Skip to 0 minutes and 8 secondsMICHAEL ANDERSON: We can think about this calculation in another way. So we were trying to work out 0.25 multiplied by 0.2. Now, it might not be immediately obvious what the answer to this question is, but we can think of a very similar question, which is 25 multiplied by 2. Now, I can do this. 25 times 2 gives us 50. And these two numbers are very similar to the numbers that we're actually working with, but they're obviously a little bit bigger. So what we're going to have to do is take this calculation, and then scale the numbers down. So if I want to go from 25 to 0.25, well, I might try dividing by 10.
Skip to 0 minutes and 49 seconds25 divided by 10 gives me 2.5. And I multiply that by 2, well, that's going to give me an answer of 5. Our answer has been scaled down by 10 as well but I'm still not quite there yet, so what I'm going to do is I'm going to repeat it. So I'm going to divide this by 10 again. It's going to be 0.25-- the value that I wanted. Now, I can pretty much see that this is going to divide by 10 as well. So I'm going to get 0.5, and that seems to work, because 0.25 times 2 gives us 0.5. And we're nearly there, but we've still got this 2 where we need to 0.2.
Skip to 1 minute and 26 secondsSo what I'm going to do this time is divide this one by 10 to get 0.2. The 0.25 stays the same. So 0.25 times 0.2-- well, I'm going to have to divide this one by 10 as well to give us 0.05. And that's our final solution.
Skip to 1 minute and 44 secondsPAULA KELLY: Which makes sense, because our previous method for the same sum, had the same answer.
Skip to 1 minute and 48 secondsMICHAEL ANDERSON: Thankfully.
Skip to 1 minute and 49 secondsPAULA KELLY: Very good news. And I like how you're consistently dividing by 10. Could you do a quicker way to get from 25 to 0.25, go straight to dividing by 100?
Skip to 2 minutes and 2 secondsMICHAEL ANDERSON: Yeah, that would definitely work, because 25 divided by 100 gives us that 0.25. And then similarly, what we do to one part of our multiplication we have to do to the answer that we had as well, so I'd divide that by 100. Quite a common mistake is for students to go, well, I'm going to divide this 25 by 10 and this 2 by 10 as well, and then only to divide the 50 by 10. Whereas actually, you have to take it one process at a time. So we've set it out like this way, so you can hopefully see quite clearly that each step were divided by 10. So each step, our answer is divided by 10 as well.
Skip to 2 minutes and 40 secondsSo let's see how this method works with a different example. I had 0.4 multiplied by 0.15. If you want to pause this video and have a go yourself now, we'll have a look at the answer in just a few seconds.
Skip to 2 minutes and 58 secondsOK, so I'm not going to go to the answer straightaway, but what I'm going to do is construct a different calculation which will hopefully give you the same answer. So if I ignore all the decimals, I get 4, and I get 15. Now, thankfully, 4 times 15 is something I can do. That's going to give us 60. So we're going to do the same kind of idea of scaling our numbers down to hopefully scale our answer down to the correct answer. So for the fourth, I'm going to divide by 10 to give me 0.4. Multiply that by 15. So I've not changed that one, but our answer is also going to be scaled down by a factor of 10.
Skip to 3 minutes and 35 secondsSo 0.4 times 15 gives us 6. Now, this is exactly the same as in our question, so I'm going to leave that one there. But I need to get 15 down to 0.15. Now, I can do this in two steps. I can divide by 10 first, and that's going to give me my 1.5. So 0.4 multiplied by 1.5 is going to give me 0.6, because I've remembered that if I'm dividing this bit by 10, I have to divide my answer by 10. And then the final step would be to scale this down one last time to be 0.15.
Skip to 4 minutes and 9 seconds0.4 multiplied by 0.15-- well, that's going to give me an answer of-- if I divide this one by 10 as well-- 0.06.
Skip to 4 minutes and 18 secondsPAULA KELLY: And then similar to how we said before, we could do one step dividing by 100, as long as we are very clear to divide our answer by 100 as well.
Skip to 4 minutes and 26 secondsMICHAEL ANDERSON: Exactly.
Multiplying decimals: scaling method
In this step, we consider an alternative method to multiplying two decimals together. In this method we scale up our problem to start with a calculation to which we do know the answer: \(25 \times 2 = 50\). Then, we systematically scale down the question and the answer until we have the desired calculation of \(0.25 \times 0.2\).
Michael notes that a common misconception with students is where both numbers being multiplied together are divided by an amount, and the student doesn’t then divide the answer by the correct amount. The amount the answer is divided by must be the product of the amounts that the two numbers are divided by. For example, if both 25 and 2 are divided by 10, then the answer must be divided by 100 (which is 10 x 10).
A common mistake made by students is that by attempting to make the method ‘more efficient’ each of the numbers in the question are scaled by a factor of ten and the solution is scaled by a factor of 10 thus creating an error.
For example: \(25 \times 2 = 50\).
If the 25 is divided by 10 and the 2 is divided by 10 and the 50 is divided by 10 to give \(2.5 \times 0.2 = 5\). This creates an error.
The equals sign can be read as ‘is the same as’.
It must be made clear that whatever is done to the statement on the left hand side of the equals sign, the exact same thing must be done to the right hand side to keep both sides the same. In the example above, if the 25 is divided by 10 and the 2 is divided by 10 the statement on the left hand side of the equals sign has been divided by 10 twice, i.e. divided by 100. This means that the statement on the right of the equals sign also has to be divided by 100 if it is to be kept the same.
\(2.5 \times 0.2 = 0.5\) is correct.
In an attempt to eliminate this error it is advisable to scale only one number on each side of the equals sign at a time.
Part way through the video, we ask you to find the product of 0.4 and 0.15. Make sure you have a go first, before watching our solution.
The game ‘Quarto’ is a good resource to use in class. You may like to give it a go. It is a game for two players so you will need to challenge a friend or colleague.
Draw a number line from zero to ten on a piece of graph paper.
Choose nine numbers. You can choose any numbers you wish but think carefully about the ones you choose. A good range of numbers works best, for example:
103, 0.4, 0.07, 25, 1.5, 5, 11, 0.5, 8.3
Each player takes it in turns to choose two numbers which they think will multiply together to give an answer between 0 and 10. Their answer can be checked by their opponent using a calculator. If the answer given is correct they mark that number, as accurately as they can, on the number line. If the declared answer is incorrect that player misses their go. The second player then chooses numbers to multiply and mark their answer on the number line using a different colour pen.
The winner is the first player to get four marks in a row with none of their opponent’s marks in between.
The game can be adapted to include division as well if desired.
Whodunnit: in this activity, students have to use their skill at multiplying and dividing decimals in order to check whether the correct answers have been given. By checking the given answers and spotting errors, students identify who committed the crime, who the victim was and who were the other two suspects, where the crime was committed and when.
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