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The prompt
The prompt
Mathematical inquiry processes: Identify and analyse structure; generate examples; reason and generalise. Conceptual field of inquiry: Multiply and divide by powers of 10; estimate; multiply decimals.
Omar Khan, a teacher of mathematics at Drayton Manor High School (Ealing, UK), devised the prompt for a mixed attainment year 8 class.
In the question, notice, and wonder phase of the inquiry, students responded to the prompt in the following ways:
I worked out that 47 x 86 equals 4042.
The three parts of the 'sum' have the same digits.
Is it still called an equation if there are two equals signs and not just one?
The 86 has been multiplied by 10 and then 100.
On the left side the '4' stands for four tens, then four units, and then four tenths.
Are they supposed to be all equal?
Misconceptions
Omar reports that students held a misconception about place value. They described the decimal point moving in the prompt. As one said, "If you move the decimal one way, then it has to move the other way on the other side." In this way the left and right movements "balanced each other out."
In introducing the concept of balance, students were applying a concept they had learned about in solving equations. The left and right movements were described as "inverses".
Omar decided to introduce the table below to explain that the digits moved rather than the decimal point. Each place to the left made the digit 10 times bigger; each place to the right made the digit a tenth of its original value.
Mathematical vocabulary
Introducing the vocabulary of multiplier and multiplicand early in the inquiry facilitates communication. It is also helpful to discuss the prompt in terms of powers of 10, particularly with a view to making a generalisation later in the inquiry.
February 2024
Chains of reasoning
Chains of reasoning
The teacher can use a chain of reasoning with inverse operations to convince students that the prompt is true.
To show that 47 x 86 = 4.7 x 860
47 x 86 = (47 x 86) ÷ 10 x 10 = (47 ÷ 10) x (86 x 10) = 4.7 x 860
Students accept the first step without much discussion because, as they will explain, "If you divide a number by 10 and then multiply the result by 10, you get back to where you started."
However, the second step, in which the order of operations is changed, often causes more controversy. Students who have learned about PEDMAS, BIDMAS, or BODMAS will insist that 47 x 86 (in the parentheses or brackets) must be calculated before any rearrangement can occur.
Those who have learned about the hierarchy of operations understand that multiplication and division are at the same level.
Therefore, we can 'fill in' the chain of reasoning with:
(47 x 86) ÷ 10 x 10 = 47 x 86 ÷ 10 x 10 = 47 ÷ 10 x 86 x 10 = (47 ÷ 10) x (86 x 10)
To show that 47 x 86 = 0.47 x 8600
47 x 86 = (47 x 86) ÷ 100 x 100 = (47 ÷ 100) x (86 x 100) = 0.47 x 8600
A small change to a chain of reasoning gives us an equivalent calculation to the ones in the prompt. For example, we can swap the multiply and divide by 10 in the first chain:
47 x 86 = (47 x 86) ÷ 10 x 10 = (47 x 10) x (86 ÷ 10) = 470 x 8.6
and swap the multiply and divide by 100 in the second chain:
47 x 86 = (47 x 86) ÷ 100 x 100 = (47 x 100) x (86 ÷ 100) = 4700 x 0.86
Lines of inquiry
Lines of inquiry
1. Change one number
Place the first equation from the prompt (47 x 86 = 4042) in the middle of a page. Students create a pattern above by multiplying the multiplicand and product by successive positive powers of 10. They extend the pattern below by dividing by powers of 10. Students can also create a pattern by changing the multiplier.
Then they generate more examples using their own equation (including numbers with three and four digits) as a starting point.
2. Change both numbers
When students change the multiplier and multiplicand, they can estimate to find the correct product. In the example, an estimate of 0.47 x 860 is 0.5 x 1000 = 500. So the product starts in the hundreds column and the first '4' represents 400.
3. General rules
As students explore more examples, they begin to develop rules to connect the operations on the multiplier and multiplicand with the operation required for the product. For example, a year 8 student made the following generalisation:
When you multiply the multiplier by 10n and the multiplicand by 10m, you multiply the product by 10n + m.
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