Larson Interactive Real-life Investigations For Heath Geometry: An Integrated Approaches
Larson Interactive Real-life Investigations For Heath Geometry: An Integrated Approaching

Big Ideas Math® Integrated Mathematics I, II, and III. Larson and Dr. Boswell began writing together in 1992. Since that time, they have authored over. Of how mathematical concepts are connected to each other and to real-life scenarios. For anyone considering switching to Saxon Math from another math curriculum, we recommend printing a free copy of a Saxon Math placement test. There are five different tests to choose from.

The content of this article is largely drawn from an Australian publication by Peter Gould that has been a source of many successful mathematics lessons for both children and student-teachers. (See publication details at end of article). It presents a style of problem-solving activity that has the potential to benefit ALL children in a class, both mathematically and socially, and is readily adaptable to most topics in mathematics curricula.

What is the Pieces of a Puzzle Approach?

Essentially, the 'Pieces of the Puzzle Approach' is a deductive reasoning activity in which each group member is given a piece of information to share with the rest of the group. The mathematical solution cannot be found without everyone's contribution. Thus, an important aspect of the approach is that the group assumes 'ownership' of the task yet individuals retain personal responsibility within the group. This style of activity was developed and used effectively by the EQUALS group in the U.S.A (1989).
To support the approach, a set of essential work rules must be established (based on the work of Burns 1992).
Rules

You are responsible for your own work and behaviour
You must be willing to help any group member who asks
You may only ask the teacher for help if everyone in the group has the same question

These rules assist in avoiding some of the pitfalls of co-operative group work. Once the children have had experience with these rules, the situation of one person dominating by collecting all the pieces of information and taking control should not occur. Similarly, a group member should not opt out of the activity and sit back and let the rest of the group solve the problem. The ability of the children to successfully follow the rules will improve with each session, and will be influenced by teacher led discussions about the purpose of each rule.
The structure of this approach is intended to provide positive opportunities for:

Risk taking - pupils are more likely ask questions of each other and put forward ideas in a small group situation, particularly with Rule 2 in place.
Mathematical language development - usually the clues for the tasks are communicated with words. The group needs to negotiate their interpretation of the mathematical vocabulary. They also must talk to each other, listen to others and explain their ideas clearly.
Peer coaching - pupils are able to clarify their understanding of mathematical concepts, correct misconceptions and test out ideas during the process of finding a collective solution to the problem.
Teacher's role as a facilitator and observer - the independence of the groups gives the teacher freedom to move around the class, observe language and strategies, interact with groups as required and assess their progress.
Effective learning - the tasks generally incorporate the manipulation of physical objects to produce a final product, which promotes the linking of verbal knowledge with visual imagery.
Mixed ability class teaching - the features listed above combine to make this approach particularly suited to use with mixed ability classes. Every child should be able to make a contribution to the problem solving process, learn from the activity and have received attention from the teacher when needed.

Larson interactive real-life investigations for heath geometry: an integrated approaching

Forming the Groups

There are, of course, many ways to organise children into groups. Unless the teacher has specific reasons for doing otherwise, a random mix method is best for this type of co-operative problem solving.

Random Mix by the Cards

Here is one way to achieve a random mix of pupils using a pack of ordinary playing cards. To form seven groups of four children, use all four card suits from one (Ace) to seven. Shuffle this pack and have each child choose a card. All the sevens form a group, all the sixes form a group and so on.

You may want to assign a role to each card suit such as:

Diamonds - team manager - responsible for assisting teacher to get group's attention at specified signal and reminding the group of the 'rules';
Spades - equipment manager - responsible for collecting and returning materials, and ensuring all group members have access to the materials;
Hearts - recorder - noting significant information, steps in solution, or preparation of finished product (such as the model, drawing or chart);
Clubs - spokesperson - responsible for reporting the group's efforts and/or solutions

Larson Interactive Real-life Investigations For Heath Geometry: An Integrated Approaches

These roles may vary according to the teacher's knowledge of the children and the particular task, but are always intended to help pass the responsibility for the task from the teacher to the children, and to ensure a spread of individual responsibility within the groups.

Lesson Structure

Introduction: This type of problem solving activity is well suited to developing and clarifying mathematical ideas that have already been introduced in other lessons. Therefore, in introducing the task to the class, the teacher can make links to previous work. If the mathematical vocabulary contained in the problem is of particular concern, then key terms should be revised.

Group work: The groups are formed and each child in a group is given one clue card. To maintain 'ownership' of the piece of information, the child may not physically give away the clue-card, but must be responsible for communicating the content to the group. Each pupil's role is now to work within his/her group to solve the puzzle, following the set of work rules. The teacher must also take care to follow these rules, and not take back responsibility for the task by interfering with the problem-solving processes or offering help before being asked by the whole group.

Larson Interactive Real-life Investigations For Heath Geometry: An Integrated Approaching

As always, it is advisable to have an extension question ready for a group that finishes before the others. It is also useful to have one or two 'extra' clues ready. These can be used to allow the inclusion of an extra group member, to give help to a group that is 'stuck', or to assist 'checking' when a group thinks it has finished the task.

Plenary - It is important for groups to report on their problem-solving processes as well as confirming the correctness of their end product. The teacher can use questions focus on particular issues and highlight points that have been observed during the session. For example:

How did your group get started?

What caused some difficulty?

Did any group do it a different way?

Examples of Problems

An important advantage of the Pieces of the Puzzle Approach is that teachers can make up their own problems to suit whatever topic is appropriate. This is done by working back-wards from the final product (such as a model) or answer (such as a number), and then designing a set of essential clues, plus a couple of extra clues. Make use of any maths resources in the school such as multi-link cubes, pattern blocks, Polydron shapes or other construc