Multiplication: Building bots
View Sequence overviewEach robot part can be combined in multiple ways.
Each robot combination is unique but related to several others.
Whole class
Building bots Slides
Each group
Robot parts pictures
Each student
Robot templates
Glue stick
Scissors
Task
Show Slide 3 of Building bots Slides to introduce today’s task and explain that students will combine the different robot parts to build different robots.
Each robot they make must have a head, a body and legs.
Pose the task: Build your own robot.
Ask students to work in groups of 3-4. Provide each group with Robot parts pictures, and provide each student with scissors, a glue stick and Robot templates to glue the robot body parts to.
Allow students time to build one robot and to compare the robots they built in their group to see how their robot is similar and how it is different to other robots made in their group.
Invite students to look at the robots the rest of the class have made and to find someone else with the same robot as them. When they find other students with the same robot, they should form themselves into a group.
Invite each group to:
- show their robot.
- notice how many different robots were made.
- explain how their robot may be related/similar to other robots in the class.
Ask students to:
- estimate how many different robots they think it might be possible to make, and explain their reasoning to a partner.
- record their estimate on the back of their robot.
Students return to their original group. Allow time for them to continue making a few more unique robots.
Multiplication as a “for each” relationship
In this task, students begin to explore combinations by building their own robots from different body parts. They choose from three different robot heads, three bodies, and three types of legs to create unique robots. Through this engaging scenario, students begin to reason about the multiplicative idea of “for each”.
For example:
- For each head, there are three possible bodies.
- For each combination of head and body, there are three possible types of legs.
As students explore “for each” relationships, they start to notice how numbers grow in a systematic way. They start to recognise that the robots can be organised based on their parts, and that this can reveal helpful structures where they can see gaps in the combination patterns and notice which robots still need to be made.
During this particular task, the teacher intentionally does not guide students toward strategies for finding all combinations yet. Instead, the teacher:
- observes how students organise their thinking.
- listens for early “for each” language (for each head, I can choose any legs…).
- encourages students to compare and discuss their robots.
- helps students articulate what they notice about patterns or missing combinations.
This positions students to be ready for Task 3, where the array structure is introduced as a tool for organising combinations.
The purpose of this task is for students to explore ways of combining the different parts and to notice how the robots each student makes are the same or different to each other.
In this task, students begin to explore combinations by building their own robots from different body parts. They choose from three different robot heads, three bodies, and three types of legs to create unique robots. Through this engaging scenario, students begin to reason about the multiplicative idea of “for each”.
For example:
- For each head, there are three possible bodies.
- For each combination of head and body, there are three possible types of legs.
As students explore “for each” relationships, they start to notice how numbers grow in a systematic way. They start to recognise that the robots can be organised based on their parts, and that this can reveal helpful structures where they can see gaps in the combination patterns and notice which robots still need to be made.
During this particular task, the teacher intentionally does not guide students toward strategies for finding all combinations yet. Instead, the teacher:
- observes how students organise their thinking.
- listens for early “for each” language (for each head, I can choose any legs…).
- encourages students to compare and discuss their robots.
- helps students articulate what they notice about patterns or missing combinations.
This positions students to be ready for Task 3, where the array structure is introduced as a tool for organising combinations.
The purpose of this task is for students to explore ways of combining the different parts and to notice how the robots each student makes are the same or different to each other.
The purpose of this Connect phase is for students to have a chance to:
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Discuss what students noticed about the robots the class made, focusing on how the robots were similar and different.
Invite students to share their estimate and explain their reasoning for it. Student suggestions may include:
- guessing.
- adding three heads + three bodies + three legs = nine possible robots.
- using each part once.
- Students may not recognise that the same part can be used in different combinations.
- using the nine different body parts to make three different robots, not realising there are more than three possible robots.
Ask: When you returned to your group, how many more different robots did you make?
Discuss whether students think that all possible robots have been made and, if not, how they might build more robots that are different to each other but are related in some way.
Developing multiplicative understanding
When solving this combination problem, students may focus on counting or adding the number of different robot parts together, rather than counting each unique combination made using each different head with each different body and each different type of legs.
Students may use additive strategies where they:
- choose one part from each set rather than making different combinations of parts. For example, they add three heads, three bodies and three types of legs together to equal nine.
- think in terms of total number of parts rather than number of different combinations that can be made. For example, they make three robots using the nine parts.
- don’t understand that they can multiply independent choices, as in 3 × 3 × 3 = 27.
Students need time to grapple with the idea that they cannot just “add the parts” to find all of the combinations. In this task students are given opportunities to estimate the number of possible robots they think they can make throughout the sequence. Over the course of the sequence they may alter their estimates as their understanding develops. By the end of the sequence, students are able to compare their various estimates, and offer reasons for the differences between them.
The “for each” idea helps students to begin to see that each robot part is connected in some way; that there is a relationship between the number of each different part and the number of unique robots that they can make.
“The capacity to think relationally lies at the heart of multiplicative thinking” (Siemon et al. 2011, p. 360).
When students use multiplicative thinking, they:
- recognise that for each robot part there are a number of possible combinations. For example, they choose a particular head then combine it with each body, then for each body they can combine each type of legs, to find that for one head there are nine unique combinations to be made.
- recognise that this process can be repeated for each different robot head, and notice that for each head, the same number of different robots can be built (for each head there are nine unique combinations).
- understand that there are equal numbers of bodies for each head, and equal numbers of legs for each body.
Students’ capacity to think relationally is fundamental to multiplicative thinking. Without the capacity to think multiplicatively, students may struggle in secondary school. Therefore, it is important that students have opportunities to use representations, such as the array, to explore, visualise and express this multiplicative structure.
References
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching mathematics: Foundations to middle years. Oxford University Press.
When solving this combination problem, students may focus on counting or adding the number of different robot parts together, rather than counting each unique combination made using each different head with each different body and each different type of legs.
Students may use additive strategies where they:
- choose one part from each set rather than making different combinations of parts. For example, they add three heads, three bodies and three types of legs together to equal nine.
- think in terms of total number of parts rather than number of different combinations that can be made. For example, they make three robots using the nine parts.
- don’t understand that they can multiply independent choices, as in 3 × 3 × 3 = 27.
Students need time to grapple with the idea that they cannot just “add the parts” to find all of the combinations. In this task students are given opportunities to estimate the number of possible robots they think they can make throughout the sequence. Over the course of the sequence they may alter their estimates as their understanding develops. By the end of the sequence, students are able to compare their various estimates, and offer reasons for the differences between them.
The “for each” idea helps students to begin to see that each robot part is connected in some way; that there is a relationship between the number of each different part and the number of unique robots that they can make.
“The capacity to think relationally lies at the heart of multiplicative thinking” (Siemon et al. 2011, p. 360).
When students use multiplicative thinking, they:
- recognise that for each robot part there are a number of possible combinations. For example, they choose a particular head then combine it with each body, then for each body they can combine each type of legs, to find that for one head there are nine unique combinations to be made.
- recognise that this process can be repeated for each different robot head, and notice that for each head, the same number of different robots can be built (for each head there are nine unique combinations).
- understand that there are equal numbers of bodies for each head, and equal numbers of legs for each body.
Students’ capacity to think relationally is fundamental to multiplicative thinking. Without the capacity to think multiplicatively, students may struggle in secondary school. Therefore, it is important that students have opportunities to use representations, such as the array, to explore, visualise and express this multiplicative structure.
References
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching mathematics: Foundations to middle years. Oxford University Press.
Explain: We can combine three different robot heads, three bodies and three types of legs to make many unique robots. Each unique robot combination is related to the other robots because they are made from different combinations of the robot parts.