# Average Number of Letters

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1 Find the average number of letters in a group of 5 names. As a group, discuss strategies to figure out how to find the average number of letters in a group of 5 names. Remember that there will be 5 groups in the end because you started with 5 names. 1 of 9

2 Suggested Grade Span Pre-K-2 Task Find the average number of letters in a group of 5 names. As a group, discuss strategies to figure out how to find the average number of letters in a group of 5 names. Remember that there will be 5 groups in the end because you started with 5 names. Alternate Versions of Task More Accessible Version: Have each person in your group put the letters of their name on a square of graph paper. Cut the squares out and use them to determine the average number of letters in 5 people s names. More Challenging Version: Conduct an investigation to determine the typical number of letters in the first and last names of students in your class. Context We have been working on averages in several areas including graphing averages in a science unit on bubbles. We have discussed and practiced averages of lower numbers and small groups of objects. This gave my students the opportunity to try averages in a different context. What This Task Accomplishes This problem gives students the opportunity to apply their knowledge of making even groups out of objects they can see and move around. For kids with uneven groups, they have a chance to physically divide whole numbers and fractions. What the Student Will Do Several kids who have had experience with averages will have suggestions like using a list of the names and/or cutting out the letters. Provide them with both so they have options to use 2 of 9

3 their own strategies. Make five even groups with cut out letters and split the leftover letters up into equal pieces for each of the five groups (halves, quarters, fifths, tenths) and call the average number rough, but close. Make five even groups and split the leftover letters into the same number as the groups so that each group gets a different size piece of the letter and call all pieces one. Make five even groups of letters (if they have a number divisible by five). Make six or seven groups so that the numbers divide up evenly. Make five groups of three or four and then split the last letters and put them in groups even though splitting was not necessary. Cross off letters on a list and rewrite them in blank spaces until the groups are even. Cross off letters on a list and rewrite some in other spaces. Time Required for Task Preparation Activities: Blocks - 20 minutes Bubbles - 30 minutes with four or five kids one at a time Problem minutes Interdisciplinary Links Science - average size in centimeters and/or inches. Language Arts - exposure to spelling of names. Social Skills - learning each other's names and how to spell them. Teaching Tips Teach fractions before this problem so they have the knowledge and language for dividing up the letters on the squares. Give a total number of letters in five names that will only need halves and quarters for the leftover letters. Hand them the same list for cutting and/or rearranging letters. Give students who cut letters pieces graph paper with squares the size of the letter pieces so that they can keep the letters organized. To prepare my students we did two activities: (1) We did an average activity with 15 blocks given to five students in a group (each gets between one to five blocks). Put all the blocks in the middle of the table. Each student takes the same number of blocks until the blocks run out. The number of blocks they have in their hand is the average number of blocks (three). Try this with a few different numbers that will end up with an even average. (2) We did a science activity measuring and recording the sizes of 10 bubbles. Kids who are eager, use a calculator to add up the 10 sizes and divide by 10 (with the guidance of a teacher). 3 of 9

4 Describe to the other students what was done and explain that a calculator is usually necessary for larger numbers. Suggested Materials Two lists of five names for each student (one for cutting and checking afterwards) Envelopes for storing letters Scissors Pencils Paper Scotch tape Possible Solutions Solutions will vary depending on the number of letters in each name. Clifford, Zach, Lizzie, Catherine, Brittany = 35, 35 divided by 5 = 7 Nathan, Toby, Nora, Meredith, Stephanie = 31, 31 divided by 5 = 6.2 Sarah, Ashley, Sam, Harper, Jacob = 25, 25 divided by 5 = 5 Grey, Maya, Michael, Zane, Jory = 23, 23 divided by 5 = 4.6 More Accessible Version Solution: Answers will vary. Look for correct reasoning and documentation. More Challenging Version Solution: Answers will vary. Look for correct reasoning and documentation. Task Specific Assessment Notes Novice This student had limited awareness of the problem. His/her organization of the problem was random and weak. S/he applied inappropriate concepts and inappropriate procedures, therefore no solution was reached. S/he put a few cut letters on the list (his/her name, and a few random 4 of 9

5 letters), folded five others and placed them in blank spaces and left about 22 letters off the list. This student did not have a clear or connected explanation of his/her solution. S/he was unable to explain his/her strategy. Apprentice This student was able to use the correct mathematical procedure to solve this problem, but could not carry out the procedure to find a correct solution. His/her strategy was partially useful. S/he lined up the different size pieces vertically in seven columns and came to the conclusion that the average was seven. This student has a basic understanding of the problem, but a lack of understanding of fractions and was unable to get close to a solution. Practitioner This student has a broad understanding of the problem. S/he was able to come close to the correct solution using fractions. S/he had three extra pieces to put in five groups. S/he placed a whole in a group, and split the other two in half and put them in the remaining groups. S/he labeled each group with either 4 1/2 or five and called the average 4 3/4 written as 4 1/3. S/he has a solution that reflects effective mathematical reasoning because s/he came up with a roughly correct average. The explanation was clear. Expert This student has a deep understanding of the problem. S/he divided up the six cut halves into the five groups and was able to express the amount in each group using correct terminology. S/he referred to the leftover half as "a little extra in each group." S/he proceeded to cut the last half into five equal pieces. S/he concluded that the average was 4 and 1/2 and 1/10, which is true. S/he applied his/her understanding of fractions and the procedures necessary to find a correct solution. His/her explanation was clear. S/he generalized from previous mathematics experience. 5 of 9

6 Novice 6 of 9

7 Apprentice 7 of 9

8 Practitioner 8 of 9

9 Expert 9 of 9

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