Do You See a Pattern? by Michael Naylor

Patterns are part of what make math so beautiful – your students might agree after these fun activities

Mathematics has been called "the science of patterns." This month, we'll look at mathematical patterns and how to use these patterns to make beautiful line art while practicing number and operation sense, geometry and measurement.

String art (Grades K-5)
Your students will be surprised at the dramatic artwork they can create if they measure carefully and follow a pattern. Ask your students to draw two line segments that meet at a right angle and place small, equally spaced marks on the line segments (3" line segments with 1/4" marks works well). With younger children, you can provide these marked line segments.

Now ask your students to sequentially connect the pairs of marks with straight lines, starting at the first mark on each segment so that the lines cross as they are shown at left.

They've just made beautiful string art! Students can now design their own arrangements of line segments.

Digit circles
A digit circle is a circle with digits 0–9 equally spaced around the outside. Fabulous patterns can be created while giving your kids practice with number operations. Make up some sheets that have several digit circles on them, or download a blackline master here. PDF 50KB

Addition circles (Grades 1-2)
Ask your students to choose an "adding number," say 3. Starting with 3, add 3 to get 6. On the digit circle, draw a line segment from 3 to 6. Now add 3 again to get 9, and make another line segment from 6 to 9. Adding 3 gives 12, but look only at the number in the ones place, 2, and connect from the 9 to the 2. Continue in this manner until you get back to the starting point.

Have your students try these with all digits from 0-9, and then compare the designs to look for similarities and differences. There are some striking patterns that emerge; for example, the designs are identical for pairs that add to 10, so 1 and 9 make the same design, as do 2 and 8, 3 and 7 and 4 and 6. The designs are created in the opposite direction, though. Ask your students to come up with ideas as to why this might be. One way to think about it is that adding 3 gives the same last digit as subtracting 7 and vice versa.

Multiplying patterns (Grades 4-8)
Choose a multiplying number, then multiply each of the digits from 0-9 by that multiplier. Draw an arrow from that digit to the number which is the last digit of the product. For example, suppose your multiplying number is 7. 1 x 7 = 7, so draw an arrow from 1 to 7. 2 x 7 = 14, so draw an arrow from 2 to 4 (4 is the last digit of 14). 3 connects to 1 (3 x 7 = 21), 4 connects to 8 (4 x 7 = 28), and so on.

Again, make designs for each of the multipliers, 0-9, and then compare the designs and look for connections. Not only are the designs surprising and beautiful, but there are also amazing similarities (as in the last activity) between pairs of designs of multipliers that sum to 10.

Problem-of-the-week (Grades 4-8)
If there are four people at a party and everyone shakes hands with everyone else, how many handshakes are there? What if there are five people? Six people? 100 people?

Encourage your students to draw pictures to help solve this problem. A string-art picture like the one shown here can really help.

Some patterns your students may find: With six people, the first person needs to shake five hands. The second person shakes four new hands (they already shook with person #1), the third person shakes three new hands, the fourth shakes two and the fifth shakes one hand.

Everyone will have shaken hands with the sixth person. The total number is 5 + 4 + 3 + 2 + 1 + 0 = 15. With 100 people, then, the total is 99 + 98 + 97 + _+ 2 + 1 + 0 = _?

Another way is to see that in a group of six people, each person shakes hands with five others for a total of 30 hands shaken. But "a handshake" is two people shaking hands, so 30 is exactly two times too many. 30 ÷ 2 = 15 handshakes. So with 100 it should be 99 x 100 ÷ 2 handshakes. Can you spot the pattern?

For blackline master click here. PDF 50KB

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.