Doubling Up

The Problem

Gill, was playing with her name and with numbers. If A = 1, B = 2, C = 4, D = 8, E = 16, F = 32, G = 64 and so on, the value of Gill’s name was 64 + 256 + 2048 + 2048 = 4416.

What is the value of your name?

Change the rules so that the square root of the value of your name 248.

Teaching sequence

1. Tell the students Gill’s story and let them find the value of some word, room, say.
2. Make sure that they understand how you find the value of a word. Get some students to share their ideas for calculating the value of letters near the end of the alphabet.
3. Ask the students to predict whose names will be the "easiest" to calculate.
4. Then ask them to find the values of their names. (That is, just their first names.) Get their partner in their group to check that they have found the right value for their name.
5. As the students solve the problem circulate asking questions that focus on their ability to find powers of numbers.
   How are you finding the value of the letters in your name?
   Is your answer reasonable? How do you know? (estimation)
   Are you convinced that you have a correct value for your name? Why?
6. Then ask the students to put themselves into groups all of whose values end in the same two digits. Get them to think about their names to see if there is a good reason why they are all in the same group.
7. In groups, the students could then find some way of ending up with a name value whose square root is 248. Can they arrange for both members of their group to have the same value using the same rules? They could be asked to find more than one way to get 248² as their name value.
8. Help the students that need it.
9. The quicker groups can go on to tackle the Extension problem.
10. Get a few groups to report on what they have done.
11. Give all students time to write down something about the answers they got. This should help them to understand what is going on.

Extension to the problem

If you substituted x for a consonant and 2x for a vowel what value would you get for your name if (i) you added the terms together to get the value and (ii) you multiplied the terms together?

Solution

The answers that you get for the first part of the question will depend upon the names of the students in the class.

The first thing to note about 248 is that it is really 248² that we want to get. Now 248² = 61504. So, if your name was Garry you would need to use some big numbers to get 61504. The trick is to put anything you like for G, A and R and then choose Y to make up the required number. There are many ways to do this.

Solution to the extension

Suppose that your name was Gill. Then in (i) you would have x + 2x + x + x = 5x. In (ii) you’d get x.2x.x.x = 2x⁴.

N.B. This problem gives lots of scope for exploration. You could find the biggest valued name, the smallest valued name, a name whose value is a whole number, names whose values are halves, etc. If you wanted to go down the route of the Extension, you could use x for consonants and 1/x for vowels.