Algebra in Action: Patterns, Generalisations and Relationships

# 4. Chocolate Boxes

Suggested Learning Intentions

• To use a pattern to find an unknown
• To describe an unknown using pronumerals
• To explore, construct and explain generalisations between two variables
• 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 pattern to find an unknown
• I can model a pattern using concrete materials
• I can use pronumerals to describe an unknown
• I can describe a relationship between two variables using an algebraic expression
• I can complete a mathematical investigation involving variables
• I can select an appropriate problem-solving strategy from a range of strategies and successfully apply it to a task
• I can use a range of manipulatives to explain and justify my solutions

Chocolate Boxes

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.

Students will work collaboratively to develop a general formula to describe the number of chocolates in a hexagonal box.

Present students with the following scenario:

Penny, Tom, and Matthew were each given a hexagonal box of chocolates as a gift. Each box of chocolates has five chocolates along each straight edge. They call it a ‘size 5 box’.

Each person eats a different number of chocolates out of their own box.

Even after eating a certain number of chocolates each person can quickly calculate that there must have been 61 chocolates in the size 5 box to begin with.

Enable students by providing them with a blank ‘chocolate box’ and counters to physically try out each idea. Guide students to recognise how many groups they can see within each diagram. Have them use familiar shapes to recognise a pattern. Provide them with the worksheet available in the Materials and texts section.

Extend students by asking them to formulate a general rule for each chocolate box.

Potential solutions include:

Each pair of students records their solutions and then share these with another pair in the class. Did they come up with the same pattern thinking? Encourage students to refine their solution based on their interactions with others. Ask them to consider refining their solution based on the observations they have made of other solutions. Students can use the worksheet available in the Materials and texts section above.

Investigate an additional scenario in which students apply their previous solutions to the number problem to determine if their rule works for a larger sized hexagonal chocolate box or if they need to refine their rule.

Provide students with the following scenario:

Penny, Tom and Matthew now each have a larger hexagonal box of chocolates. It’s a size 10 box. How many chocolates do you think are in this box?

Invite students to apply their own previous solution or a solution that they saw another group use to 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. Therefore 6 x 45 = 270, then add 1 for the centre chocolate. That makes a total of 271 chocolates.

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. 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.). 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 to share and compare their solution with others in the class. Present students who have successfully trialled a variety of strategies for both size 5 and size 10 boxes with this extended 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 to make a start on this problem by suggesting that they begin with Tom’s approach (parallelograms), and experiment with various values of n. Encourage students to draw or model potential solutions.

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 n 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 each triangle has chocolates.

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. So, 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 who require further support 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 used to find the number of chocolates for any value of n. A sample algorithm is provided here and has been embellished using n = 8.

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

There are 8 chocolates.

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

8 - 1 = 7

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

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, ...

5. Continue this process until you reach 1 x 6 = 6.

… 4 x 6 = 24, 3 x 6 = 18, 2 x 6 = 12, 1 x 6 = 6

6. Add all the products of the multiplications together.

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

6. 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:

1. The number of chocolates in each triangle is:

2. The number of chocolates in six of these triangles is:

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

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

By using algebraic skills of expanding and linear factorising, students can see, or be shown, that this expression is equivalent to:

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 reflect on the approaches they used to solve each of the problems contained within this stage. 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

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