Turbo Texting

The original idea for this lesson came from Al Overwijk. Thanks again Al!
The possible Ontario overall curriculum expectations covered in the activity:
  • Grade 10 applied:
    • graph a line and write the equation of a line from given information
  • Grade 9 applied & academic:
    • solve problems involving proportional reasoning;
    • apply data-management techniques to investigate relationships between two variables;
    • demonstrate an understanding of constant rate of change and its connection to linear relation
  • Grade 8:
    • solve problems by using proportional reasoning in a variety of meaningful contexts.
  • Grade 7:
    • demonstrate an understanding of proportional relationships using percent, ratio, and rate.
  • Grade 6:
    • demonstrate an understanding of relationships involving percent, ratio, and unit rate.

Act 1: Turbo Texting:

I started with “I was with my brother one afternoon and I needed to text my wife. After texting her, my brother informed me that I was a ‘terrible texter’. He said I was soooooo slow. I on the other hand disagreed. Then we decided to settle this once and for all—- race!!!”

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What do you notice? What do you wonder? Allow students a few minutes on their own to jot down their ideas. Then share with partners, then the class.
Here are a few questions/tasks I asked them next. I wanted to slowly build into deciding if this relationship was proportional.
  • What relationships can you see? — Number of characters in a text vs. the time to text it.
  • Create a scatter plot sketch of how the number of characters in a text affects the time to text that message.
  • How does this graph look with both texters on the same grid?
  • Who is the faster texter? Predict. How does your sketch show who is faster?
  • Kevin finishes first does that mean he is the faster texter?
  • How will we determine who is the faster texter? What will we need to see?
We took our time with these questions so we could develop and understand the relationship between characters in a text and the time to text it.

Act 2

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ME: “Use any method you choose to determine: Who is the faster texter?” I allowed them time here to work on a strategy. I watched carefully what strategies they used or didn’t use.

Seeing the different strategies gave us a nice discussion the importance understanding what rate we are determining and how to interpret it to answer the problem.

I showed this picture next:

and this piece of info…

Students completed this problem and we discussed the assumptions we needed to make.

Texting Time

How do your students compare to Jon and Kevin? Have them time each other while texting the 165 character message. Have them determine their texting speed to see who the fastest texter is in the class.

Linear Modelling

ME: “Now you may have texted that message in 18 seconds, but would you do this all of the time? Would you keep that same rate for a shorter message? Longer message? We better keep this experiment going.
I set them off to text various messages of different lengths using this handout (I modelled the handout format after Mary Bourassa’s Spegettini and Pennies handout – thanks Mary).

Click to download a copy

Students used Desmos and the regression tool to create a linear model. They used that model to predict how long it would take to text 140 characters, 200 characters, and this message: “Dear Mom and Dad I promise to never text and drive.” They finally timed themselves to compare the calculated time and the actual time.
Extension: Compare the relationship between the number of words in a message and the time to text the message. How would the equation change? Is it still proportional?

Flippity Flip, Bottle Flip!

How are all these middle schoolers/grade 9s landing these bottle flips?


Before today I hadn’t seen any of our students doing this bottle flipping thing! But I had a feeling they had all done it before. Today we started an activity with watching trick shots of bottle flips and will end with us creating and solving linear equations.

I showed this video:

My students wanted to argue that some of the tricks were fake…. but they were glued to watching. They all had tried flipping bottles before and some said they were amazing at it.

I had a full water bottle with me and asked if I could flip this. They all shouted that it was too full. I tried flipping and it was a no go. So I cracked it open and drank a few gulps. “Nope….you still won’t be able to flip that Mr. Orr — too much water still.” Again, I tried flipping it and nope. Still not even close. “Mr. Orr you probably won’t be able to flip it even if it had the perfect amount of water.”  So I took a few more swigs. “Still no good sir.” As I was chugging….someone yelled out for me to STOP! I did…..then flipped that bottle…. and…..Boom! The class was blown away!

I had them log into a simple Desmos activity that asked them to choose which bottle would be ideal for flipping.

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Almost everyone had chosen yellow.

The next slide had them moving a line to show the water level and then having them estimate how many ml would be ideal.

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Students were estimating between 100 and 200 ml.

“I think it’s 125 because that would be a quarter of the bottle. I think a quarter is the perfect amount of water.”

“I think it’s not 250ml because it has to be less than half…..but I think it’s not exactly half of that….so half of 250 is 125….but I’ll say 150ml.”

I shared all of their guesses:

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They kept asking if they were going to get to flip any bottles?? I said, “This is math class….do you think we flip bottles in math class?”

Then I broke out the bottles.

Here is the plan. We are going to have a bottle flipping contest. Rules:

  • Draw a line on your bottle where you think the ideal amount of water should be. Determine how much water to put into it in ml.
  • When you know how much water you need record it on our chart….put exactly that much water in there.
  • You must use your bottle for the contest.

Here are some pics of them working on this first part.

img_2250 img_2248 img_2247 img_2243 We had just enough time in this class to determine our volume, fill the bottle to verify it met the line, and practice flipping for about 10 minutes.

Part 2: The Contest

Students complete in five one minute trials. Recording how many “lands” they get each trial. screen-shot-2016-10-07-at-1-37-07-pm

We average those five trials to develop your “Landing” equation! Who was the winner? What does their graph look like?

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We use that equation to solve some problems. How many after ____minutes? How long will it take to make 100 lands? What does the equation look like if you have a head start of 5 lands?

I’ve modelled this lesson structure after this Paper Tossing activity and ultimately after Alex’s Card Tossing activity.

Featured Comment:

Mason:

 Well I am a middle school student and I go to chesnee middle school and I think that I just might show this to MY math teacher even though I don’t like math but you just made me want to like math. I’m in the sixth grade.

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

 

 

Double Clothesline – Solving Equations

I have always taught solving 2-step linear equations by starting with a balance scale. Having students whittle their way down to see how many marbles were in each bag was always a win for me…..in most cases.

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I valued this approach. It’s easy to visualize and it strengthens the “whatever you do to one side of an equation you do to the other” mantra we tell students when solving . However, I’ve always been left wanting more especially when we introduce solving equations with negative coefficients or even when the solution is a negative value. The balance scale kinda loses it’s effectiveness.

Using algebra tiles help fill this hole. And now…. thanks to Andrew Stadel, double clotheslines.

I was lucky enough to attend Andrew’s NCTM Annual session on Error Analysis this year. In his session he demonstrated how to use a double clothesline to solve equations. I later found this resource on his site. Watch his videos on how to use the clotheslines….they helped me piece this lesson together. Stop now and go and watch Andrew’s video on solving two step equations.

I stared as Andrew did at the NCTM session:

I put 0 on the top line and 0x on the bottom line.

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I then held up the 3x card and asked where should this go? I asked if it should go on the left or the right of zero. The students overwhelming said it needed to go on the right. “3x is more than x, so it should go more to the right, just like a number line” (Always — Sometimes — Never was going through my head at this moment but i’ll wait to talk about this with the kids until a bit later in the lesson).  Screen Shot 2016-05-16 at 12.18.16 PM

I then said “I’m going to place this 15 right above the 3x and that means equivalence. 3x is the same as 15”

Where should 9x go? You could see the some students spacing out where 9x should go. This is what I love about this method. It’s so visual and we’re forced to always think about how terms relate to each other.

I want to know what number should be above 9x. I had them draw the number lines on their desks and let them work on determining the value of 9x.

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Going around the room there were a few different types of solutions. Some students said, “3 times 3x is 9x, so 3 times 15 is 45”

Some students said, “If 3x is 15 one x is 5, so 9x is 45.” Nice. We ensured the whole class understood both of these types.

Next puzzle: I asked where to place 3x + 4…then assigned it the value of 16.

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Where should 3x be placed? It was easy to see that 3x is less than 3x + 4 so it should go to the left. Now for the amazing moment! What should be the number above?

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from the class an overwhelmingly 12 was shouted. So now what must be the value of x?

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Student: “The dividing is the easy part” We spent a few minutes here talking about why dividing 12 by 3 here makes sense.

Next Puzzle:

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Where should the 5x go? At first some students had some difficulty deciding if it should go to the left or right of 5x – 2.

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Once we settled to the right. They jumped to finishing it off to determine x.

Next Puzzle:

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Where 3x should go was a discussion. We all agreed it should be 14 down…..and where would that be? This is where the clothesline (number line) feels superior and the balance scale visual falls short. We can use the bi direction of the number line to continue working with negative values.

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What was awesome during this class was this wasn’t a big deal….the number lines seems natural!!

 

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Also watch Andrew’s example with negative coefficients.

I had students practice solving a variety of equations by drawing the cards on their handout.

 

They finally demonstrated their understanding by creating their own equation where x had to equal 4. They put their creations up around the room for the group to solve.

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I feel that the number line (clothesline) method builds a lot of great number sense. We get to reinforce our inverse operations as we build from conceptual understanding to abstract. Students’ strengthen their understanding of algebraic expressions and how those expressions relate to others.

I’m now going to investigate how to to demonstrate solving multi-step equations…. 3x + 5 = 2x + 7 using the clothesline. I’m thinking this might be a difficult task. Any ideas????

[UPDATE] – Solving equations with expression on both sides.

Since this lesson my class used the double number line to solve equations like 4x + 10 = 6x + 2. It was great to keep some continuity here while we solved harder equations.

We placed each side of the equation on separate clotheslines just like before.
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We didn’t want to re-invent a new strategy….we were great at solving equations when one line was used for numbers and the other for expressions…..so we wanted that. How can we get one line to be just numbers and one to have the expression? We subtracted 4x from both lines.  Which left us exactly where we were last class!!

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and then we subtracted 2 from both to isolate the “x-term”

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Finally dividing by 2

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Boom!

This will be our method too to solve a system of equations that are both in terms of y.

More clothesline: