Scenario: Jess teaching about the relationship between multiplication and division within the context of a lesson about solving word problems
Country: UK
Grade (student age): Year 5 (age 9-10)
Contributed by: Fay Turner, University of Cambridge, UK
Context – national, curricular, professional, other
Jess had completed a one year graduate teacher programme the previous year. The lesson took place in the second term of her first year of teaching. The curriculum guidance for England at this time (2006) gave an objective that stated pupils in year 5 should ‘understand the effect of and relationship between the four operations, and the principles (not the names) of the arithmetic laws as they apply to multiplication’. The main objective for this lesson was ‘Choose and use appropriate number operations and appropriate ways of calculating to solve problems’. The lesson began with a ‘warm-up’ in which Jess asked questions about multiplication facts relating to the 3, 4, 5 and 10 times tables. In the next part of the lesson she discussed the relationship between multiplication and division facts. The main part of the lesson involved word problems which could be solved using different operations.
Scenario
After rehearsing some multiplication facts, Jess asked her pupils to work out three times four but this time to write the calculation sentence on their individual wipe-able boards. Most of the pupils wrote ‘3 x 4 = 12’ and held them up for Jess to review. Jess directed them to leave the calculation sentence on their boards and told them that she was going to change this around. She wrote ‘12 ÷ 3 =’ and waited for a response. One pupil responded ‘four’. Jess asked how they knew this was the answer to 12 ÷ 3 and asked, ‘How is times related to divide?’ One pupil offered that “times makes things bigger and divide makes them smaller”. No other responses were given.
To provoke student responses, Jess reminded the pupils of work they had done with addition and subtraction and asked how addition was related to subtraction. A pupil responded that “add and subtract are opposites”. Other pupils then said that times and divide are also opposites. Jess pointed out that the numbers in the two calculations 3 x 4 = 12 and 12 ÷ 3 = 4 were “the same but moved around”.
Jess drew diagrams on the board to represent the multiplication and division calculations of 3 X 4 and 12 ÷ 3 as four sets of three.
OOO OOO OOO OOO
Jess wrote the calculation ‘3 x 5 =’ on the class white board and a pupil gave the answer 15 which she recorded. Under this calculation sentence Jess wrote ‘15 ÷ 5 =’ and said “fifteen divided into five groups”. Jess told the pupils she was going to give them a harder one and wrote ‘3 x 10 = 30’ on the class white board. She asked them if they could change it around to make a divide. A child volunteered and wrote ‘30 ÷ 3 = 10’ on the class white board. Jess asked if there was a different way. A pupil approached the board and Jo told her she must start with 30 because it is the biggest number. The pupil wrote ‘30 ÷ 3 = 10’. Another pupil suggested 10 ÷ 30 and Jess picked up a pile of books and asked “If I have ten books, could I share them out between thirty people?” The pupils agreed that this would not be possible. Jess wrote 4 x 6 on the board and asked the pupils to work out the answer and then rearrange the numbers to make a division calculation. The pupils gave ‘24 ÷ 6 = 4’ and ‘24 ÷4 = 6’ as possible ways of rearranging the numbers. Jess told the pupils “Multiplication can help us do our divisions because they are just the opposite way around”.
The lesson then proceeded to the next stage in which Jess introduced word problems involving different operations. These problems were:
- Four apples weigh 100g, how much does one apple weigh?
- Ten apples weigh 100g, how much does one apple weigh?
Jess addressed the second of these with the class in the final part of the lesson. She asked the pupils what operation was needed for this problem. When no one gave an ‘acceptable’ response, Jess drew a bag of ten apples on the board. She said “altogether it weighs 100g but I want to know what one weighs. How would you do it?” One pupil suggested that he added tens until he got to 100. Jess responded that this was a good strategy but “if this was a harder number some children would do it differently. How do you think they would do it?” A pupil responded “divide”. Jess expanded on this saying that they could “share the amount between each apple”.
Knowledge Quartet Coding Commentary
Contributed by: Fay Turner, University of Cambridge, UK
Knowledge Quartet Dimension: Connection
Knowledge Quartet Code: Making Connections between Procedures (MCP)
Scenario: Jess teaching about the relationship between multiplication and division within the context of a lesson about solving word problems
The National Curriculum guidance states that Year 5 pupils should learn about the relationship between operations. It is not surprising therefore that Jess addressed this with her class. However, it is of interest that she decided to spend some time on this relationship within a lesson that was primarily about using different operations and strategies for solving word problems. Jess tried to make connections between procedures in two ways. She primarily made connections between multiplication and division but she also (briefly) initiated connection-making between multiplication/division and addition/subtraction as examples of inverse pairs when she reminded pupils of the work they had done on the relationship between addition and subtraction.
The ‘inverse-of-multiplication’ has been identified as a ‘structure’ of division that might be used to solve problems (Haylock, 2006). In this structure the question 24 ÷ 4 = is recast into an equivalent question about multiplication, in this case, what number multiplied by 4 is equal to 24? The part of the lesson preceding the word problems was clearly intended to make connections between the operations of multiplication and division. Jess asked for and recorded the inverse of a number of multiplication sentences:
3x 4 = 12, 12 ÷ 3 = 4
3 x 5 = 15, 15 ÷ 5
3 x 10 = 30, 30 ÷ 3
4 x 6 = 24, 24 ÷ 6 = 4, 24 ÷ 4 = 6
Jess asked the pupils to make related calculation sentences by “rearranging the numbers” and told them that, “Multiplication can help us do our divisions because they are just the opposite way around”. Jess’ procedural focus on ‘rearranging the numbers’ led one pupil to suggest that a calculation sentence relating to ‘3 x 10 = 30’ might be ‘10 ÷ 30 =’. Although not an impossible calculation, as Jess suggested, this is not an inverse of ‘3 x 10 = 30’. The focus on rearranging numbers rather then on conceptual understanding of inverse operations led to this problematic response.
Jess’ reason for considering the relationship between multiplication and division appeared to be primarily to enable her pupils to complete division calculations by using multiplication with which they were more familiar. In a discussion following the lesson Jess said:
The two objectives were taken directly from the year 5 lesson plan, but something we work on every week, doing multiplications, because they are really, really, really, really poor … but um, because we haven’t done division so much, they are getting more confident with multiplication so if they can see it is related to multiplication then they might not get so scared about it
Since her pupils were more comfortable with multiplication Jess helped them to relate division calculations to their inverse before giving them the problems.
Jess did briefly consider the relationship at a more conceptual level when she drew a representation of 3 x 4 = 12 and 12 ÷ 3 = 4 as four groups of three dots. However she did not focus on this and the connections she made were between procedures rather than concepts. A better representation for developing conceptual understanding would have been the array (Bramby, Harries, Higgins and Suggate, 2009). Jess might have used a 3 x 4 array of dots as a representation to explain how 3 x 4 is related to 12 ÷ 3. She might also have used this representation to demonstrate the relationship between these calculations and the calculations 4 x 3 and 12 ÷ 4.
In summary, this is an example of MCP because Jess made very explicit connections between the inverse operations of multiplication and division throughout the lesson. However, the connections were procedural. She focused on rearranging numbers to make connections between a ‘simpler’ multiplication calculation and a ‘more difficult’ division calculation in order to help her pupils carry out division problems. She did not make use of appropriate representations to make the conceptual connections.
References
Barmby, P., Harries, T., Higgins, S. & Suggate, J. (2009) The array representation and primary children's understanding and reasoning in multiplication. Educational Studies in Mathematics 70(3): 217-241.
Haylock, D. (2006) Mathematics Explained for Primary Teachers, 3rd edn. London: Sage