- a V**Virtual tool to better understand the decimal place value system.**

This app allows you to explore large and small numbers and the meaning of place values interactively. It is not a classic practice app, but rather an experimental environment.

Once you understand how place values work, you have acquired one of the most important skills for teaching math. Understanding numbers and place values is the basis for many advanced topics such as non-counting arithmetic strategies, semi-written and written arithmetic, decimals, etc.

Numbers and Place Values app for iOS and iPadOS in action!

**Operation and options in the app**

- Change the individual place values by dragging your finger up and down. If you go past 9, you can see how the set overflows and 10 is bundled into the next higher place. If you move the number below 0, you can see how the set is "unbundled" from the next higher place.
- Swipe or tap the amount under a number to change it.
- At the bottom of the screen you can select how the place value set should be represented, for example as number tiles, Dienes material, Euro money and much more.
- On the right side of the screen there are some buttons that you can use to trigger activities:
- You can use the speaker to have the number read out to you.
- Using the speech bubble with a check mark, you can read out the number shown and check whether you have read it correctly.
- For example, the symbol for pulling apart numbers represents the number 142 from the numbers 100, 40 and 2.
- With the dice you can roll a random new number, which you can then read out loud or change.
- You can use the speech bubble with three dots to say a new number yourself. The number will then be displayed as soon as you speak (with a slight delay). Please make sure to speak loudly and clearly and avoid background noise.
- The question mark covers individual place values, the number is read out and you have to enter the appropriate numbers.

Then there are further customization options in the app:

- With the grey arrows on the left side of the number you can show or hide additional digits and thus change the number range. You can display and explore numbers up to 100 trillion.
- With the grey arrow to the right of the ones you can move the comma and thus also display decimal numbers / decimal numbers.
- You can change the colors of the digits by tapping the “E,” “Z,” or “H” label for the ones, tens, or hundreds and selecting a new color.
- If you swipe a number from left to right, the number will be covered. If you tap in the cover field, you can enter a solution and you will receive feedback as to whether your solution was correct.
- You can also preconfigure some displays individually in the global settings.

**Example activities**

In order for the experimental environment to effectively promote the understanding of numbers and place values, suitable tasks are required that challenge the student to explore and reflect on relationships, such as:

- Number dictation: Represent the number 3469, …
- Increase the ones by one more each time. Describe what happens when you increase the 9 by one?
- Represent the number 340. What happens if you reduce the ones by one?
- Why doesn't this work if you want to decrease the number 0 by one?
- Represent the number 5216. Expand the numbers using the expand button. What numbers does the number consist of (5000 + 200 + 10 + 6). Then think about what numbers make up the number 5316.
- Have a number read out to you. Now change one place. Can you read the number correctly? Use the voice input function to check whether you have read the correct number.
- Cover one spot (by swiping your finger from left to right). Now change another spot so that the covered spot is one (four) higher (lower). Explain how this works.

Additional question: Why can't you change the covered number with the next higher digit? - Tap the question mark button and solve some problems. Then show the teacher how many problems you solved (the number of problems is counted and can be reset by clicking on the number).
- Why does the leading zero disappear in the number 0453? Why does this not change the number? Why does the zero remain in the number 204 or 240?
- Use the app to think of an explanation as to why the number 567 and the number 765 are different even though they contain the same numbers.
- Addition and subtraction in writing: Work out the task 5498 – 2345 in writing and show each step in the app. What happens in the task 5328 – 3619? Use the app to think about why you have to write down carryovers in some places.
- …

Have fun experimenting and learning with the app!