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Summary

 

Lesson 10

  • Years: 2 - 8
  • Time: 1 - 4 lessons
  • Strands: Number

On a signal from the teacher the children form into groups of a stated size and the number of groups and the number of ungrouped children is recorded. This is played several times then the teacher poses the problem, which involves working backwards from groups/remainder data to find the original number in the class. The problem is easily understandable because the students are physically involved in setting it up. It has many extensions.

Throughout the plan, a green dart highlights how and where the process of Working Mathematically is woven into the lesson.

 

Resources

Related Lessons

  • Tables for 25

1. Introduce & play the game

Many teachers have the Working Mathematically process on charts around the room. The charts either draw attention to elements as they occur, or provide the focus for a reflection session, or both. Other teachers require the one page sheet to be pasted to the inside cover of the students' journal.

To physically involve the students in the problem, plan to use either an outdoor space or the multi-purpose room. You will also need either a tape recorder and tape (to use in the same way as in Musical Chairs), or a favourite class song which the children can sing unaccompanied.
  • You will need to record the results. This can be done by a chosen student using a notebook, or can be in public view on a board or easel.
Let's play a game which uses the number of children present today. Line up here and number off so we know how many of us there are.

Record this number on the board:

Recording the groups 1

Working in context, collecting data, recording

Now I will start the tape and you move around in this space until the music stops. When it stops I will call out a number and that is the size of the group I want you to make.
Count the number of groups and the remaining children who are not part of a group, and record.

Recording the groups 2

Repeat the activity several times and build up the record on the board.

Recording the groups 3

Recording, organising data

Physical Involvement is very important in establishing this investigation. Being part of the introductory game and seeing the data develop on the board as the game progresses, means that students readily conceive the objective of the problem solving situation when it is introduced.

Return to the classroom and examine the results, for example:

Lesson Stages

  1. Introduce & play the game
  2. Explore the game with counters
  3. Challenge
  4. Make and test hypotheses
  5. Conclusion and extension

Learning Outcomes

  • division concept - sharing
  • multiples
  • factors
  • prime numbers
  • remainders
  • Working Mathematically process

Software Contribution

The software has three options:
  1. A chosen number of buttons onto the screen, then sorts them on command into groups of a chosen size. Students should be encouraged to predict before finding the answer.
  2. The user to enter all the conditions of any Bob's Button-style problem. Once entered the user chooses the number of buttons to test one at a time - guess, check, correct.
  3. Any Bob's Buttons conditions can be entered and the software begins an organised search for solutions by trying every possible case. If a solution set exists students will see a pattern and the next level of the investigation is revealed.

Green Line

References

Visit the Task Cameo at Mathematics Centre.
Bob's Buttons

Poly Plug can be used with this lesson.
Poly Plug

Which group sizes makes it easy to know the remainder children?
Two of the results on this chart have the same remainder, what do you notice about their groupings?
Do they represent the same thing?
Discussing, understanding the problem

2. Explore the game with counters

Supply the children with sufficient counters for each pair to make groups of 29 (or whatever your focus number is for this day). Ask them to work out and record the results for all the other group sizes that could have been called.

Working in context, using standard skills, discussing, recording, understanding the problem, checking the result

Correct the students work.

This step was an eye-opener for me with some of my Year 4s. It was clear that their experience making and recording groups was far below what would be needed to tackle the actual Bob's Buttons challenge. They just needed to make and record groups for lots and lots of numbers. Fortunately, because I was using the Poly Plug, I was able to set appropriate problems for these kids and challenge the others without anyone really being aware that there were weaknesses in the room.

Bob's Buttons on the table.

I had to occupy two Year 2 classes in the hall one morning and I thought I would take a risk and use this lesson. I figured they would enjoy the game aspect and that the maths was within their grasp, so I got hold of a portable blackboard and heaps of Multilink cubes to use as pretend people and went for it with the 47 kids present. As I predicted they really liked making the groups when the music stopped and I called out the group size.

We recorded lots of our groups and remainders and then I gave out cubes to each pair and they discovered more ways of grouping within 47. There was heaps of use of the sharing concept of division.

What really stunned me though was that in the last 10 minutes I made up a challenge [...like the one below...] and asked them to use the cubes to work it out. Three groups did! I was rapt.

3. Challenge

You know, I think my memory is wearing out. I remember playing this game with a class last year, and I remember that:
  • when we made groups of 4 there were 2 children remaining, and
  • when we made groups of 5 there was 1 child remaining.
But I can't remember how many children were in the class. Could you use your counters and help me find out the number of children in the class that day?
In essence the problem that initiates this investigation is:
Find the number which produces a remainder of 2 when divided by 4 and a remainder of 1 when divided by 5.
Presented in this way the problem could automatically cut out students who are already turned off to maths. Also the type of thinking required, which involves holding two conditions true simultaneously, is complex for younger students when presented in this abstract manner. However experience shows that the same mathematics presented in the game form allows many more students to begin the investigation.

4. Make & test hypotheses

Bob's Buttons software can be used at any stage of this exploration. It can be used to focus on the specific challenge, and it can also be used to investigate, either visually or symbolically, any other pair of Bob's Buttons conditions.

The first option of the program is terrific with the little kids. I don't worry about the Bob's Buttons problem as such. We have always had a Number of the Day part of the lesson where I choose a number and they find out all they can about it with materials. Now we use the software too. The children take turns to use the computer to investigate ways our Number of the Day can make groups. They record what they find and add it to the class list.

Allow the children time to explore. Some may not need the materials. As each pair finds a solution, ask them:

Can you check it another way?
Posing a new problem, checking the result

This encourages those who used counters to make use of tables facts, and those who used tables facts to make use of counters. Suggest that there might be another solution.

How many possible class sizes are there?
Understanding the problem, posing a new problem

Eventually, the class will develop this list of possible class sizes: {6, 26, 46, 66, ...}

Making a list, seeking & seeing patterns

Do you notice anything about these numbers?
Does the pattern you see connect in any way with the numbers I remembered?
Students usually readily make the hypothesis that the 'skip' in the pattern is the product of the two group sizes in the question, ie: the product of 4 and 5 in the challenge above is 20.

Making & testing hypotheses

Encourage the students to question and explore whether this would happen with other group sizes, for example:

  • groups of 3 with 2 remainders and groups of 5 with 4 remainders
  • groups of 6 with 3 remainders and groups of 7 with 5 remainders

As this exploration continues it is likely that students will find every case they try does fit the hypothesis. This is because students seem to naturally try group sizes which have no common factor. So, if it does turn out that all cases work, encourage students to test their hypothesis on a case like:

  • groups of 6 with 5 remainders and groups of 3 with 2 remainders.

Pose a new problem

The lists of relevant class sizes for the problem in step 5 is:

{5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, ...}
{2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, ...}
and the list of common class sizes for these two sets is:
{5, 11, 17, 23, 29, ...}
Making a list, organising data

The 'skip' in this pattern is 6, and 6 is NOT the product of 6 and 3, so the earlier hypothesis has failed.

Now the hunt begins for a new hypothesis to cover the case of factors in common between the group sizes, and perhaps an hypothesis to cover all cases.

Making & testing hypotheses

Again allow sufficient time for exploration and testing. There is no reason why this investigation needs to be confined to one timetabled class lesson.

Posing a new problem

Conclusion

When students have extensively investigated this problem, it is appropriate that they should have opportunity to publish their results. This could be in the form of a class wall display, or a report from each student which could count towards an assessment portfolio. If the investigation guides have been used, these will provide a sound background for such a report.

A sample student report is provided. It links to the Extra Investigation Guide used by the student's teacher.
(See Classroom Contributions for comments from the teacher, Andrew Dunstall. Andrew designed this guide for his Year 7 class.)

Extension

When students have developed an hypothesis to cover predicting the 'skip' number in the pattern if the group sizes are known, raise the question of being able to predict the first number in the sequence of 'class sizes' which would work for BOTH sets of group sizes and remainders. For example, the Challenge problem was:

  • when we made groups of 4 there were 2 children remaining, and
  • when we made groups of 5 there was 1 child remaining.

The simplest positive answer to this question is a class size of 6. Is there a way of predicting that 6 from the numbers in the problem?

 

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