## Game Designer / CreatorEdit

- Created by Daniel Scher (dscher@icloud.com) and Scott Steketee (stek@geometricfunctions.org)

## Game SummaryEdit

Hoop Sums is a team-based game that gives children the mental math practice they need to become fluent in addition while at the same time providing elements of logic, problem solving, and fun in the form of thought-provoking puzzles.

## Players / ModeratorsEdit

- Target Age Range: 6–9
- Number of Players: 3 to 4 for each set of hula hoops
- With three players, each player is responsible for ensuring that the sum of the numbers in his or her hula hoop is correct. When played with a fourth player, that player can help any of the other three players as well as check that the final arrangement of hula hoops satisfies the necessary conditions. An adult moderator checks that the solutions proposed by the players are correct. The moderator should also be prepared to ask helpful questions and provide encouragement as needed.

## Game Set-up and ConstructionEdit

**Materials**

- Three hula hoops, each a different color (approximate cost: $12). The colors should be easily distinguishable. In these instructions, we assume the hula hoops are red, blue, and green.
- Three to four sheets of card stock or printer paper and a pen or marker (approximate cost: $3)

**Game Set-Up**

- Fold two pieces of paper in half in both directions. Carefully tear each sheet along its creases to obtain eight smaller pieces of paper. Write the numbers 1, 2, 4, 8, 16, 32, and 64 onto the font and back of seven of the pieces. Recycle the extra piece of paper.

- Make a table on a sheet of paper, as shown below, where the players can keep track of each new puzzle they create and solve. The colors listed in the columns should be those of the three hula hoops.

## How to Play / Game RulesEdit

Hoop Sums can be customized easily for children of different age levels and mathematical readiness. We recommend that newcomers to the game begin with just two of the three hula hoops and the paper sheets numbered 1, 2, 4, 8, and 16. Children who are ready to progress to more challenging versions of the game can do so after getting acquainted with this introductory version of the game.

- Distribute a red hula hoop and a blue hula hoop to the players.
- Ask two of the players to each pick at random a number between 1 and 31. Write these numbers in the table shown in Step 2 of Section 3. One number should be placed in the Red column, the other in the Blue column. For sake of illustration, we assume the numbers are 5 and 14.

- Explain to the players that the goal of the game is to place the hula hoops and the sheets of paper numbered 1, 2, 4, 8, and 16 onto the floor so that the sum of the numbers in the red hula hoop is 5 and the sum of the numbers in the blue hula hoop is 14. Both conditions must be met at the same time. It may not be necessary to use all five numbers.
- Players should work together as a team to solve the challenge. Anyone should feel free to take a hula hoop and the numbers and arrange them on the floor, but everyone is responsible for explaining his or her reasoning to the team. It is up to the team to decide how to best collaborate. For example, the team might decide that one player’s responsibility is to check that the numbers inside the red hula hoop sum to 5 while another player’s responsibility is to check the blue hoop’s sum is 14.
- In the example given here, players will likely determine that the only way to form a sum of 5 in the red hula hoop is to use the numbers 1 and 4. Similarly, the only way to form a sum of 14 in the blue hula hoop is to use the numbers 2, 4, and 8. But the number 4 is required for both sums. How can the 4 be in both the red and blue hula hoops simultaneously? Be sure to allow teams to puzzle this out themselves. [The solution is to overlap the red and blue hula hoops and place the 4 into their shared overlapping region as in the image below. This is a key insight for solving Hoop Sums challenges.]

- Children can play again by picking two new random numbers between 1 and 31 and writing them in the table shown in Step 2 of Section 3. For children who are ready to move on to a more difficult game, use the table below as guide for progressing through a series of increasingly challenging games. In Game B, for example, players still use two hula hoops, but now the number 32 is added to the collection of numbers that can be placed in the hula hoops. To create a challenge for Game B, players pick two random numbers between 1 and 63. In Game C, there are three hula hoops, six sheets of numbered paper, and players pick three random numbers between 1 and 63.

- When playing Games C and D, which use three hula hoops, it is very likely that players’ initial placement of numbers within the hoops will need to be refined. Suppose, for example, that players are solving the challenge below:

- The players begin by focusing on the red hula hoop, placing the numbered slips 1, 2, 8, and 16 into the hoop as shown below:

- To continue, the players turn to the blue hoop and determine that they’ll need the numbers 1, 4, 8, and 16 to form a sum of 29. They notice, however, that the placement of the 1, 8, and 16 within the red hula hoop does not allow them to place the blue hula hoop so that it overlaps just those three numbers and not the 2. The players rearrange the numbers within the red hula hoop and create the layout below:

- Finally, players focus on the green hula hoop and determine that they’ll need the numbers 2, 8, and 32 to form a sum of 42. Once again, however, the placement of the numbers already in the hula hoops is not going to work. The numbers must be rearranged so that the green hula hoop can overlap both the 2 and 8 but not the 1 or 16. The players ultimately create the layout below, which satisfies all three sum requirements simultaneously:

- The ongoing need to monitor and reassess the solution fosters both engagement and collaboration among team members. A team with three players might decide, for example, that each player is responsible for monitoring one of the three sums to ensure that it remains on target even as the hula hoops and numbers are repositioned on the floor. A team with four members might assign the fourth player the responsibility of checking that the final arrangement of hula hoops and numbers satisfies all of the conditions simultaneously.

- If more than one group of children is playing Hoop Sums, the teams can create challenges for each other. If there are three teams A, B, and C, then team A can create a challenge for team B, team B can create a challenge for team C, and team C can create a challenge for team A. The process of creating a Hoop Sums challenge gives children the opportunity to consider what differentiates an easy puzzle from a hard puzzle.
- Hoop Sums is well suited for tournament play. Teams of children can each be given the same challenge to solve, with the winner of a round being the team that solve the challenge fastest. In this competitive setting, teamwork is especially important.
- As a variant to using the numbers 1, 2, 4, 8, 16, 32, 64, Hoop Sums can also be played with the numbers 1, 1, 3, 3, 9, 9, 27, 27. In this case, the allowable sums range from 1 to 80. It can also be played with the numbers 1, 1, 1, 1, 5, 5, 5, 5, 25, 25, 25, 25. In this case, the allowable sums range from 1 to 124.

## Templates / DiagramsEdit

- NA

## Related Web LinksEdit

- A technology-based version of Hoop Sums (called Arranging Addends) that uses The Geometer's Sketchpad software can be downloaded from http://dynamicnumber.org/arranging_addends__target_sum_puzzles.
- An interactive web-based version of Hoop Sums can be played online at http://ow.ly/WAVF1.

## Other DetailsEdit

This work was supported in part by the National Science Foundation (NSF) Dynamic Number grant no. DRL–0918733 (www.dynamicnumber.org). Opinions and views remain the authors’and do not necessarily reflect those of the NSF. It is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License, and thus can be distributed under the same license as part of this BigLeap Challenge.