To allow the Difference Triangle resource to be used as a ready to go lesson here is a basic description of an activity you can build around it.

**Background**

As described in the resource notes on teacherLED all of the numbers in the completed triangle must be the absolute difference of the two numbers below it. The absolute difference is how far apart the numbers are on the number line. In essence it is the larger of the two numbers minus the smaller number.

**Introducing the Activity. **

Start with a difference triangle of just the first three consecutive numbers. Present this on the board and explain the rules. You can use the difference between 2 and 3 to give the top number of 1 or the difference between 1 and 3 to give a top number of 2. Next move on to the next size of triangle with a base of 3. You may want to talk about how the sequence of triangular numbers progress as a digression or just explain that we add a new bottom row of one larger than the existing bottom row.

Perhaps involve the students at the board solving this one.

It is important to not use a prepared IWB resource for this as the students need to be clear on how to use pen and paper to work on this problem as this will be all they will have. Doing it freehand illustrates this well. I let the students have anything in reason to help them solve it. Most go for pencil, paper and eraser but those who think a bit more ask for scissors so that they can more easily rearrange their triangles by making movable number tiles.

For the 4^{th} and 5^{th} triangles explain that the students must solve it for themselves. Leave the 5^{th} difference triangle resource from teacherLED on the IWB. Explain the rules of the challenge. You will have to adapt the reward currency to suit your own needs here but in my case I use minutes of free time that the students can each bank up for use at a convenient time.

I offer 15 minutes of free time as a total prize for being the first person/group to solve the 5^{th} puzzle.

Each group must show me a completed 4^{th} triangle before they can progress on to the 5^{th}.

When they feel they have completed the 5^{th} triangle they should go to the IWB and enter it on to the resource.

To prevent over-reliance on the resource they must enter the completed triangle within 1 minute of first touching the board. If they exceed this time or they find that their solution is wrong they forfeit 5 minutes from their potential winnings. This stops them using the resource rather than checking their own work.

The fifth triangle is very difficult to solve without having the top 3 numbers in place. TO encourage students to think about what they need to know and value its importance I auction off the top number for minutes from their potential winnings. The whole group will have the chance to pay 2 minutes from their potential winnings 10 minutes later but the winning bidder will get the time advantage. I repeat this after appropriate periods of time with the top 3 numbers which still leaves a tricky problem but one that with persistence and thinking about the problem will be solvable. Students don’t have to buy the numbers but they should consider that not paying the price will save reducing their potential winnings but if somebody else beats them to the solution they get nothing.

I’ve found this putting a value on the price of extra information a good thinking exercise for the students as they weigh up the costs and advantages of this decision alongside their thinking about the problem.

Even if a team reduces their potential winnings to zero they can be encouraged to continue as their win will deny other teams any winnings too.

You can set the timing of the activity by how quickly you give additional information.

**Conclusion**

As a fun conclusion to this activity when it is solved I give the students five minutes to view the finished triangle. I then present them with 15 plastic cups labelled with the numbers 1 to 15. I remove the solution from the IWB and then each student competes to be the quickest to stack the cups into the correct solution relying on memory supported by their mental arithmetic. If they work in teams each team’s final time is the mean time of each individual in the group. I use one of the online stopwatches for timing and each student starts and stops it themselves. If when they stop the timer their triangle is wrong they must restart the timer and try and correct it without audience help.

Note that the accompanying screenshot to this post gives you the solution to the 5th triangle. The next level with a base of 6 is actually impossible to do so only a particularly vindictive teacher would set it as an extension task!

Congratulations on an extremely innovative twist. The bidding system is exactly what I was looking for to bring this same problem to MathPickle teachers. Inspired!

Thank you. Glad to hear that somebody finds it useful!