Mathematical inquiry processes: Identify properties; generate examples and counter-examples. Conceptual field of inquiry: Multiplication of numbers with two and more digits; algebraic expressions.
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 equation 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.
* 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
The teacher can build on students' findings during an exploratory phase to promote a systematic approach. For example, a pair of year 7 students found 13 x 62 = 31 x 26 by "keeping the numbers low." During a class discussion, another student noticed that 12 and 13 had now appeared in examples (the former in the prompt).
This led the class to finding 14 x 82 = 41 x 28 and then to the realisation that equations with 15 and 16 are impossible within the constraints of using two-digit whole numbers. The class then split into groups to explore numbers in the twenties, thirties, forties and so on.
In another class, Yuka developed the algorithm below, which she explained to her peers.
Later in the inquiry, her classmates developed rules for choosing the 'input number' for the algorithm.
The first number (blue) cannot be 50 or above because then the red number would be a three-digit number.
The tens digit of the second number (red) cannot be odd. If it were odd, the reverse number (green) would be odd and half that number (yellow) would not be a whole number.
That means the second digit of the first number (blue) must be below five so that the second number (red) does not have an odd tens digit.
(2) Find examples with two 3- and 4-digit numbers
Yuka used her algorithm to find examples with 3-digit numbers (see her calculations below).
Another class used the observation that the equation in the prompt uses the digits 1, 2 and 4 to create these equations:
124 x 842 = 248 x 421
142 x 482 = 241 x 248
412 x 428 = 824 x 214
Students then tested permutations of 1, 2 and 3 and 2, 3 and 4.
(3) 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.
(4) 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:
(5) 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.
See the mathematical notes for more details on the lines of inquiry.
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 to find equations of the same type (see below). 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 in the prompt are 1,2 and 4.
The picture shows Nafisa's use of the algorithm to find 24 x 84 = 48 x 42.
Kelly Anne Garner invited her grade 7 students from St Michaels University School on Vancouver Island to notice the connections in the prompt.
Kelly told the class that “no observation is too small” and reports that the students went on to notice multiple number patterns embedded in the prompt.
Inquiry in the primary classroom
Questions and observations
These are the questions and comments of Amanda Klahn's grade 4 class at the Western Academy of Beijing (China). The Academy is an international PYP school with an inquiry-driven curriculum. Amanda reports that she has been trying to get more inquiry into maths lessons: "As an introduction to using prompts I worked with my more able mathematical thinkers. There was a lot of excitement as they discovered new patterns and then went on to write some great 'What if ...' questions. There was true inquiry, collaboration and challenging of each other's thinking." The board above shows a very rich set of questions and observations with which to launch an inquiry.
Amanda writes a blog about classroom inquiries called Doing Maths: From Worksheets to Wonderings.
Planning a learning path
David Aaron's year 6 class in Blackpool (UK) posed the questions and made the observations above, which represent a very impressive start to the pupils' first inquiry! David reports that one of his pupils commented: "I like this inquiry because I'm planning my own learning path." As the inquiry developed, another pupil found 31 x 21 = 651 and 13 x 12 = 156, which is a novel result and could form the basis for another inquiry. David was pleased that the class enjoyed its first taste of inquiry.
Exploring multiplication methods
GiGi Jackson, second-in-charge of the mathematics department at Castle Manor Academy (Haverhill, UK), 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.
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 use prime factors to show the prompt is true, right.)
Making connections 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." You can see pupils' initial responses to the prompt (below left), the questions they posed for inquiry (right) 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.
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 below 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.
Students' questions to generate inquiry
The questions ('True or false?') 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." The other pictures show the inquiries of groups of year 9 students. They tested a conjecture about swapping the digits, which one group posed as the question: "If I have a multiplication and then I swap the digits of each number, will I get the same amount?"
Finding all cases for two 2-digit numbers
The picture shows the development of the inquiry by a year 7 mixed attainment class over the course of a lesson. At the beginning, the students noticed a number of the properties of the equation, including the ‘digits switched’ and ‘doubled and halved’ rules. Halima explained how to use the formal column method to verify that the two sides are equal.
Nesredin speculated that, in satisfying both rules, the equation might be unique. However, after a period of exploration, students came up with 48 x 42 = 84 x 24 and 13 x 62 = 31 x 26, the former having been created by doubling the numbers in the prompt.
The lesson ended with Muzzammil explaining an algorithm for creating equations that satisfied both conditions (see the bottom right-hand corner of the picture). He had co-constructed the algorithm in discussion with the teacher.
Amending the algorithm
The algorithm was as follows:
1. Start with a number.
2. Double the original number and place it on the right-hand side of the equation.
3. Switch the digits to create a new number to place on the left hand of the equation.
4. Halve the third number and put the new number in the final empty place.
In the second lesson, students used the algorithm to create new examples that satisfied both conditions. They included equations using three- and four-digit numbers (see the picture).
Other students amended step 1 of the algorithm by specifying that the starting number had to be less than 50 because doubling a number greater than 50 leads to a three-digit number. More numbers were eliminated because, like 48, they gave an odd number for the third step (see below). This meant the fourth number could not be a whole number.
48 x ____ = ____ x ____
48 x ____ = 96 x ____
48 x 69 = 96 x ____
The inquiry ended with students presenting their examples to the rest of the class. One pair of students listed all the numbers that could lead to two-digit examples of the type in the prompt:
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
Students check their examples to verify that the left- and right-hand sides of the equations are equal.
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'.
Mike Ollerton wrote to Inquiry Maths: "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.
The poster was devised by Emma Morgan to structure students' inquiry reports. Emma blogs about using Inquiry Maths here.