"Two, four, six, eight, who do we appreciate . . . the Red Sox, the Red Sox, hurrah!" Did a coach ever have you chant this classic appreciation out to the opposing team after a game? Besides being a very sportsmanlike gesture, this is also an example of a number sequence. These kinds of patterns are everywhere in mathematics. Sequencing can be extremely simple (like the 2,4,6,8 pattern), or can be extremely difficult. Sequencing is used in basic number patterns, geometry, statistics, algebra, and beyond. Recognizing patterns in numbers is an essential skill for all students to learn.
Each day, I do a math warm up with my students before getting into the "real" lesson. I divide the whiteboard into 4 sections, and put 4 different problems in each section. Students have small notebooks, and they divide each page into four sections, just like I do on the whiteboard. They copy down the problems, and do their best to solve each one. After a few minutes, when almost everyone has finished, I call on a volunteer to come up and solve one problem at a time. The students put their thumbs up if they think the answer is correct, or their thumbs down if they think it is not correct. The "thumbs up/thumbs down" response is an important one for me, because I get a sense of how the whole class is doing with that mathematical concept. The warm up activity is a great way to get your students ready for their math lesson. Additionally, it's an excellent way to reinforce skills students already know, to practice the new skills they're working on, and a great way to introduce new concepts.
A typical day's warm up problems for upper elementary school students could be:
1. 973 X 7 =
2. Round 3,765 to the nearest: 10: ____ 100: ____ 1,000: ____
3. What is 484 divided by 4? ____
4. Complete the pattern: 3, 6, 10, 15, 21, ____
The last problem is a sequencing problem. The first thing I tell my students to do when tackling these types of problems is to identify whether the numbers are going up in value, or down. If they're going up, they will have to think in terms of addition or multiplication. If they're going down, they will have to think in terms of subtraction or division. Looking at the difference between each of the numbers is another important thing to do. Since these numbers are close together, addition is the operation to use. Once they have that figured out, the students simply have to "play" with the numbers to find the pattern. Sequencing problems provide an excellent opportunity to use one of the most important math skills there is.......mental math, or solving a problem without the use of pencil and paper. Of course, pencil and paper is fine too! But I always encourage them to try to figure out sequencing problems "in their head" first.
This is an example of how I might explain sequencing to students. First I would ask them to find the pattern embedded in the sequencing problem? Look at the difference between each of the numbers. There is a difference of 3 between 3 and 6, a difference of 4 between 6 and 10, a difference of 5 between 10 and 15, a difference of 6 between 15 and 21. So, to extend the pattern, add 7 to 21, and you get the answer.......28!
Luckily, students usually enjoy sequencing problems because they are akin to puzzles. Most students love to solve puzzles or riddles, so I am often asked to include a sequencing problem in each day's math warm up. Here's one of my favorites for you. By the way, it's tough! A student recently pointed out that there are 2 possible answers for this one. They appear at the end of the fourth lesson plan description below!
1, 2, 3, 5, 7, 11, _____
What follows are lessons that can give your students more practice with sequencing.
Math Sequencing Activities:
This wonderful kindergarten through second grade lesson provides many ways for the young learner to explore patterns and sequences in numbers. Students use pattern blocks, connecting cubes, grid paper, crayons and calculators to help them explore their numbers. This well-written lesson has many fine links, worksheets, and instructions to help you.
Students in second through fourth grades will love this lesson. They use Cheerios to complete a variety of patterns that are established on a worksheet embedded in the plan. Students also create their own number patterns, and identify the rules they used. This is another really fun lesson that uses food in a clever way.
This excellent fourth and fifth grade lesson utilizes video and the Internet. Students explore patterns in a sequence of shapes, designs, and numbers, and identify the next item or number in the series. This creative lesson uses a Cyberchase episode which requires students to solve a crime by determining a pattern.
These lessons for fifth grade and above gets into higher-level explorations of patterns and sequences in numbers. Students are challenged to complete patterns which require the use of algebraic formulas. The Fibonacci Sequence, square numbers, and Pascal's Triangle are all addressed.
Sequencing Problem: 1, 2, 3, 5, 7, 11, _____
In order to answer this question you have to think in terms of prime numbers. I have listed the first six prime numbers on the number line. So one answer could be 13. One of my students pointed out that there was a pattern found in the differences between the numbers....1,1,2,2,4......so another answer could be 15.