### PROBLEM: Joshua made a number line to show the times he started and finished his afternoon reading. The timeline below indicates his starting time and ending time. He read for an hour. What was his ending time?

- A. 3 o’clock
- B. 3:30 p.m.
- C. 4:15 p.m.
- D. 4 o’clock
- E. 4:15 a.m.

Interpreting this timeline graph to select the correct response of C involves knowing how to tell time (Goal #6) but also an understanding for units of time:

- 60 minutes is an hour
- An hour is divided into 4 quarter-hour parts of 15 minutes each. Let’s count them off: 15, 30, 45, 60 minutes

The problem also involves Goal #1: ordering numbers. However, ordering numbers on a timeline requires a different strategy than ordering counting numbers 1, 2, 3, because fractional parts of hours are involved.

### Count Discrete Quantities & Measure Continuous Quantities

The numbers 1, 2, 3, etc. are used to count and compare discrete quantities, such as number of people; number of items sold; or number of toy cars. Contrarily, time is a continuous quantity and needs to be measured, not counted. For continuous quantities, every numerical reading on the relevant measurement scale is an approximation. And for this timeline, there are many possible readings between the whole-number times of three o’clock and four o’clock: 3:01; 3:15; 3:30; and 3:45, and 3:5999. Now, don’t think I’m suggesting you talk with four-year-olds about meanings for discrete and continuous, but I’m asking you to keep this distinction in mind yourself when you talk to children about ordering numbers.

**Use number lines for representing for continuous quantities and measurements and not for representing discrete quantities and counts. On a number line, there are many, many values (mathematically speaking an infinite number) between any two counting numbers such as 1 and 2.****Use lists, tables, and other diagrams to order the counting numbers. Show in visual representations that the numbers in between the whole numbers are not being considered.**

### A Change in statement of Goal #1

At this point in the blog post, I’ll remind you that my Top-Ten List is a working draft of ten important things that third graders need to know and be able to do if they are going to attain world-class levels of proficiency by third grade. I plan to make changes, deletions, and additions to these descriptions based on comments you share with me and on new insights as we consider more world-class test questions. Recently, I received the following comment on my top-ten list from Dr. Karen Fuson, who has studied early childhood math education for many years and published widely in the field. During my doctoral program at the University of Louisville, the professors I worked with in the Early Childhood Research Center introduced me to her work. After reading the Top-Ten List on the website she suggested the following:

### “Don’t start your top-ten list with number lines as they can be confusing to children,” wrote Professor Fuson.

And taking her advice, I eliminated the word “line” from the statement for Goal #1 and added the following two sentences:

- Read, interpret, and construct visual displays of number order
- Develop mental representations of number order as well.

Math proficient students construct mental number lines for solving problems involving continuous quantities and construct mental lists, tables, etc for ordering the counting numbers when discrete quantities are involved. In our selected reading-timeline problem, a linear model of number order was presented, and the student was required to read and interpret it. Professor Karen Fuson also asked me to suggest visiting her website at www.karenfusonmath.com.

### Fractions, Clocks, and Rulers

This reading-timeline problem is absolutely packed with big math ideas! In addition to Goal #1 (ordering numbers), a basic understanding of fractions (Goal #7) and telling time (Goal #6) were also needed to find the correct answer. Furthermore, the use of a ruler-like linear scale to represent elapsed time also relates measuring lengths (Goal #8). Adults who are actively involved in the lives of young children should, throughout the preschool and early elementary years, seek opportunities to introduce fractions. They should also encourage children’s inquiries about ever-present measurement devices such as clocks and rulers.

Display for preschoolers analog clocks (with correct time, of course) and refer to analog clocks in the presence of children throughout the preschool and primary years. By introducing basics for telling time early, clocks can help preschool children build a sense and a structure for thinking about elapsed time. A clock face provides a great model (example) for partitioning circles into halves, thirds, and fourths (Goal #7). Connect sliced-pizza models of fractions with clock-face models. Here’s a poster to illustrate these ideas.

#### Paper Fraction Strips

Goal #8 involves using rulers to measure, compare, and estimate lengths (remember that every measurement is only an estimate). In my experiences, young children seem interested in rulers and other length-measuring devices. Use this interest to precede teaching ruler measurement and fractions (Goal # 5) using linear/rectangular models of paper fraction strips. This second poster illustrates how to use strips of paper for separating rectangles into halves, thirds, and fourths. These fraction strips can be used before, or simultaneously, to a child’s exploration of a one-foot ruler. Notice with the fraction strips, I used words to name the fractions instead of the symbols 1/2, 1/3, and 1/4 because those who have studied the learning of fractions say that it is better to introduce fraction symbols later (after children understand the meaning of the counting-number symbols. Also, many children and even adults are confused when comparing one-third and one-fourth. But the poster here visually closes the case for one-third being larger.

#### A Foot Ruler

This third poster shows a foot ruler marked off in inches as well as partitioned into halves, thirds, and fourths. A foot ruler is just the right size and can be used to measure the length of everyday things, including fingers and feet. Use a foot ruler to measure the length and width of a room or a rectangular rug.

### You’re finally done !

Sorry this post was so long, but as I said before, this reading-timeline problem was packed with new and important ideas that need early starts. There were four goals in all.

- For Goal # 1 (ordering numbers) we considered for the first time ordering continuous quantities such as time and lengths.
- For Goal # 6, we considered the units of time, hours and minutes.
- or Goal # 7 (geometric shapes), we considered dividing circles and rectangles into halves, thirds, and fourths.
- For Goal # 8, we considered linear measure.

Please comment and give me your thoughts and suggestions from your own math experiences and experiences with children.