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Multiplication inquiry

24 x 21 = 42 x 12
The questions and observations of a year 8 all-attainment class.
   
This prompt appears in Boris Kordemsky's The Moscow Puzzles (1956), in which Kordemsky lists the full set of equations of this type. As students invariably observe in the first phase of the inquiry, there are two key features of the equation (or 'rules') - the numbers are 'doubled and halved' and the digits are 'reversed'. In the orientation phase, students typically pose questions like the ones that follow:
    • Is the sum correct?
    • How do you multiply two 2-digit numbers?
    • One number on the right-hand side is half one on the left and the other is double.
    • The digits of the numbers on the left;hand side have been reversed to make the numbers on the right.
    • Do these rules always work?
    • Is it something to do with the digits doubling? (1, 2, and 4)
    • Are there other sums like this one?
At this stage the class might use the regulatory cards to signal the need to discuss and explain procedures for multiplying two 2-digit numbers. Once students have assured themselves that the equation is correct, they are enthusiastic to find more examples of the same type. The inquiry, therefore, is ideal for developing students' fluency with multiplication in the wider context of answering their own questions, testing their own conjectures and reaching aims they helped to establish.
   
As the inquiry develops, students quickly verify that the 'doubled and halved' rule always works, but the 'reversed' rule rarely does. To find more examples in which both rules work, the teacher can guide the class towards using - or co-construct in a class discussion - an algorithm (or set of instructions) to generate more examples. See the table below for an example (reverse, halve, reverse) and here for three other algorithms.
Start with 46* 46 x ..... = ..... x .....
Reverse 46 and place 64 on the right-hand side.  46 x ..... = 64 x .....
Halve 64 to place 32 on the left.  46 x 32 = 64 x .....
Reverse 32 and place 23 on the right.  46 x 32 = 64 x 23
* The algorithm does not work with any 2-digit start number. For example, the reverse of 56 is 65 and half of 65 is not a whole number.
 
Lines of inquiry
(1) Use a systematic method to find all equations of this type
In a classroom inquiry, a pair of year 7 students found 13 x 62 = 31 x 26 by "keeping the numbers low." During a class discussion, one student’s comment that 12 and 13 had now appeared led on to finding 14 x 82 = 41 x 28 and the realisation that using 15 is impossible within the constraints of using whole and two-digit numbers. After the teacher had highlighted the systematic method employed, the class went on to look at 21, 22, 23 and so on.
  
(2) Classify all examples by ratio
The ratios between the numbers in the prompt are 2:1 and 1:2 (halved and doubled). The example 13 x 93 = 31 x 39 shows that the ratio between numbers might be different - in this case, 1:3 and 3:1.
 
(3) Test students' conjectures
As the inquiry develops, students might make conjectures about the properties of the equations. The following two examples come from year 7 classes.
Conjecture 1: All digits are factors of the highest digit. For example, in 13 x 62 = 26 x 31, 1, 2, and 3 are factors of 6.
Conjecture 2: The sum of the digits on one side of the equation is always a multiple of 3. To support this conjecture a student wrote on the board:
 12 x 42 = 21 x 24 1 + 2 + 2 + 4 = 9
 13 x 62 = 31 x 26 1 + 2 + 3 + 6 = 12
 14 x 82 = 41 x 28 1 + 2 + 4 + 8 = 15
  
(4) Use algebra to find more examples
Write the general case as (10a + b)(10c + d) = (10b + a)(10d + c) and rearrange to find the relationship between a, b, c and d.
 
(5) Find examples with two 3-digit numbers
Building on the observation that the equation in the prompt uses the digits 1, 2 and 4, students have gone on to look at three-digit numbers, creating the following example:
124 x 842 = 248 x 421
142 x 482 = 241 x 248
412 x 428 = 824 x 214
 
See the mathematical notes for more details on the lines of inquiry.
  
Palindromic prompt
Mike Ollerton wrote: "I love this problem and for it to be a truly palindromic the calculation could read 24 x 21 = 12 x 42." The prompt in its palindromic form is more complex and leads to a diverse inquiry. Classroom trials have shown that students can become more interested in its palindromic nature than in the two 'rules'. They notice the whole (equation) to the exclusion of the parts (terms). This has led to an alternative pathway of inquiry focused on creating palindromic equations with different operations. While the pathway involves rich mathematical exploration and reasoning, the original prompt is recommended for the teacher who wants to focus, at least initially, on multiplication.

Mike Ollerton is an internationally-renowned educator who has published widely about investigations and all-attainment teaching. You can find resources and articles on his website and follow him on twitter @MichaelOllerton.
 

Resources
Guided poster Devised by Emma Morgan, a teacher of mathematics, to guide students when presenting their inquiry. Emma blogs here about using Inquiry Maths.

You can read more examples of how this inquiry has developed in the classroom on the primary section of the website.  
 
Alternative prompt
Mark Greenaway (an advanced skills teacher in Suffolk, UK) designed this prompt to encourage students to compare the product of 21 x 32 and 12 x 23. The picture shows the questions and comments from one of his classes. The comment in the top left-hand corner is intriguing and goes a long way to explaining how equations of this type 'work'. 

You can follow Mark on twitter @suffolkmaths.
Different lines of inquiry in a mixed attainment classroom
These are the questions and observations from a year 7 mixed attainment class at Haverstock School (Camden, London, UK). The students noticed the 'switch' and 'double and half' properties of the equation, verified its truth and went on to ask if there were other equations with similar properties. Through a
structured inquiry, some students consolidated their knowledge of multiplication using the formal column or grid methods. The teacher guided other students to use an algorithm (right) to find equations of the same type. In a more open inquiry towards the end of the second lesson, a small group of students looked for examples with three- and four-digit numbers. They found, for example, 124 x 842 = 421 x 248 and 1224 x 8442 = 4221 x 2448 by basing their search on the initial observation that "the digits used are 1,2 and 4." 
(The picture shows Nafisa's use of the algorithm to find 24 x 84 = 48 x 42.)

For more on the Levels of Inquiry Maths, see the post here.
  
Exploring multiplication methods through inquiry
GiGi Jackson posted these pictures on twitter. They show how her year 7 numeracy class at Castle Manor Academy (Haverhill, Suffolk, UK) responded to the prompt. GiGi reports that the students enjoyed exploring different multiplication methods to verify the prompt is true and then to create their own equations with the same properties.

GiGi is second-in-charge in the maths department at Castle Manor Academy. You can follow her on twitter @GJacksonMaths.
  
A slow start to generate fast-paced inquiry
Kelly Anne Garner used the prompt with her grade 6 class at the Frankfurt International School. Kelly reports that the students started slowly as they orientated themselves towards the prompt, but "then the questions came and didn't stop." The slow start is a necessary part of inquiry to "provide for a self-perpetuating chain reaction of interactions in the class" (Zuckerman, see this post). As Kelly explains, inquiry involves "a paradigm shift in thinking
as pupils are used to being asked the questions in math rather than asking the questions."
Students' initial questions and observations about the prompt.
  
Students use prime factors to show the prompt is true.
  
"I notice that...." Students share their ideas about the properties of the prompt.

Kelly posted the pictures on twitter. You can follow her @KellyAnneGarner.
 
Encouraging curiosity and risk taking
Carly Kaplan, a year 6 teacher at Meadowside Primary School (Burton Latimer, UK), used the prompt to encourage risk taking and curiosity. The pictures above show the initial questions and ideas from two of the pupils. For their first inquiry, the approaches show deep mathematical thinking. Carly describes how the inquiry developed: "The pupils needed guiding a bit initially as it was outside their comfort zone but some were able to start following their own line of inquiry eventually. They all wanted to just solve the calculation first which was interesting to observe." Carly had planned the inquiry to coincide with a visit from Ofsted inspectors to the school. She reports that, unfortunately, the inspectors did not visit the classroom and see the pupils' creativity and enthusiasm.

You can follow Carly on twitter @SculptingMinds. She posted more pictures from the lesson here.
  
Inquiring into the prompt
These responses to the prompt come from groups of year 9 students at Holyport College (Berkshire, UK). They test a conjecture about swapping the digits, which the second group poses as a question: "If I have a multiplication and then I swap the digits of each number, will I get the same amount?"

You can follow Holyport College mathematics department on twitter @Holyport_Maths.
  
Exploring and connecting through inquiry
Year 6 pupils at Luanda International School (Luanda, Angola) used the prompt to explore number, operations and place value. Class 6.3 commented that, "We loved engaging with this inquiry; it was exciting to find patterns and connections." Below, you can see pupils' initial responses to the prompt, the questions they posed for inquiry and a sheet that requires pupils to think about relevant procedures and concepts to support their planning. The pictures demonstrate a deep inquiry process with pupils connecting prior knowledge to develop their understanding of the mathematical structure of the prompt.

You can see other inquiries carried out by the year 6 pupils on twitter @LIS6point3.
  
Students' questions to generate inquiry
These questions come from students in years 9 and 10 at Holyport College (Berkshire, UK). The mathematics department reports that the inquiry that followed involved students in lots of multiplication, adding "it was great to see year 10 pupils taking an algebraic approach."

You can follow Holyport College mathematics department on twitter @Holyport_Maths.