It's All in the Numbers

# 5. Number Puzzles Suggested Learning Intentions

• To use recognised problem-solving strategies to solve a number puzzle

Sample Success Criteria

• I can select appropriate problem-solving strategies to help me solve a number puzzle
• I can use manipulatives to explain and justify the solutions I have found

This stage has been inspired by the Truth Tiles 1 problem on Maths300, and is reproduced with permission. Access to the problem on Maths300 requires a subscription.

These puzzles have been selected to introduce to students a specific framework used to solve problems. The ‘working mathematically’ framework is intended to replicate what mathematicians actually do: solve problems. The framework incorporates a number of steps that will be explicitly referenced throughout this task and provides a reference tool for students to use for future problem-solving tasks.

Present students with a set of numbered tiles, and the following problem: Students explore possible solutions using their tiles. You may like to provide larger tiles so that students can demonstrate, to others, the tiles being shuffled around in their search for a solution.

Before long, students may realise that there is more than one solution to this puzzle. Invite students to share their solutions and record these on the board.

Pose these challenges to students:

• How many solutions are there?
• How do you know you have found them all?

Problems with multiple solutions have been deliberately chosen and highlighted to help promote a culture of ‘working mathematically’ in the classroom. Rather than developing students who view mathematics as a set of closed problems with single answers, these problems (and others within the sequence) allow you to promote ideas of exploration, persistence, reasoning, and justification as being central to mathematics.

Show the ‘Being a Mathematician’ poster to the students, available in the Materials and texts section. Explain that to complete the set challenges, students are going to work like a mathematician.

“In real life, mathematicians solve problems, and this poster provides us with a framework which can help us in this process.”

Clarify any of the vocabulary terms students may be unfamiliar with on the poster, such as ‘conjectures’ and ‘hypotheses’.

Discuss the various steps of the process and explain that they are currently in Step 2 – the informal experimentation and exploration stage. Provide time and opportunity for students to experiment with their number tiles to find further solutions.

Some students may naturally progress to Step 3 and start to form hypotheses about the possible solutions. Enable students to reach Step 3 by providing prompts such as:

• What do you notice about the solutions you have found so far in your exploration stage?
• Are any of the three lines ‘easier’ to fill than others? Why is that so?

These prompts may help students realise that there are only two ways of using the tiles to complete the multiplication line: 2 x 3 = 6 and 2 x 4 = 8. These may lead to other hypotheses, such as:

"The 1 has to go in either the addition line or the subtraction line. If the 1 is in the addition line, it can’t go in the answer box, because none of the tiles would add to 1. So the other two numbers must be consecutive to make the addition line work, such as 4 + 1 = 5

"If the 1 is in the subtraction line, it could go in the answer box, or in the second box. This will mean that the other two numbers must be consecutive, such as 8 - 1 = 7. So, we know that either the addition line or subtraction line must have two consecutive numbers in them."

Continue to make explicit reference to the poster and the various steps that students may be working in.

As students start to form hypotheses, introduce the Problem Solving Toolbox poster (Step 4) to students.

The analogy of a toolbox is a deliberate choice to allow students to see mathematicians in the same light as a carpenter or plumber. Tools are at our disposal to help us solve problems. A carpenter may use a saw or a screwdriver from their toolbox; the Problem Solving Toolbox presents students with a range of mathematical tools to assist in our work. Students require explicit practice discerning between and using the different tools, in the same way that a plumber needs experience working with the tools at their disposal.

Discuss the various strategies presented in the toolbox and ask students which may be useful in solving this problem. Specific strategies that may be useful for this problem include:

• Break a problem into manageable parts: students determining that there are only two possible combinations for the multiplication line means they can then focus on the other two lines.
• Test all possible combinations: once students have determined the two possibilities for the multiplication line, they can then test other possible combinations.
• Make a table: as students test their combinations, they could use a table to ensure their testing is done in a systematic process.

Explain that students can use both the Problem Solving Toolbox, and their mathematical skills of addition, subtraction and multiplication (Step 5 – toolbox of basic skills) to help them find solutions to this problem.

Clarify with your students during the problem-solving process about the nature of the solutions you are finding. Resolve, as a class, whether:

• 2 x 3 = 6 is to be seen as different to 3 x 2 = 6
• 4 + 5 = 9 is to be seen as different to 5 + 4 = 9
• 8 - 7 = 1 is to be seen as different to 8 - 1 = 7

If these options are counted as different solutions, then 16 possible solutions to the problem are available. Interestingly, none of them use the multiplication 2 x 4 = 8. Proving that this is not a possible solution is an important step in the overall process of solving this problem.

The provision of the number tiles for students to shift around and ‘play’ with is a vital step in the problem-solving process. It can also help lead to the realisation that all addition statements can be rewritten as subtraction statements, which is a pivotal moment for students in the discovery of new solutions and helps underpin the conceptual understanding of the relationship between addition and subtraction.

Extend students who are ready to move into Step 6 from the Being a Mathematician poster with prompts such as:

• What if all three lines were addition?

Students may also like to create their own puzzles, using a similar structure, and explore the possible solutions.

Provide students the opportunity to publish their work (Step 7) and share their findings with others. You could ask students to prepare a poster or report with all solutions included, and the reasons why these are the only solutions possible. Your school community could be invited to engage with the original problem via the school newsletter, and the student solutions to the problem could be published in a later edition. This provides students with an authentic audience for their work and exposes them to consider the important mathematical notion of proof.

### Areas for further exploration

Step 8 of the process involves finding other problems to solve. Four further puzzles have been provided below for you and your students to select from (Truth Tiles 2, Number Tiles, Fay’s Nines and Steps).

These puzzles have been reproduced with permission from Maths300. Access to the problems on Maths300 requires a subscription.

They are presented in an approximate order of difficulty, to allow for students working at various levels to be appropriately engaged, yet each can be approached in the same manner:

• Use the tiles to find some solutions.
• How many solutions exist?
• How do you know you have found them all?

Each puzzle requires the use of numbered tiles.

Invite students to select a problem of their choosing to work on. Emphasise that the measure of success is not so much the number of solutions that can be found, but the willingness to use the framework presented to find solutions. Framing success in this way permits students to achieve success at multiple levels. For some, finding a single solution to one of these problems would be regarded as a success. This approach also permits for the task to be differentiated in that students can self-select based on interest and perceived challenge.

As students select from one of the four puzzles, divide your board up into four columns, one for each puzzle. Invite students to share any solutions found with the class by adding them to the board. The class data can assist students looking for patterns to develop their own theories and hypotheses.

For your reference, some strategies to help solve each problem have been presented.

1. Truth Tiles 2 Note: there are 24 possible solutions to Truth Tiles 2.

Students may discover different solutions that use the same four numbers.

For example, using the tiles 3, 4, 5 and 6:

5 + 4 - 3 = 6                 5 + 4 - 6 =3            6 + 3 - 4 = 5             6 + 3 - 5 = 4

4 + 5 - 3 = 6                 4 + 5 - 6 = 3           3 + 6 - 4 = 5             3 + 6 - 5 = 4

This shows 8 solutions using this set of four numbers. Students may also realise that these solutions contain pairs of numbers that add to the same value (in this case, 9).

Students may then begin searching for other sets of four numbers that can be trialled in a similar way.

This approach reflects the strategies of ‘breaking a problem into manageable parts’ and ‘looking for a pattern’.

2. Number Tiles Note: There are 168 possible solutions to Number Tiles.

As students begin to discover and share solutions, two key observations may be made about the discovered solutions:

• All of the solutions involve ‘renaming’ at some point.
• The three digits in the answer line always add to 18.

Both of these observations are correct and may open up strategic choices for students. For example, students could start by identifying a three-digit number whose digits add up to 18 (such as 954) and, using the strategy of ‘working backwards’, combined with their understanding of place value, discover a range of solutions: 3. Fay's Nines Note: There are 180 possible solutions to Fay’s Nines.

Students may recognise that the digits in the units column must add up to either 9 or 19. This provides them with smaller sets of numbers to begin trialling.

Additionally, once students recognise that the solutions will involve renaming, they may also recognise that sets of three digits that add to 8 or 18 will also be useful.

The sets of digits that add up to 9 are: 1, 2 and 6; 1, 3 and 5; and 2, 3 and 4. As these sets have at least one number in common, it will be impossible to solve the puzzle with two different sets of digits that add up to 9.

The same logic applies to sets of digits that add up to 8. There are only two distinct sets: 1, 2 and 5; and 1, 3 and 4.

The table below shows various combinations of sets of digits that could be explored. After some exploration, students may recognise that only the combinations set out in the third row of the table will provide solutions. This conclusion can be reached largely through considering the columns in which the 8 and 9 can be placed. There are five different ‘families’ of digits in the units column that can be used to find a solution. Each family has 36 different place value variations.

4. Steps Note: There are 96 possible solutions to Steps.

Students may recognise that the three ‘corner’ squares are of significance, as they each feature in two lines.

For a line total of 14, the four lines must add to 56 (4 lines of 14 each). Yet the numbers 1 to 9 only add to 45. The difference of 11 can be explained by the corner squares being counted twice. So, place three numbers in the corner squares that add to 11.

Students could begin by placing the numbers 1, 2 and 8 in the corners, and trialling various positions of the other tiles.

Refer students back to the Being a Mathematician and the Problem Solving Toolbox posters.

Facilitate a discussion whereby students can nominate strategies used from the toolbox in solving these problems. Ask students which particular strategies they felt were most useful for them during this task. Students could also identify tools from their toolbox of basic skills that were useful.

Read Justification of a solution from the Literacy in Mathematics section of the Literacy Teaching Toolkit, to support you to scaffold students' understand of how they can communicate the justifications of solutions they find to problems.

Addison, L., n.d. Strategy Toolbox poster. [Online]
[Accessed 15 March 2022].

Australian Association of Mathematics Teachers, n.d. Fay's Nines. [Online]
Available at: maths300.com/members/m300full/031lfays.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. Number Tiles. [Online]
Available at: maths300.com/members/m300full/028lnumb.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. Steps. [Online]
Available at: maths300.com/members/m300full/029lstep.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. Truth Tiles 1. [Online]
Available at: maths300.com/members/m300full/030ltrue.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. Truth Tiles 2. [Online]
Available at: maths300.com/members/m300full/168ltru2.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Lovitt, C. & Clarke, D., 2011. Being a Mathematician poster. [Online]
Available at: www.educationaldesigner.org/ed/volume1/issue4/article15/images/fig2_large.png
[Accessed 15 March 2022].

State Government of Victoria, (Department of Education and Training), 2019. Justification of a solution. [Online]
Available at: www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/lim_justificationofsolutions.aspx
[Accessed 15 March 2022].