Multiplication inequalities inquiry
Mathematical inquiry processes: Verify; rearrange; conjecture and generalise. Conceptual field of inquiry: Multiplication of positive integers; permutations.
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.
For inexperienced inquirers or for students who might find the prompt too challenging, the teacher could use the alternative prompt. There are two parts to the inequality instead of three and the prompt focuses on double-digit multiplication. If there are four digits such that a > b > c > d, does (10a + c) x (10b + d) or (10a + d) x (10b + c) give the largest product?
Investigations and inquiries
Read about the differences between investigations and inquiries in mathematics classrooms here.
Inquiries in a grade 4 classroom
Amanda Klahn's grade 4 class came up with the wonderfully mathematical observations and questions above There are a variety of rich lines of inquiry that the pupils could follow.
The inquiry was the first time the class used the regulatory cards. Amanda restricted the pupils' choice to six cards. She reports that the cards worked well, making the pupils more aware of the direction the session was taking and stopping them from defaulting to 'ask the teacher'. As the pupils become more experienced with inquiry, Amanda hopes that the cards "will help them know where to take their learning without relying on me."
These are the questions and observations of Amanda Klahn's grade 4 PYP class at the Western Academy of Beijing the year after the inquiry reported on above. 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."
At the time of carrying out the inquiries, Amanda was an IB PYP teacher at the Western Academy of Beijing (China). She writes a blog about classroom inquiries 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.
Year 7 inquiry
The comments from a year 7 class 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
Amelia O'Brien used the prompt with her grade 5 class. The pupils used the question and comment stems to initiate an open inquiry. Katherine Williams, a visitor to the class, reports that the prompt led to "lots of questions and wonderings."
In the picture (right), which shows the result of one open inquiry, the connections between the pupil's steps in reasoning are clearly presented. The inquiry combines procedural steps to ascertain if the inequality is true and observations based on a conceptual understanding.
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 by using the grid method (see picture).
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.