<p> Superb Sums</p><p>This lesson plan will deal with addition problems of a wide variety, and introduce the idea of using variables ie , and such. It also explores decimal representation subtly. </p><p>Resources  Calculators may prove useful.  1 Activities sheet per student 1. Mixed-up Multiplication (5 mins) a) In the product shown to the right, the letters and represent different digits from 1 to 9. What are and ?</p><p>Solution: a) If we expand out the multiplication (exactly as if we were doing long multiplication), we see that must end in a zero. As cannot be a zero, this means that must equal 5. So now our multiplication is . Expanding this out we get</p><p>Isolating we get</p><p>Hence</p><p>So and .</p><p>2. Austere Addition (10 mins) a) In the summation to the right, the letters and are unknown. If the sum is to work out, what must they be?</p><p>Solutions: a) Solution 1; Trial and Error: Note that this solution is quite long, especially compared to solution 2. By looking at the first column, we see that must be 1 or 2. However we know that a 1 must “carry over” from the second column, otherwise both and would have to be zero. So can’t be 2, ie . If , then by looking at the last column, or . By trial and error, neither of these work. If , then by looking at the 3rd column (and remembering that the largest carry over we can get from the last column is 2) we can say that . Hence, we can find the solution . Other values of don’t work because on inspection, they are too small. Solution 2; Decimal Representation: By using decimal representation, we see that ( In other words,</p><p>Hence, , otherwise the left hand side is too large. Subtracting: So is at most 8. Bringing B over we see that</p><p>If , this is the same as saying . Then:</p><p> which is impossible as the largest the left hand side can be is 108. So . Hence:</p><p>So and Note: Students should be shown both solutions! They are two different techniques (brute force and decimal representation) and one is as important as the other.</p><p>3. Playing with digits (5 mins) a) Find all 3 digit numbers which become 9 times smaller when you erase their middle digit. Hint: Let the 3 digit number be .</p><p>Solutions: 1. The trick is to rephrase our question into an equation, like we did with the last question. Think about it carefully and you should get</p><p>We can simplify this to:</p><p>Dividing across by 2:</p><p>As must be a 1 digit number and must be a multiple of 5 (by observing the right hand side of the above equation), then C must equal 5. Hence</p><p>So to get our values of and , we simply run through the different possible expressions. Our 4 answers are and . Why isn’t an answer?</p><p>4. Activities Sheet  Attempt to answer all the questions on the activities sheet. Activities Sheet - Superb Sums</p><p>1. What is the sum of the digits of ? No calculators allowed!</p><p>2. In the following addition, each letter stands for a different digit from 1- 9. Find the digit corresponding to each letter.</p><p>3. Find the five digit numbers whose digits are reversed on multiplying by 4.</p><p>4. Find all five digit numbers whose digits are reversed on multiplying by 9.</p><p>5. In the following addition, each letter stands for a different digit from 0- 9. Find the digit corresponding to each letter.</p>