Multiplication: Building bots
View Sequence overviewThe multiplicative structure in “for each” problems can be represented in systematic ways to find the total number of combinations.
Whole class
Building bots Slides
Each student
A3 paper
Robot parts pictures
Glue stick
Robot templates
Build
Revise: For each head there were three bodies and three sets of legs; for each body there were three heads and three sets of legs; and for each set of legs there were three bodies and three heads.
Show Slide 17 of Building bots Slides.
Discuss with students what they think will happen if the number of robot body parts is different.
Show Slide 18.
Ask:
- What would happen if you built different robots using two heads, two bodies and two types of legs?
- Can you estimate how many different combinations can be made?
- This question allows you to identify students who suggest six combinations, as they add the number of parts, in the same way as many students at the beginning of the sequence likely put forward nine.
Students turn and talk to the person next to them to clarify how they made their estimation with another student. They need to discuss:
- what mathematical thinking helped them make their estimate
- what they already know which can help them find the number of unique robots.
- how they could clearly represent their mathematical thinking for others.
When students have had time to reflect on ways to approach the problem, they can move on to the next activity.
Representing mathematical understanding
To think multiplicatively, students need to be able to communicate their thinking in a variety of ways, such as using words, models, drawings and symbols (Siemon & Breed 2006).
Throughout this sequence, students experience various ways to determine the number of robot combinations, such as using trial and error, organising the robots to find missing combinations, systematic thinking and using numbers to calculate. Students are encouraged to use representations to both make sense of mathematical problems as well as to communicate their understanding to others (Cai & Lester 2005).
This task focuses students on using the array as a representation to organise combinations to find every possibility. The array structure supports students to experience the “for each” idea as they keep one body part constant and find all combinations for this. Multiplicative thinking is embedded within this representation as students are able to see when they have found every combination for each part.
References
Cai, J., & Lester Jr, F. K. (2005). Solution representations and pedagogical representations in Chinese and US classrooms. The Journal of Mathematical Behavior, 24(3-4), 221-237.
Siemon, D., & Breed, M. (2006). Assessing multiplicative thinking using rich tasks. In Annual Conference of the Australian Association for Research in Education.
To think multiplicatively, students need to be able to communicate their thinking in a variety of ways, such as using words, models, drawings and symbols (Siemon & Breed 2006).
Throughout this sequence, students experience various ways to determine the number of robot combinations, such as using trial and error, organising the robots to find missing combinations, systematic thinking and using numbers to calculate. Students are encouraged to use representations to both make sense of mathematical problems as well as to communicate their understanding to others (Cai & Lester 2005).
This task focuses students on using the array as a representation to organise combinations to find every possibility. The array structure supports students to experience the “for each” idea as they keep one body part constant and find all combinations for this. Multiplicative thinking is embedded within this representation as students are able to see when they have found every combination for each part.
References
Cai, J., & Lester Jr, F. K. (2005). Solution representations and pedagogical representations in Chinese and US classrooms. The Journal of Mathematical Behavior, 24(3-4), 221-237.
Siemon, D., & Breed, M. (2006). Assessing multiplicative thinking using rich tasks. In Annual Conference of the Australian Association for Research in Education.
Provide each student with a sheet of A3 paper, Robot parts pictures, Robot templates and a glue stick.
Show students Slide 19.
Pose the activity: Select the number of different robot body parts you want to use. Predict how many unique robots you think you will be able to make with that number of heads, bodies and legs. Record your solution in any way you choose, to clearly show your mathematical thinking.
Ask students to record their prediction on their sheet before they start. They can show their thinking in any way they choose, but they must clearly show how they know they have found every possible robot combination.
This Build task may be used to assess student understanding; ensure students work independently if you use it in this way.
Students may repeat the task using a different number of heads, bodies and sets of legs.
Allow students time to record their work however they choose. Encourage students to express their ideas using number sentences, to show they can generalise.
Indicators of student understanding
As students combine the robot parts in a variety of ways, they demonstrate their understanding.
Students may:
- record a prediction for the possible number of robot combinations, and justify this prediction.
- represent robot combinations which include:
- a smaller number of parts to combine (two heads, two bodies, two legs).
- different numbers of parts (two heads, three bodies, two legs).
- record robot combinations as number sentences, for example:
- two heads, two bodies, two legs as $(2 + 2) + (2 + 2) = 8$ or $(2 \times 2) + (2 \times 2) = 8$
- two heads, three bodies, two legs as $(2 + 2 + 2) + (2 + 2 + 2) = 12$ or $(3 \times 2) + (3 \times 2) = 12$
- organise information in a structured array or table so no combinations are repeated or skipped.
- Group robots by shared features (same head, same body, same legs).
- Represent how changing one attribute affects the combination.
- Describe patterns across rows, columns, or layers (for example, each head should pair with each body and each type of legs).
- Explain how incomplete patterns help to discern missing robots.
- represent mathematical thinking about the problem in more than one way, such as an array, or repeated addition, or multiplication number sentences, etc.
As students combine the robot parts in a variety of ways, they demonstrate their understanding.
Students may:
- record a prediction for the possible number of robot combinations, and justify this prediction.
- represent robot combinations which include:
- a smaller number of parts to combine (two heads, two bodies, two legs).
- different numbers of parts (two heads, three bodies, two legs).
- record robot combinations as number sentences, for example:
- two heads, two bodies, two legs as $(2 + 2) + (2 + 2) = 8$ or $(2 \times 2) + (2 \times 2) = 8$
- two heads, three bodies, two legs as $(2 + 2 + 2) + (2 + 2 + 2) = 12$ or $(3 \times 2) + (3 \times 2) = 12$
- organise information in a structured array or table so no combinations are repeated or skipped.
- Group robots by shared features (same head, same body, same legs).
- Represent how changing one attribute affects the combination.
- Describe patterns across rows, columns, or layers (for example, each head should pair with each body and each type of legs).
- Explain how incomplete patterns help to discern missing robots.
- represent mathematical thinking about the problem in more than one way, such as an array, or repeated addition, or multiplication number sentences, etc.