This problem involves addition of 1- and 2-digit numbers, and multiples of 5 and 10. Students are encouraged to see the efficiency of multiplication over repeated addition. Attributing number values to letters, and developing an efficient equation to represent the solution for solving the problem is early algebra.
Penina is playing with her name and with numbers. She lets all the consonants equal 10 and all the vowels equal 5. So the value of Penina’s name is
10 + 5 + 10 + 5 + 10 + 5 = 45.
What is the value of your name?
Can you find at least 5 names that have a value of 30?
Penina's name goes even, odd, even, odd, even, odd. What other names have an even, odd or odd, even pattern?
What is the biggest value that a name of six letters can have? What is the biggest value that you can actually find?
- Together read the problem. Have them find the value of their own name and check with a partner that their value is correct.
- Ask for suggestions of how they will record their ideas.
- Set a time and have them work on the problem individually or in pairs.
- As the students solve the problem ask questions that focus on their understanding of the addition of fives and tens.
How are you adding these numbers?
How do you know that you are correct?
What can you tell me about the answers that you get? (end in 5 or 0)
- Call time and have groups report their findings. Have them demonstrate how they recorded their ideas. Some may use repeated addition or may skip count. Notice if students have used multiplication and build on this for the extension.
- Explain that Penina wants to find a quicker (more efficient) way to show the value of her name.
- Look at Penina's example again: 10 + 5 + 10 + 5 + 10 + 5 = 45. Have students identify the letters with the value 10 (Pnn), and those with value 5 (e,i,a)
With discussion have students see that 10 + 10 + 10 = 3 x 10.
3 x 10 = 3 times the consonant value, which can be written as 3c. Have students suggest how we could write the value for Penina's vowels. Agree on 3v.
Write 3c + 3v = 45 ( 3 x 10 + 3 x 5 = 45) and explain that Penina now has a quicker way to show the value of her name.
- Have each student write an equation for their own name using c and v and check with a partner that their equation is correct.
- Share some and check before posing the extension problem.
Extension to the problem
Penina writes an equation for her name: 3c + 3v = 45. What does she mean?
Write an equation for your name?
For what 5 names might this equation be true? 3c + 2v = 40
For what 5 names might this equation be true? 2c + 3v = 35
Have students write their own equation and find names for which it is true.
The value of student names will depend on those of your students.
A name with a value of 30 will have four letters two of which are consonants (eg. Fetu, Hana), have five letters one of which is a consonant (eg. Amaia,) or have six letters with no consonants (improbable).
Odd, even names will have a vowel, consonant pattern (eg. Elijah, Oliver). Even odd names will have a consonant vowel pattern (eg. Tane, Wiremu, Mila).
With six letters the biggest value that you can get is 60. However, it is not very likely that they will be able to find a name that has no vowels that is six letters long. So 55 is perhaps the best that the students will be able to find. An example of this is Myrtle.
Solution to the Extension:
The equation 3c + 2v = 40 will be true for names such as Ethan, Hazel, Rangi, Anika, Grace
The equation 2c + 3v = 35 will be true for names such as Tiare, Olive, Hoani, Aroha, Moana