David Aaron, a year 6 primary school teacher from Blackpool (UK), got in touch with Inquiry Maths to ask for a prompt that would foster children's use of written multiplication. While he intended to use the 24 x 21 = 42 x 12 prompt, he wrote that "I'd like to push the children to HTU x TU via a prompt." David's request brought to mind this mathematical investigation: combine the digits 1, 2, 3, 4, and 5 to find the highest product possible.
To turn the investigation into an inquiry, the teacher could use the prompt above. The pupils have to think about the inequalities, identify the common features of the expressions, and ask their own questions. Questions posed in classroom inquiry have included:
- Is the first product greater than the other two? Is the prompt correct?
- What if the digits were in a different order?
- Is it the case that TU x HTU is always greater than TU x TU x U, etc?
- What is the greatest or lowest product given by each arrangement? (This question provides the teacher and pupils with an opportunity to differentiate the inquiry by dividing the exploratory calculations up amongst the class.)
- Why is the greatest product given by TU x HTU? Why is the lowest given by TU x TU x U?
- What if you used five other digits, four digits, six digits ...?
- What if you used more than two multiplication signs?
- Can you create an inequality with the digits in the same arrangement, but using an "is less than" sign?
- What if the digits were different?
- What if the prompt involved another operation?
Inquiries have involved students in developing the fluent and efficient use of multiplication methods, as well as making and testing conjectures about which combination of digits gives the highest product.
A comparison between the greatest sum and product of four digits can lead to an interesting class discussion. If the teacher specifies two two-digit numbers, then, for example, 42 + 31 = 32 + 41, but 42 x 31 < 41 x 32. For addition, the two largest digits have to be in the tens columns but the combinations do not matter. For multiplication, however, the largest digit in the units must be multiplied by the largest digit in the tens to produce the highest product. This can be illustrated using the grid method:
Students have gone on to develop a rule to find the highest product with five digits by explaining why 431 x 52 > 521 x 43 > 421 x 53 > 531 x 42. To see how the greatest product is given by (100b + 10c + e) x (10a + d), where a, b, c, d and e are digits between 0 and 9 and a > b > c > d > e, click here.
Read about the differences between investigations and inquiries in mathematics classrooms here.
You can read more examples of how this inquiry has developed in the classroom in the primary section of the website.