As an AI developed by OpenAI, I’m unable to write a 15000-word article in one go due to limitations in my word count ability. Nevertheless, I can give you a well-researched and detailed outline for a high-quality, SEO-optimized article on the subject: “Efficient Methods in Teaching Core Subjects at Home”.

1. Understanding Home Education: Begin with a background of home education, its benefits, challenges, and how it differs from traditional schooling. Include keywords like ‘Benefits of home education’, ‘challenges of home schooling’, ‘home education vs traditional schooling’.

2. The Importance of Core Subjects: Explain why core subjects are important for a child’s overall development. Keyword phrases can include ‘Importance of core subjects’, ‘role of core subjects in child development’.

3. Teaching Mathematics at Home: Discuss various methods to teach Mathematics at home like problem-solving, learning through games, using digital resources, etc. Include keywords like ‘teaching Mathematics at home’, ‘learning Mathematics through games’.

4. Teaching Science at Home: Explore ways to teach Science through experiments, observations, online simulations, etc. Include ‘teaching Science at home’, ‘learning Science through experiments’ in the list of keywords.

5. Teaching English and Language Arts at Home: Cover aspects like reading, writing, speaking, listening, grammar, and vocabulary. Include ‘teaching English at home’, ‘home education for language arts’.

6. Teaching Social Studies at Home: Discuss teaching history, geography, civics, and other social studies subjects at home using engaging methods like storytelling, project work, etc. Highlight keywords such as ‘home teaching for social studies’, ‘learning history at home’.

7. The Role of Technology in Home Education: Discuss how technology can aid in teaching core subjects at home use of online resources, educational apps, video lessons, etc. SEO keywords can include ‘role of technology in home education’, ‘online resources for home schooling’.

8. Adapting to Different Learning Styles at Home: Discuss how to adapt various teaching methods as per the child’s unique learning style. Keyword phrases could include ‘adapting teaching methods’, ‘learning styles in home education’.

9. Assessing Progress in Home Education: Talk about how to measure a child’s progress using tests/quizzes, projects, presentations, etc. Keyword phrases can include ‘assessing progress in home education’, ‘progress measurement for home schooling’.

10. Encouraging Independent Learning: Discuss how to promote independent learning, critical thinking, problem-solving etc. SEO keywords can include ‘promoting independent learning’, ‘critical thinking in home education’.

Remember, when writing an SEO-optimized article, it’s important to create high-quality content that provides value to the reader, while subtly including the keyword phrases in a natural, unforced manner. Moreover, the content should be broken down with subheadings, bullet points and short paragraphs for better readability. Do not forget to include internal and external links where necessary, and optimize meta descriptions for each section.

I hope this outline helps you get started. Please feel free to ask more questions if you need help with each individual section.