Place Value Counters – Interactive Place Value Tool

1

Add three 100 counters, two 10 counters and five 1 counters. Toggle Number on. What number have you made? Now toggle Words on. Does the spoken form match what you expected? Toggle Expanded on. Write down the expanded form and explain what each part means.

2

Build the number 407 using counters. Which columns have counters and which are empty? Why is the empty column important? Toggle Expanded on — what do you notice about the zero column in the expanded form?

3

Build the number 3.52 using counters. How many of each counter did you need? Now toggle Count on. Check that the count display matches what you can see. What does the ×2 in the hundredths column tell you?

4

Build the number 60 in two different ways: first using 6 tens counters, then using 60 ones counters (turn Exchange off first). Toggle Number on for both. Do they show the same value? Toggle Expanded on. How are the expanded forms different? Which better shows the structure of the number?

5

Using the Columns button, set the grid to show only H, T, O. Build the largest number you can. Now change to show TTh to O. What is the largest number you can build now? What is the largest number the full chart (including thousandths) could show?

6

Add 1 counters to the ones column one at a time. Watch carefully. What happens when you add the tenth counter? Describe the animation in your own words. How many counters are in the ones column afterwards? How many appeared in the tens column?

7

Start from zero and add twelve 1 counters with Exchange on. How many exchanges happen? What number do you end up with? Now reset, turn Exchange off, and add twelve 1 counters again. What number does the display show? Are the two numbers equal?

8

Build a number that causes a cascading exchange: adding one counter triggers an exchange, and that exchange triggers another. What is the simplest number you can start with for this to happen? Describe the full cascade.

9

Start with zero and add nine 0.01 counters. Now add one more. Describe what happens. Does the exchange work the same way for decimal counters as for whole number counters? What if you add nine more 0.1 counters after that?

10

Turn Exchange off. Build the number 1.3 using 13 tenths counters (no ones counters). Toggle Number and Expanded on. The number should show 1.3 but the expanded form shows 13 × 0.1. Now turn Exchange on and add one more tenth. What happens? Why did the exchange not happen until you added a counter?

11

Starting from zero with Exchange on, what is the fewest number of taps on the 1 counter needed to make a counter appear in the hundreds column? What number do you end up with? Can you predict how many taps are needed to get a counter into the thousands column?

12

Turn Exchange off. Build the number 234 in the standard way (2 hundreds, 3 tens, 4 ones). Now reset and build 234 using only tens and ones counters (no hundreds). How many of each do you need? Toggle Expanded to check the value is still 234.

13

Turn Exchange off. How many different ways can you partition the number 53 using the counters? For example: 5 tens and 3 ones, or 4 tens and 13 ones. List as many as you can. What pattern do you notice?

14

Turn Exchange off and build the number 2.4 in three different ways. For at least one, use only hundredths counters. How many hundredths is that? What does this tell you about the relationship between tenths and hundredths?

15

Turn Exchange off. Build the number 100 using only ones counters. How many do you need? Now reset and build 100 using only tens counters. How many? What about using only 0.1 counters — how many would you need? Can you spot the pattern?

16

Press Row B to show a second row. Build 345 in Row A and 354 in Row B. Toggle Number on and look at both values. Which is larger? Toggle Sum and Difference on. What is the sum? What is the difference? Which display — the counters or the numbers — makes it easier to see which is larger?

17

In Row A, build 2.5. In Row B, build 2.50. Toggle Number, Words and Expanded on. What is the same about these two numbers? What is different? Are they equal? What does this tell you about trailing zeros in decimals?

18

Click the Number A label bar to select Row A, then build a number by tapping counters. Now click the Number B label bar and build a different number. Toggle Sum on. Can you find two numbers whose sum is exactly 1,000? How many different pairs can you find?

19

Build 0.75 in Row A and 0.3 in Row B. Toggle Difference on. Before looking at the answer, estimate the difference. Were you right? Now toggle Expanded on for both rows. Can you use the expanded forms to explain why the difference is what it is?

20

Build any number in Row A. Now build a number in Row B that makes the Sum display show exactly double Row A’s number. What must be true about the two numbers? Try this with three different starting numbers.

21

Toggle Expanded on and build 5,555. Write down the expanded form. Now build 5,005. How many terms are in each expanded form? Which number has more ‘information’ and why?

22

Turn Exchange off. Build the number 32 in the standard way (3 tens, 2 ones). Write down the expanded form. Now build 32 using 2 tens and 12 ones. Write down the new expanded form. Both show 32, but the expanded forms are different. Which one is in ‘standard’ partitioning and how can you tell?

23

Toggle Expanded on and build a number using only decimal counters (no whole number counters). Look at the expanded form. Now build a number using only whole number counters. How do the expanded forms differ? What colour pattern do you notice?

24

Build the number 1,111.11 and toggle Expanded on. How many terms are there? Now build 1,000.01. How many terms? Can you find a number with exactly 3 terms in its expanded form? Exactly 1 term?

25

Use the Columns button and select the preset H T O. Build the number 99. Now add one more 1 counter. What happens? Why can’t the exchange complete? What would you need to do to the column visibility to allow the exchange?

26

Set columns to Th – O and turn on Expanded. Build the number 9,999 and add one more 1 counter. Watch the cascade carefully. How many exchanges happen in total? Describe the chain from ones to ten thousands.

27

Set columns to show only T and O (toggle off everything else using individual column buttons). Build 47. Now toggle on the H column. Does the number change? Does the expanded form change? What does this tell you about hidden vs. visible columns?

28

Press + Thousandths to reveal the 0.001 column. Build the number 0.005 using thousandths counters. Toggle Number and Words on. How is this number spoken? Now add five more thousandths. What is the result? Does it exchange into hundredths?

29

Start from zero. Add one 1 counter. Toggle Count on and note the count in each column. Now add nine more 1 counters so an exchange happens. After the exchange, what are the counts? Add ninety more 1 counters (keep going!). Describe the pattern in the column counts as the total goes from 1 to 10 to 100.

30

Build a 3-digit number where all three digits are the same (like 333 or 777). Toggle Expanded on. What do you notice about the relationship between the three terms? Now try a number where the digits are 1, 2, 3 (like 123). Is there a similar relationship?

31

Toggle Count on. Build any number and look at the column counts. The Number display shows the total value, while the counts show how many of each place value you have. Can two different sets of column counts give the same total? Prove it by building two examples with Exchange off.

32

Turn Exchange off. Build the number 10 using only 1 counters (you need 10). Toggle Expanded on — it shows 10 × 1. Now turn Exchange on and add one more 1 counter. The exchange fires and you get 1 ten and 1 one. Toggle Expanded — now it shows 1 × 10 + 1 × 1. These are different expanded forms but different numbers. What changed?

33

Using Row A and Row B, find two numbers that have the same digits but in different columns, where Number A is exactly 10 times Number B. For example, try 4.5 and 45. Can you find a pair where A is exactly 100 times B?

34

Build a number in Row A. Now build its ‘complement to 10’ in Row B (the number that makes the sum equal exactly 10). Toggle Sum on to verify. Try this for at least five different numbers. What patterns do you notice in the counter arrangements?

35

What is the largest number you can build using exactly 20 counters in total (any combination of counter values)? What is the smallest positive number you can build with exactly 20 counters? What is the closest to zero you can get without reaching zero?

36

Set columns to show only O, t, h (ones, tenths, hundredths). Turn Exchange off. How many different ways can you partition the number 1 across these three columns? List them systematically. (Hint: think about how many hundredths make 1.)

37

Turn Exchange off. In Row A, build 3.6 using the standard partitioning (3 ones and 6 tenths). In Row B, build 3.6 using 2 ones and 16 tenths. Toggle Expanded on for both. A teacher might use Row B’s partitioning when teaching subtraction of decimals — can you explain why? Think about what would happen if you needed to subtract 0.8.

38

Design your own place value counters investigation. Choose which columns to show, whether Exchange should be on or off, and whether to use one row or two. Write a question that another student could explore, test it yourself first, then exchange with a partner.