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21 x 534 > 21 x 53 x 4 > 2 x 1534

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.
An alternative prompt

Mathematical notes
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 

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Students' questions and observations
These are the questions and observations of Amanda Klahn's grade 4 PYP class at the Western Academy of Beijing. The students are thinking about the properties of the inequality and whether the solutions are in proportion. They also propose to change the prompt (including the digits and the operation) and then consider whether the change continues to fulfill the inequality in the prompt or, indeed, another condition set by the pupils. For example, one asks "Can we change the order so it is true for 'less than'?" Amanda reports that the prompt generated great excitement in the class with a pupil exclaiming that, "My brain is bursting with ideas."

You can follow Amanda on twitter @KlahnAmanda. She writes a blog about the mathematical inquiries of her class called Doing Maths: From Worksheets to Wonderings.
Generating inquiry through students' questions
These are the initial responses of a year 7 class at Haverstock School (Camden, UK). Students describe the meaning of the prompt and verify that the inequality is correct. They suggest using other digits and reversing the inequality. That the class had been involved in inquiries before is evident in the students' selection of a regulatory card. Asked to suggest the direction of the inquiry, many pairs of students chose two complementary cards (as indicated by the colour of the ticks in the picture below). One pair, for example, proposed to practise a procedure by finding more examples. Others specified how they would change the prompt during a period of exploration and one pair set their own aim to find the highest product with five digits.
As the inquiry went into the second lesson, students began to think about reversing the inequality. Many thought that four digits arranged __ __ x __ __ would always give a greater product then the same digits arranged __ x __ __ __. The first example supported the conjecture: 59 x 87 > 5 x 987. However, this was quickly followed by a counter-example: 17 x 42 < 1 x 742. The examples led students to explore how the size and order of the digits affected the inequality. 

You can follow the maths department at Haverstock School on twitter @havamaths.
Year 7 inquiry
The comments from a year 7 class (middle ability) show the students' attempts to make sense of the prompt, either by commenting on a feature of the prompt they notice or, in one case, by trying to make a conjecture about the general relationships. Other than asking about the calculator button, the students did not spontaneously ask questions about the mathematics underlying the prompt. After the teacher led a discussion based on the 'what-if-not?' strategy, students came up with this list of questions:
The final question was particularly novel. The student who asked the question went on to find these inequalities:
215 x 34 > 215 x 3 x 4 > 2 x 1 x 53 x 4
215 + 34 > 215 + 3 + 4 > 2 + 1 + 53 + 4.
You can read more examples of how this inquiry has developed in the classroom in the primary section of the website.