.

yours in productive struggle,

Mark

It is a good visual. It goes along with linear word problems like the taxi that cost x dollars per mile plus an initial cost, or the health club with a joining fee and a monthly. Having something more visual is useful.

(peaks - 1) x 4 sticks + 6 initial sticks

The guesses above are good

I stand by my guesses above, though I saw the pattern in a different way in terms of what is changing and what is staying the same.

I saw that each peak is made up of 4 sticks, plus two more sticks for the left corner at the end. So my function was 4(# of peaks) + 2.

Another way I saw it was to imagine the full 6 stick triangle for each peak minus all of the pairs of sticks for the corners we need to leave out. Something like:  6(# of peaks) - 2(# of peaks - 1).

 

 

Mark,

How could you adapt this problem to make it more challenging?  I have a group of some high-level learners and need to find challenging tasks for them to complete ideas? thoughts? perspectives?

Thanks for participating,

Brooke

Hey Brooke,

I've done a bunch of work having students go from visual patterns to tables and descriptions and rules and graphs, but something I've always wanted to try is to give students a function, table, or graph and then ask them to create the first few terms of a visual pattern that reflects it.

Or also for students to design their own visual patterns and then make predictions about their own sequence.

Has anyone else done either of those types of activities with their students?

Another thing I've done is to show students a visual pattern and ask them what they notice and what math questions they can ask. This way they generate a bunch of different questions with a pretty wide range of difficulty. Then I usually let them choose the few they want to work on.

yours in productive struggle, 

Mark

Mark,

I love the idea of asking learners to design their own visual patterns!  I have asked learners to look for patterns within numbers or situations but hadn't thought about asking to design a visual pattern.  What a great idea!  Has anyone else tried this in your class?  What a great way to create engagement!

Thanks for the idea!

Brooke

To add more challenge I often propose a few what if scenarios to see how students extend their experience to other factors. In this example, I may ask...

  1. What if the mountains were shaped more like hexagons (or pick any other polygon). How would they over lap and would there be an expression that matched that pattern? 
  2. How many different shapes could you find patterns for other than triangles?
  3. How would the formula change if each side of our triangle mountains were 3 toothpicks long? Could you then predict what 50 length sides would produce for a pattern?
  4. Are there other patterns in life that follow the 4n+2 sequence? 
  5. Design your own pattern with the same manipulatives, and challenge your peers to find your pattern. 

Just a few extensions that come to mind. Interacting with the specific learner often triggers other variations or themes. 

Other ways people offer more challenge? 

 

Hi all,

I love these pattern problems. I agree with all that has been shared thus far. There are several ways to approach this problem but I like what how John and Mark approached it as I too saw the first mountain as 6 and the next after that having 2 less toothpicks. Thanks Eric to links for more patterns!

1st mountain peak  = total toothpicks : 6 toothpicks or 6+ 4(0)

2nd mountain = total toothpicks: 6 + 4(1)

3rd mountain = total toothpicks: 6 + 4 + 4 or 6 + 4(2)

4th mountain = total toothpicks: 6 + 4 + 4 + 4 or 6 + 4(3)

Etc

So I see the relationship for each mountain is 6 + 4(n-1)

Simplifying 6 + 4n – 4 or 4n + 2 where n is how many peaks there are.

So Mark your answers fit my expression. That is for 5 peaks it takes 22 toothpicks, for 8 peaks 34 toothpicks, and for 10 peaks 42 toothpicks.

 

 

 

 

Interesting question. I would use this visual with one small change. I would add one more peak to show the pattern more clearly. My students often think I am trying to trick them and they can may the third peak by adding it between the 2 existing peaks and the pattern would change. Starting with a clearer pattern would be useful.

There was another comment about changing the shape. That is also good, as it can be simply starting with a square and adding 3 sticks to add a second square and so on.