Multiplication: Building bots
View Sequence overviewArrays make the multiplicative structure in “for each” problems visible by showing combinations as rows and columns.
Whole class
Building bots Slides
Each group
Robot parts pictures
Each student
3 sheets of Robot templates
Glue stick
Scissors
Task
Revise: In the last task we found that it is helpful to simplify the problem by focusing on finding the number of combinations for just one robot body part.
Show students Slide 10 of the Building bots Slides and discuss how they might organise the robots shown on the slide to find the missing combinations.
Ask them how they can be sure to have found all the combinations for a robot focusing on one body part, for example a square yellow head.
Pose the task: How could you sort and organise these robots to show which robots are missing?
Ask students to work in groups of 3-4. Provide each group with a set of Robot parts and glue stick. Provide each student with scissors, a glue stick and three sheets of Robot templates to glue the robot body parts to.
Each group selects a specific body part (e.g. square yellow head) and makes as many different robots as they can, using that part.
Ask students to estimate how many unique robots they think they can make for the specific body part they selected as their focus.
Ask:
- What did you find helpful when looking for every combination for this body part?
- Can students express how they worked to keep track of each combination they made?
- Can students reason systematically and methodically?
- Do they use patterns to support their thinking?
- Can students explain why the array is helpful?
- Can you explain this part a bit more? (point to a section needing clarification)
- Can students explain that for each head (orange, green and blue) there are nine possible combinations, and that this means there are three groups of nine combinations in total?
- Can they see that this same structure can be seen if they start with each body, or each type of legs (i.e. for each body, there are nine possible combinations)?
Spotlight: Highlight student work that systematically organises the robots to make it easy to see the missing combinations, including:
- work that focuses on keeping a single feature constant.
- work that uses arrays to see the patterns.
- work that shows all nine robots for one body part.
You may show students Slide 11 to demonstrate how reorganising the combinations for the yellow headed robot makes it easier to see what’s missing.
Pose the task: Organise the robots so that you can find all possible robot combinations for each of the other body parts.
Allow groups who have not found all combinations for their specific body part some time to complete this task, prior to continuing to find other robot combinations.
The multiplicative structure of the array
The array is a powerful representation that simultaneously organises a whole quantity as a number of rows and columns which can be seen at once.
There are two ways an array can be understood (Siemon, Beswick, Brady, Clark, Faragher, & Warren, 2011):
- as a count of equal groups e.g. 1 three, 2 threes, 3 threes etc. (additive thinking). This interpretation draws on skip counting and repeated addition of equal groups.
- as a change in size of group e.g. 3 ones, 3 twos, 3 threes etc. (multiplicative thinking). 3 ones are increased by a factor of 3 to become 3 times as many, or triple. 3 of/times anything, or something by 3 represents the “for each” idea.
Understanding the array as a change in the size of group focuses repeated multiplication of different size groups using the same factor rather than repeated addition of the same size group.
This task sequence has been designed to support the shift from repeated addition to repeated multiplication. The array is used to support this idea, because an array can be used to focus on finding every combination “for each” set of body parts. Here, it is useful to focus on one set to simplify the problem and to be systematic about finding every combination.
Students use the array to systematically combine one type of head with each of the three bodies, and to combine each of these with each of the three different types of legs.
The rows and columns structure of the array enables students to clearly see the robot head which does not change, and for each body, the legs that do change.
Students notice the step-by-step, methodical way in which only one body part of the robot changes. This helps them to see how each combination is related to the others.
This can be recorded as a multiplication equation:
$$3 \times 3 = 9$$
Therefore, for robots with a yellow head, there are nine different combinations that can be made. The same thinking can be applied to combinations for the two other robot heads.
References
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching mathematics: Foundations to middle years. Oxford University Press.
The array is a powerful representation that simultaneously organises a whole quantity as a number of rows and columns which can be seen at once.
There are two ways an array can be understood (Siemon, Beswick, Brady, Clark, Faragher, & Warren, 2011):
- as a count of equal groups e.g. 1 three, 2 threes, 3 threes etc. (additive thinking). This interpretation draws on skip counting and repeated addition of equal groups.
- as a change in size of group e.g. 3 ones, 3 twos, 3 threes etc. (multiplicative thinking). 3 ones are increased by a factor of 3 to become 3 times as many, or triple. 3 of/times anything, or something by 3 represents the “for each” idea.
Understanding the array as a change in the size of group focuses repeated multiplication of different size groups using the same factor rather than repeated addition of the same size group.
This task sequence has been designed to support the shift from repeated addition to repeated multiplication. The array is used to support this idea, because an array can be used to focus on finding every combination “for each” set of body parts. Here, it is useful to focus on one set to simplify the problem and to be systematic about finding every combination.
Students use the array to systematically combine one type of head with each of the three bodies, and to combine each of these with each of the three different types of legs.
The rows and columns structure of the array enables students to clearly see the robot head which does not change, and for each body, the legs that do change.
Students notice the step-by-step, methodical way in which only one body part of the robot changes. This helps them to see how each combination is related to the others.
This can be recorded as a multiplication equation:
$$3 \times 3 = 9$$
Therefore, for robots with a yellow head, there are nine different combinations that can be made. The same thinking can be applied to combinations for the two other robot heads.
References
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching mathematics: Foundations to middle years. Oxford University Press.
Spotlight
A “spotlight” is used here as an opportunity for the whole class to pause and for you to actively guide and support student learning. The teacher asks students probing questions to help explain their thinking and justify the strategies they used to organise the robots to be able to see the missing combinations.
The teacher might ask:
- What thinking did you use to get started?
- What did you already know which helped you begin?
- Explain how organising the robots in this way helps you to find each different robot?
- How do you know for certain that you have found every combination for this robot body part?
- Would you use this strategy to make robots using a different robot part? Why/why not?
The spotlight offers opportunity for students to see the work of others in the class. It invites questions, prompts discussion, deepens understanding, and encourages them to reflect on their own work and that of other students. It facilitates them to construct their own meaning through their experiences and mathematical discourse.
A “spotlight” is used here as an opportunity for the whole class to pause and for you to actively guide and support student learning. The teacher asks students probing questions to help explain their thinking and justify the strategies they used to organise the robots to be able to see the missing combinations.
The teacher might ask:
- What thinking did you use to get started?
- What did you already know which helped you begin?
- Explain how organising the robots in this way helps you to find each different robot?
- How do you know for certain that you have found every combination for this robot body part?
- Would you use this strategy to make robots using a different robot part? Why/why not?
The spotlight offers opportunity for students to see the work of others in the class. It invites questions, prompts discussion, deepens understanding, and encourages them to reflect on their own work and that of other students. It facilitates them to construct their own meaning through their experiences and mathematical discourse.
The purpose of this Connect phase is for students to have a chance to:
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Invite students to discuss how close their estimates were to the actual number of robots they made.
- What thinking did they use to help them make their estimate?
- How was their reasoning similar/different to what they did to make their first estimate?
Show Slide 12 and discuss:
- how students were able to find some other robot combinations.
- how they know when they have found them all.
- how many robots they found for each head (or body or legs).
Show Slides 13-14 and discuss what students notice. Some suggestions may include:
- the robot groups can be added together to get closer to the total number of robots.
- there is a repeated pattern (this might be seen as repeated addition, or three groups of nine robots).
- once we have found the total number of different robots that can be made using one head, that tells us the total number of different robots that can be made for each head.
- for each head there are three bodies and for each body there are three types of legs.
- the numbers of different robot parts can be multiplied together to find the total number of possible combinations.
Ask: If nine robots can be made for each head, how many robots can be made in total?
Encourage students to notice that the total number of robots can be thought of as three groups of nine (as illustrated in Slide 15).
Show Slide 15.
Explain: For each head, there are nine possible combinations. Because this is true for every head, the total number of robots is three groups of nine. This same structure can be seen starting from each body, or each set of legs.