Algebra 2 Applications Lesson Plan Outline
Lesson Title: Arithmetic Series Lesson 2
Washington State Algebra 2 Standard Name(s) and Number(s):
A2.7.C Find the terms and partial sums of arithmetic and geometric series and
the infinite sum for geometric series.
Learning objective:
Students will be able to find the partial sum of an arithmetic series.
Vocabulary: arithmetic series, sigma notation, index of summation
Prerequisite Skills:
Evaluate expressions and understand arithmetic sequences.
Vocabulary: arithmetic sequence, term, common difference.
Material for Students:
Small whiteboard if available
Teaching Aids:
Estimated Time For Completion:
1-2 periods
References:
Gauss story (http://www.education2000.com/demo/demo/botchtml/arithser.htm)
Hansel and Gretel story (Wikipedia)
I.
PREASSESSMENT:
a. This lesson follows the arithmetic sequence lesson.
II.
INTRODUCTION:
a. Asking the students to find the sum from 1-100 as a class opener.
b. Pair: Students share their methods with a partner for few minutes.
c. Class: Discuss different strategies that the students use to find the sum.
Is there a more effective way to find the sum?
III.
LESSON:
a. Story about Gauss.
According to an old story, one day Gauss and his classmates were asked to
find the sum of these first hundred counting numbers. All the other students
in the class began by adding two numbers at a time, starting from the first
term. This is a natural reaction. Also, it is a valid approach, although it is not
the most efficient method, as will be seen. But Gauss found a quicker way.
First, he wrote the sum twice, one in an ordinary order and the other in a
reverse order:
1 + 2 + 3 + 4 + . . . + 99 + 100
100 + 99 + . . . + 4 + 3 + 2 + 1
By adding vertically, each pair of numbers adds up to 101:
1
100
+
+
2
99
+
+
3
98
+
+
...
...
+
+
98
3
+
+
99
2
+
+
100
1
101
+
101
+
101
+
...
+
101
+
101
+
101
Since there are 100 of these sums of 101, the total is 100 101 = 10,100.
Because this sum 10,100 is twice the sum of the numbers 1 through 100, we
have:
1 + 2 + 3 + . . . + 98 + 99 + 100 = 100 101 / 2 = 5050.
b. Using the logical above, assist students to develop a formula 𝑠𝑛 =
𝑛
2
(𝑎1 + 𝑎𝑛 ) for the sum of any arithmetic series. What does each term
represent?
c. Discuss the differences between arithmetic sequence and arithmetic
series.
IV.
APPLICATION
a. Find 𝑆𝑛 for each arithmetic series. Depends on the students in class,
more practice questions may be required and can be found in the
textbook.
1. 𝑎1 = 9, 𝑎𝑛 = 90, 𝑛 = 10
10
(9 + 90) = 495
𝑠10 =
2
2. 𝑎1 = 4, 𝑎𝑛 = 100, 𝑛 = 25
𝑠25 = 1300
3. 𝑎1 = −5, 𝑎𝑛 = −26, 𝑛 = 8
𝑠8 = −124
b. Find the sum of each arithmetic series. These questions require one
additional step to find n first.
1. 5 + 11 + 17 + ⋯ + 101
= 901
2. 34 + 30 + 26 + ⋯ + 2
= 162
c. Application
Hansel and Gretel are the young children of a poor woodcutter. They
have an evil step-mother who convinces the father to abandon the
children in the woods as there is not enough food to feed the whole
family. Hansel, aware of the plan, leaves a trail of pebbles back to the
house so he and his sister can find their way back home. The step-mother
is angry and locks the two children up for the night with only a loaf of
bread and water. (Wikipedia)
The next night, the woodcutter attempts the same plan again;
fortunately Hansel recalls the path forks 10 times as he walked into the
woods. Since he will be using bread crumbs this time, he decided to
leave 5 pieces of bread at the first fork, 10 at the second, 15 at the third
and so on. He figures there will be more animals as he walks deeper into
the woods and he will have better chance of finding his way back if he
leaves more bread crumbs as he goes. Please help Hansel to figure out
how many piece of bread crumbs he needs for the entire trip. (Ans: 275
pieces of bread crumbs)
V.
VI.
ASSESSMENT
a. Formative assessment throughout the class. Practices above can be done
on a small write board.
b. Exit slip: Think of a way that would help you remember arithmetic
sequences and series.
EXTENSIONS
a. Sigma notation (Σ), what is it? What other Greek alphabet do we use in
mathematics? How is Sigma notation used to find the sum of arithmetic
series?
b. Why is Sigma notation a useful way to write a series?
c. Depend on availability; it would be wonderful to show the students how
to find the partial sum of an arithmetic series on a graphing calculator.
Based on: CURRICULUM GUIDE FOR PROFESSIONAL TECHNICAL COLLEGE
INSTRUCTORS
TEACHING & FACILITATING LEARNING - LEVEL I