Ropes of Different Thickness and Equal Length - Knot

When we tie a knot in a rope we use up a bit of that rope. I took two ropes of different thicknesses and asked my students to guess too low, too high and best guess for how much rope would get used up if we tied a knot.

Groups then collected data to get how many centimeters per knot.

 As a class we took each group rates and averaged them to get a class rate per knot. We decided on the question shown on this whiteboard.

 

And some student work to solve this problem.

 

 

And some of my work to consolidate with the students how to use the equations to solve this problem.

 

 

 

 

 So we got 30 knots and a length of 4 meters when both ropes would have the same length.

This is what it looked like when we tied the 30 knots in each of the ropes.

 

And here is a close up of the lengths.

 

 

 Great discussion at this point about our model and how averaging the entire classes rates made it better. Students loved this.

Of course I could knot resist and we decided to look at figure 8 knots.

It took a little while for groups to figure out figure 8 knots. Data collection from various groups.

The class average.

  And some student work.

 I was knot prepared to do 60 knots.

 

 Knotted up for equal lengths!!!!! 

Card Tossing

PEDAGOGY
There are many pedagogical moves in this lesson that I tried for the first time. Bare with me.

First after hearing Peter Liljedahl and Michael Pruner talk at the Canadian Mathematics Educators Forum I tried this lesson with visible random groupings (VRG) of size three and the students did their work on vertical non-permanent surfaces (VNPS). These are two ideas that Peter and Michael presented at CMEF 14 that I thought made sense and could transform my teaching.

First visible random groupings- the idea is that every day students see you make the groups randomly. Hopefully this will break down social barriers. Hopefully this will enable students to collaborate. Hopefully this will put students on the edge.

Vertical non-permanent surfaces. Non-Permanent allows students to make mistakes and take risks when working on a problem or activity. Vertical makes the work visible for all to see. Knowledge can easily be shared around the room. One pen / marker / chalk - person writing cannot write their own thoughts. Hopefully this will help with collaboration within the classroom.

More about VRG and VNPS at another post. (I think these two ideas are awesome!!!)

THE IDEA
Shout out to my buds in the Limestone District School Board who triggered the idea for this activity. I would not of thought of it without them. Great idea. Hopefully they like what I did with it.

DAY 1 DATA COLLECTION



On the first day students folded up their box and grabbed  some cards. We tossed cards for 30 second intervals and counted how many we made from 4 feet (this was an easy distance to choose as the tiles on the floor were one foot by one foot-next time I think I will make it 5 feet- 4 feet might of been a little too close). Students recorded their data and near the end of the period calculated their rate in cards per second.



Here is my chart:

My rate ended up being 2.05 cards per second. Any takers? The dimensions of the boxes were

 L 16 5/8 inches by W 12 5/8 inches by H 12 5/8 inches and were purchased at U-Haul. The students rates varied anywhere from 0.36 cards per second to 1.81 cards per second. Students loved the data collection part of this activity.

After school I calculated a reasonable card advantage I would give each student depending on their rate so that a game against me would take anywhere from 30-45 seconds.

Here are the advantages:

0.3 to 0.4 cards per second gets a 60 card advantage

0.4 to 0.5 cards per second gets a 50 card advantage

0.5 to 0.75 cards per second gets a 45 card advantage

0.75 to 1 cards per second gets a 40 card advantage

1 to 1.25 cards per second gets a 35 card advantage

1.25 to 1.40 cards per second gets a 30 card advantage

1.40 to 1.50 cards per second gets a 25 card advantage

1.50 to 1.65 cards per second gets a 20 card advantage

1.65 to 1.80 cards per second gets a 15 card advantage

And a shout out to Ahmed who had a 1.81 cards per second average. I gave him a 10 card advantage. His rate was the best rate of the 27 students who collected data. (two classes) 

DAY 2

What is the longest game you could play against Mr. O and still win the game given a certain number of cards advantage? (as stated above depending on your rate)

Students came into class and experienced visible random groupings for the first time. We then huddled up and I set up two boxes at four feet and asked for a volunteer.

Me: "Ok my rate of making cards is 2.05 cards per second and Nabil's rate is 0.84 cards per second. If we have a card tossing competition (Oh Oh could I possibly be World Card Tossing Champion!!! WCTC?) how long does the game need to be for me to win?"

Class: ........ "You would win right away- wouldn't you?"

Me: "Right because I am better. Watch - let's play a 10 second game."

We do this and I clearly win.

Me: "Ok so how could we give Nabil an advantage?

Nabil: "Why don't you move back. Throw from further."

Me: "Ok that would be an advantage but....I didn't collect my rate from that distance."

Me: "Any other suggestions for an advantage?"

Class: ........

I walk over to his empty box. Count 30 cards in front of the students and dump them in.

Class: "Oh ya. That would be a huge advantage."

Nabil and I stand back getting ready to toss.

Me: "So now there is a length of game where I would catch up to Nabil but not quite beat him. In other words - there is time for the game for Nabil to win but Mr. O would be super close. The fans would be going wild!"

The class totally gets it.

At this point the groups are sent to their vertical non-permanent services. They are instructed to use the person's rate in their group who is the best card tosser. I come around and tell each group their card advantage. I ask them to figure out the length of time for the competition so that it will be super exciting.

Here are some pictures of some group's work:

Group A

Group B

Group C

Group D



Group's worked on this problem for the entire period. Engagement was awesome. Using vertical non-permanent surfaces allowed me to quickly assess where groups were at and help them move forward. Most groups attacked the task with trial and error at first. I would then prompt them to use a graph, equations and the graphing calculator, or an algebraic approach. Knowledge passed around the room from group to group - a huge advantage to using vertical non-permanent surfaces.

DAY 3 COMPETITION DAY

Groups were asked to go back to their vertical boards from yesterday and put up an algebraic solution to the problem. Here is an example:



Once all groups had a length of time for the game to be close and the number of cards we could expect in the box, we found a time keeper and played our matches.

Here is a head start:

And here is a video of me versus one of the students.



The results were awesome. We played games to the length of time the students predicted with their models and then compared the number of actual cards in the boxes compared to the expected number of cards.

My first period class had these results:

And my afternoon class had these results:

THIS ACTIVITY WAS AWESOME!

If you try it out I would love to hear about it. Comments?











When does y1=m1x+b1 meet y2=m2x+b2? Plastic (Beer) cups versus Styrofoam (Coffee) cups

Disclaimer
This one has been around for a while and I don't know where I saw it first. But thanks. And thanks to the #MTBoS


Set Up

Students worked with a partner on this one. They were given a measuring tape and 10 beer cups and 10 styrofoam cups. Here is what I wrote on the board.

What went down?

For the Models

Some groups picked up one styrofoam cup measured the lip, used that for m (slope) and the distance to the lip as b (y intercept). For the plastic cups they placed one inside the other measured the increase in height and used that for m and subtracted that m from the height of the cup to get b. Not a bad strategy but a small error in measuring creates an inaccurate answer.

Other groups made measurements for 1, 2, 3, ...., 10 cups and made a table of values for each stack ( one cup = 12 cm, 2 cups = 12.5 cm etc....). They then put the data into a graphing calculator and did a regression to generate y=mx+b. Good strategy. Some of these groups included the data point 0 cups is 0 cm and of course created inaccurate models.

Other groups had no idea where to start. They couldn't imagine that they could answer the question with only 10 cups of each type. They looked at other groups to see what they were doing. They even went over and asked them - because I was not telling them what to do.

Other groups measured the height of the 10 cups for each stack and then used this to predict. When I visited these groups I just took two stacks of 10 of the same cup and stacked them in front of them. As they watched the stacks slide inside of each other they had their aha moment.

I circulated and tried to get all groups moving in the right direction.

For the Solution

Once groups had the models they solved the question in a variety of ways. 

Some used the graphing calculator with all the usual problems. How do I find where they meet? How come I can't see both lines? Some of these groups, once they had the point of intersection, could not articulate what it meant.

Other groups used the equations and solved for the point of intersection algebraically. Some of these groups needed help with this.


Others just found the number of cups and not the height.

Other groups decided to just extend their tables until the heights got close to the same. I explained to these groups the inefficiency of this strategy.

Results

Here are the results:

The actual solution is on the bottom line.

Here is a photo of it:

Solving Systems with Manipulatives

In the Grade 10 applied Math course in Ontario students are required to solve systems of two equations in two unknowns. The overall expectation in the curriculum guide reads " By the end of the course students will solve systems of two linear equations, and solve related problems that arise from realistic situations."

First and foremost let me address 'by the end of the course":

1) OK so by the end of the course- not at the end of the unit on solving systems - with that in mind I will emphasize that I cycle the curriculum by doing activities as I have written about here.

2) By slowly building the idea of solving a system throughout the course students gain control of their learning: solving systems by trial and error; using manipulatives to solve systems like Ax+By=C; solving y=mx+b systems with a table, a graph, a graphing calculator, and eventually with the equations; solving systems like Ax+By=C with equations.

3) Because it is spaced out (like me) the students build confidence and adapt the growth mindset that we want our students to have.

So how about the manipulatives????

Setting the Scene
Mr. "O" walks into a candy store and buys 3 JubJubs and 4 Smarties for 26 cents, you go into the same store and buy 2 JubJubs and 7 Smarties for 24 cents. How much does it cost for one JubJub and one Smartie at this candy store?

Bring in the Manipulatives
Students are given manipulatives to represent JubJubs and Smarties and lots of Pennies. Then they assign pennies to JubJubs and Smarties until it "works".

It looks like this.

Letting them Struggle / Explore /Play

We do a few so that all students can experience some success and I choreograph the learning.

Here are some other photos of other questions. I know you can figure out the questions!

Eventually we have done a few and this is what the board looks like.

The students tried #4 for a long time until someone screamed "I have tried everything! This one doesn't work!"

Ah Ha!

Then I gave them this one.

Mr. "O" walks into a candy store and buys 6 JubJubs and 2 Smarties for 22 cents, you go into the same store and buy 3 JubJubs and 1 Smartie for 11 cents. How much does it cost for one JubJub and one Smartie at this candy store?

It was great to listen to them share their answers. Here is what the board looked like on that one.

Create your Own

Next I asked students to create their own example. Here are a few samples.

Thoughts:

1) Students loved this and found it easy once they got the hang of it.

2) It took a while for some students to realize that the prices could not be different for themselves and Mr. "O".

3) I loved that all students could do this activity. I can't imagine starting with the algebra and doing elimination now that I have tried this with manipulatives. We will get to the algebraic solution  "by the end of the course".

4) They enjoyed creating their own examples.

If you try this or have tried something like this I would love to here about it.

Stay the course!