Teaching Arithmetic Progressions and Series to Grade 7 involves explaining the concept of a sequence, and how to find its different components.

The following are the steps on how we deliver this lesson to our Grade 7 students:

1. Introduce the concept of a sequence: Start by explaining what a sequence is - a list or ordered set of numbers. Emphasize that the numbers in a sequence are arranged in a specific order.

2. Define arithmetic sequence: Explain that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. This constant is called the common difference, represented by the letter 'd'.

3. Provide examples and non-examples: Show students several examples of arithmetic sequences and emphasize how the common difference is constant. Also, present them with non-examples of sequences that do not have a constant difference.

4. Find the common difference: Teach students how to find the common difference by subtracting any term from its previous term. Explain that this value remains constant throughout the arithmetic sequence.

5. Determine the nth term: Explain that the nth term of an arithmetic sequence is represented by the letter 'n' and can be found using the formula: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.

6. Identify specific terms: Teach students how to find specific terms in the sequence by substituting the position of the term into the formula. For example, if they need to find the 5th term, they would substitute n = 5 into the formula.

7. Find the sum of a series: Introduce the concept of a series, which is the sum of all the terms in a sequence. Teach students how to find the sum of an arithmetic series using the formula: Sn = [n/2] * (a1 + an), where Sn is the sum of the first n terms.

8. Practice problems: Provide students with a variety of arithmetic sequence and series problems to solve. Start with simple examples and gradually increase the difficulty level.

9. Real-life applications: Discuss real-life applications of arithmetic sequences and series, such as calculating interest rates, predicting future population growth, and analyzing patterns in data sets.

10. Review and reinforcement: Review the concepts covered and provide additional practice problems for students to solidify their understanding. Encourage students to explain their reasoning and steps taken to solve problems.

11. Assessment: Evaluate student understanding through quizzes, tests, or other forms of assessment. Provide feedback and offer support where needed.

By following these steps, the students can effectively learn arithmetic sequences and series, ensuring that they have a strong grasp of the concepts and can apply them confidently in various mathematical scenarios.