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Investigation Questions
Use these alongside the tool above. Drag manipulatives onto the canvas, use the place value grid, drawing tools and multiple slides to explore. Long-press to duplicate, tap counters to flip, and tap dice to re-roll.
1
Add a ten-frame to the canvas and use the + button to fill it with 7 counters. Without counting, can you immediately see how many empty spaces remain? How does the two-row structure of the ten-frame help you see this?
2
Place two ten-frames on the canvas. Fill the first with 8 counters and the second with 5 counters. Using the pen tool, draw an arrow to show how you would move counters from the second frame to complete the first. How many counters move? How many are left in the second frame? What is the total? How does this strategy help with adding numbers that bridge 10?
3
Make three different ten-frames that each show a number less than 10. Ask a partner to look at all three and estimate the total without counting individual counters. How close was their estimate? What strategies did they use? Does the arrangement of full rows and part-rows help?
4
Fill a ten-frame completely (10 counters) and place a second empty ten-frame beside it. Now fill the second frame with 3 counters. You have modelled the number 13. How would you model 20? What about 17? What is the largest number you can show using exactly three ten-frames?
5
Place a ten-frame with 6 counters on one slide and a ten-frame with 6 counters on another slide. Even though both show 6, can you arrange the counters differently using the + and − buttons? Does the arrangement change how quickly you recognise the number? What arrangement is easiest to read at a glance, and why?
6
Add single cubes (ones) to the canvas one at a time by tapping the tray. Drag each new cube onto the stack that forms. What happens when the stack reaches 10? Now tap the ⇌ exchange button. Describe what happens. How does the ten-rod that appears compare to the stack of 10 cubes?
7
Use Dienes blocks to build the number 234 on the canvas. How many hundred flats, ten-rods and ones do you need? Now use the ✂ split button to break one of the ten-rods into individual cubes. How many of each piece do you have now? Has the total value changed? Why might a teacher do this when teaching subtraction?
8
Build the number 100 using only single cubes. Stack groups of 10, exchange each stack for a ten-rod, then when you have 10 ten-rods, can you exchange for a hundred flat? Turn on the place value grid (PV button). Drag each type of piece into its correct column. Does seeing the pieces in columns make the value clearer?
9
Place a hundred flat on the canvas. Use ⇌ to exchange it for 10 ten-rods. Now pick one ten-rod and exchange it for 10 ones. How many pieces do you have in total? What is the value? A student says ‘I have more pieces now so the number must be bigger.’ Are they correct? Explain why the number of pieces and the value are different things.
10
Build 143 using Dienes blocks. Now imagine you need to subtract 27. Which pieces do you need to exchange before you can take away 7 ones and 2 tens? Use the split and exchange buttons to perform the regrouping, then use the bin to remove the pieces being subtracted. What number remains? Use the pen to record your working on the canvas.
11
Investigate: how many different ways can you represent the number 32 using Dienes blocks? For example, 3 ten-rods and 2 ones, or 2 ten-rods and 12 ones. List all the possibilities. What pattern do you notice? How many ways could you represent 45?
12
Add 8 yellow counters to the canvas. Tap 3 of them to flip them to red. You now have a model of 8 split into two groups. How many yellow and how many red? Record the number sentence with the pen. Now flip one more counter. What is the new split? How many different ways can you split 8 into two groups by flipping different numbers of counters?
13
Place 10 counters on the canvas, all yellow. Tap counters one at a time to flip them red, and after each flip, record the split (e.g. 9 yellow + 1 red, 8 + 2, 7 + 3…). How many different number bonds to 10 can you make? When you reach 5 + 5 and keep going, what do you notice? Are 3 + 7 and 7 + 3 the same split or different?
14
Place 6 counters on the canvas. Flip some so that there are more red than yellow. Without counting exactly, ask a partner to predict: are there more red or more yellow? Can they tell just by looking? Now arrange the counters in a line, alternating colours. Is it still easy to tell which colour has more?
15
A teacher places 12 counters on the canvas and says ‘I want exactly one-quarter of the counters to be red.’ How many should you flip? Check by flipping that many. Now the teacher says ‘Make one-third red.’ How many? What fractions of 12 give a whole number of counters? What fractions are impossible with exactly 12 counters?
16
Place 20 counters on the canvas, all yellow. Now tap each counter and, before you flip it, predict: will it land red or yellow? (Pretend each tap is a coin flip.) After flipping all 20, how many are red and how many are yellow? Is it exactly 10 and 10? Repeat the experiment on a new slide. Do you get the same result? What does this tell you about probability?
17
Place a Numicon 5 piece and a Numicon 3 piece on the canvas side by side. Which is taller? By how many holes? Now place a Numicon 1 piece next to them. Can you arrange all three in order from shortest to tallest? What does the height of each piece tell you about the number it represents?
18
Place a Numicon 7 on the canvas. Now add a Numicon 3 and rotate it using the rotate button. Drag the 3 piece on top of the 7 piece so it covers the empty spaces. Does the 3 fill the gap in the 7 exactly? Together, do the two pieces make a complete rectangle? What do the two numbers add up to? Try this with 6 and 4, then 8 and 2. What do all these pairs have in common?
19
Place all ten Numicon pieces (1 through 10) on the canvas. Pair them up so that each pair makes 10 (use the pen to draw lines between partners). How many pairs did you make? Is there a piece left over? Which one? What is special about that number in relation to 10?
20
Place a Numicon 10 piece on the canvas. It forms a complete 2×5 rectangle with no gaps. Now place a Numicon 9 on top of it. What do you see? There is one hole in the 10 that the 9 does not cover. What does this tell you about the difference between 10 and 9? Try placing other pieces on top of the 10. The uncovered holes always show what number?
21
Place a Numicon 4 and a Numicon 4 side by side. Together, do they make the same shape as a Numicon 8? Now try two 3s compared with a 6, and two 5s compared with a 10. Does doubling a Numicon piece always give you the corresponding even number? What does this tell you about even numbers?
22
Look at the Numicon pieces for 1, 3, 5, 7 and 9. What do they all have in common? (Hint: look at the shape — they all have an L-shaped notch.) Now look at 2, 4, 6, 8 and 10. What is different about their shape? What is the mathematical name for each group? Why do the odd pieces have a notch?
23
Turn on the place value grid using the PV button in the toolbar. You should see four columns labelled Th, H, T, O. Now add place value counters: drag two 100 counters into the H column, four 10 counters into the T column, and three 1 counters into the O column. What number have you built? Use the pen to write the number above the grid.
24
With the PV grid on, build the number 305 using place value counters. Which column is empty? Why is the empty column important for reading the number correctly? What would happen to the value if you accidentally placed the 5 counter in the T column instead of the O column?
25
Build the number 47 using place value counters in the PV grid. Now build 74 on a second slide. Look at both numbers. They use the same digits but in different columns. Which is larger? By how much? What does this show about the importance of which column a digit sits in?
26
Build a number in the PV grid using exactly 5 counters in total (any combination of 1s, 10s and 100s). What is the largest number you can make with exactly 5 counters? What is the smallest? How many different numbers can you make? List them systematically.
27
Build the number 130 in the PV grid using one 100 counter and three 10 counters. Now build 130 a different way: thirteen 10 counters and no 100 counters. Both represent the same value. Which arrangement makes the value clearer? When might a teacher use the non-standard arrangement? (Hint: think about subtraction.)
28
From the arrow cards tray, add a 300 card, a 50 card and a 7 card to the canvas. Drag the 50 card on top of the 300 card so they snap together, then snap the 7 on top. What three-digit number have you built? Now peel the cards apart by dragging. What does each card show? How does this reveal the expanded form of the number?
29
Use arrow cards to build 406. You will need a 400 card, a blank (or no tens card), and a 6 card. Can you build 406 without a tens card? What happens to the zero in the tens place? Compare this with building 460. How do the two numbers look different when made from arrow cards?
30
Build the number 555 using arrow cards (500 + 50 + 5). Stack them together. Each card shows the digit 5, but the cards are different lengths. Why? What does the length of each card represent? Pull them apart and use the pen to label the value of each card.
31
Turn on the PV grid. Build a number using arrow cards and place the stacked result in the O column. Now build the same number using place value counters in the correct columns. Which representation makes the place value structure clearer — the arrow cards or the counters in columns? Discuss with a partner. In what situations might each be more useful?
32
Place a dark green Cuisenaire rod (6) on the canvas. Now find combinations of smaller rods that are exactly the same length. Can you make 6 using two rods? Three rods? How many different two-rod combinations make 6? Use the pen to record each combination.
33
Place one rod of each colour from 1 to 10 on the canvas. Arrange them as a staircase, smallest to largest. What does the staircase look like? Now take the white rod (1) and see how many whites fit alongside the red (2). How many fit alongside the green (3)? The orange (10)? What does this tell you about each rod’s value?
34
Use Cuisenaire rods to investigate: is 3 + 4 the same as 4 + 3? Place a light green rod (3) next to a purple rod (4) in one line. On a new line, place a purple rod then a light green rod. Are the two lines the same total length? What property of addition does this demonstrate? Try other pairs. Does it always work?
35
Place an orange rod (10) on the canvas. Now place a yellow rod (5) underneath it. How much of the orange rod is covered? What fraction of 10 is 5? Try with a red rod (2) underneath the orange. What fraction of 10 is 2? Can you find a rod that is exactly one-quarter of the orange rod?
36
Build a rod train (end-to-end line) that is exactly 10 long using only red rods (2). How many do you need? Now build another train that is 10 long using only yellow rods (5). How many? What about using light green rods (3) — can you make exactly 10? Why or why not? Which rod values divide exactly into 10?
37
Look at the coins in the tray. Notice that the 20p and 50p coins are not circles — they are heptagonal (7-sided), and the £1 coin is dodecagonal (12-sided). Place one of each coin on the canvas. Can you sort them into groups by shape? Which coins are circular, which are heptagonal, and which is dodecagonal? Why do you think the Royal Mint makes coins different shapes?
38
Using coins, make exactly 50p in as many different ways as you can. Use a different slide for each combination. How many ways did you find? What is the fewest number of coins needed to make 50p? What is the most coins you could use?
39
Place one of each coin on the canvas (1p, 2p, 5p, 10p, 20p, 50p, £1, £2). What is the total value of one complete set? Now duplicate each coin (long-press). What is the total of two complete sets? Can you predict the total for three complete sets without adding every coin?
40
A shopkeeper needs to give 83p in change. Using the fewest possible coins, which coins should they use? Build the combination on the canvas. Now try giving 83p using the most possible coins (using only real UK coins). How many more coins does this take?
41
Add a die to the canvas. Tap it to re-roll. Roll it 30 times (use tally marks drawn with the pen to track results). Which number came up most often? Which came up least? Did each number appear roughly the same number of times? What would you expect to happen if you rolled 600 times?
42
Place two dice on the canvas. Tap both to roll them, then add the two values together. Record the total. Repeat this 20 times. Which totals appeared most often? Which totals never appeared? What is the smallest possible total from two dice? The largest? Why is a total of 7 more likely than a total of 2?
43
Place two dice on the canvas. Roll both and find the difference between the two values (larger minus smaller). What is the largest possible difference? What is the smallest? Can the difference ever be zero? Roll 20 times and keep a tally of the differences. Which difference is most common? Can you explain why?
44
Place three dice on the canvas. Roll all three and multiply the values together. What is the largest product you could possibly get? What is the smallest? Is it possible to get every number from 1 to 216? Roll 10 times and record your products. Are most of your results odd or even? Why?
45
Turn on the PV grid. Add the digits 3, 7 and 5 to the canvas. Arrange them in the H, T and O columns to make the largest possible three-digit number. What is it? Now rearrange them to make the smallest possible three-digit number. How did you decide where to place each digit?
46
Using the digits 1, 2 and 3 (one of each), how many different three-digit numbers can you make? Use a different slide for each arrangement. List them all in order from smallest to largest. How many are there? Can you predict how many different numbers you could make with four digits?
47
Represent the number 23 in five different ways, each on a separate slide: (1) using Dienes blocks, (2) using place value counters in the PV grid, (3) using arrow cards, (4) using Cuisenaire rods, and (5) using digits. Which representation shows the place value structure most clearly? Which is quickest to build? When might a teacher choose one over another?
48
Use Dienes blocks to build 54 on one slide and Numicon pieces that show 5 and 4 on another. Both slides involve the digits 5 and 4, but they show very different things. What does the Dienes model show that the Numicon model does not? What does the Numicon model show that the Dienes model does not? Discuss with a partner which manipulative you would choose for different topics.
49
Use the drawing tools and any manipulatives to create a poster on one slide that explains the number 37 to a younger student. Include at least three different representations. Use the highlighter to label the tens and ones. Save your poster as an image. What did you choose to include, and why?
50
Using two-colour counters: a bag contains 10 counters. You flip each one and count the reds. A student says ‘I will always get 5 red and 5 yellow because it’s fair.’ Test this claim by running the experiment across five different slides (10 counters each, flip them all). Record the number of reds each time. Does the student’s claim hold? What range of results did you get? What would you say to the student?
51
Using Dienes blocks: start with 1 hundred flat. Your task is to exchange your way down to having only single cubes. How many exchanges does it take in total? How many individual cubes do you end up with? Now go the other way: start with all single cubes and exchange your way back to 1 hundred flat. Is the number of exchanges the same in both directions?
52
Challenge: using any combination of manipulatives, build a model that represents a calculation and its answer. For example, use Cuisenaire rods to show 3 + 4 = 7, or ten-frames to show 8 + 5 = 13. Use the pen to write the number sentence. Can you create a subtraction? A model that shows a number bond to 20? Save your best example as an image.
53
Design your own manipulatives investigation. Choose one or two types of manipulative, decide on a mathematical focus (place value, number bonds, comparison, probability, or something else), and write three questions that another student could explore. Test them yourself first across different slides, then exchange with a partner. Which of your questions led to the most interesting discoveries?