Chapter 2 - Number Sequences, Lesson 1 – Arithmetic Sequences

Most math books I’ve seen start out reviewing things the student “should” know. Boring, and often frustrating for students who aren’t on top of the details, and have to deal right off with being behind their peers. Jacobs had a brilliant idea when he started out with something broader – how do we think when we’re doing math? As we go through Chapter 2, let’s try to think about when we’re using inductive reasoning (looking for patterns) and when we’re doing deductive reasoning (showing something is true step by step).

Number sequences are right at the end in a few courses I teach, and I’m always excited when we get to this material. I think it’s more fun than a lot of the topics I teach. This lesson is the first time that I’ve felt the 3rd edition was substantially better than the 1st. It asks the student lots more questions about the ideas (there’s over twice as many problems in Set I). In both editions, he’s telling us patterns before giving us a chance to find them for ourselves. It would be fun to have more lead up before he shows us how to find the 60th term or the sum of the first 60 terms.

Notation: He also gives more notation in the 3rd edition. He uses a t with a little n after it to represent the nth term, and calls it t-sub-n. This is the standard notation (except that I think most texts use a-sub-n, which isn’t as helpful). He also uses the standard phrase “common difference” to represent the number that’s always added. (One thing he does differently from most texts is to just have spaces between the numbers, where I’ve always seen sequences written with commas. Both ways work fine. I just wanted to let you know these will look a bit different elsewhere.)

Here’s a fun set of problems from the 3rd edition:
Find the missing terms:
10 _ 70
10 _ _ 70
10 _ _ _ 70
10 _ _ _ _ 70
10 _ _ _ _ _ 70

In the first edition, he warned that the problems weren’t in order of difficulty and that some were tricky. He left the trickiest ones out of the 3rd edition. I want to show you two of them, and ask what you think. They’re tricky for very different reasons.
1. Find the missing terms: 7 _ _ _
2. Is this an arithmetic sequence? Explain. 1/6 1/3 ½ 2/3 5/6 …

In Set III, he shows you a shortcut for finding the sum of a sequence. He gives different shortcuts in the 1st and 3rd editions, but they both work. Would someone with each edition explain how the shortcut works? Then we can think together about why two different ways work.

One other thing I think would be fun to do together is make up stories that go with sequences. Here’s mine: My son has 102 hot wheels cars in a box. He gets 5 more each month. How many cars in the box at the beginning of each month? 102 107 112 117 122 …

Most math books I’ve seen start out reviewing things the student “should” know. Boring, and often frustrating for students who aren’t on top of the details, and have to deal right off with being behind their peers. Jacobs had a brilliant idea when he started out with something broader – how do we think when we’re doing math? As we go through Chapter 2, let’s try to think about when we’re using inductive reasoning (looking for patterns) and when we’re doing deductive reasoning (showing something is true step by step).

Number sequences are right at the end in a few courses I teach, and I’m always excited when we get to this material. I think it’s more fun than a lot of the topics I teach. This lesson is the first time that I’ve felt the 3rd edition was substantially better than the 1st. It asks the student lots more questions about the ideas (there’s over twice as many problems in Set I). In both editions, he’s telling us patterns before giving us a chance to find them for ourselves. It would be fun to have more lead up before he shows us how to find the 60th term or the sum of the first 60 terms.

Notation: He also gives more notation in the 3rd edition. He uses a t with a little n after it to represent the nth term, and calls it t-sub-n. This is the standard notation (except that I think most texts use a-sub-n, which isn’t as helpful). He also uses the standard phrase “common difference” to represent the number that’s always added. (One thing he does differently from most texts is to just have spaces between the numbers, where I’ve always seen sequences written with commas. Both ways work fine. I just wanted to let you know these will look a bit different elsewhere.)

Here’s a fun set of problems from the 3rd edition:

Find the missing terms:

10 _ 70

10 _ _ 70

10 _ _ _ 70

10 _ _ _ _ 70

10 _ _ _ _ _ 70

In the first edition, he warned that the problems weren’t in order of difficulty and that some were tricky. He left the trickiest ones out of the 3rd edition. I want to show you two of them, and ask what you think. They’re tricky for very different reasons.

1. Find the missing terms: 7 _ _ _

2. Is this an arithmetic sequence? Explain. 1/6 1/3 ½ 2/3 5/6 …

In Set III, he shows you a shortcut for finding the sum of a sequence. He gives different shortcuts in the 1st and 3rd editions, but they both work. Would someone with each edition explain how the shortcut works? Then we can think together about why two different ways work.

One other thing I think would be fun to do together is make up stories that go with sequences. Here’s mine: My son has 102 hot wheels cars in a box. He gets 5 more each month. How many cars in the box at the beginning of each month? 102 107 112 117 122 …

- Notes by Sue VanHattum