King of the Castle

The game that precedes this activity provides good practice in cumulative addition, but its real purpose is to challenge students to find and use effective strategies. The game is easy to set up. It is best to have the students play the game a few times before they attempt the questions in the activity. This will give them a chance to become familiar with the structure of the cumulative addition as they go up the pyramid.
Avoid telling the students the best strategy, but if they need hints, try questions like these: “Have you tried putting the numbers in different places at the start?”
“Is there anything the same about the position of the small numbers in those patterns that give the largest number?”
When they realise that the highest number at the top is achieved by having the largest numbers in the middle and the smallest numbers on the outside, make sure that they discuss why this makes a larger result than other ways.
They should realise that the numbers in the centre are added to both left and right sides above them, whereas the numbers on the end are added only one way, so the centre numbers are used as addends more times than the end ones.
The game is extended to more layers in question 2 of the activity. Further extensions can occur by using larger numbers as well as by adding more layers.

Answers to Activity

Game
A game using addition strategies
Activity
1. a. Answers may vary. The best strategy is to put the largest numbers in the middle squares so that they can be added to both the left and the right.
b. This time, the best strategy is to put the largest numbers on the outside.
2. Answers will vary. To get the highest possible number in the top layer, you need to have the 8 in the centre of the bottom row and the lowest numbers (4 and 5) on the outside. Possible orders are: 4, 7, 8, 6, 5; 5, 7, 8, 6, 4; 4, 6, 8, 7, 5; 5, 6, 8, 7, 4. With all these options, you get 109 in the top layer.