Place Value Columns

0 to 9,999,999.999

1,000,000

HTh

100,000

TTh

10,000

1,000

100

•

0.1

0.01

0.001

Number

Multiply & Divide — digits shift left or right

Add & Subtract — digits change within their column

History

Investigation Questions

Use these alongside the tool above. Enter numbers, multiply and divide by powers of 10, add and subtract, and toggle Expanded Form and Words to explore.

Enter the number 34 and press × 10. Watch the animation. Which direction do the digits move? Now press × 10 again. Describe what has happened to each digit’s position since you started.

Enter 5.7 and press × 10. What is the new number? Now press ÷ 10 to go back. If you toggle Expanded Form on, describe how the expanded form changes when you multiply by 10.

Enter 4.05 and press × 100. What happens to the zero? Does it move? Turn on the Expanded Form to check. Why is it important that the zero moves too, even though it’s a zero?

Enter 360 and press ÷ 10, then ÷ 10 again. What happens to each digit? At what point does the number become a decimal? Toggle Words on — does the spoken form make sense?

Start with the number 7. Press × 10 three times. Now press ÷ 1000 once. Do you end up back where you started? Explain why using what you know about the shift animation.

Enter 0.8 and press × 10. Now reset, enter 0.8 and press ÷ 0.1 instead. What do you notice? Try the same comparison with × 100 and ÷ 0.01. What can you conclude about multiplying by a decimal?

Enter 450 and press × 0.1. What happens? Now reset, enter 450, and press ÷ 10. Compare the two results. What do you notice? Can you explain why?

Press × 0.01 on the number 3,000. Watch the direction of the arrows. Which direction do the digits shift? Is this the direction you would expect for multiplication? Discuss.

Enter the number 56 and predict: what will × 0.1 give? What will ÷ 0.1 give? Check your predictions. Which one makes the number bigger, and why is this surprising?

Start with 2.5. Press ÷ 0.1. What is the result? Now press ÷ 0.01 on 2.5. Can you explain why dividing by a number less than 1 makes the answer bigger?

Use the tool to complete this sentence with as many operations as you can: ‘× 0.1 gives the same result as ___’. Now do the same for ‘÷ 0.01 gives the same result as ___’. Use the History to prove your answers.

Enter 397 and press + 1. Watch the cascade animation carefully. Describe what happens column by column, starting from the Ones. Why does more than one digit change?

Enter 999 and press + 1. How many columns change? Now enter 909 and press + 1. How many columns change this time? What determines whether a carry cascades?

Enter 5.99 and press + 0.01. Describe the cascade. Now enter 5.09 and press + 0.01. Why does the second one not cascade?

Start at 1,000 and subtract 1 using the − 1 button. What is the result? Turn on Expanded Form. How does the expanded form change? Why does subtracting 1 from 1,000 change every single digit?

Use only the + 1 button and the + 0.1 button to build the number 3.5 from zero. What is the minimum number of button presses? Is there a faster way using different buttons?

Enter 456. Now use only the subtract buttons (any of them) to reach exactly 0. What is the fewest number of presses you need? Can someone else beat your score?

Click the ▲ arrow above the Tens column five times, starting from 0. What number do you have? Now click the ▲ arrow above the Hundreds column once. What is the total now? Describe how the stepper arrows relate to the + 10 and + 100 buttons.

Enter 290 and click the ▲ arrow above the Tens column. What happens? Why does the Hundreds digit change when you only pressed the Tens stepper?

Start at 0 and try to build the number 3.14 using only the stepper arrows. Which columns do you need to use, and how many clicks in each?

Turn on Expanded Form. Enter 4,502. Write down the expanded form. Now press × 10. Write down the new expanded form. What happened to each part?

Turn on both Expanded Form and Words. Enter 0.072. Read the number in words. Now press × 10. Read it again. Press × 10 once more. How does the spoken form change each time? At what point does ‘point’ disappear from the words?

Enter 10,010. Turn on Words. Read it aloud. Now enter 10,100. How are the two numbers different in words? What does this tell you about the importance of zero as a placeholder?

Turn on Expanded Form and enter 300.03. How many terms appear in the expanded form? Now enter 333.33. How many terms appear? Which number has more ‘information’ in its expanded form, and why?

Enter a number where the Words display says ‘and’ in the middle (e.g., one hundred and five). Find at least three different numbers where the word ‘and’ appears. What do they all have in common?

Enter any 3-digit number. Press × 10 then ÷ 10. Do you always get back to the start? Now try × 10 then × 0.1. Is this the same? Use the History to compare the chains of operations.

Start with 5. What is the fewest number of button presses (using any multiply or divide buttons) needed to reach 0.005? What about reaching 5,000,000? Which is harder, and why?

Enter 123.456. Look at the digit 4. Turn on Expanded Form. What is the value of the 4? Now press × 10. What is the value of the 4 now? How many times bigger is it? Press × 10 again. Can you describe a rule for what happens to a digit’s value when the whole number is multiplied by 10?

Create a number where every digit is different (e.g., 1,234.567). Press ÷ 10. Which digit crosses the decimal point? What was its value before, and what is its value after? What is the ratio of those two values?

Enter 0.5 and repeatedly press × 10. How many presses before the tool stops you? Now start at 5 and repeatedly press ÷ 10. How many presses before it stops? Why are these limits different (or the same)?

Start at 3.6. Using exactly 2 operations (any buttons), can you reach 360? How many different pairs of operations work? Use the History to record each successful route.

Start at 5,000. Using only the divide buttons (÷ 10, ÷ 100, ÷ 1000, ÷ 0.1, ÷ 0.01), can you reach 0.5? What is the smallest number of presses needed?

Can you find a starting number where pressing + 1 causes a cascade that changes exactly 4 digits? What about exactly 5 digits? What is the maximum number of digits that can change from a single + 1?

Enter 0.001 and press × 10 repeatedly. After each press, toggle Words on and read the number aloud. At what point does the number stop having ‘point’ in its name? Keep going — what is the largest number you can reach?

Start with any number. Your challenge: reach exactly 100 using a combination of multiply/divide buttons AND add/subtract buttons. What is the most creative route you can find? Compare with a partner — did anyone find a route using all four types of operation?