3 DIGIT FASCINATION

 Students need to be encouraged to investigate problems for patterns, rather than just have exercises which have single answers.

Too often maths Teaching is about asking Closed Questions, that is a question that has just one answer.  We need to change this to Open Questions, where students investigate and come up with various answers which they have to Justify.

A simple way to start with Open Questions is to start with the "answer" and ask what is the question/start?     e.g. The answer is 36 what is the question?

The beauty of this is that it caters for all abilities in a mixed ability class or group.  (One child might suggest "what is one more than 35?" while another, "what two numbers produce a product of 36?" and another, "What is The Square Root of 36?"

What a rich discussion could eventuate as all responses are shared and discussed

 Try this with your class, allow calculators if necessary, as the process is the important issue here

• Choose three single digits all different

• Using the three digits make all the two-digit numbers.

• Add the 6 two digit numbers together

• Add the original 3 single digits

• Divide the sum of the 6 pairs by the sum of the 3 digits

• What did you get? Why do you think this happens?

• Try with another set of 3 digits. Does the same thing happen?

• Does it work with 2 digits? 5 digits? 6 digits?

• What is the “maths behind this”? Let the numbers be a and b and c

NUMBER CHAINS

                                                     ABOVE IS A NUMBER CHAIN

                                                              How was is created?





•    Finish the chain?    When does it stop?

•    Investigate with other numbers as the starting point.  What do you notice?
            1.    If the number is even, divide by 2
            2.    If a number is odd, multiply by 3 and add 1

•    With large numbers use a calculator.

•     What happens if you change the rules:
            e.g. Change rule 2 to read : If a number is odd multiply it by 3 and subtract 1




•     Make a report about your investigations and share it with your friends.

A Few of My Discoveries About Using Math Games in the Classroom

From: Justin   http://www.mathfilefoldergames.com/


When I first started to work on the use of math games in the classroom, I was amazed at what I began to see happening! Here are a few of my discoveries about games where children can learn and practice math:

• Many of the games lead students to talk mathematics.
• Games forced students to justify their reasoning.
• Games put pressure on players to work mentally.
• Games did not define the way in which a problem had to be solved or worked out.
• Students began to explore and learn new strategies by working and talking with each other as they played.
• A game could often be played at more than one level allowing the teacher to differentiate instruction.

Games and Assessment
 
Teachers who observe and interact with children while they are playing math games can diagnose a wide variety of their mathematical strengths and weaknesses. In assessing learning through math games, teachers' concerns are not just confined to the children's levels of factual knowledge. Rather, they may also note, record, and analyze the following:

• reasoning and problem-solving skills,
• the forms of children's responses,
• the processes that children employ in solving problems and arriving at answers,
• children's patterns of persistence and curiosity, and
• their ability to work with peers, adults, and a variety of resources.

In addition, the recording sheets that children produce while playing games can be placed in assessment portfolios, where they can be of great value to children, teachers, and parents.

Finally, games provide children with a powerful way of assessing their own mathematical abilities. The immediate feedback children receive from their peers while playing games can help them evaluate their mathematical concepts and algorithms and revise inefficient, inadequate, or erroneous ones.

Good games evaluate children's progress. They provide feedback so that teachers, parents, and the child know what they have done well and what they need to practice.

"Family Maths is based on repeatable activities which are played by families.  Usually a school, church group or cultural group invites a facilitator to run a session to introduce the games and then the families take copies of the games/activities home so that they can be played a number of times, problem-solving, thinking about strategies talking maths.
A week or two later another session is held and further games and activities are taught, with reflection on successes or otherwise of the previous ones.
Ideally 4 sessions should be held as this gives families a bank of activities to play/experiment with on a regular basis.

One New Zealand School introduced "Maths Back Packs"  These were reasonably cheap 'Back Packs' each with a repeatable math activity/game inside. The students were assigned a Maths Back Pack for a week and when they were returned Parent Volunteers checked to make sure the game board and playing pieces were there or replaced ready for the next student to take home.

The attitude change by students, towards maths, is so noticeable to teachers and the community.  The games are not the sometimes boring traditional maths skill type practice or learning but activities that were fun, enjoyable, problem based and challenging."

Consider introducing games as a 'Home-School' activity instead of the usual homework activities and say to the students, "this is homework for you and your parents to play together and then to report back about how you enjoyed them and what mum and/or dad thought about this 'homework'?"

'Learning Mathematics Through Games Series: 4. from Strategy Games'

From: http://nrich.maths.org/  4th in a series

The first three articles in this series can be found on My Blog (Why Games?), (Types of Games) and (Creating Your Own Games).

There are several educationally useful ways of incorporating games into mathematics lessons.
Games can be used as lesson or topic starters that introduce a concept that will then be dealt with
in other types of activities. Some games can be used to explore mathematical ideas or develop
mathematical skills and processes and therefore be a main component of a lesson. Perhaps the
most common use of games is for practice and consolidation of concepts and skills that have
already been taught. Yet another way to use games is to make them the basis for mathematical
investigations.

Basic Strategy Games

Basic strategy games are particularly suitable as starting points for investigations because players
instinctively to try to discover a winning strategy, and usually the best way to do this is to analyse
the outcomes of series of 'moves'. With a little encouragement from the teacher, a mathematical
investigation is born. A few questions at the appropriate time will open up the task for the children
and lead to some good quality mathematical thinking.

For example, a basic version of the ancient game of NIM can be used to start an investigation.

          NIM
         Make a pile of seven counters. Two players each take turns to remove either one or two
         counters from the pile. The player left with last counter is the loser .

 

Invite the children to play the game several times and they are sure to begin searching for winning
strategies without being prompted. Ask whether it matters whom goes first and encourage them to record moves. Opponents will soon become partners in investigation as they test their theories.

The teacher's role then is to get the children to explain and justify their strategy and so 'teach' or
convince someone else. Now the game has been mastered it will no longer be enjoyable. It is time
for a "What if??" question.

What happens if you start the game with a different number of counters? (A series of key numbers
will emerge, as well as some interesting observations about odds and evens and multiples).

Problem-Solving Skills

Analysing a game in this way will typically engage the student in some highly desirable problem
solving strategies and processes: -

• being systematic,
• transforming information, (e.g. inventing a method for recording moves), 
• searching for patterns,
• applying mathematics (calculations, algebra),
• manipulating variables,
• working backwards, simplifying the problem,
• hypothesising and testing, and
• generalising (perhaps even producing a formula)

What if you can take a different number of counters away?

Variations of the Same Game

A way to take the investigation further and hence the mathematics further is to introduce a
variation of the game and search for winning strategies.

          NIM 3, 4, 5
          Make a row of 3 counters, a row of 4 and a row of 5. Two players each take turns to remove
          any number of counters from a particular row. The player left with the last counter is the      
          loser (or winner, as agreed at the start).

It might be helpful to suggest simplifying this game to a configuration of seven counters.

This may help them realise the importance of groups of two of four in the analysis.

It is often the case that problems and puzzles that appear to be quite different have a very similar
underlying mathematical structure. This can also be the case for strategy games.

          SLIDE (Linear NIM)

          Place a counter on each of the four coloured squares. Two players take turns to move any
          counter one, two or three spaces, until they reach the end of the track and are removed. No
          jumping is allowed. The winner (or loser as agreed) is the person left with sliding the last
         counter of the track. 

To analyse the game, it is helpful to start by playing it with only one counter, then two, then three.
Clear strategies can be found with one counter, but the introduction of other counters allows
blocking, which complicates the moves.

How can what has been discovered about these games be used t o create new challenging games?

The NIM game can also be extended into a two-dimensional game board.

          MINIM (2-D NIM)

          Place twenty-five counters on the game board as shown. Players take turns to remove one   

          or more counters that are side-by-side (no spaces between) on a straight line. The last player
          to take a counter is the loser .

Though complete analysis is too difficult, continuous scoring will help focus attention on early
moves. (1 point for each counter removed, minus 5 for the last counter). Encourage the children to
think backwards form the final move to discover helpful strategies towards the end of the game.

More Games?

All the games published monthly on the NRICH site are accompanied by questions that prompt
mathematical thinking and investigations.

An excellent source for groups of related strategy games is a book called "Strategy Games" by R.
Sheppard and J. Wlikinson. It is published by Tarquin and available through their website.