100 Square Explorer — Fractions, Decimals & %

1

Shade 1 cell on Grid A. What fraction of the whole square is shaded? Now shade 1 more cell at a time, toggling the Fraction reveal on after every 10 cells. What do you notice about the denominator of the fraction each time? Why is it always 100?

2

Shade exactly 50 cells on Grid A. Toggle the Decimal reveal on. What decimal is shown? Now clear the grid and shade exactly 25 cells. What decimal appears this time? Without shading, predict the decimal for 75 cells. Check by shading and revealing.

3

Use the Row Totals toggle to show the numbers 10, 20, 30… down the right side of the grid. Shade cells row by row until you reach the second row total label. How many cells have you shaded? What fraction, decimal and percentage does this represent? Reveal all to check.

4

Toggle the Tenths boundary lines on. The grid now shows thicker lines dividing it into horizontal bands of 10 cells. Shade exactly one complete band. Toggle the Fraction reveal. What fraction is shown? Now shade two complete bands. What fraction is shown now? Describe the connection between the number of complete bands and the fraction.

5

Without shading any cells, look at the empty grid with Row Totals showing. A student says: ‘The number at the end of each row tells you what percentage is shaded if you shade up to that row.’ Do you agree? Shade cells to test with at least three different row boundaries. Explain why this works.

6

Use the dice button (⚤) to generate a random fill on Grid A. Before revealing anything, estimate: is the shaded amount closer to a quarter, a half, or three-quarters? Now reveal all four values. How close was your estimate? What visual clues helped you — the row totals, the tenths lines, or something else?

7

Shade 10 cells on Grid A. Toggle the Fraction reveal on. What fraction is shown? Now toggle the Simplified reveal on as well. Is the simplified form different from the fraction? What is the simplified form of 10/100? Explain in your own words what ‘simplifying a fraction’ means, using what you see on the two reveal pills.

8

Shade cells to show each of these fractions in turn, using the Fraction and Simplified reveals to check: 1/2, 1/4, 3/4, 1/5, 2/5, 3/5. For each, how many cells did you shade? Which fractions already appear in their simplest form on the Fraction reveal? Which ones simplify to something different?

9

Shade 6 cells and reveal the Fraction and Simplified pills. What does the Fraction pill show? What does Simplified show? Now shade 4 more cells (10 total). Reveal again. Both fractions simplify to the same value. Can you find two other different shadings that both simplify to 1/10? What must be true about a number of cells for its fraction to simplify to 1/10?

10

Shade 33 cells. Reveal the Fraction pill. The tool shows 33/100. Now reveal Simplified. Does 33/100 simplify? To check your answer, ask yourself: do 33 and 100 share any common factors? What is the highest common factor of 33 and 100? Now try 35 cells. Does that fraction simplify? What about 36 cells?

11

Using only the quick-jump buttons (10, 25, 50, 75, 100), shade each value in turn and reveal the Simplified fraction each time. Write down all five simplified fractions. What do the denominators tell you about these landmark numbers? Without using the tool, predict the simplified form for 20 cells and 40 cells. Check by shading.

12

Hide all reveals. Shade some cells on Grid A so that the Simplified fraction is 3/4. How many cells did you shade? Now shade a different number of cells so that Simplified is still 3/4 — is this possible? What does this tell you about how many cells give a fraction that simplifies to 3/4?

13

Shade 7 cells and reveal the Decimal pill. What decimal is shown? Is it the same as 7 ÷ 100? Confirm using a calculator or mental division. Now shade 70 cells and reveal the Decimal again. What has changed? Write a general rule: if you shade n cells, what is the decimal always equal to?

14

Shade 1 cell and reveal the Percentage pill. What percentage is shown? Now shade 10 cells, then 100 cells, revealing the percentage each time. What do you notice? A student claims: ‘The number of shaded cells always equals the percentage.’ Is this always true? Explain why.

15

Shade 33 cells. Reveal Decimal and Percentage side by side. What do you notice about the relationship between the two values? Now try 8 cells. Is the relationship the same? Write a rule connecting the decimal and the percentage for any shading on this grid.

16

Toggle the Tenths lines on. Shade exactly two complete tenths bands (the first 20 cells). Reveal Decimal and Percentage. Now shade three complete bands (30 cells) and reveal again. Predict, without shading: what will the decimal and percentage be for five complete bands? What is the connection between tenths bands and the decimal representation?

17

Use the dice button to generate three different random fills on separate attempts. For each, hide all reveals first, then reveal only the Percentage pill. Ask a partner to look at the percentage and shade what they think is the correct number of cells on Grid B. Then reveal Grid A’s Fraction to check. What strategies does your partner use to convert a percentage to a cell count?

18

Shade 12 cells. Reveal the Percentage pill. Is the result a whole number? Now shade 15 cells, then 18 cells. Which of these give a whole-number percentage? What must be true about the number of cells for the percentage to be a whole number? (Hint: think about what makes a fraction with denominator 100 simplify cleanly.)

19

Turn on both Tenths boundary lines and Row Totals. Shade exactly 15 cells and reveal all four values. The fraction shows 15/100 and the decimal shows 0.15. Notice that the 15th cell sits in the second tenths band. The row total for that row shows 20. How many cells of the second band are shaded? What fraction of the second band is that? How does this relate to the digit after the decimal point in 0.15?

20

With Row Totals on, shade exactly to the boundary of each row in turn (10, 20, 30… up to 100). After each fill, reveal the Decimal. Record all ten decimals. What pattern do you notice? What does this tell you about the connection between tenths and the 100 square?

21

Shade 35 cells. With Tenths lines on, count: how many complete tenths bands are shaded? How many extra cells are in the next partial band? Look at the Decimal reveal: 0.35. What is the connection between the number of complete bands, the extra cells, and the two decimal digits after the point?

22

Turn on Row Totals. The labels 10, 20, 30… appear next to each row. A student says these are the cumulative totals of shaded cells if every row were full. Another student says they just show the multiples of 10. Who is right? How do these row totals help you read off a fraction quickly without counting every cell?

23

Use the Tenths lines to shade exactly 1 complete tenths band plus 5 extra cells (15 total). Reveal the Fraction, Simplified, Decimal and Percentage. Now shade 1 complete band plus 3 extra cells (13 total). Which of the four reveals changes, and which stay the same structure? What does a shading that is exactly a whole number of tenths look like on all four reveals?

24

Shade 25 cells on Grid A and 50 cells on Grid B. Look at the ‘vs’ divider between the two grids. Which grid shows more? Reveal the Simplified fraction on both. What is the simplest way of writing the relationship between the two values? Now adjust Grid B so that it shows exactly twice the value of Grid A. What happens to the ‘vs’ divider if both grids show the same value?

25

Use the dice button on Grid A and Grid B to generate two random fills. Before revealing anything, look at the two grids and decide which shows more. Was it easy to tell? Now reveal the Percentage on both. Which is larger? By how much? What visual feature of the grid helped you estimate without looking at the reveals?

26

Set Grid A to show 1/3 of the square and Grid B to show 1/4. (Hint: use the quick-jump buttons and adjust.) Reveal the Decimal on both. Which decimal is larger? Now look at the grids: you shaded more cells on Grid A. Does the grid with more cells shaded always show the larger fraction? Can you find a counterexample using fractions with different denominators?

27

Shade 40 cells on Grid A. Now shade a number of cells on Grid B such that A + B = 100 (they sum to one whole square). Reveal the Fraction on both. What do you notice about the two fractions? What is the mathematical relationship between a fraction and its complement to 1? Now try A = 72 and find the complement. Does the pattern hold?

28

Set Grid A to 60 cells and Grid B to 60 cells. Watch the ‘vs’ divider. What does it show? Now reveal all four values on both grids. Are they identical? Try three more pairs where both grids are equal. Does the divider always respond? Why might it be useful in a classroom for the divider to change when both values are the same?

29

Using both grids and the ‘vs’ divider, find two different numbers of cells (one on each grid) that give the same decimal value. Is this possible if both values must be between 1 and 99 cells? What must be true about two shadings for them to have the same decimal? What does this tell you about equivalent fractions?

30

Shade 50 cells on Grid A. Reveal the Simplified fraction. Now shade cells on Grid B so that Grid B’s Simplified fraction also equals 1/2. You must use a different number of cells from Grid A. Is this possible? Write down all the numbers of cells (from 1 to 100) whose Simplified fraction is 1/2. What do these numbers have in common?

31

Reveal only the Simplified pill on Grid A. Use the quick-jump buttons to set Grid A to each of: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 cells. After each jump, note the simplified fraction. Which of these give the same simplified fraction as another value in the list? Write down every pair of equivalent fractions you find.

32

Shade 25 cells on Grid A (simplified: 1/4). Without using reveals, shade Grid B to also show a fraction equivalent to 1/4. Check using the Simplified reveal. How many different shadings equivalent to 1/4 can you find in the range 1–100 cells? Is there a pattern in the numbers (25, 50, 75, 100…)? Why does 1/3 have fewer equivalents visible on a 100-square than 1/4 does?

33

Turn off all reveals on Grid A. Shade a mystery number of cells. Tell your partner only the Simplified fraction (e.g. ‘it simplifies to 3/5’). Can your partner shade Grid B to show an equivalent fraction? How many correct shadings are possible? Now try a simplified fraction that has only one equivalent on the 100-square. What fractions have exactly one equivalent?

34

Compare 1/3 and 1/4 using both grids. For 1/3, there is no whole number of cells between 1 and 100 that gives exactly 1/3. Use the Decimal reveal to see what the closest shadings give. Which number of cells gives a decimal closest to 0.333…? A student says: ‘You can’t show 1/3 on a 100-square without rounding.’ Do you agree? What does this tell you about which fractions have exact representations as a number of hundredths?

35

Shade 100 cells on Grid A so the entire grid is filled. Reveal the Fraction pill. What does it show? Now press the + 1 whole button. A second shaded square appears to the left. Look at the Fraction reveal: the numerator is now 200 and the denominator is still 100. What does 200/100 equal? Why is the whole square shown separately?

36

Press + 1 whole and then shade 35 cells on the main grid. Reveal all four values. The Fraction pill shows 135/100. The Simplified pill shows a mixed number. What is the mixed number? What do the two parts of the mixed number represent visually on the screen? Try 1 whole and 50 cells, then 1 whole and 75 cells. What pattern do you notice in the mixed numbers?

37

Shade 100 cells and press + 1 whole. Now shade 0 extra cells. Toggle Fraction and Simplified. What is shown? Is 100/100 the same as 1 whole? Now shade 100 cells again so you have 1 whole + 100/100. What is the Simplified value? What would happen if you could press + 1 whole a second time? What number would that represent?

38

Press + 1 whole and shade 20 cells. Reveal Simplified: the mixed number is 1 1/5. Now clear the extra cells, shade 40 cells, and reveal: 1 2/5. Continue with 60 and 80 cells. Write down the four mixed numbers. These are all 1 and some number of fifths. Can you predict the mixed number for 1 whole and 90 cells? And for 1 whole and 15 cells? Check both.

39

Switch on Challenge mode using the 🎯 Challenge button. A target appears in the amber panel above the grids. Set the filter to Fractions only. Read the target and shade the correct number of cells on the target grid before pressing Check. What did you shade, and how did you work it out? Was the first target straightforward? Which fractions are easiest to show and why?

40

Set the Challenge filter to Decimals. Generate several targets. For each decimal target, describe your strategy: do you count cells directly, use the row totals, use the tenths lines, or convert the decimal to a fraction first? Which strategy is fastest? Compare your approach with a partner’s.

41

Set the Challenge filter to % only. A target appears such as 35%. Before shading, predict: how many cells will you need? Press Check after shading. If you get it wrong, what does the feedback tell you? Work through five percentage targets and keep a tally: how many did you get right on the first attempt? Which percentages are hardest to shade accurately, and why?

42

Switch the Challenge to Grid B and set it to Fractions. While Challenge mode is running on Grid B, use Grid A freely to help you think. For example, if the target is 3/20, you might use Grid A to work out that 3/20 = 15/100. Describe how you used Grid A as a ‘working space’. Can you find a strategy using Grid A that works for any fraction target?

43

Work through challenges from all three types (Fractions, Decimals, %) until you earn a streak of three in a row (🔥 appears after three consecutive correct answers). Record which types you found easiest and hardest to convert mentally. A student says: ‘Percentage targets are easiest because you just count to that many cells.’ Do you agree? Are there any percentage targets where this approach is difficult?

44

Use the dice button to generate a random fill on Grid A. Reveal all four values. Now shade Grid B with the ‘complement’ of Grid A — the number of cells that, added to Grid A’s fill, makes 100. Reveal all four values on Grid B. What is the relationship between Grid A’s decimal and Grid B’s decimal? Between Grid A’s percentage and Grid B’s? Will these relationships always hold, whatever the dice generates? Explain why.

45

Shade 1 cell, then 4, then 9, then 16, then 25 cells on Grid A in turn (clear and re-shade each time). After each shading, reveal the Simplified fraction. These are the square numbers. What do you notice about which of them give a fraction that can be simplified, and which cannot? What determines whether a square number of cells simplifies with 100?

46

Investigate the ‘multiples of 4’ pattern: shade 4, 8, 12, 16… cells, revealing the Simplified fraction each time. Record every simplified fraction you find. How many distinct simplified fractions appear before the pattern repeats? At what number of cells does the fraction simplify back to 1/25? Can you describe the complete cycle?

47

Set Grid A to 30 cells and Grid B to 70 cells. Reveal the Decimal on both. What do you notice about the two decimals? Now try Grid A = 45 and Grid B = 55. And Grid A = 18 and Grid B = 82. Write a general rule about the decimals of two shadings that sum to 100 cells. Can you prove this rule must always hold without using the tool?

48

A student claims: ‘If you double the number of shaded cells, the fraction doubles too.’ Test this claim using at least five different starting values. Is the student right? Now test: ‘If you halve the cells, the percentage halves.’ For which starting values does halving cells give a whole-number percentage? What must be true about a number for its half to also be a whole-number percentage?

49

Working on Grid A, shade 24 cells and reveal all four values. Now shade 48 cells (double), then 12 cells (half). Record the Simplified fraction each time. A student notices that 24, 48 and 12 all share the factor 4 with 100. Does every multiple of 4 give a simplified fraction with denominator 25? Test your conjecture with 4, 8, 36, 44, 52. Write a rule.

50

How many different numbers of cells (from 1 to 99) give a Simplified fraction with denominator 4? List them all. How many give a denominator of 5? Of 10? Of 20? Of 25? Of 50? Which denominator appears most often? What determines which denominators are possible on a 100-square?

51

Using both Grid A and Grid B, find two shadings that are not equal but whose Simplified fractions are equivalent (e.g. one simplifies to 1/2 and the other also). How many such pairs exist for each simplified fraction? Now use the dice button repeatedly to generate random pairs. How often do both grids produce the same Simplified fraction by chance? Estimate the probability — what fraction of all possible pairs share a simplified form?

52

Investigate which numbers of cells produce a Simplified fraction with an odd numerator and an odd denominator. List at least ten examples. What do the unsimplified fractions (n/100) have in common? (Hint: 100 = 4 × 25. Think about which factors n shares with 100.) Can you find all such values between 1 and 100 systematically?

53

Use the print button to print a sheet of six blank 100-squares with answer boxes. On each grid, design your own challenge: shade a specific number of cells, write the target (e.g. ‘Shade 3/5’) in the margin, and leave the answer boxes blank for a partner to fill in without the tool. Include at least one fraction that does NOT have an exact representation on the 100-square. How will your partner handle that? What is the closest they can get?

54

Design your own 100 Square Explorer investigation. Choose a focus (equivalent fractions, the connection between decimals and tenths, comparing values on two grids, or something you have discovered), decide which reveals and overlays to use, and write three questions that another student could explore. Test your questions yourself using the tool first, then exchange with a partner. Which of your questions led to the most interesting or surprising discoveries?