Polygon Pile Up

When it comes to angles involving parallel lines, triangles, and other polygons I’ve always assumed my grade 9 applied students “get this”. I’ve felt that angles were an easy topic. I guess I thought this because most students seem pretty happy when solving angle problems and for the most part being doing pretty well on assessments. However, this year I noticed two inadequacies that I am trying to address.

  1. Most of my students didn’t actually know what an angle measurement of 65 degrees really means.
  2. They have a hard time determining what information is needed when solving multi-step angle problems. Lack of a good strategy.

Addressing #1

When having students determine angles in triangles almost all of them knew that all three angles should add to 180 degrees. The trouble came when I saw some answers like this (from more than one student). 

What bothered me was the location of the 40. I wondered why outside the triangle? I pressed this student for more info. I asked him to draw me any right triangle and label the three angles.

 

Hmmm…I asked him to point to one of the angles. He pointed to where he labeled the 85. What I found is that this student was mixing up length measurements with rotational measurements and he was not alone.

I found a great activity to hit this head on. Laser Challenge from Desmos worked wonders to get my students to understand and experience rotational measurements. Students have to enter values to rotate the laser and mirror to hit targets.

My students “felt” what 60 degrees is. Experiencing that rotation made all the difference to clear up what we were actually measuring. When second semester rolled around and my new crop of kids came in we started with this activity right away.

Addressing #2

Most of our students struggle with solving complex problems where they have to think of a strategy. Before I gave them something like this,

I wanted to them to experience what information would be useful to know first. I decided to turn the problem around and inside out.

I gave them this.

I wanted them to think backwards….just like we need to do sometimes when solving longer problems. On the “easy” side most filled in 3 angles in the quadrilateral. What was great was that prepared them to think what we could leave out for the harder one. This simpler diagram challenged my class to think, plan, and strategize!

It was great to do this before we introduced this puzzle Jim Roesch, Kristyn Wilson, and myself created:

[There is a video embedded here — Can’t see it? Click through to the post page]

Here is the puzzle

Click to download a PDF copy to print.

And to really challenge yourself or your students here is a blank one. Can you fill it in so it’s “hard” to determine that indicated angle? What is the least amount of info you can give to bring out the most amount of thinking? Share them out! 

 

Really Big Lights – A math problem

Here’s a really big problem you can work on with your students this holiday season.

Act 1:
Show them this video and ask: What do you notice? What do you wonder?

After allowing them to voice their noticing and wonderings guide them to wonder: How big is that new light? How many times bigger is the big light compared to the old light? How many Really Big Lights would you have to put up to cover the same length as last year?

Act 2: Here are some images to help make some conclusions:

Guess: How long is the big light? How many times longer is the big light than the small light?

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Reveal:

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Guess: How many small lights are in one string that stretches 15 feet?

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Reveal:

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Work together to determine how many Really Big Lights would replace the string of 50 lights? What assumptions will you make?

Act 3: Reveal

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Why might your calculated answer be different from the answer shown?

If you had 50 Really Big lights how long would could they reach? How many cars could you put in that garage?

Grab all files for this activity

You can see more info about the lights over at http://reallybiglights.com/

Pentomino Puzzles

A few years ago I was introduced to a series of activities (through my then districts math consultant) that builds a driving need for students to createscreen-shot-2016-09-30-at-8-14-39-am, simplify, and solve linear equations. I used the activity for a few years in a row while I taught grade 9 academic. Since then I had forgotten all about it (funny how that goes) UNTIL NOW!

The activity ran as a series of challenge puzzles around Pentominoes and a giant hundred grid chart.

Activity 1: Explore

Ask students in groups to choose this tile and place it on the hundreds chart so that it covers a sum of 135. The task seems so simple to start but unpacks some great math.

Allow them to determine this sum anyway they like.

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I circulate and listen to their strategies. I give them very little feedback at this point. After a few minutes I choose some of those groups I heard interesting strategies to share..then let any other group share out their strategy.

img_2198Activity 2: Keep Exploring

I have them use the same tile and try again. Place the tile so that it covers a sum of 420. Listen to those strategies! Most groups that didn’t have a strategy before will try to adopt a strategy they heard last round. At this point most students will catch the strategy “If I divide the sum by 5, being like the average then I should have the middle number in the shape.”

This is where I stop and have a formal discussion as to why dividing by 5 here works? Will this always work? Will this always work with other shapes? What other shapes will this work with then?

We formalize the strategy.

Our big problem to start is not knowing where to place the tile. Let’s say I label the middle square n. What will the square immediately to the right of n always be? The left? The top? The bottom? Have them check this out by placing the tile repeatedly back on the grid.

Now let’s add all of those expressions up

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The middle square must be a multiple of 5!!! I have them try this strategy out by throwing out another sum and have them place the tile.

Look at another tile!

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We go back and outline that we could have chosen a different square to label n. Which results in a new equation and solves for different value…..but results in the same placement of the tile!!

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We continue by me having them select different tiles, giving them sums, having them create equations and solving them. I love how hands-on this lesson is. Holding the tiles adds some “realness” which I feel drives the need to solve these equations.

However,

this year when I remembered this activity I wasn’t sure I still had the tiles kicking around (I found them later). I immediately made a digital version with Explain Everything.

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The digital version gives each student their own copy and while working in groups can chat about what strategy worked and what didn’t. Before on the paper version….only one student could hold the tile. Also, when students have to voice their strategy through Explain Everything they have to have careful thought. They think about the words they want to use. We this careful thought they get to make their thinking visible for me!

One new addition to the activity I get to make here is that they can create their own pentomino…..and then their own puzzle to share with their classmates.

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Since then I also created the activity with some help from the team over at Desmos

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Click to access and rune the teacher.desmos.com activity

I love their new conversation tools….I get to pause the class and discuss when needed!

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Students can even sketch their new tile and create an expression to match! screen-shot-2016-09-30-at-9-24-03-am

 

Desmos even added some nice extension questions. Love it! screen-shot-2016-09-30-at-9-24-23-am

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In the future the next time I explore this lesson I see a blend of hands on tiles with digital support. I think having the best of both worlds here can pack a powerful 1-2-3-4-5 punch!

Pick your favourite!

Download the Explain Everything Pentomino Puzzles .xpl file. 

Access the Desmos Activity

 

 

Pumpkin Time-Bomb Activity

Last year around this time I shared out a Google Form for classes to record measurements around their pumpkins and make them explode! I shared that form on Twitter so that we could crowd source as many pumpkins as we could to make the sample size large enough. I was pretty shocked at how many schools from North America took on Pumpkin Time-bomb. By the time Halloween was over the spreadsheet had over 90 entries. That’s over 90 pumpkins exploded in the name of math and data collection.

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This coming week let’s add to the data and use the it in our classroom to discuss: Scatterplots, Trends, Correlation strong, weak, no-correlation, lines of best fit, correlation coefficient, etc.

Here’s a sample lesson you could use on the day you make your pumpkin explode.

Generate Curiosity

Play this video which shows Jimmy placing rubber bands around his pumpkin.

How many rubber bands will make the pumpkin explode?
Have students write down a guess that is too low. Too high. Then estimate their best guess.

Show the Act 3 Video

Now Bring out your pumpkin for the class to see! Have them predict how many rubber bands it will take before it will explode. Repeat the estimation process. Have them save their guess till the end of class.

Making A Model

Throw out the question: “What measurements of the pumpkin changes how many rubber bands are used?” Let your students brainstorm a list of variables. Have a discussion on variables & relationships. Write all the variables on the board they come up with. Narrow down the list to items that are measurable with the pumpkin we have in the class. What affects the explosion the most? Height, diameter — circumference, thickness of the wall?

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Have them choose a variable that they feel should have a relationship with the number of rubber bands. Fill out the prediction part of the handout.

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Click here to grab a copy of the prediction handout

As a class measure all variables needed. Write them on the board for all to see.

Analyzing Data

Give students the link to the spreadsheet of all the pumpkins to date (You should copy and paste the data to your own sheet so you can filter/sort the results and share that sheet out to your students.)
Discuss with your students the lack of consistency in the selection of rubber bands from all over the country. How can we minimize this variable skewing our results? Filter the data with your students(or before hand) showing one type of rubber band (Most common is a rubber band of length 8.65 cm). This will only show all the pumpkins that have been destroyed using that type of band.
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Get your students to grab the data that relates to their relationship.

For example:
If Kristen chose the relationship Circumference vs. Rubber bands she should copy and paste the circumference column and the rubber bands column into a new sheet side by side. Then copy and paste all that data into the pre-made Desmos File.
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She can adjust the scale in Desmos as needed. Have her move the movable point and drop it where she thinks your class’ pumpkin will lie. Or you can have her find the line of best fit to help predict how many rubber bands it will take. Either way we want her to predict with more accuracy.

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So Kristen would predict that if her circumference was 90.5 cm then it will take 272 rubber bands to blow up the pumpkin!

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Now if Kristen chose a variable that it was clear there is no relationship then you get to have a discussion about correlation vs. no correlation. Have her choose new variables to predict on.

Once everyone in the class has a new prediction start wrapping bands around that pumpkin (You may want to start this as early as possible).

Watch your pumpkin explode and give congratulations to the student who predicted closest to the actual number of rubber bands.

Don’t forget to enter all your data to the sheet by filling out this form (you can also use the form to show the videos to the class).

[Updated] – You can use this Desmos Activity Builder Activity to facilitate the lessson. It includes only data for Diameter and Circumference.

[Updated] – You can grab a copy of the spreadsheet to save in your Google Drive. From here you can modify. 

From Oct 30. 2015

A few pumpkins from 2014 & 2015