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In Key Stage 3 mathematics, students learn about multiplying and dividing numbers by 0.1 and 0.001. This can be a difficult concept for some students to grasp, but it is important for them to understand in order to be successful in later mathematics courses.

One way to help students understand multiplying and dividing by a power of ten is to use the place value table. The place value table shows the value of each digit in a number, and it can be used to track the movement of the numbers when multiplying or dividing by a power of ten.

The following is a summary of the key teaching points for this lesson:

- The place value table can be used to help students understand multiplying and dividing by a power of ten.
- The decimal point is fixed in a place value table.
- When multiplying by a negative power of ten, we move the numbers to the left.
- When dividing by a negative power of ten, we move the numbers to the right.

When students approach multiplying and dividing by 0.1, 0.01, and 0.001, common misconceptions often arise. Many believe that numbers “become smaller” when multiplied by these decimals, which can result in incorrect solutions. Others may add zeros rather than shift digits in the appropriate direction on the place value table. Addressing these misconceptions early is crucial for building a robust foundation in the subject.

Students begin by revisiting multiplication and division by 10, 100, and 1000. Through matching exercises, they mentally connect numbers to their corresponding calculations.

**Key Questions:**

- What happens to a digit when we multiply it by 100?
- How does dividing by 10 affect the position of a number on the place value table?

Students learn to multiply and divide using negative powers of ten. By appealing to their intuition, teachers can help students grasp that dividing by a negative power of ten shifts numbers to the left, while multiplying does the opposite.

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**Key Questions:**

- When we multiply by 0.01, how many places does our number move, and in which direction?
- How does the position of a digit change when dividing by 0.1 on the place value table?

In the independent practice section, students will be presented with a similar set of questions as the previous slide. They will need to work through these questions independently to consolidate their learning. The teacher can use this as an opportunity to check their progress and feedback. A differentiated independent learning worksheet is included for early finishers.

**Key Questions:**

- Are there any patterns you notice when multiplying or dividing by 0.1, 0.01, or 0.001?
- How does using the place value table help in ensuring your answers are accurate?

The plenary will act to consolidate students’ learning by giving them a range of questions that includes an introduction to standard form. Students will complete this on mini-whiteboards and present their solutions, with working to the teacher, so the teacher can check progress and feedback.

- How does the concept of multiplying and dividing by 0.1 and 0.01 relate to standard form?
- Can you explain, using your mini-whiteboard, the process of multiplying a number by 0.001?

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