The X Factor: Using Algebra to Solve Problems

# 4. Chocolate Boxes

Suggested Learning Intentions

• To complete a mathematical investigation involving variables
• To successfully apply a range of problem-solving strategies to a given task

Sample Success Criteria

• I can use a range of problem-solving strategies to help me solve a problem
• I can model a problem using materials
• I can identify and justify which strategies are most useful
• I can justify my solution using a range of manipulatives

This stage has been inspired by the Christmas Chocolates problem on the nRich website, reproduced with permission of the University of Cambridge, all rights reserved. You are encouraged to access the Teachers’ Resources and Solutions section of this problem prior to using this stage in the classroom.

Provide students with the Initial problem sheet from the Materials and texts section above, while telling the following story:

Penny, Tom and Matthew were each given a hexagonal box of chocolates as a gift. The box of chocolates has five chocolates along each straight edge and is called a 'size 5' box. Each of them eats a particular number of chocolates out of their box. They are all able to work out that there must have been 61 chocolates in the size 5 box to begin with. What strategies do you think each of them used to get to this answer?

Provide students some time to look at the three different diagrams on the Initial problem sheet. Support students further if needed by providing them with blank templates that can be used to shade in the chocolates eaten by each person. These are accessible via the Teachers’ Resources section of this problem on the nRich website.

• What do you notice about the position of the chocolates that each person ate first?
• Which chocolates do you think that each person will eat next?

Ask students to share the strategies that they believe Penny, Tom, and Matthew used to work out the number of chocolates in the size 5 box. The nRich task provides sample explanations for the three different strategies.

Invite students to contribute other strategies they may have discovered. It is important to validate any student who simply counted the chocolates one-by-one.

Emphasise that the purpose of pattern recognition (which is what Penny, Tom and Matthew used) is to enable us to complete these types of calculations more efficiently – but counting them is still a useful starting place. Counting the chocolates in rows can lead to a valid pattern being recognised and then used later.

Once students have had an opportunity to see how different strategies work, present them with a second problem:

Penny, Tom and Matthew now each have a larger hexagonal box of chocolates – it’s a 'size 10' box.  How many chocolates could be in this box?

Students apply any of the strategies discussed earlier to try and solve this problem.

Sample student responses may include:

• Penny’s approach: There are six triangles in the hexagon.  Each of them has a base of 9 chocolates.  Each of those triangles has 45 chocolates in it. So, 6 x 45 equals 270, and then add 1 for the centre chocolate, and there are 271 chocolates in total.
• Tom’s approach:  There are three parallelograms in the hexagon.  Each parallelogram is made up of 10 rows of 9 chocolates, so there are 90 chocolates in each parallelogram.  So, 3 x 90 equals 270 and then add 1 for the centre chocolate, which gives 271 chocolates in total.
• Matthew’s approach:  If there are 10 chocolates on each side, then I can see 6 rows of 9 chocolates in the outside layer. This will give me 54 chocolates in the outer layer. Each layer afterwards will have 6 less in it (6 rows of 8 chocolates, then 6 rows of 7 chocolates, etc.). Therefore, the total number of chocolates will be equal to 54 + 48 + 42 + 36 + 30 + 24 + 18 + 12 + 6 and then 1 for the middle, which equals to 271.

Provide opportunities for students who have used different strategies to compare their answers, so they can see that multiple strategies can lead to the same answer.

Present students who have successfully trialled a variety of strategies for both size 5 and size 10 boxes with this problem:

'Can you determine a rule that would help you quickly work out how many chocolates are in a box with a side length of n chocolates?'

Enable students requiring further support to make a start on this problem by suggesting that they begin with Tom’s approach (parallelograms), and experiment with various values of n

Model the use of a table as a way of helping students explore the relationship between the size of the box and the number of chocolates in it. Use the data collected from the size 5 and size 10 boxes as a starting point for the table.

Using this approach, students can develop the generalised rule for the number of chocolates in a box of size as 3n(n - 1) +1 .

Ask students to explain the significance of the different parts of this rule, or to annotate the rule with an explanation. A sample response may be:

'The n(n - 1) represents the number of chocolates in one of the parallelograms. The 3 reflects the three parallelograms within the box, so 3n(n - 1) represents the number of chocolates in the three parallelograms, and the plus 1 on the end is for the chocolate in the middle.'

Ask students to develop a generalisation using Penny’s approach (triangles).

Students who have successfully understood Tom’s ‘parallelogram’ method may be able to visualise that the triangle is half of the parallelogram. Their response may be:

We know that the number of chocolates in a parallelogram is equal to:

Each triangle is half of a parallelogram, so the number of chocolates in each triangle is:

There are 6 triangles, so we need to multiply:

and then add 1 for the chocolate in the middle. So the total number of chocolates is equal to:

Demonstrate that this generalisation, obtained using Penny’s method, is in fact equivalent to the generalisation obtained using Tom’s method.

Students who are familiar with the triangular numbers (1, 3, 6, 10…) could use these as part of their generalisation, as the number of chocolates in each triangle is a triangular number. A sample response could be:

'Subtract 1 from the side length of the box. If the box is a size 9 box, subtract 1 from 9, which gives you 8. The 8th triangular number will tell you how many chocolates are in each of the triangles. The 8th triangular number is 36, so there are 36 chocolates in each triangle. Multiply 36 by 6, because there are 6 triangles, and then add 1 for the chocolate in the centre.'

While this generalisation isn’t written as a symbolic rule, it is still a form of algebraic reasoning, as it can work for any size box, and relies on the recognition of a pattern.

Enable students by providing them a list and/or visual representation of the triangular numbers and encourage them to make links between this list and the chocolate boxes.

### Areas for Further Exploration

Using Matthew's approach to develop a generalised algebraic rule is a far more challenging task. However, students could be asked to explore Matthew's approach and write an algorithm that could be sued to find the number of chocolates for any value of n. A sample algorithm is provided here, modelled using n = 8:

Step 1: Count the number of chocolates on each side of the box.

There are 8 chocolates.

Step 2: Subtract 1 from this number. This represents the number of chocolates in each row around the outside of the box.

8 - 1 = 7

Step 3: Multiply this number by 6, because there are 6 rows around the outside of the box. The answer will represent the total number of chocolates in the outside layer.

7 x 6 = 42

Step 4: Continue to subtract 1 and multiply this number by 6. Each of these numbers represents the number of chocolates in each layer, as we move from the outside of the box towards the inside.

6 x 6 = 36, 5 x 6 = 30, 4 x 6 = 24, ...

Step 5: Continue this process until you reach 1 x 6 = 6

3 x 6 = 18, 2 x 6 = 12, 1 x 6 = 6

Step 6: Add the products of the multiplications together.

42 + 36 + 30 + 24 + 18 + 12 + 6 = 168

Step 7: Add 1 to represent the chocolate in the middle.

168 + 1 = 169

Extend students by asking them to write a generalisation using triangles with a base of n (rather than n - 1). This is quite challenging, as it will require them to subtract the 'overlapped' rows that are counted twice. A sample explanation for the development of this generalisation is provided below.

The number of chocolates in each triangle is:

The number of chocolates in six of these triangles is:

Then, you need to subtract 6 rows of n chocolates, as they have been counted twice. Therefore, the number of chocolates will be:

Finally, add one chocolate back in for the centre chocolate.

Using the algebraic skills of expanding and linear factorising, demonstrate to students that this expression is equivalent to 3n(n - 1) + 1.

The purpose of this part of the stage is for students to make links between the task and a set of recognised problem-solving strategies. You may have your own set of problem-solving strategies that you would like to use instead of the set referenced below.

Introduce students to the Problem Solving Strategies poster. The poster presents a range of strategies that students can apply to problem solving tasks.

Ask students to read the list of strategies in the toolbox and identify any that they used during this task. Ask students which strategies they felt were most useful for them during this task.

The specific problem-solving strategies that students have used in Chocolate Boxes may include:

• Draw a diagram
• Make a model
• Make a table
• Look for a pattern
• Test all possible combinations
• Seek an exception
• Break the problem into manageable parts

Students could prepare an investigative report outlining the strategies used by Tom, Penny, and Matthew, as well as any additional generalisations they developed.

Addison, L., n.d. Strategy Toolbox poster. [Online]